Therefore

$$
\begin{align*}
\operatorname{Hom}_{U(t)}(\lambda \otimes_L A_I/\mathfrak{m}^k, J_{B \cap L_I}(J_{P_I}(\Pi^{\mathrm{la}})_\lambda)) & \\
&\simeq J_{B \cap L_I}(\operatorname{Hom}_{U(t)}(A_I/\mathfrak{m}^k, \operatorname{Hom}_{U(\mathfrak{I}^{\mathrm{ss}})}(L_I(\lambda), J_{P_I}(\Pi^{\mathrm{la}})))) \\
&= J_{B \cap L_I}(\operatorname{Hom}_{U(\mathfrak{I}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, J_{P_I}(\Pi^{\mathrm{la}}))) \\
&\simeq J_{B \cap L_I}(J_{P_I}(\operatorname{Hom}_{U(\mathfrak{p}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, \Pi^{\mathrm{la}}))) \\
&\simeq J_B(\operatorname{Hom}_{U(\mathfrak{g})}(\widetilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k, \Pi^{\mathrm{la}}))
\end{align*}
$$

where the first isomorphism comes from [Wu, Lemm. 5.3]. The claim now follows from the fact that the source of this chain of isomorphisms is the dual (of the global sections) of $i_{k,*}i_k^*t_\lambda^*\mathcal{M}_{\Pi}^{I,\lambda}$ and the target is the dual of $\mathcal{M}_{\Pi}(M)$. $\square$

# 4 Quasi-trianguline local deformation rings

Let $F$ be a finite extension of $\mathbb{Q}$. We keep the notation of section 3 but we specialize ourselves to the case $\underline{G} = \operatorname{Res}_{(F \otimes_{\mathbb{Q}} \mathbb{Q}_p)/\mathbb{Q}_p}(\operatorname{GL}_{n,F \otimes_{\mathbb{Q}} \mathbb{Q}_p}) \simeq \coprod_{v|p} \operatorname{Res}_{F_v/\mathbb{Q}_p} \operatorname{GL}_{n,F_v}$. We fix $\underline{B}$ the upper triangular Borel subgroup and $\underline{T}$ the diagonal torus. It is therefore sufficient to choose $L$ a finite extension of $\mathbb{Q}_p$ splitting all the $F_v$. We point out that, though the field $L$ of coefficients is the same as in the preceding section, the group $\underline{G}$ in this section should be considered as the Langlands dual group of the group in section 3.

Let $\Sigma_F$ be the set of embeddings of $F$ in $L$. This set can be decomposed as $\Sigma_F = \coprod_{v|p} \Sigma_{F_v}$, where $\Sigma_{F_v}$ is the set of $\mathbb{Q}_p$-linear embeddings of $F_v$ into $L$ and where the index set is the set of places $v$ of $F$ that divide $p$. We have a decomposition

$$
\mathfrak{g} \simeq ( \bigoplus_{\tau \in \Sigma_F} \operatorname{Lie}(\underline{G}) \otimes_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p, \tau} L ) \simeq \bigoplus_{\tau \in \Sigma_F} \operatorname{Lie}(\operatorname{GL}_{n,L}).
$$

Let $\Delta$ be the set of simple roots of $\underline{G}_L$ with respect to $\underline{B}_L$. Then

$$
\Delta = \coprod_{\tau \in \Sigma_F} \Delta_{\tau}, \quad \Delta_{\tau} = \{\alpha_{1,\tau}, \dots, \alpha_{n-1,\tau}\}
$$

where $\alpha_{1,\tau}, \dots, \alpha_{n-1,\tau}$ are the simple roots of the copy of $\operatorname{Lie}(\operatorname{GL}_{n,L})$ corresponding to $\tau$. For $I \subset \Delta$ we denote $\underline{P}_I$ the standard parabolic subgroup of $\underline{G}_L$ corresponding to $I$.

## 4.1 Local models

Let $\tilde{\mathfrak{g}} := \underline{G}_L \times_{\underline{B}_L} \mathfrak{b}$ be the Grothendieck-Springer resolution of $\mathfrak{g}$ (which is considered as a scheme over $L$ not just as a vector space in this section). We have a closed embedding $\tilde{\mathfrak{g}} \hookrightarrow \underline{G}_L/\underline{B} \times_{\mathfrak{g}} \mathfrak{g}$ given by $(g\underline{B}, X) \mapsto (b\underline{B}, \operatorname{Ad}(g)X)$ and set

$$
X := \tilde{\mathfrak{g}} \times_{\mathfrak{g}} \tilde{\mathfrak{g}} \subset \underline{G}_L/\underline{B}_L \times_{\mathfrak{g}} \underline{G}_L/\underline{B}_L.
$$
More generally if $I \subset \Delta$, we set

$$ \tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 := \underline{G}_L \times_{\underline{P}_I} (\mathfrak{z}_I \oplus \mathfrak{n}_I) $$

where we recall that $\underline{P}_I$ is the parabolic subgroup of $\underline{G}$ associated to $\Delta$ and $\mathfrak{p}_I$ is its Lie algebra. Moreover, we write $\mathfrak{z}_I$ for the center of $\mathfrak{p}_I$ and $\mathfrak{n}_I$ for its unipotent radical. Again we consider all these $L$-vector spaces as $L$-schemes. We have also a closed embedding $\tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 \hookrightarrow \underline{G}_L/\underline{P}_I \times \mathfrak{g}$ given by $(g\underline{P}_I, X) \mapsto (g\underline{P}_I, \mathrm{Ad}(g)X)$ and we set

$$ X_{\mathfrak{p}_I} := \tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 \times_{\mathfrak{g}} \tilde{\mathfrak{g}} \hookrightarrow \underline{G}_L/\underline{P}_I \times_{\mathfrak{g}} \underline{G}_L/\underline{B}_L. $$

In particular we have $X_b = X$. There scheme $X_{\mathfrak{p}_I}$ decomposes into irreducible components as follows:

$$ X_{\mathfrak{p}_I} = \bigcup_{w \in W_I \setminus W} X_{\mathfrak{p}_I, w} \subset \underline{G}_L/\underline{P}_I \times_{\mathfrak{g}} \underline{G}_L/\underline{B}. $$

Here $X_{\mathfrak{p}_I, w}$ is the closure of on open subset $V_{\mathfrak{p}_I, w} \subset X_{\mathfrak{p}_I}$, which is by definition the preimage of the $\underline{G}$-orbit of $\underline{G} \cdot (1, \tilde{w}) \subset \underline{G}/\underline{P}_I \times \underline{G}/\underline{B}$, where $\tilde{w} \in W$ is a lift of $w \in W_I \setminus W$ (see [BD, Cor. 5.2.2] for details). In this paper we need to control the singularities of $X_{\mathfrak{p}_I}$. Even though, for our purpose, the result of [BHS19, Rk. 4.1.6] would be sufficient, we mention the following more general result.

**Proposition 4.1.** Let $w \in W$. Then $X_w$ is smooth if, and only if, $w$ is a product of distinct simple reflections.

*Proof.* We note that the natural action

$$ t \cdot (g\underline{B}, h\underline{B}, N) = (g\underline{B}, h\underline{B}, tN) $$

of $\mathbb{G}_m$ on $X$ by scaling on the $\mathfrak{g}$-factor extends to an action of the monoid $\mathbb{A}^1$. This action obviously preserves each $X_w$. As the singular locus is closed the non-singular locus, if non-empty, contains a point of the form $(g\underline{B}h\underline{B}, 0)$ We will thus prove the previous proposition using [BHS19] Proposition 2.5.3 (ii).

We first assume that $w$ is a product of distinct simple reflections. In this case it is enough to prove that

a) $\bar{U}_w$ is smooth in $\underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$;

b) $t^{ww'^{-1}}$ has codimension $\mathrm{lg}(w) - \mathrm{lg}(w')$ in $t$ for all $w' \le w$ for Bruhat ordering (with $\mathrm{lg}$ the Bruhat length).

By Fan's Theorem [BL00, Theorem 7.2.14], if $w$ is a product of distinct simple reflections, then $\bar{U}_w$ is smooth and a) is true. Thus we only need to prove b). For $w \in W$, let us introduce

$$ \ell(w) := \min\{k \ge 0 \mid w = r_1 \dots r_k, r_k \in W \text{ a reflection}\} $$

(we recall that reflection is an element of the form $s_\alpha$ where $\alpha \in \Phi$ is a root, but not necessarily a simple root). By [Car72, Lemma 2] and [BHS17a, Lemma 2.7] we have $\ell(w) = \dim_L t - \dim_L t^w = d_w$ (in the notations of [BHS17a]).
More generally if $I \subset \Delta$, we set

$$ \tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 := \underline{G}_L \times_{\underline{P}_I} (\mathfrak{z}_I \oplus \mathfrak{n}_I) $$

where we recall that $\underline{P}_I$ is the parabolic subgroup of $\underline{G}$ associated to $\Delta$ and $\mathfrak{p}_I$ is its Lie algebra. Moreover, we write $\mathfrak{z}_I$ for the center of $\mathfrak{p}_I$ and $\mathfrak{n}_I$ for its unipotent radical. Again we consider all these $L$-vector spaces as $L$-schemes. We have also a closed embedding $\tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 \hookrightarrow \underline{G}_L/\underline{P}_I \times \mathfrak{g}$ given by $(g\underline{P}_I, X) \mapsto (g\underline{P}_I, \mathrm{Ad}(g)X)$ and we set

$$ X_{\mathfrak{p}_I} := \tilde{\mathfrak{g}}_{\mathfrak{p}_I}^0 \times_{\mathfrak{g}} \tilde{\mathfrak{g}} \hookrightarrow \underline{G}_L/\underline{P}_I \times_{\mathfrak{g}} \underline{G}_L/\underline{B}_L. $$

In particular we have $X_b = X$. There scheme $X_{\mathfrak{p}_I}$ decomposes into irreducible components as follows:

$$ X_{\mathfrak{p}_I} = \bigcup_{w \in W_I \setminus W} X_{\mathfrak{p}_I, w} \subset \underline{G}_L/\underline{P}_I \times_{\mathfrak{g}} \underline{G}_L/\underline{B}. $$

Here $X_{\mathfrak{p}_I, w}$ is the closure of on open subset $V_{\mathfrak{p}_I, w} \subset X_{\mathfrak{p}_I}$, which is by definition the preimage of the $\underline{G}$-orbit of $\underline{G} \cdot (1, \tilde{w}) \subset \underline{G}/\underline{P}_I \times \underline{G}/\underline{B}$, where $\tilde{w} \in W$ is a lift of $w \in W_I \setminus W$ (see [BD, Cor. 5.2.2] for details). In this paper we need to control the singularities of $X_{\mathfrak{p}_I}$. Even though, for our purpose, the result of [BHS19, Rk. 4.1.6] would be sufficient, we mention the following more general result.

**Proposition 4.1.** Let $w \in W$. Then $X_w$ is smooth if, and only if, $w$ is a product of distinct simple reflections.

*Proof.* We note that the natural action

$$ t \cdot (g\underline{B}, h\underline{B}, N) = (g\underline{B}, h\underline{B}, tN) $$

of $\mathbb{G}_m$ on $X$ by scaling on the $\mathfrak{g}$-factor extends to an action of the monoid $\mathbb{A}^1$. This action obviously preserves each $X_w$. As the singular locus is closed the non-singular locus, if non-empty, contains a point of the form $(g\underline{B}h\underline{B}, 0)$ We will thus prove the previous proposition using [BHS19] Proposition 2.5.3 (ii).

We first assume that $w$ is a product of distinct simple reflections. In this case it is enough to prove that

a) $\bar{U}_w$ is smooth in $\underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$;

b) $t^{ww'^{-1}}$ has codimension $\mathrm{lg}(w) - \mathrm{lg}(w')$ in $t$ for all $w' \le w$ for Bruhat ordering (with $\mathrm{lg}$ the Bruhat length).

By Fan's Theorem [BL00, Theorem 7.2.14], if $w$ is a product of distinct simple reflections, then $\bar{U}_w$ is smooth and a) is true. Thus we only need to prove b). For $w \in W$, let us introduce

$$ \ell(w) := \min\{k \ge 0 \mid w = r_1 \dots r_k, r_k \in W \text{ a reflection}\} $$

(we recall that reflection is an element of the form $s_\alpha$ where $\alpha \in \Phi$ is a root, but not necessarily a simple root). By [Car72, Lemma 2] and [BHS17a, Lemma 2.7] we have $\ell(w) = \dim_L t - \dim_L t^w = d_w$ (in the notations of [BHS17a]).
Claim 4.2. If w is a product of distinct simple reflections, we have

$$ \ell(ww'^{-1}) = \ell(w) - \ell(w') = \lg(w) - \lg(w') $$

for all $w' \le w$.

If Claim 4.2 is true, we have $\ell(ww'^{-1}) = \dim t - \dim t^{ww'^{-1}} = \lg(w) - \lg(w')$ thus Proposition 2.5.3 of [BHS19] applies and $X_w$ is smooth. We now prove the claim. The second equality of the claim is a consequence of [Car72, Lemma 3] as $w$ and $w'$ are products of pairwise distinct simple reflections. Indeed, a product of pairwise distinct simple reflexions $s_1...s_k$ is always a composition of reflections $s_i$ along vectors $v_i$ such that $v_1, ..., v_k$ are linearly independent. Thus [Car72, Lemma 3] implies $\ell(w) = \lg(w)$ and $\ell(w') = \lg(w')$.

We write $w' = s_{i_1}...s_{i_k}$ and $w = t_1...t_b$ as reduced expressions of pairwise distinct simple roots such that there exists $a_1 \le ... \le a_k$ satisfying $t_{a_j} = s_{i_j}$. For $a_t \le j < a_{t+1}$ let $r_j$ denote the reflection $r_j := s_{i_1}...s_{i_t}t_{j}s_{i_t}...s_{i_1}$. We then have

$$ \begin{aligned} ww'^{-1} &= t_1 \dots t_b s_{i_k} \dots s_{i_1} \\ &= t_1 \dots t_{a_1-1} \underbrace{[s_{i_1} t_{a_1+1} s_{i_1}]}_{r_{a_1+1}} \dots \underbrace{[s_{i_1} t_{a_2-1} s_{i_1}]}_{r_{a_2-1}} \underbrace{[s_{i_1} s_{i_2} t_{a_2+1} s_{i_2} s_{i_1}]}_{r_{a_2+1}} \\ &\quad \dots \underbrace{[s_{i_1} \dots s_{i_k} t_{a_k+1} s_{i_k} \dots s_{i_1}]}_{r_b} \dots \underbrace{[s_{i_1} \dots s_{i_k} t_b s_{i_k} \dots s_{i_1}]}_{r_b} \\ &= t_1 \dots t_{a_1-1} r_{a_1+1} \dots r_{a_2-1} r_{a_2+1} \dots r_b. \end{aligned} $$

In particular, $\ell(ww'^{-1}) \le \lg(w) - \lg(w') = \ell(w) - \ell(w')$. Now Claim 4.2 follows from Claim 4.3. Let $w \in W$ and $w'$ be a product of distinct simple reflections. Then $\ell(ww'^{-1}) \ge \ell(w) - \ell(w') = \ell(w) - \lg(w')$.

We now prove Claim 4.3. By induction on the number of simple reflexions appearing in $w'$, it is enough to prove $\ell(\overline{ws}) \ge \ell(w) - 1$ when $w' = s$ is a simple reflexion. Note that for any $w$ we have $\dim_L t^{ws} \cap t^s \ge \dim_L t^{ws} - 1$ as $t^s$ is a hyperplane in $t$. Moreover, $t^{ws} \cap t^s = t^w \cap t^s \subset t^w$. Thus $\dim t^w \ge \dim t^{ws} - 1$. Using $\ell(w) = \dim t - \dim t^w$ we hence find

$$ \ell(w) \le \ell(\overline{ws}) + 1. $$

Thus $\ell(\overline{ws}) \ge \ell(w) - 1$, which proves Claim 4.3.

We now prove the converse, i.e. that $X_w$ is singular, if $w$ is not a product of distinct simple reflections. We hence assume that $w$ is not a product of distinct simple reflections.

It is enough (but actually equivalent) to prove that $X_w$ is singular at $(\underline{B}, \underline{B}, 0)$. We will use Mowlavi's results [Mow23]. The pair $(1, w)$ is a good pair ([Mow23]), and thus [Mow23, Theorem 6] applies. Hence [Mow23, Proposition 3.2.2] gives an exact formula for the tangent space at $x = (\underline{B}, \underline{B}, 0) \in (X_w \cap V_1)(L)$. This can be rewritten as

$$ \begin{aligned} \dim_L T_x X_w &= \dim_L T_{\pi(x)} \overline{U_w} - d_w + \dim_L t + \lg(w_0) \\ &> \dim \underline{B} + \lg(w) - \lg(w) + \dim_L t + \lg(w_0), \end{aligned} $$
Claim 4.2. If w is a product of distinct simple reflections, we have

$$ \ell(ww'^{-1}) = \ell(w) - \ell(w') = \lg(w) - \lg(w') $$

for all $w' \le w$.

If Claim 4.2 is true, we have $\ell(ww'^{-1}) = \dim t - \dim t^{ww'^{-1}} = \lg(w) - \lg(w')$ thus Proposition 2.5.3 of [BHS19] applies and $X_w$ is smooth. We now prove the claim. The second equality of the claim is a consequence of [Car72, Lemma 3] as $w$ and $w'$ are products of pairwise distinct simple reflections. Indeed, a product of pairwise distinct simple reflexions $s_1 \dots s_k$ is always a composition of reflections $s_i$ along vectors $v_i$ such that $v_1, \dots, v_k$ are linearly independent. Thus [Car72, Lemma 3] implies $\ell(w) = \lg(w)$ and $\ell(w') = \lg(w')$.

We write $w' = s_{i_1} \dots s_{i_k}$ and $w = t_1 \dots t_b$ as reduced expressions of pairwise distinct simple roots such that there exists $a_1 \le \dots \le a_k$ satisfying $t_{a_j} = s_{i_j}$. For $a_t \le j < a_{t+1}$ let $r_j$ denote the reflection $r_j := s_{i_1} \dots s_{i_t} t_j s_{i_t} \dots s_{i_1}$. We then have

$$ \begin{aligned} ww'^{-1} &= t_1 \dots t_b s_{i_k} \dots s_{i_1} \\ &= t_1 \dots t_{a_1-1} \underbrace{[s_{i_1} t_{a_1+1} s_{i_1}]}_{r_{a_1+1}} \dots \underbrace{[s_{i_1} t_{a_2-1} s_{i_1}]}_{r_{a_2-1}} \underbrace{[s_{i_1} s_{i_2} t_{a_2+1} s_{i_2} s_{i_1}]}_{r_{a_2+1}} \\ &\quad \dots \underbrace{[s_{i_1} \dots s_{i_k} t_{a_k+1} s_{i_k} \dots s_{i_1}]}_{r_b} \dots \underbrace{[s_{i_1} \dots s_{i_k} t_b s_{i_k} \dots s_{i_1}]}_{r_b} \\ &= t_1 \dots t_{a_1-1} r_{a_1+1} \dots r_{a_2-1} r_{a_2+1} \dots \dots r_b. \end{aligned} $$

In particular, $\ell(ww'^{-1}) \le \lg(w) - \lg(w') = \ell(w) - \ell(w')$. Now Claim 4.2 follows from Claim 4.3. Let $w \in W$ and $w'$ be a product of distinct simple reflections. Then $\ell(ww'^{-1}) \ge \ell(w) - \ell(w') = \ell(w) - \lg(w')$.

We now prove Claim 4.3. By induction on the number of simple reflexions appearing in $w'$, it is enough to prove $\ell(\overline{ws}) \ge \ell(w) - 1$ when $w' = s$ is a simple reflexion. Note that for any $w$ we have $\dim_L t^{ws} \cap t^s \ge \dim_L t^{ws} - 1$ as $t^s$ is a hyperplane in $t$. Moreover, $t^{ws} \cap t^s = t^w \cap t^s \subset t^w$. Thus $\dim t^w \ge \dim t^{ws} - 1$. Using $\ell(w) = \dim t - \dim t^w$ we hence find

$$ \ell(w) \le \ell(\overline{ws}) + 1. $$

Thus $\ell(\overline{ws}) \ge \ell(w) - 1$, which proves Claim 4.3.

We now prove the converse, i.e. that $X_w$ is singular, if $w$ is not a product of distinct simple reflections. We hence assume that $w$ is not a product of distinct simple reflections.

It is enough (but actually equivalent) to prove that $X_w$ is singular at $(\underline{B}, \underline{B}, 0)$. We will use Mowlavi's results [Mow23]. The pair $(1, w)$ is a good pair ([Mow23]), and thus [Mow23, Theorem 6] applies. Hence [Mow23, Proposition 3.2.2] gives an exact formula for the tangent space at $x = (\underline{B}, \underline{B}, 0) \in (X_w \cap V_1)(L)$. This can be rewritten as

$$ \begin{aligned} \dim_L T_x X_w &= \dim_L T_{\pi(x)} \overline{U_w} - d_w + \dim_L t + \lg(w_0) \\ &> \dim \underline{B} + \lg(w) - \lg(w) + \dim_L t + \lg(w_0), \end{aligned} $$
as $w$ is not a product of distinct simples so $\lg(w) > d_w$ ([BHS17a] Lemma 2.7), and where we use the notation $^1d_w = \dim_L t - \dim_L t^w$. Thus

$$ \dim_L T_x X_w > 2 \dim \underline{B} + \dim_L t = \dim \underline{G}_L = \dim X_w, $$

i.e. $X_w$ is not smooth at $x$.

□

We write $X_I$ for the inverse image of $X_{p_I}$ under the canonical projection $\underline{G}_L/\underline{B}_L \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L \to \underline{G}_L/\underline{P}_I \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L$. This scheme can also be defined as

$$ X_I := (\underline{G}_L \times_{\underline{B}_L} (\mathfrak{z}_I \oplus \mathfrak{n}_I)) \times_{\mathfrak{g}} \tilde{\mathfrak{g}}, $$

in particular $X_\emptyset = X$. The map $X_I \to X_{p_I}$ is a $\underline{P}_I/\underline{B}_L$-torsor and thus is projective and smooth. We deduce that we have a decomposition in irreducible components

$$ X_I = \bigcup_{w \in W_I \setminus W} X_{I,w}, $$

where each $X_{I,w} \to X_{p_I,w}$ is projective and smooth. Moreover, we have a closed embedding $X_I \hookrightarrow X$ induced by the closed embedding $\mathfrak{z}_I \oplus \mathfrak{n}_I \hookrightarrow \mathfrak{b}$, and this induces a closed embedding $X_{I,w} \hookrightarrow X_{w^{\max}}$, as each fiber of $X_I \to X_{p_I}$ over a point in $V_{p_I,w}$ contains a (dense) open subset consisting of points that lie in the Schubert cell $\underline{G}_L(1, w^{\max}) \subset \underline{G}/\underline{B} \times \underline{G}/\underline{B}$.

**Lemma 4.4.** *The schemes $X_I$ and $X_{p_I}$ are generically reduced.*

*Proof.* As $X_I$ is smooth over $X_{p_I}$, it suffices to prove the claim for $X_{p_I}$. For $w \in W$, let $U_w = \underline{G}_L(1, w) \subset \underline{G}_L/\underline{P}_I \times \underline{G}_L/\underline{B}$ and let $V_w \subset X_{p_I}$ be the inverse image of $U_w$. It follows from [BD, Prop. 5.2.1] that the $V_w$ are smooth $L$-schemes, and they all have the same dimension. As they also cover $X_{p_I}$, their generic points are the generic points of the irreducible components of $X_{p_I}$. This shows that $X_{p_I}$ is generically reduced. □

Recall that we have two maps $\kappa_1, \kappa_2 : X \to t$ (see [BHS19, §2.3]) defined by $\kappa_i(g_1\underline{B}, N, g_2\underline{B}) = g_i^{-1}Ng_i(\text{mod } \mathfrak{n})$. By construction, the image of $\kappa_{1|X_I}$ lands in $\mathfrak{z}_I$ and the map $\kappa_{1|X_I}$ factors through $X_{p_I}$. This provides a commutative diagram

$$ \begin{tikzcd} X_I \arrow[r] \arrow[d, "\Theta_I"] & X_{p_I} \arrow[d] \\ \mathfrak{z}_I \times_{t/W} t \arrow[u] \arrow[r] & \mathfrak{z}_I \end{tikzcd} $$

where $\Theta_I$ is the restriction of the map $(\kappa_1, \kappa_2)$ to $X_I$.

The following result is the analogue of [BHS19, Lem. 2.5.1] in our context, with analogous proof.

<footnote n="1">¹see [BHS19] just before Proposition 4.1.5</footnote>
as $w$ is not a product of distinct simples so $\lg(w) > d_w$ ([BHS17a] Lemma 2.7), and where we use the notation $^1d_w = \dim_L t - \dim_L t^w$. Thus

$$ \dim_L T_x X_w > 2 \dim \underline{B} + \dim_L t = \dim \underline{G}_L = \dim X_w, $$

i.e. $X_w$ is not smooth at $x$.

□

We write $X_I$ for the inverse image of $X_{p_I}$ under the canonical projection $\underline{G}_L/\underline{B}_L \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L \to \underline{G}_L/\underline{P}_I \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L$. This scheme can also be defined as

$$ X_I := (\underline{G}_L \times_{\underline{B}_L} (\mathfrak{z}_I \oplus \mathfrak{n}_I)) \times_{\mathfrak{g}} \tilde{\mathfrak{g}}, $$

in particular $X_\emptyset = X$. The map $X_I \to X_{p_I}$ is a $\underline{P}_I/\underline{B}_L$-torsor and thus is projective and smooth. We deduce that we have a decomposition in irreducible components

$$ X_I = \bigcup_{w \in W_I \setminus W} X_{I,w}, $$

where each $X_{I,w} \to X_{p_I,w}$ is projective and smooth. Moreover, we have a closed embedding $X_I \hookrightarrow X$ induced by the closed embedding $\mathfrak{z}_I \oplus \mathfrak{n}_I \hookrightarrow \mathfrak{b}$, and this induces a closed embedding $X_{I,w} \hookrightarrow X_{w^{\max}}$, as each fiber of $X_I \to X_{p_I}$ over a point in $V_{p_I,w}$ contains a (dense) open subset consisting of points that lie in the Schubert cell $\underline{G}_L(1, w^{\max}) \subset \underline{G}/\underline{B} \times \underline{G}/\underline{B}$.

**Lemma 4.4.** *The schemes $X_I$ and $X_{p_I}$ are generically reduced.*

*Proof.* As $X_I$ is smooth over $X_{p_I}$, it suffices to prove the claim for $X_{p_I}$. For $w \in W$, let $U_w = \underline{G}_L(1, w) \subset \underline{G}_L/\underline{P}_I \times \underline{G}_L/\underline{B}$ and let $V_w \subset X_{p_I}$ be the inverse image of $U_w$. It follows from [BD, Prop. 5.2.1] that the $V_w$ are smooth $L$-schemes, and they all have the same dimension. As they also cover $X_{p_I}$, their generic points are the generic points of the irreducible components of $X_{p_I}$. This shows that $X_{p_I}$ is generically reduced. □

Recall that we have two maps $\kappa_1, \kappa_2 : X \to t$ (see [BHS19, §2.3]) defined by $\kappa_i(g_1\underline{B}, N, g_2\underline{B}) = g_i^{-1}Ng_i(\text{mod } \mathfrak{n})$. By construction, the image of $\kappa_{1|X_I}$ lands in $\mathfrak{z}_I$ and the map $\kappa_{1|X_I}$ factors through $X_{p_I}$. This provides a commutative diagram

$$ \begin{tikzcd} X_I \arrow[r] \arrow[d, "\Theta_I"] & X_{p_I} \arrow[d] \\ \mathfrak{z}_I \times_{t/W} t \arrow[u] \arrow[r] & \mathfrak{z}_I \end{tikzcd} $$

where $\Theta_I$ is the restriction of the map $(\kappa_1, \kappa_2)$ to $X_I$.

The following result is the analogue of [BHS19, Lem. 2.5.1] in our context, with analogous proof.

<footnote n="1">¹see [BHS19] just before Proposition 4.1.5</footnote>
**Lemma 4.5.** The irreducible components of $\mathfrak{z}_I \times_{\mathrm{t}/W} t$ are the $(T_{I,w})_{w \in W_I \setminus W}$ where

$$ T_{I,w} = \{(z, \mathrm{Ad}(w^{-1})(z)) \mid z \in \mathfrak{z}_I\}. $$

Moreover, the irreducible component $X_{I,w}$ (resp. $X_{\mathfrak{p}_I,w}$) is the unique component of $X_I$ (resp. $X_{\mathfrak{p}_I}$) whose image under $\Theta_I$ (resp. $\Theta_{\mathfrak{p}_I}$) dominates $T_{I,w}$.

**Remark 4.6.** For future use, we make the following notational convention: When $F = \mathbb{Q}$, we have $\underline{G}_L = \mathrm{GL}_{n,L}$, we will use the notations $X_n$, $X_{n,I}$, $X_{n,I,w}$ etc. for the schemes $X$, $X_I$, $X_{I,w}$ etc.

## 4.2 Partially de Rham deformation rings

For each place $v|p$ of $F$, we fix $r_v : \mathrm{Gal}_{F_v} \to \mathrm{GL}_n(L)$ a framed $\varphi$-generic Hodge–Tate regular crystalline representation, that we assume that the $(\varphi, \Gamma)$-module $D_{\mathrm{rig}}(r_v)$ associated to $r_v$ is crystalline $\varphi$-generic with regular Hodge–Tate type in the sense of [HMS, §3.3&§3.4]. We also fix a refinement $\mathcal{R}_v = (\varphi_1, \dots, \varphi_n) \in L^n$ of $r_v$ (see loc. cit.). We will use the notation $r = (r_v)_{v|p}$ and $\mathcal{R} = (\mathcal{R}_v)_{v|p}$ and say that $r$ is $\varphi$-generic Hodge–Tate regular and that $\mathcal{R}$ is a refinement of $r$.

Let $\mathcal{C}_L$ be the category of local artinian $L$-algebras. Fix $v|p$ a place of $F$. Let $\mathcal{X}_{r_v}^{\square}$ be the groupoid over $\mathcal{C}_L$ of deformations of $r_v$. It is represented by a formal scheme over $L$ that we also denote by $\mathcal{X}_{r_v}^{\square}$ by abuse of notation. We recall from [BHS19, 3.6] that, given the refinement $\mathcal{R}_v$, the groupoid of trianguline deformations of $\mathcal{M}_{\bullet,v}$ is representable by a closed formal subscheme $\mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}} \subset \mathcal{X}_{r_v}^{\square}$. Here $\mathcal{M}_{\bullet,v}$ the $(\varphi, \Gamma)$-module over $\mathcal{R}_{K,L}[1/t]$ obtained from $D_{\mathrm{rig}}(r_v)$ by inverting $t$ which is equipped with the unique triangulation corresponding to the refinement $\mathcal{R}_v$. We set $W_v = W_{\mathrm{dR}}(D_{\mathrm{rig}}(r_v)[1/t])$ and $W_{\bullet,v} = W_{\mathrm{dR}}(\mathcal{M}_{\bullet,v})$ and let $X_{W_v, W_{\bullet,v}}$ denote the groupoid of deformations of $(W_v, W_{\bullet,v})$ as defined in [BHS19, §3.3].

Fix a finite subset $I_v \subset \Delta_v$. For an object $A$ of $\mathcal{C}_L$, we define $X_{W_v, W_{\bullet,v}}^{P_{I_v}}(A)$ to be the subset of all $(W_A, W_{A,\bullet}) \in X_{W_v, W_{\bullet,v}}(A)$ such that for any $\tau \in \Sigma_{F_v}$ and $\alpha_{i,\tau} \in \Delta_\tau \setminus I_v$, the $\mathbf{B}_{\mathrm{dR}}^+$-representation $W_{A,i} \otimes_{K,\tau} L/W_{A,j+1} \otimes_{K,\tau} L$ is de Rham, where $j$ is the largest integer $< i$ such that $\alpha_{\tau,j} \notin I$ (and $j = 0$ if $i$ is the smallest integer such that $\alpha_{i,\tau} \notin I$). It is obvious from the definition that $X_{W_v, W_{\bullet,v}}^{P_{I_v}}$ is a subgroupoid of $X_{W_v, W_{\bullet,v}}$.

For an object $A$ of $\mathcal{C}_L$ and $r_A \in X_{r_v, \mathcal{R}_v}^{\mathrm{qtri}}(A)$, we denote by $\mathcal{M}_{A,\bullet}$ the unique triangulation of $D_{\mathrm{rig}}(r_A)$ lifting $\mathcal{M}_{\bullet,v}$. We say that $r_A$ is $\underline{P}_I$-de Rham if

$$ (W_{\mathrm{dR}}(r_A), W_{\mathrm{dR}}(\mathcal{M}_{A,\bullet})) \in X_{W_v, W_{\bullet,v}}^{P_{I_v}}(A) $$

(see [Wu, Def. 3.10]). It now follows from [Wu, Lemm. 3.11] that this functor is representable by a closed formal subscheme of $\mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}}$ that we denote $\mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \mathrm{qtri}}$. More precisely, we have an isomorphism of groupoids

$$ \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \mathrm{qtri}} \simeq \mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}} \times_{X_{W_v, W_{\bullet,v}}} X_{W_v, W_{\bullet,v}}^{P_{I_v}}. $$
**Lemma 4.5.** The irreducible components of $\mathfrak{z}_I \times_{\mathrm{t}/W} t$ are the $(T_{I,w})_{w \in W_I \setminus W}$ where

$$T_{I,w} = \{(z, \mathrm{Ad}(w^{-1})(z)) \mid z \in \mathfrak{z}_I\}.$$

Moreover, the irreducible component $X_{I,w}$ (resp. $X_{\mathfrak{p}_I,w}$) is the unique component of $X_I$ (resp. $X_{\mathfrak{p}_I}$) whose image under $\Theta_I$ (resp. $\Theta_{\mathfrak{p}_I}$) dominates $T_{I,w}$.

**Remark 4.6.** For future use, we make the following notational convention: When $F = \mathbb{Q}$, we have $\underline{G}_L = \mathrm{GL}_{n,L}$, we will use the notations $X_n$, $X_{n,I}$, $X_{n,I,w}$ etc. for the schemes $X$, $X_I$, $X_{I,w}$ etc.

## 4.2 Partially de Rham deformation rings

For each place $v|p$ of $F$, we fix $r_v : \mathrm{Gal}_{F_v} \to \mathrm{GL}_n(L)$ a framed $\varphi$-generic Hodge–Tate regular crystalline representation, that we assume that the $(\varphi, \Gamma)$-module $D_{\mathrm{rig}}(r_v)$ associated to $r_v$ is crystalline $\varphi$-generic with regular Hodge–Tate type in the sense of [HMS, §3.3&§3.4]. We also fix a refinement $\mathcal{R}_v = (\varphi_1, \dots, \varphi_n) \in L^n$ of $r_v$ (see loc. cit.). We will use the notation $r = (r_v)_{v|p}$ and $\mathcal{R} = (\mathcal{R}_v)_{v|p}$ and say that $r$ is $\varphi$-generic Hodge–Tate regular and that $\mathcal{R}$ is a refinement of $r$.

Let $\mathcal{C}_L$ be the category of local artinian $L$-algebras. Fix $v|p$ a place of $F$. Let $\mathcal{X}_{r_v}^{\square}$ be the groupoid over $\mathcal{C}_L$ of deformations of $r_v$. It is represented by a formal scheme over $L$ that we also denote by $\mathcal{X}_{r_v}^{\square}$ by abuse of notation. We recall from [BHS19, 3.6] that, given the refinement $\mathcal{R}_v$, the groupoid of trianguline deformations of $\mathcal{M}_{\bullet,v}$ is representable by a closed formal subscheme $\mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}} \subset \mathcal{X}_{r_v}^{\square}$. Here $\mathcal{M}_{\bullet,v}$ the $(\varphi, \Gamma)$-module over $\mathcal{R}_{K,L}[1/t]$ obtained from $D_{\mathrm{rig}}(r_v)$ by inverting $t$ which is equipped with the unique triangulation corresponding to the refinement $\mathcal{R}_v$. We set $W_v = W_{\mathrm{dR}}(D_{\mathrm{rig}}(r_v)[1/t])$ and $W_{\bullet,v} = W_{\mathrm{dR}}(\mathcal{M}_{\bullet,v})$ and let $X_{W_v, W_{\bullet,v}}$ denote the groupoid of deformations of $(W_v, W_{\bullet,v})$ as defined in [BHS19, §3.3].

Fix a finite subset $I_v \subset \Delta_v$. For an object $A$ of $\mathcal{C}_L$, we define $X_{W_v, W_{\bullet,v}}^{P_{I_v}}(A)$ to be the subset of all $(W_A, W_{A,\bullet}) \in X_{W_v, W_{\bullet,v}}(A)$ such that for any $\tau \in \Sigma_{F_v}$ and $\alpha_{i,\tau} \in \Delta_\tau \setminus I_v$, the $\mathbf{B}_{\mathrm{dR}}^+$-representation $W_{A,i} \otimes_{K,\tau} L/W_{A,j+1} \otimes_{K,\tau} L$ is de Rham, where $j$ is the largest integer $< i$ such that $\alpha_{\tau,j} \notin I$ (and $j = 0$ if $i$ is the smallest integer such that $\alpha_{i,\tau} \notin I$). It is obvious from the definition that $X_{W_v, W_{\bullet,v}}^{P_{I_v}}$ is a subgroupoid of $X_{W_v, W_{\bullet,v}}$.

For an object $A$ of $\mathcal{C}_L$ and $r_A \in X_{r_v, \mathcal{R}_v}^{\mathrm{qtri}}(A)$, we denote by $\mathcal{M}_{A,\bullet}$ the unique triangulation of $D_{\mathrm{rig}}(r_A)$ lifting $\mathcal{M}_{\bullet,v}$. We say that $r_A$ is $\underline{P}_I$-de Rham if

$$(W_{\mathrm{dR}}(r_A), W_{\mathrm{dR}}(\mathcal{M}_{A,\bullet})) \in X_{W_v, W_{\bullet,v}}^{P_{I_v}}(A)$$

(see [Wu, Def. 3.10]). It now follows from [Wu, Lemm. 3.11] that this functor is representable by a closed formal subscheme of $\mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}}$ that we denote $\mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \mathrm{qtri}}$. More precisely, we have an isomorphism of groupoids

$$\mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \mathrm{qtri}} \simeq \mathcal{X}_{r_v, \mathcal{R}_v}^{\mathrm{qtri}} \times_{X_{W_v, W_{\bullet,v}}} X_{W_v, W_{\bullet,v}}^{P_{I_v}}.$$
Fix an $L \otimes_{\mathbb{Q}_p} F_v$-basis $\alpha_v$ of $W_v^{\text{Gal}_K}$ and let $X_{W_v}^{\square}$ be the groupoid of deformations of the pair $(W_v, \alpha_v)$. We set

$$X_{W_v, W_{\bullet, v}}^{\square} = X_{W_v}^{\square} \times_{X_{W_v}} X_{W_v, W_{\bullet, v}} \\ X_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} = X_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}} \times_{X_{W_v}} X_{W_v}^{\square}.$$

As the map $\chi_{r_v, \mathcal{R}_v}^{\text{qtri}} \to X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet, v}}$ is formally smooth by [BHS19, Cor. 3.5.6], we deduce that the map $\chi_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet, v}}^{P_{I_v}, \square}$ is formally smooth as well.

If $I = \coprod_{v|p} I_v \subset \Delta$ and if $\alpha = (\alpha_v)_{v|p}$ is fixed, we set $\chi_{r, \mathcal{R}}^{I-\text{qtri}} := \coprod_{v|p} \chi_{r_v, \mathcal{R}_v}^{I_v-\text{qtri}}$ and $\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} := \coprod_{v|p} \chi_{r_v, \mathcal{R}_v}^{I_v-\text{qtri}, \square}$.

We consider the point

$$x_{\text{pdR}} := (g\underline{B}_L, 0, h\underline{B}_L) \in X_I(L) \subset (\underline{G}_L/\underline{B}_L \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L)(L), \quad (6)$$

where $g \in \underline{G}(L)$ (resp. $h$) is the matrix sending the standard flag (corresponding to our fixed basis $\alpha$) of $\coprod_{v|p} W_v^{\text{Gal}_K}$ to the complete flag $\coprod_{v|p} W_{\text{dR}}(\mathcal{M}_{\bullet, v})^{\text{Gal}_K}$ (resp. to the Hodge flag). We deduce the following result (see [BD, §6.3] in a slightly different context):

**Theorem 4.7.** There exists a diagram of formal L-schemes with formally smooth maps

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}} \leftarrow \chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \xrightarrow{f} \hat{X}_I, x_{\text{pdR}}$$

*Proof.* Let $I = \coprod_{v \in S_p} I_v$, with $I_v \subset \Delta_v$ for $v \in S_p$. Note that we have a decomposition $X_I \simeq \coprod_{v \in S_p} X_{I_v}$ where $X_{I_v}$ is the L-scheme defined in the same way as $X_I$ but for the group $\text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n, F_v}$. We also write $x_{\text{pdR}} = (x_{\text{pdR},v})_{v \in S_p}$ where $x_{\text{pdR},v}$ is the image of $x_{\text{pdR}}$ in $X_{I_v}$. We just have to check that the groupoid

$$X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet, v}}^{P_{I_v}} \times_{X_{W_v}} X_{W_v}^{\square}$$

is represented by the completion of $X_{I_v}$ at $x_{\text{pdR},v}$. This can be checked easily as in the proof of [Wu, Lemm. 3.11] using [BHS19, Cor. 3.1.9 &Thm. 3.2.5]. □

We finally note that the map $\kappa_1$ from above induces a map of formal schemes $\kappa_1 : \hat{X}_{I, x_{\text{pdR}}} \to \hat{\mathfrak{S}}_I$, where $\hat{\mathfrak{S}}_I$ is the completion of $\mathfrak{S}_I$ at 0, and thus a map

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \to \hat{\mathfrak{S}}_I.$$

This maps factors into a map of formal schemes $\kappa_1 : \chi_{r, \mathcal{R}}^{I-\text{qtri}} \to \hat{\mathfrak{S}}_I$.

For $w \in W$ such that $x_{\text{pdR}} \in X_{I,w}(L)$, we denote by $\chi_{r, \mathcal{R}}^{I-\text{qtri}, w}$ the schematic image of

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \times_{\hat{X}_{I, x_{\text{pdR}}}} \hat{X}_{I, w, x_{\text{pdR}}} \to \chi_{r, \mathcal{R}}^{I-\text{qtri}}$$
Fix an $L \otimes_{\mathbb{Q}_p} F_v$-basis $\alpha_v$ of $W_v^{\text{Gal}_K}$ and let $X_{W_v}^{\square}$ be the groupoid of deformations of the pair $(W_v, \alpha_v)$. We set

$$X_{W_v, W_{\bullet,v}}^{\square} = X_{W_v}^{\square} \times_{X_{W_v}} X_{W_v, W_{\bullet,v}}$$
$$\chi_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} = \chi_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}} \times_{X_{W_v}} X_{W_v}^{\square}.$$

As the map $\chi_{r_v, \mathcal{R}_v}^{\text{qtri}} \to X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet,v}}$ is formally smooth by [BHS19, Cor. 3.5.6], we deduce that the map $\chi_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet,v}}^{P_{I_v}, \square}$ is formally smooth as well.

If $I = \coprod_{v|p} I_v \subset \Delta$ and if $\alpha = (\alpha_v)_{v|p}$ is fixed, we set $\chi_{r, \mathcal{R}}^{I-\text{qtri}} := \coprod_{v|p} \chi_{r_v, \mathcal{R}_v}^{I_v-\text{qtri}}$ and $\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} := \coprod_{v|p} \chi_{r_v, \mathcal{R}_v}^{I_v-\text{qtri}, \square}$.

We consider the point

$$x_{\text{pdR}} := (g\underline{B}_L, 0, h\underline{B}_L) \in X_I(L) \subset (\underline{G}_L/\underline{B}_L \times \mathfrak{g} \times \underline{G}_L/\underline{B}_L)(L), \quad (6)$$

where $g \in \underline{G}(L)$ (resp. $h$) is the matrix sending the standard flag (corresponding to our fixed basis $\alpha$) of $\coprod_{v|p} W_v^{\text{Gal}_K}$ to the complete flag $\coprod_{v|p} W_{\text{dR}}(\mathcal{M}_{\bullet,v})^{\text{Gal}_K}$ (resp. to the Hodge flag). We deduce the following result (see [BD, §6.3] in a slightly different context):

**Theorem 4.7.** There exists a diagram of formal L-schemes with formally smooth maps

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}} \leftarrow \chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \xrightarrow{f} \hat{X}_I, x_{\text{pdR}}$$

*Proof.* Let $I = \coprod_{v \in S_p} I_v$, with $I_v \subset \Delta_v$ for $v \in S_p$. Note that we have a decomposition $X_I \simeq \coprod_{v \in S_p} X_{I_v}$ where $X_{I_v}$ is the L-scheme defined in the same way as $X_I$ but for the group $\text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n, F_v}$. We also write $x_{\text{pdR}} = (x_{\text{pdR},v})_{v \in S_p}$ where $x_{\text{pdR},v}$ is the image of $x_{\text{pdR}}$ in $X_{I_v}$. We just have to check that the groupoid

$$X_{W_v^+} \times_{X_{W_v}} X_{W_v, W_{\bullet,v}}^{P_{I_v}} \times_{X_{W_v}} X_{W_v}^{\square}$$

is represented by the completion of $X_{I_v}$ at $x_{\text{pdR},v}$. This can be checked easily as in the proof of [Wu, Lemm. 3.11] using [BHS19, Cor. 3.1.9 &Thm. 3.2.5]. □

We finally note that the map $\kappa_1$ from above induces a map of formal schemes $\kappa_1 : \hat{X}_{I, x_{\text{pdR}}} \to \hat{\mathfrak{S}}_I$, where $\hat{\mathfrak{S}}_I$ is the completion of $\mathfrak{S}_I$ at 0, and thus a map

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \to \hat{\mathfrak{S}}_I.$$

This maps factors into a map of formal schemes $\kappa_1 : \chi_{r, \mathcal{R}}^{I-\text{qtri}} \to \hat{\mathfrak{S}}_I$.

For $w \in W$ such that $x_{\text{pdR}} \in X_{I,w}(L)$, we denote by $\chi_{r, \mathcal{R}}^{I-\text{qtri}, w}$ the schematic image of

$$\chi_{r, \mathcal{R}}^{I-\text{qtri}, \square} \times_{\hat{X}_{I, x_{\text{pdR}}}} \hat{X}_{I, w, x_{\text{pdR}}} \to \chi_{r, \mathcal{R}}^{I-\text{qtri}}$$
and by $\overline{\mathcal{X}}_{r,\mathcal{R}}^{\text{qtri}}$ (resp. $\overline{\mathcal{X}}_{r,\mathcal{R}}^{I-\text{qtri},w}$) the schematic inverse image of $\{0\}$ under $\kappa_1$ in $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri}}$ (resp. $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri},w}$).

The schemes $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri}}$ and $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri},w}$ are formal spectra of complete local noetherian rings that we denote by $R_{r,\mathcal{R}}^{I-\text{qtri}}$ and $R_{r,\mathcal{R}}^{I-\text{qtri},w}$. It follows from the constructions that moreover $R_{r,\mathcal{R}}^{I-\text{qtri},w}$ is an integral local ring.

# 5 Global construction

Let $F$ be a totally real number field and let $E/F$ be a totally imaginary CM extension of number fields, in particular $[E:F] = 2$. We assume that all places of $F$ dividing $p$ are unramified and split in $E/F$ and denote by $S_p$ the set of places above $p$ in $F$. We fix a set $\Sigma$ of places of $E$ dividing $p$ such that, for each place $v \in S_p$, there is exactly one place of $\Sigma$ above $v$. Let $U$ be a unitary group in $n$ variables for $E/F$ that we regard, via Weil restriction, as an algebraic group over $\mathbb{Q}$. We assume that $U(\mathbb{R})$ is compact and and that $U_{\mathbb{Q}_p}$ is quasi-split. This implies in particular that there exists an isomorphism $U_{\mathbb{Q}_p} \simeq \prod_{v \in S_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v}$ that we fix from now on. From now we note $\underline{G} = U_{\mathbb{Q}_p}$ identified with $\prod_{v \in S_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v}$ via this fixed isomorphism and we use notations of section 3, i.e. $L$ is the choice of a field of coefficients that is assumed to be big enough so that all embeddings of $E$ (equivalently of $F$) in $\overline{\mathbb{Q}}_p$ factor through $L$. Moreover, $\underline{B} \subset \underline{G}_L$ is the Borel subgroup of upper triangular matrices, $\underline{T} \subset \underline{B}$ is the maximal torus of diagonal matrices, $\underline{N}$ is the unipotent radical of $\underline{B}$ etc.

## 5.1 Classical and p-adic automorphic forms

We write $T = \underline{T}(\mathbb{Q}_p) \simeq (\prod_{v \in S_p} F_v^\times)^n$ and let $T_0 \simeq (\prod_{v \in S_p} \mathcal{O}_{F_v}^\times)^n \subset T$ denote its maximal compact subgroup. We denote by $\hat{T}$ (resp. $\hat{T}_0$) the rigid analytic spaces over $L$ parametrizing the continuous characters of $T$ (resp. of $T_0$) and recall from 5 that there is a weight map

$$ \text{wt} : \hat{T}_0 \to t^* $$

with values in the dual Lie algebra $t^*$ of $\underline{T}$ (considered as a rigid space over $L$). We will often, by abuse of notation, also write wt for the composition of wt with the canonical projection $\hat{T} \to \hat{T}_0$. Recall that we had identified $X^*(\underline{T})$ with a $\mathbb{Z}$-lattice in $t^*$. Often we will identify $X^*(\underline{T})$ with $\mathbb{Z}^{n[F:\mathbb{Q}]}$.

**Definition 5.1.** Let $\delta \in \hat{T}$ (resp. $\in \hat{T}_0$) be a character.

(i) The weight of $\delta$ is the image $\text{wt}(\delta)$ under the weight map.

(ii) The character $\delta$ is called *algebraic weight* if $\text{wt}(\delta) \in X^*(\underline{T}) \subset t^*$.

(iii) The character $\delta$ is called *algebraic* if it is of the form

$$ \delta_{\underline{k}} : (z_1 \otimes 1, \dots, z_n \otimes 1) \mapsto \prod_{\tau} (\tau(z_1)^{k_1^{\tau}} \cdots \tau(z_n)^{k_n^{\tau}}) $$
and by $\overline{\mathcal{X}}_{r,\mathcal{R}}^{\text{qtri}}$ (resp. $\overline{\mathcal{X}}_{r,\mathcal{R}}^{I-\text{qtri},w}$) the schematic inverse image of $\{0\}$ under $\kappa_1$ in $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri}}$ (resp. $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri},w}$).

The schemes $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri}}$ and $\mathcal{X}_{r,\mathcal{R}}^{I-\text{qtri},w}$ are formal spectra of complete local noetherian rings that we denote by $R_{r,\mathcal{R}}^{I-\text{qtri}}$ and $R_{r,\mathcal{R}}^{I-\text{qtri},w}$. It follows from the constructions that moreover $R_{r,\mathcal{R}}^{I-\text{qtri},w}$ is an integral local ring.

# 5 Global construction

Let $F$ be a totally real number field and let $E/F$ be a totally imaginary CM extension of number fields, in particular $[E:F] = 2$. We assume that all places of $F$ dividing $p$ are unramified and split in $E/F$ and denote by $S_p$ the set of places above $p$ in $F$. We fix a set $\Sigma$ of places of $E$ dividing $p$ such that, for each place $v \in S_p$, there is exactly one place of $\Sigma$ above $v$. Let $U$ be a unitary group in $n$ variables for $E/F$ that we regard, via Weil restriction, as an algebraic group over $\mathbb{Q}$. We assume that $U(\mathbb{R})$ is compact and and that $U_{\mathbb{Q}_p}$ is quasi-split. This implies in particular that there exists an isomorphism $U_{\mathbb{Q}_p} \simeq \prod_{v \in S_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v}$ that we fix from now on. From now we note $\underline{G} = U_{\mathbb{Q}_p}$ identified with $\prod_{v \in S_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v}$ via this fixed isomorphism and we use notations of section 3, i.e. $L$ is the choice of a field of coefficients that is assumed to be big enough so that all embeddings of $E$ (equivalently of $F$) in $\overline{\mathbb{Q}}_p$ factor through $L$. Moreover, $\underline{B} \subset \underline{G}_L$ is the Borel subgroup of upper triangular matrices, $\underline{T} \subset \underline{B}$ is the maximal torus of diagonal matrices, $\underline{N}$ is the unipotent radical of $\underline{B}$ etc.

## 5.1 Classical and p-adic automorphic forms

We write $T = \underline{T}(\mathbb{Q}_p) \simeq (\prod_{v \in S_p} F_v^\times)^n$ and let $T_0 \simeq (\prod_{v \in S_p} \mathcal{O}_{F_v}^\times)^n \subset T$ denote its maximal compact subgroup. We denote by $\hat{T}$ (resp. $\hat{T}_0$) the rigid analytic spaces over $L$ parametrizing the continuous characters of $T$ (resp. of $T_0$) and recall from 5 that there is a weight map

$$ \text{wt} : \hat{T}_0 \to \mathfrak{t}^* $$

with values in the dual Lie algebra $\mathfrak{t}^*$ of $\underline{T}$ (considered as a rigid space over $L$). We will often, by abuse of notation, also write wt for the composition of wt with the canonical projection $\hat{T} \to \hat{T}_0$. Recall that we had identified $X^*(\underline{T})$ with a $\mathbb{Z}$-lattice in $\mathfrak{t}^*$. Often we will identify $X^*(\underline{T})$ with $\mathbb{Z}^{n[F:\mathbb{Q}]}$.

**Definition 5.1.** Let $\delta \in \hat{T}$ (resp. $\in \hat{T}_0$) be a character.

(i) The weight of $\delta$ is the image $\text{wt}(\delta)$ under the weight map.

(ii) The character $\delta$ is called of algebraic weight if $\text{wt}(\delta) \in X^*(\underline{T}) \subset \mathfrak{t}^*$.

(iii) The character $\delta$ is called algebraic if it is of the form

$$ \delta_{\underline{k}} : (z_1 \otimes 1, \dots, z_n \otimes 1) \mapsto \prod_{\tau} (\tau(z_1)^{k_1^{\tau}} \cdots \tau(z_n)^{k_n^{\tau}}) $$
for some $\underline{k} = (k_1^\tau, \dots, k_n^\tau)_{\tau:F \hookrightarrow L} \in \mathbb{Z}^{n[F:\mathbb{Q}]}$. It is called dominant algebraic if $\underline{k} \in X^*(\underline{T})^+$, i.e. if $k_1^\tau \ge \dots \ge k_n^\tau$ for all $\tau$.

Note that $\underline{k} \mapsto \delta_{\underline{k}}$ defines a section of the weight map over the algebraic weights, and we use this map to identify $X^*(\underline{T})$ with a subset of $\hat{T}$ (resp. $\hat{T}_0$).

Let $K^p \subset U(\mathbb{A}^{\infty,p})$ be a compact open subgroup, called a *tame level* that we assume to be of the form $\prod_{\ell \ne p} K_\ell$ where $K_\ell$ is a compact open subgroup of $U(\mathbb{Q}_\ell)$. Let $I_p$ be the Iwahori subgroup of $G = \mathcal{G}(\mathbb{Q}_p) = U(\mathbb{Q}_p)$ with respect to our choice of $\mathcal{B}$. For any compact open $K_p \subset U(\mathbb{Q}_p)$ we consider the Shimura set

$$ \mathrm{Sh}_{K^p K_p} := U(\mathbb{Q}) \backslash U(\mathbb{A}^{\infty}) / K^p K_p. $$

As $U(\mathbb{R})$ is compact, this is indeed a finite set of points.

**Definition 5.2.** The completed cohomology of the tower $(\mathrm{Sh}_{K^p K_p})_{K_p \subset U(\mathbb{Q}_p)}$ of Shimura sets is:

$$ \Pi := \Pi^\circ \otimes_{\mathcal{O}_L} L, \quad \text{with} \quad \Pi^\circ := \varprojlim_{\substack{n \\ \mathcal{N} \in \mathbb{N}(\mathbb{Q}_p)}} \varprojlim_{K_p} H^0(\mathrm{Sh}_{K^p K_p}, \mathcal{O}_L / \pi_L^n), $$

see [Eme06b].

The completed cohomology is an $L$-Banach space endowed with a continuous action of $U(\mathbb{Q}_p)$. This space is naturally identified with the space of continuous functions

$$ f : U(\mathbb{Q}) \backslash U(\mathbb{A}^{\infty}) / K^p \to L. \qquad (7) $$

We denote $\Pi^{\mathrm{la}}$ the subspace of locally analytic vectors in $\Pi$ for $U(\mathbb{Q}_p)$. This is the subspace of functions in (7) which are locally analytic. As $\Pi^{\mathrm{la}}$ is a locally analytic representation, there is a natural $U(\mathfrak{g})$-action on $\Pi^{\mathrm{la}}$ obtained by deriving the $G = \mathcal{G}(\mathbb{Q}_p)$-action. Here, as above, we write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{b}, \mathfrak{t}, \mathfrak{n}$ for the Lie algebras of the Borel $B$, of the torus $T$ and of the unipotent radical $N$ of $B$.

**Definition 5.3.** The space of overconvergent $p$-adic automorphic forms of tame weight $K^p$ is the space

$$ S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n = \varinjlim_{N_0 \subset \mathcal{N}(\mathbb{Q}_p)} (\Pi^{\mathrm{la}})^{N_0}, $$

where $N_0$ varies among the compact open subgroups of $\mathcal{N}(\mathbb{Q}_p)$. Given a weight $\kappa \in \mathfrak{t}^*$, the space of overconvergent $p$-adic automorphic forms of tame weight $K^p$ and weight $\kappa$ is the eigenspace

$$ S_{\kappa}^{\dagger}(K^p) \subset S^{\dagger}(K^p) $$

of eigenvalue $\kappa$ for the $U(\mathfrak{t})$-action.

Denote by $\mathbb{T}(K^p) := \mathbb{Z}[K^p \backslash U(\mathbb{A}^{\infty,p})/K^p]$ the Hecke algebra of Hecke operators over $\mathbb{Z}$ of tame level $K^p$. Then $\mathbb{T}(K^p)$ acts by convolution on $S^\dagger(K^p)$ and $S_\kappa^\dagger(K^p)$. Let $S$ be a finite set of prime numbers containing $p$ and all the $\ell$ such that $K_\ell$ is not hyperspecial. The subalgebra $\mathbb{T}^S := \bigotimes_{\ell \notin S} \mathbb{T}_\ell \subset \mathbb{T}(K^p)$ is commutative.
for some $\underline{k} = (k_1^\tau, \dots, k_n^\tau)_{\tau:F \hookrightarrow L} \in \mathbb{Z}^{n[F:\mathbb{Q}]}$. It is called dominant algebraic if $\underline{k} \in X^*(\underline{T})^+$, i.e. if $k_1^\tau \ge \dots \ge k_n^\tau$ for all $\tau$.

Note that $\underline{k} \mapsto \delta_{\underline{k}}$ defines a section of the weight map over the algebraic weights, and we use this map to identify $X^*(\underline{T})$ with a subset of $\hat{T}$ (resp. $\hat{T}_0$).

Let $K^p \subset U(\mathbb{A}^{\infty,p})$ be a compact open subgroup, called a *tame level* that we assume to be of the form $\prod_{\ell \ne p} K_\ell$ where $K_\ell$ is a compact open subgroup of $U(\mathbb{Q}_\ell)$. Let $I_p$ be the Iwahori subgroup of $G = \mathcal{G}(\mathbb{Q}_p) = U(\mathbb{Q}_p)$ with respect to our choice of $\mathcal{B}$. For any compact open $K_p \subset U(\mathbb{Q}_p)$ we consider the Shimura set

$$ \mathrm{Sh}_{K^p K_p} := U(\mathbb{Q}) \backslash U(\mathbb{A}^\infty) / K^p K_p. $$

As $U(\mathbb{R})$ is compact, this is indeed a finite set of points.

**Definition 5.2.** The completed cohomology of the tower $(\mathrm{Sh}_{K^p K_p})_{K_p \subset U(\mathbb{Q}_p)}$ of Shimura sets is:

$$ \Pi := \Pi^\circ \otimes_{\mathcal{O}_L} L, \quad \text{with} \quad \Pi^\circ := \varprojlim_n \varinjlim_{K_p} H^0(\mathrm{Sh}_{K^p K_p}, \mathcal{O}_L / \pi_L^n), $$

see [Eme06b].

The completed cohomology is an $L$-Banach space endowed with a continuous action of $U(\mathbb{Q}_p)$. This space is naturally identified with the space of continuous functions

$$ f : U(\mathbb{Q}) \backslash U(\mathbb{A}^\infty) / K^p \to L. \qquad (7) $$

We denote $\Pi^{\mathrm{la}}$ the subspace of locally analytic vectors in $\Pi$ for $U(\mathbb{Q}_p)$. This is the subspace of functions in (7) which are locally analytic. As $\Pi^{\mathrm{la}}$ is a locally analytic representation, there is a natural $U(\mathfrak{g})$-action on $\Pi^{\mathrm{la}}$ obtained by deriving the $G = \mathcal{G}(\mathbb{Q}_p)$-action. Here, as above, we write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{b}, \mathfrak{t}, \mathfrak{n}$ for the Lie algebras of the Borel $B$, of the torus $T$ and of the unipotent radical $N$ of $B$.

**Definition 5.3.** The space of overconvergent $p$-adic automorphic forms of tame weight $K^p$ is the space

$$ S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n = \varinjlim_{N_0 \subset \mathcal{N}(\mathbb{Q}_p)} (\Pi^{\mathrm{la}})^{N_0}, $$

where $N_0$ varies among the compact open subgroups of $\mathcal{N}(\mathbb{Q}_p)$. Given a weight $\kappa \in \mathfrak{t}^*$, the space of overconvergent $p$-adic automorphic forms of tame weight $K^p$ and weight $\kappa$ is the eigenspace

$$ S_{\kappa}^{\dagger}(K^p) \subset S^{\dagger}(K^p) $$

of eigenvalue $\kappa$ for the $U(\mathfrak{t})$-action.

Denote by $\mathbb{T}(K^p) := \mathbb{Z}[K^p \backslash U(\mathbb{A}^{\infty,p}) / K^p]$ the Hecke algebra of Hecke operators over $\mathbb{Z}$ of tame level $K^p$. Then $\mathbb{T}(K^p)$ acts by convolution on $S^\dagger(K^p)$ and $S_\kappa^\dagger(K^p)$. Let $S$ be a finite set of prime numbers containing $p$ and all the $\ell$ such that $K_\ell$ is not hyperspecial. The subalgebra $\mathbb{T}^S := \bigotimes_{\ell \notin S} \mathbb{T}_\ell \subset \mathbb{T}(K^p)$ is commutative.
**Definition 5.4.** Let

$$ \underline{T}(\mathbb{Q}_p)^+ := \{ \operatorname{diag}(a_1^v, \dots, a_n^v)_v \in \underline{T}(\mathbb{Q}_p) \mid v(a_1^v) \ge \dots \ge v(a_n^v), \forall v \in S_p \}. $$

The Atkin–Lehner ring $\mathcal{A}(p)$ is the sub-algebra of $\mathbb{Z}[\underline{T}(\mathbb{Q}_p)]$ generated by the elements $t \in \underline{T}(\mathbb{Q}_p)^+$.

Let $\delta : T \to L^\times$ be a continuous character. Then we can extend $\delta$ to a character $\mathcal{A}(p) \to L$ whose restriction to $T^+$ is given by $\delta$. By abuse of notation we still write $\delta$ for this character of $\mathcal{A}(p)$.

Note that there is a cofinal system of compact open subgroups $N_0 \subset N = \underline{N}(\mathbb{Q}_p)$ such that $tN_0t^{-1} \subset N_0$ for all $t \in T^+$. We hence can define a Hecke action of $\mathcal{A}(p)$ on $S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n$ by letting $t \in \underline{T}(\mathbb{Q}_p)^+$ act on $f \in (\Pi^{\mathrm{la}})^{N_0}$ via

$$ [t]f := \left( x \mapsto \frac{1}{[N_0 : tN_0t^{-1}]} \sum_{n \in N_0/tN_0t^{-1}} f(xnt) \right), $$

where $N_0$ is a sufficiently small compact open subgroup of $N$ such that $f \in (\Pi^{\mathrm{la}})^{N_0}$ and such that $tN_0t^{-1} \subset N_0$.

Let $\mathbb{T}$ be the commutative algebra $\mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$. Definition 5.4 provides a structure of $\mathbb{T}$-module on $S^\dagger(K^p)$ and $S_\kappa^\dagger(K^p)$.

**Definition 5.5.** An overconvergent $p$-adic automorphic form $f \in S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n$ is called a *finite slope eigenvector* for the $\mathcal{A}(p)$-action if, for any $t \in \underline{T}(\mathbb{Q}_p)^+$, there exists $a_t \in L^\times$ such that

$$ [t]f = a_t f. $$

More generally $f$ is of *finite slope* for the $\mathcal{A}(p)$-action if for all $t \in \underline{T}(\mathbb{Q}_p)^+$, there exists a polynomial $P \in L[X]$ such that $P(0) \neq 0$ and $P([t])f = 0$.

Given a continuous character $\delta : T \to L^\times$, we write $S^\dagger(K^p)[\delta]$ for the eigenspace with respect to the $\mathcal{A}(p)$-action of eigensystem $\delta : \mathcal{A}(p) \to L$. Note, that by definition this eigensystem is automatically of finite slope and of weight $\kappa = \mathrm{wt}(\delta)$. Moreover, the $\mathcal{A}(p)$-action on $S^\dagger(K^p)[\delta]$ uniquely extends to an action of $\mathbb{Z}[T(\mathbb{Q}_p)]$.

**Remark 5.6.** An overconvergent automorphic form of tame level $K^p$ with eigenvalue $\delta : T \to L^\times$ for the Hecke-action at $p$ (i.e. for the action of the Atkin–Lehner ring) is thus the same as a locally analytic function

$$ f : U(\mathbb{Q}) \backslash U(\mathbb{A}^\infty) / K^p \to L, $$

such that there exists a compact open subgroup $N_0 \subset \underline{N}(\mathbb{Q}_p)$ so that, for all $g \in U(\mathbb{A}^\infty)$, $t \in T_0$, $n \in N_0$,

$$ f(gtn) = \delta(t)f(g), $$

and such that moreover, for all $t \in \underline{T}(\mathbb{Q}_p)^+$, $[t]f = \delta(t)f$.
**Definition 5.4.** Let

$$ \underline{T}(\mathbb{Q}_p)^+ := \{ \operatorname{diag}(a_1^v, \dots, a_n^v)_v \in \underline{T}(\mathbb{Q}_p) \mid v(a_1^v) \ge \dots \ge v(a_n^v), \forall v \in S_p \}. $$

The Atkin–Lehner ring $\mathcal{A}(p)$ is the sub-algebra of $\mathbb{Z}[\underline{T}(\mathbb{Q}_p)]$ generated by the elements $t \in \underline{T}(\mathbb{Q}_p)^+$.

Let $\delta : T \to L^\times$ be a continuous character. Then we can extend $\delta$ to a character $\mathcal{A}(p) \to L$ whose restriction to $T^+$ is given by $\delta$. By abuse of notation we still write $\delta$ for this character of $\mathcal{A}(p)$.

Note that there is a cofinal system of compact open subgroups $N_0 \subset N = \underline{N}(\mathbb{Q}_p)$ such that $tN_0t^{-1} \subset N_0$ for all $t \in T^+$. We hence can define a Hecke action of $\mathcal{A}(p)$ on $S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n$ by letting $t \in \underline{T}(\mathbb{Q}_p)^+$ act on $f \in (\Pi^{\mathrm{la}})^{N_0}$ via

$$ [t]f := \left( x \mapsto \frac{1}{[N_0 : tN_0t^{-1}]} \sum_{n \in N_0/tN_0t^{-1}} f(xnt) \right), $$

where $N_0$ is a sufficiently small compact open subgroup of $N$ such that $f \in (\Pi^{\mathrm{la}})^{N_0}$ and such that $tN_0t^{-1} \subset N_0$.

Let $\mathbb{T}$ be the commutative algebra $\mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$. Definition 5.4 provides a structure of $\mathbb{T}$-module on $S^\dagger(K^p)$ and $S_\kappa^\dagger(K^p)$.

**Definition 5.5.** An overconvergent $p$-adic automorphic form $f \in S^\dagger(K^p) = (\Pi^{\mathrm{la}})^n$ is called a *finite slope eigenvector* for the $\mathcal{A}(p)$-action if, for any $t \in \underline{T}(\mathbb{Q}_p)^+$, there exists $a_t \in L^\times$ such that

$$ [t]f = a_t f. $$

More generally $f$ is of *finite slope* for the $\mathcal{A}(p)$-action if for all $t \in \underline{T}(\mathbb{Q}_p)^+$, there exists a polynomial $P \in L[X]$ such that $P(0) \neq 0$ and $P([t])f = 0$.

Given a continuous character $\delta : T \to L^\times$, we write $S^\dagger(K^p)[\delta]$ for the eigenspace with respect to the $\mathcal{A}(p)$-action of eigensystem $\delta : \mathcal{A}(p) \to L$. Note, that by definition this eigensystem is automatically of finite slope and of weight $\kappa = \mathrm{wt}(\delta)$. Moreover, the $\mathcal{A}(p)$-action on $S^\dagger(K^p)[\delta]$ uniquely extends to an action of $\mathbb{Z}[T(\mathbb{Q}_p)]$.

**Remark 5.6.** An overconvergent automorphic form of tame level $K^p$ with eigenvalue $\delta : T \to L^\times$ for the Hecke-action at $p$ (i.e. for the action of the Atkin–Lehner ring) is thus the same as a locally analytic function

$$ f : U(\mathbb{Q}) \backslash U(\mathbb{A}^\infty) / K^p \to L, $$

such that there exists a compact open subgroup $N_0 \subset \underline{N}(\mathbb{Q}_p)$ so that, for all $g \in U(\mathbb{A}^\infty)$, $t \in T_0$, $n \in N_0$,

$$ f(gtn) = \delta(t)f(g), $$

and such that moreover, for all $t \in \underline{T}(\mathbb{Q}_p)^+$, $[t]f = \delta(t)f$.
**Definition 5.7.** The space of classical automorphic forms of tame level $K^p$ is the subspace $S^{\text{cl}}(K^p) = (\Pi^{\text{cl}})^n$ of $S^\dagger(K^p) = (\Pi^{\text{la}})^n$ of elements which are $K_p$-finite for some (resp. any) compact open $K_p \subset U(\mathbb{Q}_p)$.

We note that this subspace is stable under the action of $\mathbb{T}$.

For any character $\chi^S: \mathbb{T}^S \to L$, we let $\Pi[\chi^S]$ (resp. $S^\dagger(K^p)[\chi^S]$, resp. $S^{\text{cl}}(K^p)[\chi^S]$) denote the subspace of $\chi^S$-eigenvectors for $\mathbb{T}^S$ in $\Pi$ (resp. $S^\dagger(K^p)$, resp. $S^{\text{cl}}(K^p)$). If $\delta: T \to L$ is a character of $T$ (defining a character of $\mathcal{A}(p)$) and if $\chi = \chi^S \otimes \delta$ is the corresponding character of $\mathbb{T} = \mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$, we write $S^\dagger(K^p)[\chi]$ etc. for the corresponding eigenspace.

Let $\mathfrak{m}$ be a maximal ideal in $\mathbb{T}^S$. We then define

$$ \Pi_{\mathfrak{m}} := \Pi_{\mathfrak{m}}^{\circ} \otimes_{\mathcal{O}_L} L, \quad \text{where} \quad \Pi_{\mathfrak{m}}^{\circ} := \varinjlim_{n} (\Pi^{\circ} / \pi_L^n \Pi^{\circ})_{\mathfrak{m}}. $$

As there are only finitely many maximal ideals $\mathfrak{m}$ of $\mathbb{T}^S$ such that $(\Pi^{\circ}/\pi_L\Pi^{\circ})_{\mathfrak{m}}$ is nonzero, the space $\Pi_{\mathfrak{m}}$ is a topological direct summand of $\Pi$ stable under the actions of $U(\mathbb{Q}_p)$ and $\mathbb{T}$.

Recall that if $\mathfrak{m}$ is a maximal ideal (whose residue field is assumed to equal $k_L$) such that $\Pi_{\mathfrak{m}}$ is non zero, then we may associate to $\mathfrak{m}$ a continuous representation $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ which is conjugate autodual, and unramified away from $S$. Such representations $\bar{\rho}$ are called modular (see for example [BHS17b, §2.4]).

## 5.2 Patching the completed cohomology

We fix a maximal ideal $\mathfrak{m} \subset \mathbb{T}^S$ such that $\Pi_{\mathfrak{m}} \neq 0$ is non-zero and denote by $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ the corresponding modular Galois representation. For each place $v$ of $F$ which splits in $E$ we write

$$ \bar{\rho}_v := \bar{\rho}|_{\text{Gal}_{E\bar{v}}}, $$

for a choice of $\tilde{v}|v$ of $E$. From now on we assume that, for $v \in S$, the place $v$ splits in $E/F$, we make a fixed choice $\tilde{v}|v$ as before such that $\tilde{v} \in \Sigma$ if $v|p$, and denote $\tilde{S} = \{\tilde{v}|v \in S\}$ so that $\tilde{S}$ is in bijection with $S$ and contains $\Sigma$. For $v \in S$ we write $R_{\tilde{\rho}_v}^{\square}$ for the universal lifting (i.e. framed deformation) ring of $\bar{\rho}_v$ and define

$$ R_{\tilde{\rho}_v}^{\square} \to \bar{R}_{\tilde{\rho}_v}^{\square} $$

to be the maximal reduced $\mathbb{Z}_p$-flat quotient.

*Remark 5.8.* If $v|p$ we have in fact, by the main results of [BIP23], $\bar{R}_{\tilde{\rho}_v}^{\square} = R_{\tilde{\rho}_v}^{\square}$. Using the main result of [DHKM24] we find that the same applies to places $v \nmid p$, as the deformation rings $R_{\tilde{\rho}_v}^{\square}$ may be identified with versal rings to the moduli space of L-parameters. We still keep the notations introduced above in order to be consistent with the notations from the references for the patching construction below.
**Definition 5.7.** The space of classical automorphic forms of tame level $K^p$ is the subspace $S^{\text{cl}}(K^p) = (\Pi^{\text{cl}})^n$ of $S^\dagger(K^p) = (\Pi^{\text{la}})^n$ of elements which are $K_p$-finite for some (resp. any) compact open $K_p \subset U(\mathbb{Q}_p)$.

We note that this subspace is stable under the action of $\mathbb{T}$.

For any character $\chi^S: \mathbb{T}^S \to L$, we let $\Pi[\chi^S]$ (resp. $S^\dagger(K^p)[\chi^S]$, resp. $S^{\text{cl}}(K^p)[\chi^S]$) denote the subspace of $\chi^S$-eigenvectors for $\mathbb{T}^S$ in $\Pi$ (resp. $S^\dagger(K^p)$, resp. $S^{\text{cl}}(K^p)$). If $\delta: T \to L$ is a character of $T$ (defining a character of $\mathcal{A}(p)$) and if $\chi = \chi^S \otimes \delta$ is the corresponding character of $\mathbb{T} = \mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$, we write $S^\dagger(K^p)[\chi]$ etc. for the corresponding eigenspace.

Let $\mathfrak{m}$ be a maximal ideal in $\mathbb{T}^S$. We then define

$$ \Pi_{\mathfrak{m}} := \Pi_{\mathfrak{m}}^{\circ} \otimes_{\mathcal{O}_L} L, \quad \text{where} \quad \Pi_{\mathfrak{m}}^{\circ} := \varinjlim_{n} (\Pi^{\circ} / \pi_L^n \Pi^{\circ})_{\mathfrak{m}}. $$

As there are only finitely many maximal ideals $\mathfrak{m}$ of $\mathbb{T}^S$ such that $(\Pi^{\circ}/\pi_L\Pi^{\circ})_{\mathfrak{m}}$ is nonzero, the space $\Pi_{\mathfrak{m}}$ is a topological direct summand of $\Pi$ stable under the actions of $U(\mathbb{Q}_p)$ and $\mathbb{T}$.

Recall that if $\mathfrak{m}$ is a maximal ideal (whose residue field is assumed to equal $k_L$) such that $\Pi_{\mathfrak{m}}$ is non zero, then we may associate to $\mathfrak{m}$ a continuous representation $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ which is conjugate autodual, and unramified away from $S$. Such representations $\bar{\rho}$ are called modular (see for example [BHS17b, §2.4]).

## 5.2 Patching the completed cohomology

We fix a maximal ideal $\mathfrak{m} \subset \mathbb{T}^S$ such that $\Pi_{\mathfrak{m}} \neq 0$ is non-zero and denote by $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ the corresponding modular Galois representation. For each place $v$ of $F$ which splits in $E$ we write

$$ \bar{\rho}_v := \bar{\rho}|_{\text{Gal}_{E\bar{v}}}, $$

for a choice of $\tilde{v}|v$ of $E$. From now on we assume that, for $v \in S$, the place $v$ splits in $E/F$, we make a fixed choice $\tilde{v}|v$ as before such that $\tilde{v} \in \Sigma$ if $v|p$, and denote $\tilde{S} = \{\tilde{v}|v \in S\}$ so that $\tilde{S}$ is in bijection with $S$ and contains $\Sigma$. For $v \in S$ we write $R_{\tilde{\rho}_v}^{\square}$ for the universal lifting (i.e. framed deformation) ring of $\bar{\rho}_v$ and define

$$ R_{\tilde{\rho}_v}^{\square} \to \bar{R}_{\tilde{\rho}_v}^{\square} $$

to be the maximal reduced $\mathbb{Z}_p$-flat quotient.

*Remark 5.8.* If $v|p$ we have in fact, by the main results of [BIP23], $\bar{R}_{\tilde{\rho}_v}^{\square} = R_{\tilde{\rho}_v}^{\square}$. Using the main result of [DHKM24] we find that the same applies to places $v \nmid p$, as the deformation rings $R_{\tilde{\rho}_v}^{\square}$ may be identified with versal rings to the moduli space of L-parameters. We still keep the notations introduced above in order to be consistent with the notations from the references for the patching construction below.
We denote by $R_{\bar{\rho},S}$ the quotient of $R_{\bar{\rho}}$ corresponding to the deformation problem

$$S = (E/F, S, \tilde{S}, \mathcal{O}_L, \bar{\rho}, \varepsilon^{1-n} \delta_{E/F}^n, \{\overline{R_{\bar{\rho}_v}}\}_{v \in S})$$

in the notations of [CHT08, §2.3], where $\delta_{E/F}: \mathrm{Gal}_F \to \{\pm 1\}$ is the quadratic character associated to $E/F$, and

$$R^{\mathrm{loc}} := \widehat{\otimes}_{v \in S} \overline{R_{\bar{\rho}_v}}^{\square}.$$

We assume the following (strong) Taylor-Wiles hypothesis

**Hypothesis 5.9.** 1. $p > 2$;

2. the extension $E/F$ is unramified and $E$ does not contain a (non-trivial) $p$-th root $\zeta_p$ of 1;

3. the group $U$ is quasi-split at all finite places of $\mathbb{Q}$;

4. the level $K^p$ is chosen such that $K_v$ is hyperspecial whenever the finite place $v$ of $F$ is inert in $E$;

5. the representation $\bar{\rho}|_{\mathrm{Gal}_{E(\zeta_p)}}$ is adequate.

By [CEG$^+$16] sections 2.7,2.8, (see also [BHS17b, Théorème 3.5]), we have the following data.

**Proposition 5.10.** There exist

1. an integer $g \ge 1$;

2. a continuous, admissible, unitary $R_\infty$-representation $\Pi_\infty$ of $U(\mathbb{Q}_p)$ over $L$, where

$$R_\infty := R^{\mathrm{loc}}[[x_1, \dots, x_g]];$$

3. a local map of local rings $S_\infty := \mathcal{O}_L[[y_1, \dots, y_t]] \to R_\infty$ with

$$t = g + \dim R^{\mathrm{loc}} - [F^+ : \mathbb{Q}] \frac{n(n+1)}{2}$$

and a local map of local rings $R_\infty \to R_{\bar{\rho},S}$ such that

(i) there exists an $\mathcal{O}_L$-lattice $\Pi_\infty^0 \subset \Pi_\infty$ stable by $U(\mathbb{Q}_p)$ and $R_\infty$ such that

$$(\Pi_\infty^0)' = \mathrm{Hom}_{\mathcal{O}_L}(\Pi_\infty^0, \mathcal{O}_L),$$

is a projective $S_\infty[[K_p]]$-module of finite type (via $S_\infty \to R_\infty$) for some (equivalently all) compact open subgroup $K_p \subset U(\mathbb{Q}_p)$;
We denote by $R_{\bar{\rho},S}$ the quotient of $R_{\bar{\rho}}$ corresponding to the deformation problem

$$S = (E/F, S, \tilde{S}, \mathcal{O}_L, \bar{\rho}, \varepsilon^{1-n} \delta_{E/F}^n, \{\overline{R_{\bar{\rho}_v}}\}_{v \in S})$$

in the notations of [CHT08, §2.3], where $\delta_{E/F}: \mathrm{Gal}_F \to \{\pm 1\}$ is the quadratic character associated to $E/F$, and

$$R^{\mathrm{loc}} := \widehat{\otimes}_{v \in S} \overline{R_{\bar{\rho}_v}}^{\square}.$$

We assume the following (strong) Taylor-Wiles hypothesis

**Hypothesis 5.9.** 1. $p > 2$;

2. the extension $E/F$ is unramified and $E$ does not contain a (non-trivial) $p$-th root $\zeta_p$ of 1;

3. the group $U$ is quasi-split at all finite places of $\mathbb{Q}$;

4. the level $K^p$ is chosen such that $K_v$ is hyperspecial whenever the finite place $v$ of $F$ is inert in $E$;

5. the representation $\bar{\rho}|_{\mathrm{Gal}_{E(\zeta_p)}}$ is adequate.

By [CEG$^+$16] sections 2.7,2.8, (see also [BHS17b, Théorème 3.5]), we have the following data.

**Proposition 5.10.** *There exist*

1. an integer $g \ge 1$;

2. a continuous, admissible, unitary $R_\infty$-representation $\Pi_\infty$ of $U(\mathbb{Q}_p)$ over $L$, where

$$R_\infty := R^{\mathrm{loc}}[[x_1, \dots, x_g]];$$

3. a local map of local rings $S_\infty := \mathcal{O}_L[[y_1, \dots, y_t]] \to R_\infty$ with

$$t = g + \dim R^{\mathrm{loc}} - [F^+ : \mathbb{Q}] \frac{n(n+1)}{2}$$

and a local map of local rings $R_\infty \to R_{\bar{\rho},S}$ such that

(i) there exists an $\mathcal{O}_L$-lattice $\Pi_\infty^0 \subset \Pi_\infty$ stable by $U(\mathbb{Q}_p)$ and $R_\infty$ such that

$$(\Pi_\infty^0)' = \mathrm{Hom}_{\mathcal{O}_L}(\Pi_\infty^0, \mathcal{O}_L),$$

is a projective $S_\infty[[K_p]]$-module of finite type (via $S_\infty \to R_\infty$) for some (equivalently all) compact open subgroup $K_p \subset U(\mathbb{Q}_p)$;
(ii) the map $R_{\infty} \to R_{\bar{\rho},S}$ induces an isomorphism

$$R_{\infty}/\mathfrak{a}R_{\infty} \simeq R_{\bar{\rho},S},$$

of local noetherian $O_L$-algebras and an isomorphism of continuous admissible unitary $R_{\infty}/\mathfrak{a}R_{\infty}$-representations of $U(\mathbb{Q}_p)$ on $L$

$$\Pi_{\infty}[\mathfrak{a}] \simeq \Pi_{\mathfrak{m}},$$

where $\mathfrak{a} = (y_1, \dots, y_t)$ denotes the augmentation ideal of $S_{\infty}$,

It is a direct consequence of this proposition that the $R_{\infty}$-representation $\Pi_{\infty}$ of $U(\mathbb{Q}_p)$ satisfies Hypothesis 3.15. We note that the same applies to a slightly more general context:

**Lemma 5.11.** Let $V$ be a finite dimensional algebraic representation of $U(\mathbb{Q}_p)$ over $L$. Then the $R_{\infty}$-Banach representation $\Pi_{\infty} \otimes_L V$ satisfies Hypothesis 3.15.

*Proof.* As $\Pi_{\infty}$ satisfies Hypothesis 3.15, for any open pro-$p$-subgroup $H$ of $U(\mathbb{Q}_p)$ there exists an isomorphism of $\mathbb{Z}_p^t \times H$-representations $\Pi_{\infty|\mathbb{Z}_p^t \times H} \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, L)^m$ for some $m \ge 1$. But then

$$(\Pi_{\infty} \otimes_L V)|_{\mathbb{Z}_p^t \times H} \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, V)^m \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, L)^{m \dim_L V}. \quad \square$$

In the remainder of this paper we will use the following notations: we set

$$\mathcal{X}^p := \mathrm{Spf}(\widehat{\otimes}_{v \in S \setminus S_p} \bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}} \simeq \prod_{v \in S \setminus S_p} \mathrm{Spf}(\bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}},$$

where $\mathbb{U}^g := \mathrm{Spf}(\mathcal{O}_L[[x_1, \dots, x_g]])^{\mathrm{rig}}$ is an open polydisc. Moreover, we set

$$\mathcal{X}_{\bar{\rho}_p} := \mathrm{Spf}(\widehat{\otimes}_{v \in S_p} R_{\bar{\rho}_v}^{\square})^{\mathrm{rig}},$$

$$\mathcal{X}_{\infty} := \mathrm{Spf}(R_{\infty})^{\mathrm{rig}} \simeq \mathcal{X}^p \times \mathcal{X}_{\bar{\rho}_p} \times \mathbb{U}^g.$$

By construction the space $\mathcal{X}_{\infty}$ contains $\mathcal{X}_{\bar{\rho},S} = (\mathrm{Spf} R_{\bar{\rho},S})^{\mathrm{rig}}$ as a closed subspace. For a point $x = (x^p, x_p, z) \in \mathcal{X}_{\infty}(L)$ and a place $v$ of $F$ dividing $p$, we denote by $\rho_{x,v}$ the framed representation $\mathrm{Gal}_{F_v} \to \mathrm{GL}_n(L)$ associated to $x$. Finally we write $\rho_{x,p}$ for the family of representations $(\rho_{x,v})_{v|p}$.

# 6 Patching functors

In this section, we keep notations and conventions of section 5. In particular, we have $\underline{G} \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \mathrm{Res}_{F_v/\mathbb{Q}_p} \mathrm{GL}_{n,F_v})$ which is an algebraic group over $L$ and we consider
(ii) the map $R_{\infty} \to R_{\bar{\rho},S}$ induces an isomorphism

$$R_{\infty}/\mathfrak{a}R_{\infty} \simeq R_{\bar{\rho},S},$$

of local noetherian $O_L$-algebras and an isomorphism of continuous admissible unitary $R_{\infty}/\mathfrak{a}R_{\infty}$-representations of $U(\mathbb{Q}_p)$ on $L$

$$\Pi_{\infty}[\mathfrak{a}] \simeq \Pi_{\mathfrak{m}},$$

where $\mathfrak{a} = (y_1, \dots, y_t)$ denotes the augmentation ideal of $S_{\infty}$,

It is a direct consequence of this proposition that the $R_{\infty}$-representation $\Pi_{\infty}$ of $U(\mathbb{Q}_p)$ satisfies Hypothesis 3.15. We note that the same applies to a slightly more general context:

**Lemma 5.11.** Let $V$ be a finite dimensional algebraic representation of $U(\mathbb{Q}_p)$ over $L$. Then the $R_{\infty}$-Banach representation $\Pi_{\infty} \otimes_L V$ satisfies Hypothesis 3.15.

*Proof.* As $\Pi_{\infty}$ satisfies Hypothesis 3.15, for any open pro-$p$-subgroup $H$ of $U(\mathbb{Q}_p)$ there exists an isomorphism of $\mathbb{Z}_p^t \times H$-representations $\Pi_{\infty|\mathbb{Z}_p^t \times H} \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, L)^m$ for some $m \ge 1$. But then

$$(\Pi_{\infty} \otimes_L V)|_{\mathbb{Z}_p^t \times H} \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, V)^m \simeq \mathcal{C}(\mathbb{Z}_p^t \times H, L)^{m \dim_L V}. \quad \square$$

In the remainder of this paper we will use the following notations: we set

$$\mathcal{X}^p := \mathrm{Spf}(\widehat{\otimes}_{v \in S \setminus S_p} \bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}} \simeq \prod_{v \in S \setminus S_p} \mathrm{Spf}(\bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}},$$

where $\mathbb{U}^g := \mathrm{Spf}(\mathcal{O}_L[[x_1, \dots, x_g]])^{\mathrm{rig}}$ is an open polydisc. Moreover, we set

$$\mathcal{X}_{\bar{\rho}_p} := \mathrm{Spf}(\widehat{\otimes}_{v \in S_p} R_{\bar{\rho}_v}^{\square})^{\mathrm{rig}},$$

$$\mathcal{X}_{\infty} := \mathrm{Spf}(R_{\infty})^{\mathrm{rig}} \simeq \mathcal{X}^p \times \mathcal{X}_{\bar{\rho}_p} \times \mathbb{U}^g.$$

By construction the space $\mathcal{X}_{\infty}$ contains $\mathcal{X}_{\bar{\rho},S} = (\mathrm{Spf} R_{\bar{\rho},S})^{\mathrm{rig}}$ as a closed subspace. For a point $x = (x^p, x_p, z) \in \mathcal{X}_{\infty}(L)$ and a place $v$ of $F$ dividing $p$, we denote by $\rho_{x,v}$ the framed representation $\mathrm{Gal}_{F_v} \to \mathrm{GL}_n(L)$ associated to $x$. Finally we write $\rho_{x,p}$ for the family of representations $(\rho_{x,v})_{v|p}$.

# 6 Patching functors

In this section, we keep notations and conventions of section 5. In particular, we have $\underline{G} \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \mathrm{Res}_{F_v/\mathbb{Q}_p} \mathrm{GL}_{n,F_v})$ which is an algebraic group over $L$ and we consider
the associated categories $\mathcal{O}$, $\mathcal{O}_{\text{alg}}^{\infty}$ and $\tilde{\mathcal{O}}_{\text{alg}}$ as in section 2 (for the choice of the upper triangular Borel subgroup $\tilde{\mathcal{B}}$).

We fix once and for all a point $x \in \mathcal{X}_{\infty}(L)$ such that $x$ maps to the origin in $(\text{Spf } S_{\infty})^{\text{rig}}$ (i.e. the point defined by the augmentation ideal of $S_{\infty}$) and we denote by $\hat{R}_{\infty,x}$ the completed local ring of $\mathcal{X}_{\infty}$ at $x$.

## 6.1 Locally analytic patching functors

We fix a smooth and unramified character $\varepsilon : \underline{T}(\mathbb{Q}_p) \to L^{\times}$ and consider $\varepsilon$ as a point of $\hat{T}$.

By Lemma 5.11, we can apply Corollary 3.18 to the admissible locally analytic representation $\Pi_{\infty}^{\text{la}}$, and obtain a functor

$$ \begin{gathered} \mathcal{O}_{\text{alg}}^{\infty} \to \text{Coh}(\mathcal{X}_{\infty} \times \hat{T}) \\ M \mapsto \mathcal{M}_{\Pi_{\infty}}(M). \end{gathered} $$

**Definition 6.1.** For $M \in \mathcal{O}_{\text{alg}}^{\infty}$ we define

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) := \mathcal{M}_{\Pi_{\infty}}(M)_{x, \varepsilon} $$

to be the stalk of $\mathcal{M}_{\Pi_{\infty}}(M)$ at $(x, \varepsilon)$.

It follows from Proposition 3.17 that $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is a Cohen-Macaulay $\hat{R}_{\infty,x}$-module and is follows from Theorem 3.14 that the functor $M \mapsto \mathcal{M}_{\infty, x, \varepsilon}(M)$ is exact.

**Remark 6.2.** We also have the following description:

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) \simeq (\text{Hom}_{U(\mathfrak{g})}(M, \Pi_{\infty}^{\text{la}}[\mathfrak{m}_{x}^{\infty}])^{N_0}[\mathfrak{m}_{\varepsilon}^{\infty}])' $$

where $\mathfrak{m}_{\varepsilon}$ is the maximal ideal of $\mathcal{A}(p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p = \mathbb{Q}_p[\underline{T}(\mathbb{Q}_p)^{+}]$ corresponding to the character $\varepsilon$ and $\mathfrak{m}_x$ is the maximal ideal of $\hat{R}_{\infty}[1/p]$ corresponding to $x$.

**Remark 6.3.** Note that we have two $U(t)$-module structures on $\mathcal{M}_{\infty, x, \varepsilon}(M)$: The first one comes from the nilpotent $U(t)$-module structure on $M$ as in section 2.2. The second one comes from the action of $U(t)$ induced from the locally analytic $T$-structure on $\Pi_{\infty}^{\text{la}}$. It is a tautological consequence of the construction, but we point out that these two actions coincide.

**Definition 6.4.** Let $I \subset \Delta$ be a finite subset of simple roots and let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$. Then we define

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) := \varinjlim_{n} \mathcal{M}_{\infty, x, \varepsilon}(M/\mathfrak{m}_{I}^{n}). $$

**Proposition 6.5.** The functor $M \mapsto \mathcal{M}_{\infty, x, \varepsilon}(M)$ is exact on $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ and for each $M \in \tilde{\mathcal{O}}_{\text{alg}}^{I}$ the $\hat{R}_{\infty,x}$-module $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is finitely generated and Cohen-Macaulay of dimension $t + \dim_K \mathfrak{I}$. Moreover $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is flat over $U(\mathfrak{I})$.
the associated categories $\mathcal{O}$, $\mathcal{O}_{\text{alg}}^{\infty}$ and $\tilde{\mathcal{O}}_{\text{alg}}$ as in section 2 (for the choice of the upper triangular Borel subgroup $\tilde{\mathcal{B}}$).

We fix once and for all a point $x \in \mathcal{X}_{\infty}(L)$ such that $x$ maps to the origin in $(\text{Spf } S_{\infty})^{\text{rig}}$ (i.e. the point defined by the augmentation ideal of $S_{\infty}$) and we denote by $\hat{R}_{\infty,x}$ the completed local ring of $\mathcal{X}_{\infty}$ at $x$.

## 6.1 Locally analytic patching functors

We fix a smooth and unramified character $\varepsilon : \underline{T}(\mathbb{Q}_p) \to L^{\times}$ and consider $\varepsilon$ as a point of $\hat{T}$.

By Lemma 5.11, we can apply Corollary 3.18 to the admissible locally analytic representation $\Pi_{\infty}^{\text{la}}$, and obtain a functor

$$ \begin{gathered} \mathcal{O}_{\text{alg}}^{\infty} \to \text{Coh}(\mathcal{X}_{\infty} \times \hat{T}) \\ M \mapsto \mathcal{M}_{\Pi_{\infty}}(M). \end{gathered} $$

**Definition 6.1.** For $M \in \mathcal{O}_{\text{alg}}^{\infty}$ we define

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) := \mathcal{M}_{\Pi_{\infty}}(M)_{x, \varepsilon} $$

to be the stalk of $\mathcal{M}_{\Pi_{\infty}}(M)$ at $(x, \varepsilon)$.

It follows from Proposition 3.17 that $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is a Cohen-Macaulay $\hat{R}_{\infty,x}$-module and is follows from Theorem 3.14 that the functor $M \mapsto \mathcal{M}_{\infty, x, \varepsilon}(M)$ is exact.

**Remark 6.2.** We also have the following description:

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) \simeq (\text{Hom}_{U(\mathfrak{g})}(M, \Pi_{\infty}^{\text{la}}[\mathfrak{m}_{x}^{\infty}])^{N_0}[\mathfrak{m}_{\varepsilon}^{\infty}])' $$

where $\mathfrak{m}_{\varepsilon}$ is the maximal ideal of $\mathcal{A}(p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p = \mathbb{Q}_p[\underline{T}(\mathbb{Q}_p)^{+}]$ corresponding to the character $\varepsilon$ and $\mathfrak{m}_x$ is the maximal ideal of $R_{\infty}[1/p]$ corresponding to $x$.

**Remark 6.3.** Note that we have two $U(t)$-module structures on $\mathcal{M}_{\infty, x, \varepsilon}(M)$: The first one comes from the nilpotent $U(t)$-module structure on $M$ as in section 2.2. The second one comes from the action of $U(t)$ induced from the locally analytic $T$-structure on $\Pi_{\infty}^{\text{la}}$. It is a tautological consequence of the construction, but we point out that these two actions coincide.

**Definition 6.4.** Let $I \subset \Delta$ be a finite subset of simple roots and let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$. Then we define

$$ \mathcal{M}_{\infty, x, \varepsilon}(M) := \varinjlim_{n} \mathcal{M}_{\infty, x, \varepsilon}(M/\mathfrak{m}_{I}^{n}). $$

**Proposition 6.5.** The functor $M \mapsto \mathcal{M}_{\infty, x, \varepsilon}(M)$ is exact on $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ and for each $M \in \tilde{\mathcal{O}}_{\text{alg}}^{I}$ the $\hat{R}_{\infty,x}$-module $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is finitely generated and Cohen-Macaulay of dimension $t + \dim_K \mathfrak{I}$. Moreover $\mathcal{M}_{\infty, x, \varepsilon}(M)$ is flat over $U(\mathfrak{I})$.
Proof. Let $\hat{S}_{\infty}$ be the completion of $S_{\infty}[1/p]$ along the maximal ideal generated by the augmentation ideal $\mathfrak{a}$ of $S_{\infty}$. Moreover, we write $\hat{U}_I$ for the completion of $U(\mathfrak{z}_I)$ at the maximal ideal $\mathfrak{m}_I$.

By exactness of the functor $\mathcal{M}_{\infty,x,\varepsilon}$, we have

$$ \mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^{n+1})/\mathfrak{m}_I^n \simeq \mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^n) $$

for any $n \ge 1$. It follows from Theorem 3.17 that $\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I)$ is a finite projective $\hat{S}_{\infty}$-module. We denote its rank by $r \ge 0$. The exactness of $\mathcal{M}_{\infty,x,\varepsilon}$ implies that $\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^n)$ is a finite projective $\hat{S}_{\infty} \otimes_L U(\mathfrak{z}_I)/\mathfrak{m}_I^n$-module of rank $r$ and it follows that $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a finite projective $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-module of rank $r$. As the action of $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$ factors through $\hat{R}_{\infty,x}$ we deduce the result. The exactness of the functor $\mathcal{M}_{\infty,x,\varepsilon}$ is a consequence of the exactness of $\mathcal{M}_{\infty,x,\varepsilon}$ restricted to $\mathcal{O}_{\text{alg}}^{\infty}$ and the fact that each system $(\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^n))_n$ satisfies the Mittag-Leffler condition.

Let $\underline{t} = (t_1, \dots, t_m)$ be a regular sequence generating the maximal ideal of $U(\mathfrak{z}_I)_{\mathfrak{m}_I}$. This is also a regular sequence generating the maximal ideal of the completion $\hat{U}_I$. By exactness of the functor $\hat{S}_{\infty} \otimes_L -$ on strict exact sequences of Fréchet $L$-algebras, the sequence $\underline{t}$ is $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-regular. As $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a finite free $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-module, the sequence $\underline{t}$ is $\mathcal{M}_{\infty,x,\varepsilon}(M)$-regular. This is equivalent to flatness over $U(\mathfrak{z}_I)_{\mathfrak{m}_I}$. $\square$

## 6.2 A factorization property

We use the spaces and notations introduced in section 4. A point $x \in \mathcal{X}_{\infty}(L)$ is said to be crystalline $\varphi$-generic and Hodge-Tate regular if for all $v|p$ the representation $\rho_{x,v}$ is crystalline $\varphi$-generic and Hodge-Tate regular. Let $x = (\rho^p, \rho_p, z) \in \mathcal{X}_{\infty}(L)$ be such a $\varphi$-generic Hodge-Tate regular point. We fix a refinement $\mathcal{R}$ of $\rho_p$.

Recall that $\underline{G} \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v})$. If $I$ is a set of simple roots of $\underline{G}$, we set

$$ \mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri}} := \hat{\mathcal{X}}^{p}_{\rho^p} \times \mathcal{X}_{\rho_p,\mathcal{R}}^{I-q\text{tri}} \times \hat{\mathbb{U}}^g. $$

This is a closed subscheme of $(\hat{\mathcal{X}}_{\infty})_x$ and we write $\hat{R}_{\infty,x} \to R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$ the corresponding quotient map. Moreover, for $w \in W$, we set

$$ \mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri},w} := \hat{\mathcal{X}}^{p}_{\rho^p} \times \mathcal{X}_{\rho_p,\mathcal{R}}^{I-q\text{tri},w} \times \hat{\mathbb{U}}^g. $$

If $\mathcal{R} = (\varphi_{1,v}, \dots, \varphi_{n,v})_{v|p} \in \prod_{v|p} (L^\times)^n$, we define $\delta_{\mathcal{R}}$ to be the smooth unramified character of $T$ defined by

$$ (x_{1,v}, \dots, x_{n,v})_{v|p} \mapsto \prod_{v|p} \prod_i (\varphi_{i,v}^{v_{F_v}(x_{i,v})} q_v^{i-n}) $$

where $q_n$ denotes the cardinality of the residue field of $F_v$. We use the notation $\mathcal{M}_{\infty,x,\mathcal{R}} := \mathcal{M}_{\infty,x,\delta_{\mathcal{R}}}$. The goal of this section is to prove the following result.
Proof. Let $\hat{S}_{\infty}$ be the completion of $S_{\infty}[1/p]$ along the maximal ideal generated by the augmentation ideal $\mathfrak{a}$ of $S_{\infty}$. Moreover, we write $\hat{U}_I$ for the completion of $U(\mathfrak{z}_I)$ at the maximal ideal $\mathfrak{m}_I$.

By exactness of the functor $\mathcal{M}_{\infty,x,\varepsilon}$, we have

$$ \mathcal{M}_{\infty, x, \varepsilon}(M/\mathfrak{m}_I^{n+1})/\mathfrak{m}_I^n \simeq \mathcal{M}_{\infty, x, \varepsilon}(M/\mathfrak{m}_I^n) $$

for any $n \ge 1$. It follows from Theorem 3.17 that $\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I)$ is a finite projective $\hat{S}_{\infty}$-module. We denote its rank by $r \ge 0$. The exactness of $\mathcal{M}_{\infty,x,\varepsilon}$ implies that $\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^n)$ is a finite projective $\hat{S}_{\infty} \otimes_L U(\mathfrak{z}_I)/\mathfrak{m}_I^n$-module of rank $r$ and it follows that $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a finite projective $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-module of rank $r$. As the action of $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$ factors through $\hat{R}_{\infty,x}$ we deduce the result. The exactness of the functor $\mathcal{M}_{\infty,x,\varepsilon}$ is a consequence of the exactness of $\mathcal{M}_{\infty,x,\varepsilon}$ restricted to $\mathcal{O}_{\text{alg}}^{\infty}$ and the fact that each system $(\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_I^n))_n$ satisfies the Mittag-Leffler condition.

Let $\underline{t} = (t_1, \dots, t_m)$ be a regular sequence generating the maximal ideal of $U(\mathfrak{z}_I)_{\mathfrak{m}_I}$. This is also a regular sequence generating the maximal ideal of the completion $\hat{U}_I$. By exactness of the functor $\hat{S}_{\infty} \otimes_L -$ on strict exact sequences of Fréchet $L$-algebras, the sequence $\underline{t}$ is $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-regular. As $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a finite free $\hat{S}_{\infty} \boxtimes_L \hat{U}_I$-module, the sequence $\underline{t}$ is $\mathcal{M}_{\infty,x,\varepsilon}(M)$-regular. This is equivalent to flatness over $U(\mathfrak{z}_I)_{\mathfrak{m}_I}$. $\square$

## 6.2 A factorization property

We use the spaces and notations introduced in section 4. A point $x \in \mathcal{X}_{\infty}(L)$ is said to be crystalline $\varphi$-generic and Hodge–Tate regular if for all $v|p$ the representation $\rho_{x,v}$ is crystalline $\varphi$-generic and Hodge–Tate regular. Let $x = (\rho^p, \rho_p, z) \in \mathcal{X}_{\infty}(L)$ be such a $\varphi$-generic Hodge–Tate regular point. We fix a refinement $\mathcal{R}$ of $\rho_p$.

Recall that $\underline{G} \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_{n,F_v})$. If $I$ is a set of simple roots of $\underline{G}$, we set

$$ \mathcal{X}_{\infty, x, \mathcal{R}}^{I-q\text{tri}} := \hat{\mathcal{X}}^{p}_{\rho^p} \times \mathcal{X}_{\rho_p, \mathcal{R}}^{I-q\text{tri}} \times \hat{\mathbb{U}}^g. $$

This is a closed subscheme of $(\tilde{\mathcal{X}}_{\infty})_x$ and we write $\hat{R}_{\infty,x} \to \mathcal{R}^{I-q\text{tri}}$ the corresponding quotient map. Moreover, for $w \in W$, we set

$$ \mathcal{X}_{\infty, x, \mathcal{R}}^{I-q\text{tri},w} := \hat{\mathcal{X}}^{p}_{\rho^p} \times \mathcal{X}_{\rho_p, \mathcal{R}}^{I-q\text{tri},w} \times \hat{\mathbb{U}}^g. $$

If $\mathcal{R} = (\varphi_{1,v}, \dots, \varphi_{n,v})_{v|p} \in \prod_{v|p} (L^\times)^n$, we define $\delta_{\mathcal{R}}$ to be the smooth unramified character of $T$ defined by

$$ (x_{1,v}, \dots, x_{n,v})_{v|p} \mapsto \prod_{v|p} \prod_i (\varphi_{i,v}^{v_{F_v}(x_{i,v})} q_v^{i-n}) $$

where $q_n$ denotes the cardinality of the residue field of $F_v$. We use the notation $\mathcal{M}_{\infty,x,\mathcal{R}} := \mathcal{M}_{\infty,x,\delta_{\mathcal{R}}}$. The goal of this section is to prove the following result.
**Theorem 6.6.** Let $x \in \mathcal{X}_{\infty}(L)$ be a $\varphi$-generic Hodge-Tate regular crystalline point and let $\mathcal{R}$ be a refinement of $x$. Then, for any $M \in \mathcal{O}_{\text{alg}}^{\infty,I}$, the $\hat{R}_{\infty,x}$-module $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is killed by the kernel of the map $\hat{R}_{\infty,x} \to \hat{R}_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$. Equivalently its support is contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

*Proof.* This is a consequence of Proposition 3.20, Proposition 2.14 and Corollary 6.11 which will be proved below. □

We will prove the auxiliary statements in (the proof of) this theorem by making use of variants of the construction of eigenvarieties. More precisely, for a subset $I \subset \Delta$, a character $\lambda \in X^*(\underline{T})_I^+$ (dominant with respect to $\underline{P}_I$) and an algebraic representation $V$ of $\underline{G}$ we will consider the scheme-theoretic supports

$$ \begin{aligned} \mathcal{E}_{\infty}^{I}(\lambda) &= \text{supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}) \subset \mathcal{X}_{\infty} \times \hat{T} \\ \mathcal{E}_{\infty}^{I}(\lambda, V) &= \text{supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}) \subset \mathcal{X}_{\infty} \times \hat{T}, \end{aligned} $$

where $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ respectively $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are the coherent sheaves associated to $J_{I,\lambda}(\Pi_{\infty}^{\text{la}})'$ respectively to $J_{I,\lambda}((\Pi_{\infty} \otimes_L V)^{\text{la}})'$ (see section 3.4 for the notation). We will link the completions of $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ at points $(x, \delta) \in \mathcal{X}_{\infty} \times \hat{T}$ to the quasi-trianguline deformation rings of section 4. This is done in two steps: we first show that the set-theoretic support of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ is contained in the (quasi-)trianguline locus (see the proof of Proposition 6.7). We then prove that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ is reduced (see the proof of Proposition 6.9). The proof of the latter statement follows the usual argument in the case of eigenvarieties, see e.g. [BHS17b, Corollaire 3.12 and Corollaire 3.20]: the general properties of eigenvarieties (deduced from the fact that the sheaves $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are locally finite projective over $(\text{Spf } S_{\infty})^{\text{rig}} \times \hat{T}_0$ imply that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ have no embedded components. Hence it is enough to produce on each of their irreducible components a point $y$ such that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ are reduced in a neighborhood of $y$. By the same projectivity argument as above, the point $y$ can be chosen so that the weight map to $\hat{T}_0$ is smooth at this point. Reducedness then boils down to checking that the Hecke operators (that generate the local ring of $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ at $y$) act semi-simply on the fiber of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ over $\hat{T}_0$ which in turn follows from the fact that Hecke-operators act semi-simply on spaces of classical automorphic forms. We now give the details of these arguments.

Let $\delta = (\delta_{1,v}, \dots, \delta_{n,v})_{v|p} \in \hat{T}(L)$ be a parameter for a quasi-triangulation of $x$ at $p$, i.e. the trianguline filtration of the $(\varphi, \Gamma)$-module $D_{\text{rig}}^{\dagger}(\rho_v)[1/t]$ over $\mathcal{R}_{K,L}[1/t]$ has graded pieces $\mathcal{R}_{K,L}(\delta_{i,v})[1/t]$. As $x$ is Hodge-Tate regular, there is a natural map

$$ \omega_{\delta} : \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri}} \to \hat{T}_{\delta}^{\wedge}, $$

mapping a deformation at $p$ of the $(\varphi, \Gamma)$-module $D_{\text{rig}}^{\dagger}(\rho_v)[1/t]$, equipped with its trianguline filtration, to its parameter (see e.g. [BHS19, eq (3.15)]). If $\delta$ is locally algebraic
**Theorem 6.6.** Let $x \in \mathcal{X}_{\infty}(L)$ be a $\varphi$-generic Hodge-Tate regular crystalline point and let $\mathcal{R}$ be a refinement of $x$. Then, for any $M \in \mathcal{O}_{\text{alg}}^{\infty,I}$, the $\hat{R}_{\infty,x}$-module $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is killed by the kernel of the map $\hat{R}_{\infty,x} \to \hat{R}_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$. Equivalently its support is contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

*Proof.* This is a consequence of Proposition 3.20, Proposition 2.14 and Corollary 6.11 which will be proved below. □

We will prove the auxiliary statements in (the proof of) this theorem by making use of variants of the construction of eigenvarieties. More precisely, for a subset $I \subset \Delta$, a character $\lambda \in X^*(\underline{T})_I^+$ (dominant with respect to $\underline{P}_I$) and an algebraic representation $V$ of $\underline{G}$ we will consider the scheme-theoretic supports

$$ \begin{aligned} \mathcal{E}_{\infty}^{I}(\lambda) &= \text{supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}) \subset \mathcal{X}_{\infty} \times \hat{T} \\ \mathcal{E}_{\infty}^{I}(\lambda, V) &= \text{supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}) \subset \mathcal{X}_{\infty} \times \hat{T}, \end{aligned} $$

where $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ respectively $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are the coherent sheaves associated to $J_{I,\lambda}(\Pi_{\infty}^{\text{la}})'$ respectively to $J_{I,\lambda}((\Pi_{\infty} \otimes_L V)^{\text{la}})'$ (see section 3.4 for the notation). We will link the completions of $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ at points $(x, \delta) \in \mathcal{X}_{\infty} \times \hat{T}$ to the quasi-trianguline deformation rings of section 4. This is done in two steps: we first show that the set-theoretic support of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ is contained in the (quasi-)trianguline locus (see the proof of Proposition 6.7). We then prove that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ is reduced (see the proof of Proposition 6.9). The proof of the latter statement follows the usual argument in the case of eigenvarieties, see e.g. [BHS17b, Corollaire 3.12 and Corollaire 3.20]: the general properties of eigenvarieties (deduced from the fact that the sheaves $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are locally finite projective over $(\text{Spf } S_{\infty})^{\text{rig}} \times \hat{T}_0$ imply that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ have no embedded components. Hence it is enough to produce on each of their irreducible components a point $y$ such that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ are reduced in a neighborhood of $y$. By the same projectivity argument as above, the point $y$ can be chosen so that the weight map to $\hat{T}_0$ is smooth at this point. Reducedness then boils down to checking that the Hecke operators (that generate the local ring of $\mathcal{E}_{\infty}^{I}(\lambda)$ resp. $\mathcal{E}_{\infty}^{I}(\lambda, V)$ at $y$) act semi-simply on the fiber of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ over $\hat{T}_0$ which in turn follows from the fact that Hecke-operators act semi-simply on spaces of classical automorphic forms. We now give the details of these arguments.

Let $\delta = (\delta_{1,v}, \dots, \delta_{n,v})_{v|p} \in \hat{T}(L)$ be a parameter for a quasi-triangulation of $x$ at $p$, i.e. the trianguline filtration of the $(\varphi, \Gamma)$-module $D_{\text{rig}}^{\dagger}(\rho_v)[1/t]$ over $\mathcal{R}_{K,L}[1/t]$ has graded pieces $\mathcal{R}_{K,L}(\delta_{i,v})[1/t]$. As $x$ is Hodge-Tate regular, there is a natural map

$$ \omega_{\delta} : \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri}} \to \hat{T}_{\delta}^{\wedge}, $$

mapping a deformation at $p$ of the $(\varphi, \Gamma)$-module $D_{\text{rig}}^{\dagger}(\rho_v)[1/t]$, equipped with its trianguline filtration, to its parameter (see e.g. [BHS19, eq (3.15)]). If $\delta$ is locally algebraic
of the form $\delta = \lambda\delta_R$ for $\lambda \in X^*(T)$ and some smooth character $\delta_R \in \hat{T}(L)$, we shift the previous map to get

$$ \omega = t_{-\lambda}\omega_{\delta}: \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \hat{T}_{\delta_R}^{\wedge} $$

which only depends on the chosen refinement. This induces a map

$$ i \times \omega : \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \widehat{\mathcal{X}}_{\infty,x} \times \hat{T}_{\delta_R}^{\wedge}, $$

or equivalently, a homomorphism $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,R}^{I-qtri}$.

**Proposition 6.7.** Let $\lambda \in X^*(\underline{T})_I^+$ be a weight dominant with respect to $\underline{P}_I$. The $\hat{R}_{\infty,x}$-module $M_{\infty,x,R}(\widetilde{M}_I(\lambda))$ is annihilated by the kernel of $\hat{R}_{\infty,x} \to R_{\infty,x,R}^{I-qtri}$. More precisely, $M_{\infty,x,R}(\widetilde{M}_I(\lambda))$ is an $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge}$-module and annihilated by the kernel of

$$ \hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,R}^{I-qtri}. $$

*Proof.* It follows from Proposition 3.20 and the definition of $M_{\infty,x,R}(\widetilde{M}_I(\lambda))$ that

$$ M_{\infty,x,R}(\widetilde{M}_I(\lambda)) = (t_{\lambda}^{*} M_{\Pi_{\infty}}^{I,\lambda})_{(x,\delta_{R})}^{\wedge} $$

as an $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge}$-module. It is thus enough to show that the completion of $M_{\Pi_{\infty}}^{I,\lambda}$ at the point $(x, \lambda\delta_R) \in \mathcal{X}_{\infty}(L) \times \hat{T}(L)$ is supported at the closed subspace

$$ i \times w_{\delta}: \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \widehat{\mathcal{X}}_{\infty,x} \times \hat{T}_{\delta_{\lambda}\delta_{R}}^{\wedge}. $$

We closely follow the proof of [Wu, Prop. 5.13]. Let us write $\mathcal{E}_{\infty} \subset \mathcal{X} \times \hat{T}$ for the scheme-theoretic support of the coherent sheaf defined by $J_B(\Pi_{\infty})'$. By [Wu, 5.4] this contains $\mathcal{E}_{\infty}^I(\lambda)$ as a closed subspace. As in the proof of [Wu, Prop. 5.13] we consider a proper birational map $f: \mathcal{E}_{\infty}' \to \mathcal{E}_{\infty}$ such that the universal $(\varphi, \Gamma)$-module over $\mathcal{E}_{\infty}'$ has a quasi-triangulation, and write $\mathcal{E}_{\infty}''$ for the preimage of $\mathcal{E}_{\infty}^I(\lambda)$ in $\mathcal{E}_{\infty}'$. Let $Y \subset \mathcal{E}_{\infty}''$ be the Zariski closed reduced subspace of $\mathcal{E}_{\infty}''$ whose points are exactly the points of $\mathcal{E}_{\infty}''$ where the universal filtered $(\varphi, \Gamma)$-module over $\mathcal{R}[1/t]$ is $P_I$-de Rham. As in [Wu], the existence of $Y$ is a consequence of [Wu, Prop. A.10]. It follows that for any $y \in Y$ lying above $(x, \delta_R)$ the map

$$ \hat{Y}_y \to \mathcal{X}_{\infty} \times \hat{T} $$

factors through $\mathcal{X}_{\infty,y,R_y}^{I-qtri}$. Let $U \subset \mathcal{E}_{\infty}^I(\lambda)$ be an affinoid open subset containing $x$ and a Zariski dense subset of points which are de Rham (and in particular $P_I$-de Rham) and trianguline with parameter given by $\mathcal{E}_{\infty}^I(\lambda) \to \hat{T}$. Such a neighborhood exists by [Wu, Prop. 5.11 & 5.12]. We deduce that $f(Y) \supset U$ and hence $f^{-1}(U) \subset Y$ and we conclude as in the proof of [BHS19, Prop. 3.7.2] (see the erratum in [BD]) that the map

$$ \hat{U}_{x,\lambda\delta_R} \to \mathcal{X}_{\infty} \times \hat{T} $$

factors through $\mathcal{X}_{\infty,x,R}^{I-qtri}$.

□
of the form $\delta = \lambda\delta_R$ for $\lambda \in X^*(T)$ and some smooth character $\delta_R \in \hat{T}(L)$, we shift the previous map to get

$$ \omega = t_{-\lambda}\omega_{\delta}: \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \hat{T}_{\delta_R}^{\wedge} $$

which only depends on the chosen refinement. This induces a map

$$ i \times \omega: \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \widehat{\mathcal{X}}_{\infty,x} \times \hat{T}_{\delta_R}^{\wedge}, $$

or equivalently, a homomorphism $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,R}^{I-qtri}$.

**Proposition 6.7.** Let $\lambda \in X^*(T)_I^+$ be a weight dominant with respect to $\underline{P}_I$. The $\hat{R}_{\infty,x}$-module $M_{\infty,x,R}(\tilde{M}_I(\lambda))$ is annihilated by the kernel of $\hat{R}_{\infty,x} \to R_{\infty,x,R}^{I-qtri}$. More precisely, $M_{\infty,x,R}(\tilde{M}_I(\lambda))$ is an $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge}$-module and annihilated by the kernel of

$$ \hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,R}^{I-qtri}. $$

*Proof.* It follows from Proposition 3.20 and the definition of $M_{\infty,x,R}(\tilde{M}_I(\lambda))$ that

$$ M_{\infty,x,R}(\tilde{M}_I(\lambda)) = (t_{\lambda}^{*} M_{\Pi_{\infty}}^{I,\lambda})_{(x,\delta_{\mathcal{R}})}^{\wedge} $$

as an $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge}$-module. It is thus enough to show that the completion of $M_{\Pi_{\infty}}^{I,\lambda}$ at the point $(x, \lambda\delta_R) \in \mathcal{X}_{\infty}(L) \times \hat{T}(L)$ is supported at the closed subspace

$$ i \times w_{\delta}: \mathcal{X}_{\infty,x,R}^{\text{qtri}} \to \widehat{\mathcal{X}}_{\infty,x} \times \hat{T}_{\delta_{\lambda}\delta_{\mathcal{R}}}^{\wedge}. $$

We closely follow the proof of [Wu, Prop. 5.13]. Let us write $\mathcal{E}_{\infty} \subset \mathcal{X} \times \hat{T}$ for the scheme-theoretic support of the coherent sheaf defined by $J_B(\Pi_{\infty})'$. By [Wu, 5.4] this contains $\mathcal{E}_{\infty}^I(\lambda)$ as a closed subspace. As in the proof of [Wu, Prop. 5.13] we consider a proper birational map $f: \mathcal{E}_{\infty}' \to \mathcal{E}_{\infty}$ such that the universal $(\varphi, \Gamma)$-module over $\mathcal{E}_{\infty}'$ has a quasi-triangulation, and write $\mathcal{E}_{\infty}''$ for the preimage of $\mathcal{E}_{\infty}^I(\lambda)$ in $\mathcal{E}_{\infty}'$. Let $Y \subset \mathcal{E}_{\infty}''$ be the Zariski closed reduced subspace of $\mathcal{E}_{\infty}''$ whose points are exactly the points of $\mathcal{E}_{\infty}''$ where the universal filtered $(\varphi, \Gamma)$-module over $\mathcal{R}[1/t]$ is $P_I$-de Rham. As in [Wu], the existence of $Y$ is a consequence of [Wu, Prop. A.10]. It follows that for any $y \in Y$ lying above $(x, \delta_R)$ the map

$$ \hat{Y}_y \to \mathcal{X}_{\infty} \times \hat{T} $$

factors through $\mathcal{X}_{\infty,y,R_y}^{I-qtri}$. Let $U \subset \mathcal{E}_{\infty}^I(\lambda)$ be an affinoid open subset containing $x$ and a Zariski dense subset of points which are de Rham (and in particular $P_I$-de Rham) and trianguline with parameter given by $\mathcal{E}_{\infty}^I(\lambda) \to \hat{T}$. Such a neighborhood exists by [Wu, Prop. 5.11 & 5.12]. We deduce that $f(Y) \supset U$ and hence $f^{-1}(U) \subset Y$ and we conclude as in the proof of [BHS19, Prop. 3.7.2] (see the erratum in [BD]) that the map

$$ \hat{U}_{x,\lambda\delta_R} \to \mathcal{X}_{\infty} \times \hat{T} $$

factors through $\mathcal{X}_{\infty,x,R}^{I-qtri}$.

□
**Corollary 6.8.** Let $V$ be an algebraic representation of $\mathcal{G}$, then

$$ \mathcal{M} = \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(\lambda) \otimes_L V) $$

is annihilated by some power of the kernel of $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri}}$.

*Proof.* We recall that

$$ \widetilde{M}_I(\lambda) \otimes_L V = U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda)) \otimes_L V \cong U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda) \otimes V_{|P_I}) $$

and that $V_{|P_I}$ is an extension of algebraic irreducible representations of $\mathcal{L}_I$. Exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (see Proposition 6.5) implies that the $\hat{R}_{\infty,x}$-module $\mathcal{M}$ is an extension of $\hat{R}_{\infty,x,\mathcal{R}}$-module of the form $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(\mu))$ for $\mu \in X^*(\mathcal{L})_I^+$. We deduce the result from Proposition 6.7. $\square$

**Proposition 6.9.** Let $V$ be an algebraic representation of $\mathcal{G}$. Then the schematic support $\mathcal{E}_{\infty}^I(\lambda, V)$ of the coherent sheaf associated to $J_{I,\lambda}((\Pi_{\infty} \otimes_L V)^{\mathrm{la}})'$ is reduced.

*Proof.* We follow closely the proof of [BHS17b, Cor. 3.20] replacing, where it is needed, some arguments by results of [Wu]. To simplify notations we just write $\mathcal{E} = \mathcal{E}_{\infty}^I(\lambda, V)$ and $\mathcal{M} = \mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ for the remainder of this proof.

Let $\mathcal{N}$ be the radical ideal of $\mathcal{O}_{\mathcal{E}}$. Assume that $\mathcal{N} \neq 0$ and let $x \in \mathcal{E}$ be a point in the support of $\mathcal{N}$. Let $\hat{T}_{\lambda}^{\circ}$ be the preimage of $\lambda_{|t \cap \mathfrak{l}_I^{ss}} \in (t \cap \mathfrak{l}_I^{ss})^*$ under the map

$$ \hat{T} \to t^* \to (t \cap \mathfrak{l}_I^{ss})^*, $$

where the first map is the weight map (5). According to [Wu, §5.4] there exists an open affinoid neighborhood $U$ of $x$ and an open affinoid subset $W \subset \hat{T}_{\lambda}^{\circ} \times \mathrm{Spf}(S_{\infty})^{\mathrm{rig}}$ such that $\Gamma(U, \mathcal{M})$ is a finite free $\mathcal{O}(W)$-module (such a data exists according to the results of [Wu, §5.4]). Then $\Gamma(U, \mathcal{N})$ is the radical ideal of $\mathcal{O}(U)$. Moreover, as $\mathcal{O}(U) = \Gamma(U, \mathcal{O}_{\mathcal{E}})$ is a sub-$\mathcal{O}(W)$-module of $\mathrm{End}(\Gamma(U, \mathcal{M}))$ (by the same argument as in the proof of Theorem 3.17 respectively of [BHS17b, Prop. 3.11]), the same is true for $\Gamma(U, \mathcal{N})$. Therefore $\Gamma(U, \mathcal{N})$ is a torsion free $\mathcal{O}(W)$-module and its support has the same dimension as $W$ and hence contains an irreducible component $U_0$ of $U$. As a consequence the support of $\mathcal{N}$ contains an admissible open subset of $\mathcal{E}$. As the support of $\mathcal{N}$ is also a closed analytic subset of $\mathcal{E}$, it follows from [Con99, Lemm. 2.2.3] that the support of $\mathcal{N}$ contains an irreducible component of $\mathcal{E}$. It hence suffices to produce on each irreducible component of $\mathcal{E}$ a point $y$ such that $\mathcal{E}$ is reduced in a neighborhood of $y$.

By [Wu, Prop. 5.11] every irreducible component of $\mathcal{E}$ contains a point with algebraic weight.

Therefore we fix a point $x \in \mathcal{E}(L)$ with integral weight $\lambda' \in \hat{T}_{\lambda}^{\circ}$. Let $U$ be an open affinoid neighborhood of $x$ and $W \subset \hat{T}_{\lambda}^{\circ} \times \mathrm{Spf}(S_{\infty})^{\mathrm{rig}}$ an open affinoid open subset such that $M = \Gamma(U, \mathcal{M})$ is a direct factor of $\mathcal{O}(W) \hat{\otimes}_L J_{B_I}(J_{P_I}(\Pi_{\infty} \otimes_L V)_{\lambda}')$. Let $A = \mathcal{O}(W)$
**Corollary 6.8.** Let $V$ be an algebraic representation of $\mathcal{G}$, then

$$ \mathcal{M} = \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(\lambda) \otimes_L V) $$

is annihilated by some power of the kernel of $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T},\delta_R}^{\wedge} \to R_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri}}$.

*Proof.* We recall that

$$ \widetilde{M}_I(\lambda) \otimes_L V = U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda)) \otimes_L V \cong U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda) \otimes V_{|P_I}) $$

and that $V_{|P_I}$ is an extension of algebraic irreducible representations of $\mathcal{L}_I$. Exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (see Proposition 6.5) implies that the $\hat{R}_{\infty,x}$-module $\mathcal{M}$ is an extension of $\hat{R}_{\infty,x,\mathcal{R}}$-module of the form $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(\mu))$ for $\mu \in X^*(\mathcal{L})_I^+$. We deduce the result from Proposition 6.7. $\square$

**Proposition 6.9.** Let $V$ be an algebraic representation of $\mathcal{G}$. Then the schematic support $\mathcal{E}_{\infty}^I(\lambda, V)$ of the coherent sheaf associated to $J_{I,\lambda}((\Pi_{\infty} \otimes_L V)^{\mathrm{la}})'$ is reduced.

*Proof.* We follow closely the proof of [BHS17b, Cor. 3.20] replacing, where it is needed, some arguments by results of [Wu]. To simplify notations we just write $\mathcal{E} = \mathcal{E}_{\infty}^I(\lambda, V)$ and $\mathcal{M} = \mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ for the remainder of this proof.

Let $\mathcal{N}$ be the radical ideal of $\mathcal{O}_{\mathcal{E}}$. Assume that $\mathcal{N} \neq 0$ and let $x \in \mathcal{E}$ be a point in the support of $\mathcal{N}$. Let $\hat{T}_{\lambda}^{\circ}$ be the preimage of $\lambda_{|t \cap \mathfrak{l}_I^{\mathrm{ss}}} \in (t \cap \mathfrak{l}_I^{\mathrm{ss}})^*$ under the map

$$ \hat{T} \to t^* \to (t \cap \mathfrak{l}_I^{\mathrm{ss}})^*, $$

where the first map is the weight map (5). According to [Wu, §5.4] there exists an open affinoid neighborhood $U$ of $x$ and an open affinoid subset $W \subset \hat{T}_{\lambda}^{\circ} \times \mathrm{Spf}(S_{\infty})^{\mathrm{rig}}$ such that $\Gamma(U, \mathcal{M})$ is a finite free $\mathcal{O}(W)$-module (such a data exists according to the results of [Wu, §5.4]). Then $\Gamma(U, \mathcal{N})$ is the radical ideal of $\mathcal{O}(U)$. Moreover, as $\mathcal{O}(U) = \Gamma(U, \mathcal{O}_{\mathcal{E}})$ is a sub-$\mathcal{O}(W)$-module of $\mathrm{End}(\Gamma(U, \mathcal{M}))$ (by the same argument as in the proof of Theorem 3.17 respectively of [BHS17b, Prop. 3.11]), the same is true for $\Gamma(U, \mathcal{N})$. Therefore $\Gamma(U, \mathcal{N})$ is a torsion free $\mathcal{O}(W)$-module and its support has the same dimension as $W$ and hence contains an irreducible component $U_0$ of $U$. As a consequence the support of $\mathcal{N}$ contains an admissible open subset of $\mathcal{E}$. As the support of $\mathcal{N}$ is also a closed analytic subset of $\mathcal{E}$, it follows from [Con99, Lemm. 2.2.3] that the support of $\mathcal{N}$ contains an irreducible component of $\mathcal{E}$. It hence suffices to produce on each irreducible component of $\mathcal{E}$ a point $y$ such that $\mathcal{E}$ is reduced in a neighborhood of $y$.

By [Wu, Prop. 5.11] every irreducible component of $\mathcal{E}$ contains a point with algebraic weight.

Therefore we fix a point $x \in \mathcal{E}(L)$ with integral weight $\lambda' \in \hat{T}_{\lambda}^{\circ}$. Let $U$ be an open affinoid neighborhood of $x$ and $W \subset \hat{T}_{\lambda}^{\circ} \times \mathrm{Spf}(S_{\infty})^{\mathrm{rig}}$ an open affinoid open subset such that $M = \Gamma(U, \mathcal{M})$ is a direct factor of $\mathcal{O}(W)\hat{\otimes}_L J_{B_I}(J_{P_I}(\Pi_{\infty} \otimes_L V)_{\lambda}')$. Let $A = \mathcal{O}(W)$
and $B = \mathcal{O}(U)$. Then $M$ is a finitely generated $B$-module and a finite projective $A$-module. Let $C > 0$ and $C' > 0$ as in the proof of [Wu, Prop. 5.11]. We set $Z \subset W$ be the subset of algebraic character $\delta_{\lambda'}$ such that, for any simple root $\alpha \notin I$, $\langle \lambda' + \nu, \alpha \rangle > C'$ for any $\nu$ weight of $V^{\vee}$. This is a Zariski dense subset of $W$. Then for $z = \delta_{\lambda'}\delta_{\text{sm}}$ with $\delta_{\text{sm}}$ a smooth character, using Proposition 3.20, we see that the $B$-module $M_z = M \otimes k(z)$ is a direct factor of $J_B(\text{Hom}(M_I(\lambda'), \Pi_{\infty} \otimes_L V))'$. Let $(x, \delta) \in U$ be a point above $z$, i.e. $\delta = \delta_{\lambda'}\delta_{\text{sm}}$, then arguing as in loc. cit., we have $\text{Hom}_G(\mathcal{F}_B^G(N \otimes_L V^{\vee}, \delta_{\text{sm}}\delta_B^{-1}), \Pi_{\infty}[\mathfrak{p}_x]) = 0$ for any subquotient $N$ of $M_I(\lambda')$ different from $L(\lambda')$. This implies that $M_z$ is actually a quotient of $J_B(\text{Hom}_{U(\mathfrak{g})}(L(\lambda') \otimes_L V^{\vee}, \Pi_{\infty}))$ which is isomorphic to a finite direct sum of $J_B(\text{Hom}_{U(\mathfrak{g})}(L(\mu), \Pi_{\infty}))$ with $\mu$ dominant. The proof of [BHS17b, Cor. 3.20] shows that the global sections of the coherent sheaf associated to each $J_B(\text{Hom}_{U(\mathfrak{g})}(L(\mu), \Pi_{\infty}))'$ on $U \cap \kappa^{-1}(\{\delta_{\lambda'}\})$ is a semisimple $B$-module. This concludes the proof. $\square$

**Corollary 6.10.** *The rigid analytic space $\mathcal{E}_{\infty}^{I}(\lambda)$ is reduced.*

*Proof.* This is Proposition 6.9 with $V$ the trivial representation. $\square$

**Corollary 6.11.** Let $V$ be an irreducible algebraic representation of $G$. Then the $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T}^{\wedge}_{\delta_R}}$-module $M_{\infty,x,R}(\tilde{M}_I(\lambda) \otimes_L V)$ is killed by the kernel of the map

$$ \hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T}^{\wedge}_{\delta_R}} \to R^{I-q\text{tri}}_{\infty,x,R}. $$

*Proof.* By Proposition 6.9, the support of the module $M_{\infty,x,R}(\tilde{M}_I(\lambda) \otimes_L V)$ is reduced for any $\lambda \in X^*(\mathcal{T})$ dominant with respect to $\mathcal{P}_I$ and any algebraic representation $V$ of $G$. Therefore the result follows from Corollary 6.8. $\square$

## 6.3 Bi-module structure on the patched functor

Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}^{I}$. As seen in section 2.2, there is a natural structure of $A = U(\mathfrak{t})_{\mathfrak{m}}$-module on $M$ which provides, by functoriality, the structure of an $A$-module on $M_{\infty,x,R}(M)$. This $A$-module structure extends to an action of the completion $\hat{A}$ of $A$ with respect to the maximal ideal $\mathfrak{m}$. We recall from Remark 6.3 that this action coincides with the structure of an $\hat{A}$-module on $M_{\infty,x,R}(M)$ induced from the $T$-action on $\Pi_{\infty}$.

On the other hand, the ring $R_{\infty,x,R}^{\text{qtri}}$ also carries a structure of an $\hat{A}$-module induced from the map $\kappa_1$ defined in section 4.1. This gives a further structure of an $\hat{A}$-module on the $R_{\infty,x,R}^{\text{qtri}}$-module $M_{\infty,x,R}(M)$. We will show that these $\hat{A}$-module structures agree.

For $a \in \hat{A}$, we denote by $a$ (resp. $\tilde{a}$) the endomorphism of $M_{\infty,x,R}(M)$ defined by the first (resp. second) action. Note that if $M$ is an object of $\tilde{\mathcal{O}}_{\text{alg}}$, then $M_{\infty,x,R}(M)$ is a finite free $\hat{A} \hat{\otimes}_L \hat{S}_{\infty}$-module for the first $\hat{A}$-module structure by the proof of Proposition 6.5. Thus it is $A$-torsion free (since $\hat{A}$ is domain).
and $B = O(U)$. Then $M$ is a finitely generated $B$-module and a finite projective $A$-module. Let $C > 0$ and $C' > 0$ as in the proof of [Wu, Prop. 5.11]. We set $Z \subset W$ be the subset of algebraic character $\delta_{\lambda'}$ such that, for any simple root $\alpha \notin I$, $\langle \lambda' + \nu, \alpha \rangle > C'$ for any $\nu$ weight of $V^\vee$. This is a Zariski dense subset of $W$. Then for $z = \delta_{\lambda'}\delta_{sm}$ with $\delta_{sm}$ a smooth character, using Proposition 3.20, we see that the $B$-module $M_z = M \otimes k(z)$ is a direct factor of $J_B(\mathrm{Hom}(M_I(\lambda'), \Pi_\infty \otimes_L V))'$. Let $(x, \delta) \in U$ be a point above $z$, i.e. $\delta = \delta_{\lambda'}\delta_{sm}$, then arguing as in loc. cit., we have $\mathrm{Hom}_G(\mathcal{F}_B^G(N \otimes_L V^\vee, \delta_{sm}\delta_B^{-1}), \Pi_\infty[\mathfrak{p}_x]) = 0$ for any subquotient $N$ of $M_I(\lambda')$ different from $L(\lambda')$. This implies that $M_z$ is actually a quotient of $J_B(\mathrm{Hom}_{U(g)}(L(\lambda') \otimes_L V^\vee, \Pi_\infty))$ which is isomorphic to a finite direct sum of $J_B(\mathrm{Hom}_{U(g)}(L(\mu), \Pi_\infty))$ with $\mu$ dominant. The proof of [BHS17b, Cor. 3.20] shows that the global sections of the coherent sheaf associated to each $J_B(\mathrm{Hom}_{U(g)}(L(\mu), \Pi_\infty))'$ on $U \cap \kappa^{-1}(\{\delta_{\lambda'}\})$ is a semisimple $B$-module. This concludes the proof. $\square$

**Corollary 6.10.** *The rigid analytic space $\mathcal{E}_\infty^I(\lambda)$ is reduced.*

*Proof.* This is Proposition 6.9 with $V$ the trivial representation. $\square$

**Corollary 6.11.** Let $V$ be an irreducible algebraic representation of $G$. Then the $\hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T}^\wedge_{\delta_R}}$-module $M_{\infty,x,R}(\tilde{M}_I(\lambda) \otimes_L V)$ is killed by the kernel of the map

$$ \hat{R}_{\infty,x} \otimes \mathcal{O}_{\hat{T}^\wedge_{\delta_R}} \to R_{\infty,x,R}^{I-q\mathrm{tri}}. $$

*Proof.* By Proposition 6.9, the support of the module $M_{\infty,x,R}(\tilde{M}_I(\lambda) \otimes_L V)$ is reduced for any $\lambda \in X^*(T)$ dominant with respect to $P_I$ and any algebraic representation $V$ of $G$. Therefore the result follows from Corollary 6.8. $\square$

## 6.3 Bi-module structure on the patched functor

Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^\infty$ or $\tilde{\mathcal{O}}_{\mathrm{alg}}^I$. As seen in section 2.2, there is a natural structure of $A = U(t)_m$-module on $M$ which provides, by functoriality, the structure of an $A$-module on $M_{\infty,x,R}(M)$. This $A$-module structure extends to an action of the completion $\hat{A}$ of $A$ with respect to the maximal ideal $\mathfrak{m}$. We recall from Remark 6.3 that this action coincides with the structure of an $\hat{A}$-module on $M_{\infty,x,R}(M)$ induced from the $T$-action on $\Pi_\infty$.

On the other hand, the ring $R_{\infty,x,R}^{q\mathrm{tri}}$ also carries a structure of an $\hat{A}$-module induced from the map $\kappa_1$ defined in section 4.1. This gives a further structure of an $\hat{A}$-module on the $R_{\infty,x,R}^{q\mathrm{tri}}$-module $M_{\infty,x,R}(M)$. We will show that these $\hat{A}$-module structures agree.

For $a \in \hat{A}$, we denote by $a$ (resp. $\tilde{a}$) the endomorphism of $M_{\infty,x,R}(M)$ defined by the first (resp. second) action. Note that if $M$ is an object of $\tilde{\mathcal{O}}_{\mathrm{alg}}$, then $M_{\infty,x,R}(M)$ is a finite free $\hat{A} \hat{\otimes}_L \hat{S}_\infty$-module for the first $\hat{A}$-module structure by the proof of Proposition 6.5. Thus it is $A$-torsion free (since $\hat{A}$ is domain).
**Lemma 6.12.** For any $a \in \tilde{A}$ and any $M$ in $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, there is an equality

$$a = \tilde{a} \in \text{End}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$$

*Proof.* If $M = \tilde{M}(\mu) \otimes_{U(t)} U(t)/m^n$ for some $\mu \in X^*(\underline{T})$, this is a consequence of [BHS17b, Thm. 3.21], the commutative diagram [BHS19, (3.30)] and Remark 6.3. This implies that for any $\mu \in X^*(\underline{T})$, we have $a = \tilde{a}$ on $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu))$.

Now we consider the general case. It follows from Proposition 2.14 that it is sufficient to prove the equality $\tilde{a} = a$ when $M = \tilde{M}(\mu) \otimes_L V$ for $\mu \in X^*(\underline{T})$ dominant and $V$ a finite dimensional $U(\mathfrak{g})$-module. Let $(\text{Fil}_i)$ be an increasing filtration of $\tilde{M}(\mu) \otimes_L V$ such that $\text{Fil}_i / \text{Fil}_{i-1} \simeq \tilde{M}(\mu_i)$ where $\mu_1, \dots, \mu_d \in X^*(\underline{T})$ and $d = \dim_L V$ (such a filtration exists by [Soe92, Lem. 8]). Let $K$ denote the fraction field of $A$. It follows from Proposition 2.12 that we have a decomposition of $U(\mathfrak{g})_K$-modules

$$(\tilde{M}(\mu) \otimes_L V) \otimes_A K \simeq \bigoplus_{i=1}^d \tilde{M}(\mu_i)_K$$

splitting the filtration $(\text{Fil}_i \otimes_A K)$. Let $p_i \in \text{End}_{U(\mathfrak{g})_K}((\tilde{M}(\mu) \otimes_L V) \otimes_A K)$ be the projector on $\tilde{M}(\mu_i)_K$. As

$$\text{End}_{U(\mathfrak{g})_K}((\tilde{M}(\mu) \otimes_L V) \otimes_A K) \simeq \text{End}_{U(\mathfrak{g})}((\tilde{M}(\mu) \otimes_L V)) \otimes_A K$$

by [Soe92, Thm. 5], there exists, for each $1 \le i \le d$, a nonzero element $q_i \in A$ such that $q_i p_i$ actually restricts to an endomorphism of $\tilde{M}(\mu) \otimes_L V$. We set $q = q_1 \cdots q_r$ and $\alpha_i = q p_i$. Then the $\alpha_i$ are endomorphisms of $\tilde{M}(\mu) \otimes_L V$ that stabilize the filtration $\text{Fil}_\bullet$. As each $\text{Fil}_i / \text{Fil}_{i-1}$ is a free $A$-module, the endomorphisms $\alpha_i$ induce the zero endomorphism of $\text{Fil}_{i-1}$ and $\tilde{M}(\mu) \otimes_L V / \text{Fil}_i$ and the multiplication by $q$ on $\text{Fil}_i / \text{Fil}_{i-1}$.

In order to simplify notations we set

$$M_{\infty} = \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu) \otimes_L V), \\ \text{Fil}_i M_{\infty} = \mathcal{M}_{\infty,x,\mathcal{R}}(\text{Fil}_i).$$

By construction, for each $i$ the endomorphism $\alpha_i$ induces an $R_{\infty,x}$-linear endomorphisms of $\text{Fil}_j M_{\infty}$ for all $j$. By exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$, the family $(\text{Fil}_i M_{\infty})$ is a filtration of $M_{\infty}$ and $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty} \simeq \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu_i))$ for any $i$, so that $a$ and $\tilde{a}$ induces the same endomorphism of $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty}$. Finally, for $1 \le i \le d$, we denote by $M_{\infty}^{(i)} = \alpha_i(\text{Fil}_i M_{\infty})$ the image of the $i$-th filtration step under $\alpha_i$. It follows from the properties of $\alpha_i$ that

* $M_{\infty}^{(i)} \subset \text{Fil}_i M_{\infty}$;

* the quotient $\text{Fil}_i M_{\infty} / (\text{Fil}_{i-1} M_{\infty} + M_{\infty}^{(i)})$ is killed by $q$;

* $M_{\infty}^{(i)}$ is isomorphic to a quotient of $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty}$.
**Lemma 6.12.** For any $a \in \tilde{A}$ and any $M$ in $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, there is an equality

$$a = \tilde{a} \in \text{End}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$$

*Proof.* If $M = \tilde{M}(\mu) \otimes_{U(t)} U(t)/m^n$ for some $\mu \in X^*(\underline{T})$, this is a consequence of [BHS17b, Thm. 3.21], the commutative diagram [BHS19, (3.30)] and Remark 6.3. This implies that for any $\mu \in X^*(\underline{T})$, we have $a = \tilde{a}$ on $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu))$.

Now we consider the general case. It follows from Proposition 2.14 that it is sufficient to prove the equality $\tilde{a} = a$ when $M = \tilde{M}(\mu) \otimes_L V$ for $\mu \in X^*(\underline{T})$ dominant and $V$ a finite dimensional $U(\mathfrak{g})$-module. Let $(\text{Fil}_i)$ be an increasing filtration of $\tilde{M}(\mu) \otimes_L V$ such that $\text{Fil}_i / \text{Fil}_{i-1} \simeq \tilde{M}(\mu_i)$ where $\mu_1, \dots, \mu_d \in X^*(\underline{T})$ and $d = \dim_L V$ (such a filtration exists by [Soe92, Lem. 8]). Let $K$ denote the fraction field of $A$. It follows from Proposition 2.12 that we have a decomposition of $U(\mathfrak{g})_K$-modules

$$(\tilde{M}(\mu) \otimes_L V) \otimes_A K \simeq \bigoplus_{i=1}^d \tilde{M}(\mu_i)_K$$

splitting the filtration $(\text{Fil}_i \otimes_A K)$. Let $p_i \in \text{End}_{U(\mathfrak{g})_K}((\tilde{M}(\mu) \otimes_L V) \otimes_A K)$ be the projector on $\tilde{M}(\mu_i)_K$. As

$$\text{End}_{U(\mathfrak{g})_K}((\tilde{M}(\mu) \otimes_L V) \otimes_A K) \simeq \text{End}_{U(\mathfrak{g})}((\tilde{M}(\mu) \otimes_L V)) \otimes_A K$$

by [Soe92, Thm. 5], there exists, for each $1 \le i \le d$, a nonzero element $q_i \in A$ such that $q_i p_i$ actually restricts to an endomorphism of $\tilde{M}(\mu) \otimes_L V$. We set $q = q_1 \cdots q_r$ and $\alpha_i = q p_i$. Then the $\alpha_i$ are endomorphisms of $\tilde{M}(\mu) \otimes_L V$ that stabilize the filtration $\text{Fil}_\bullet$. As each $\text{Fil}_i / \text{Fil}_{i-1}$ is a free $A$-module, the endomorphisms $\alpha_i$ induce the zero endomorphism of $\text{Fil}_{i-1}$ and $\tilde{M}(\mu) \otimes_L V / \text{Fil}_i$ and the multiplication by $q$ on $\text{Fil}_i / \text{Fil}_{i-1}$.

In order to simplify notations we set

$$M_{\infty} = \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu) \otimes_L V),$$

$$\text{Fil}_i M_{\infty} = \mathcal{M}_{\infty,x,\mathcal{R}}(\text{Fil}_i).$$

By construction, for each $i$ the endomorphism $\alpha_i$ induces an $R_{\infty,x}$-linear endomorphisms of $\text{Fil}_j M_{\infty}$ for all $j$. By exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$, the family $(\text{Fil}_i M_{\infty})$ is a filtration of $M_{\infty}$ and $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty} \simeq \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(\mu_i))$ for any $i$, so that $a$ and $\tilde{a}$ induce the same endomorphism of $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty}$. Finally, for $1 \le i \le d$, we denote by $M_{\infty}^{(i)} = \alpha_i(\text{Fil}_i M_{\infty})$ the image of the $i$-th filtration step under $\alpha_i$. It follows from the properties of $\alpha_i$ that

* $M_{\infty}^{(i)} \subset \text{Fil}_i M_{\infty}$;

* the quotient $\text{Fil}_i M_{\infty} / (\text{Fil}_{i-1} M_{\infty} + M_{\infty}^{(i)})$ is killed by $q$;

* $M_{\infty}^{(i)}$ is isomorphic to a quotient of $\text{Fil}_i M_{\infty} / \text{Fil}_{i-1} M_{\infty}$.
Therefore, we have $\tilde{a} = a$ on $M_{\infty}^{(i)}$ for any $a \in \hat{A}$ and the quotient of $M_{\infty}$ by the sum of the $M_{\infty}^{(i)}$ is killed by $q^d$. As $M_{\infty}$ is A-torsion free it follows that $\tilde{a} = a$. $\square$

Let $\xi : Z(\mathfrak{g}) \to U(\mathfrak{t})$ be the Harish-Chandra map as recalled in section 2.4. As in loc.cit. we write $t_{\nu}$ for the unique endomorphism of $U(\mathfrak{t})$ mapping $x \in \mathfrak{t}$ to $t_{\nu}(x) = x + \nu(x)$.

Let $h = (h_{1,\tau,v} < \dots < h_{n,\tau,v})_{\tau,v} \in X^*(\underline{T})$ be the weight corresponding to the Hodge-Tate weights of $\rho_x = (\rho_v)_{v|p}$ and let $\delta'_G = (0, -1, -2, \dots, 1-n)_{\tau,v} \in X^*(\underline{T})$ be fixed central shift of the half sum of the positive roots $\delta_G \in X^*(\underline{T}) \otimes \mathbb{Q}$. We have a map

$$ \kappa_2 : \hat{A} = \widehat{U(\mathfrak{t})}_m \to R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

induced from the map $\kappa_2$ of section 4.1 and we define the L-algebra homomorphism

$$ \alpha = \kappa_2 \circ t_{h-\delta'_G} \circ \xi : Z(\mathfrak{g}) \to R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

As in [DPS, Def. 4.23], we define, for any $v|p$, an L-algebra homomorphism

$$ \zeta_{\tilde{\rho}_v}^C : Z(\mathrm{Lie}(\mathrm{Res}_{F_v/\mathbb{Q}_p} \mathrm{GL}_n)) \to R_{\tilde{\rho}_v}^{\square, \mathrm{rig}} $$

where $\tilde{\rho}_v$ is the universal family of Galois representations over $R_{\tilde{\rho}_v}^{\square, \mathrm{rig}}$. After completion at $\rho_v$ and taking the tensor product over all $v|p$, we obtain an L-algebra homomorphism

$$ \zeta^C : Z(\mathfrak{g}) = \bigotimes_{v|p} Z(\mathrm{Lie}(\mathrm{Res}_{F_v/\mathbb{Q}_p} \mathrm{GL}_n)) \to R_{\rho_p}^{\square} \twoheadrightarrow R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

Note that the definition of $\zeta_{\tilde{\rho}_v}^C$ from $\rho_v$ depends on a choice of a central shift of $\delta_G$ (see the discussion ending [DPS, §4.7]). We choose it equal to $\delta'_G$. More concretely $\zeta^C$ is characterized by the following property. This is the unique continuous homomorphism such that, for any local artinian L-algebra and any local homomorphism $f : R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$, corresponding to $\rho_A = (\rho_{A,v} : \mathrm{Gal}_{F_v} \to \mathrm{GL}_n(A))_{v|p}$, the composition map $Z(\mathfrak{g}) \xrightarrow{\zeta^C} R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$ is $Z(\mathfrak{g}) \xrightarrow{\xi} U(\mathfrak{t}) \xrightarrow{t_{\nu-\delta'_G}} k(x)$ where

$$ \nu \in \mathrm{Hom}_L(U(\mathfrak{t})^W, A) \simeq \mathrm{Hom}_L(U(\mathfrak{t}^*)^W, A) \simeq \mathrm{Hom}_L(U(\mathfrak{g}^*)^{\underline{G}_L}, A) $$

is the map induced by the conjugacy class of the Sen operators

$$ (\Theta_{\mathrm{Sen}, \rho_{A,v}})_{v|p} \in (\mathfrak{g} \otimes_L A). $$

**Proposition 6.13.** *The homomorphisms $\zeta^C$ and $\alpha$ defined above coincide.*

*Proof.* It is sufficient to prove that for any local artinian L-algebra $A$ and any map $f : R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$, we have $f \circ \zeta^C = f \circ \alpha$. Note that the map $f$ gives rise to a family $(\rho_{A,v})_{v|p}$ of local Galois representations. It follows from [BHS19, Lem. 3.7.5] that, for any embedding $\tau : F_v \hookrightarrow L$, the $\tau$-part of the Sen polynomial of $\rho_v$ is $\prod_{i=1}^n (X - (h_{i,\tau} + \nu_{i,\tau}))$ where $(\nu_{i,\tau}) \in \mathrm{Hom}_L(\mathfrak{t}, A)$ corresponds to $f \circ \kappa_2 : U(\mathfrak{t}) \to A$. The result is then a direct comparison of the definitions of $\alpha$ and $\zeta^C$. $\square$
Therefore, we have $\tilde{a} = a$ on $M_{\infty}^{(i)}$ for any $a \in \hat{A}$ and the quotient of $M_{\infty}$ by the sum of the $M_{\infty}^{(i)}$ is killed by $q^d$. As $M_{\infty}$ is A-torsion free it follows that $\tilde{a} = a$. $\square$

Let $\xi : Z(\mathfrak{g}) \to U(\mathfrak{t})$ be the Harish-Chandra map as recalled in section 2.4. As in loc.cit. we write $t_{\nu}$ for the unique endomorphism of $U(\mathfrak{t})$ mapping $x \in \mathfrak{t}$ to $t_{\nu}(x) = x + \nu(x)$.

Let $h = (h_{1,\tau,v} < \dots < h_{n,\tau,v})_{\tau,v} \in X^*(\underline{T})$ be the weight corresponding to the Hodge-Tate weights of $\rho_x = (\rho_v)_{v|p}$ and let $\delta'_G = (0, -1, -2, \dots, 1-n)_{\tau,v} \in X^*(\underline{T})$ be fixed central shift of the half sum of the positive roots $\delta_G \in X^*(\underline{T}) \otimes \mathbb{Q}$. We have a map

$$ \kappa_2 : \hat{A} = \widehat{U(\mathfrak{t})}_m \to R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

induced from the map $\kappa_2$ of section 4.1 and we define the L-algebra homomorphism

$$ \alpha = \kappa_2 \circ t_{h-\delta'_G} \circ \xi : Z(\mathfrak{g}) \to R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

As in [DPS, Def. 4.23], we define, for any $v|p$, an L-algebra homomorphism

$$ \zeta_{\tilde{\rho}_v}^C : Z(\text{Lie}(\text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_n)) \to R_{\tilde{\rho}_v}^{\square, \text{rig}} $$

where $\tilde{\rho}_v$ is the universal family of Galois representations over $R_{\tilde{\rho}_v}^{\square, \text{rig}}$. After completion at $\rho_v$ and taking the tensor product over all $v|p$, we obtain an L-algebra homomorphism

$$ \zeta^C : Z(\mathfrak{g}) = \bigotimes_{v|p} Z(\text{Lie}(\text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_n)) \to R_{\rho_p}^{\square} \twoheadrightarrow R_{\rho_p, \mathcal{R}}^{\text{qtri}} $$

Note that the definition of $\zeta_{\tilde{\rho}_v}^C$ from $\rho_v$ depends on a choice of a central shift of $\delta_G$ (see the discussion ending [DPS, §4.7]). We choose it equal to $\delta'_G$. More concretely $\zeta^C$ is characterized by the following property. This is the unique continuous homomorphism such that, for any local artinian L-algebra and any local homomorphism $f : R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$, corresponding to $\rho_A = (\rho_{A,v} : \text{Gal}_{F_v} \to \text{GL}_n(A))_{v|p}$, the composition map $Z(\mathfrak{g}) \xrightarrow{\zeta^C} R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$ is $Z(\mathfrak{g}) \xrightarrow{\xi} U(\mathfrak{t}) \xrightarrow{t_{\nu-\delta'_G}} k(x)$ where

$$ \nu \in \operatorname{Hom}_L(U(\mathfrak{t})^W, A) \simeq \operatorname{Hom}_L(U(\mathfrak{t}^*)^W, A) \simeq \operatorname{Hom}_L(U(\mathfrak{g}^*)^{\underline{G}_L}, A) $$

is the map induced by the conjugacy class of the Sen operators

$$ (\Theta_{\text{Sen}, \rho_{A,v}})_{v|p} \in (\mathfrak{g} \otimes_L A). $$

**Proposition 6.13.** *The homomorphisms $\zeta^C$ and $\alpha$ defined above coincide.*

*Proof.* It is sufficient to prove that for any local artinian L-algebra $A$ and any map $f : R_{\rho_p, \mathcal{R}}^{\text{qtri}} \to A$, we have $f \circ \zeta^C = f \circ \alpha$. Note that the map $f$ gives rise to a family $(\rho_{A,v})_{v|p}$ of local Galois representations. It follows from [BHS19, Lem. 3.7.5] that, for any embedding $\tau : F_v \hookrightarrow L$, the $\tau$-part of the Sen polynomial of $\rho_v$ is $\prod_{i=1}^n (X - (h_{i,\tau} + \nu_{i,\tau}))$ where $(\nu_{i,\tau}) \in \operatorname{Hom}_L(\mathfrak{t}, A)$ corresponds to $f \circ \kappa_2 : U(\mathfrak{t}) \to A$. The result is then a direct comparison of the definitions of $\alpha$ and $\zeta^C$. $\square$
For each element $M$ of the category $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, there is a natural homomorphism of $L$-algebras $Z(\mathfrak{g}) \to \text{End}(M)$. By functoriality of $\mathcal{M}_{\infty,x,\mathcal{R}}$, this gives a map

$$z : Z(\mathfrak{g}) \to \text{End}_{\tilde{\mathcal{R}}_{\infty,x}}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$$

The following result tells us that this map factors through $R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

**Corollary 6.14.** For any $x \in Z(\mathfrak{g})$, the element $z(x)$ is the multiplication by $\alpha(x) \otimes 1 \in R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

*Proof.* This is a consequence of Proposition 6.13 and of [DPS, Thm. 9.27]. $\square$

*Remark 6.15.* Recall that $h = (h_{1,\tau,v} < \cdots < h_{n,\tau,v})_{\tau,v}$ denotes the weight corresponding to the Hodge–Tate weights of $\rho$. Let $\lambda := w_0(h) - \delta'_G \in X^*(\mathcal{T})$, which is still a dominant character. Recall that $t_{-\delta_G} \circ \xi$ has image contained in $U(\mathfrak{t})^W$. Hence we have

$$t_{h-\delta'_G} \circ \xi = t_h \circ \text{Ad}(w_0) \circ t_{-\delta'_G} \circ \xi = \text{Ad}(w_0) \circ t_{w_0(h)} \circ t_{-\delta'_G} \circ \xi = \text{Ad}(w_0) \circ t_{\lambda} \circ \xi.$$

Therefore

$$\text{Id} \otimes \alpha = (\text{Id} \otimes \text{Ad}(w_0)) \circ h_{\lambda} : A \otimes_L Z(\mathfrak{g}) \to A \otimes_{Aw} A,$$

where $h_{\lambda}$ is the map defined in section 2.4.

## 6.4 Computation of a support

Now we can prove our main result of this section concerning the support of the patched functor applied to a generalized Verma module respectively applied to its dual.

**Theorem 6.16.** Let $x \in \mathcal{X}_{\infty}(L)$ be a point whose associated Galois representation is crystalline, $\varphi$-generic and Hodge–Tate regular. Let $\mathcal{R}$ be a refinement of $x$. Let $h = (h_{1,\tau} < \cdots < h_{n,\tau})_{\tau:F \hookrightarrow L} \in X^*(\mathcal{T})$ be the character given by the Hodge–Tate weights of $\rho_x$. Let $\delta'_G = \det \frac{1-n}{2} \delta_G = (0, -1, \dots, 1-n)_{\tau:F \hookrightarrow L} \in X^*(\mathcal{T})$, where $\delta_G$ is the half sum of the positive roots, and define $\lambda := w_0(h) - \delta'_G \in X^*(\mathcal{T})^+$. Then, for $I \subset \Delta$ and $w \in W$, the schematic supports of $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda))$ and $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee})$ are either $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri},w^{\min}w_0}$ or empty.

*Proof.* Let $M$ be $\tilde{M}_I(w^{\min} \cdot \lambda)$ or $\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee}$. As $R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$ is generically reduced and equi-dimensional by Lemma 4.4 and as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is Cohen–Macaulay of dimension $\dim R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$, its schematic support is reduced and is a union of irreducible components of $\text{Spec } R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$, i.e. it is a union of $\text{Spec } R_{\infty,x,\mathcal{R}}^{I-q\text{tri},w'}$ for some $w' \in W$.

By Proposition 2.15, the module $M$ is annihilated by $I_w \subset A_I \otimes_L Z(\mathfrak{g})$. This implies in particular that the action of $A_I \otimes_L Z(\mathfrak{g})$ on $M$ factors through $h_{\lambda}$. By functoriality, this gives rise to a structure of an $A_I \otimes_{Aw} A$-module on $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. Note that the
For each element $M$ of the category $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, there is a natural homomorphism of $L$-algebras $Z(\mathfrak{g}) \to \text{End}(M)$. By functoriality of $\mathcal{M}_{\infty,x,\mathcal{R}}$, this gives a map

$$z : Z(\mathfrak{g}) \to \text{End}_{\tilde{\mathcal{R}}_{\infty,x}}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$$

The following result tells us that this map factors through $R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

**Corollary 6.14.** For any $x \in Z(\mathfrak{g})$, the element $z(x)$ is the multiplication by $\alpha(x) \otimes 1 \in R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$.

*Proof.* This is a consequence of Proposition 6.13 and of [DPS, Thm. 9.27]. $\square$

*Remark 6.15.* Recall that $h = (h_{1,\tau,v} < \dots < h_{n,\tau,v})_{\tau,v}$ denotes the weight corresponding to the Hodge–Tate weights of $\rho$. Let $\lambda := w_0(h) - \delta'_G \in X^*(\mathcal{T})$, which is still a dominant character. Recall that $t_{-\delta_G} \circ \xi$ has image contained in $U(\mathfrak{t})^W$. Hence we have

$$t_{h-\delta'_G} \circ \xi = t_h \circ \text{Ad}(w_0) \circ t_{-\delta'_G} \circ \xi = \text{Ad}(w_0) \circ t_{w_0(h)} \circ t_{-\delta'_G} \circ \xi = \text{Ad}(w_0) \circ t_{\lambda} \circ \xi.$$

Therefore

$$\text{Id} \otimes \alpha = (\text{Id} \otimes \text{Ad}(w_0)) \circ h_{\lambda} : A \otimes_L Z(\mathfrak{g}) \to A \otimes_{Aw} A,$$

where $h_{\lambda}$ is the map defined in section 2.4.

## 6.4 Computation of a support

Now we can prove our main result of this section concerning the support of the patched functor applied to a generalized Verma module respectively applied to its dual.

**Theorem 6.16.** Let $x \in \mathcal{X}_{\infty}(L)$ be a point whose associated Galois representation is crystalline, $\varphi$-generic and Hodge–Tate regular. Let $\mathcal{R}$ be a refinement of $x$. Let $h = (h_{1,\tau} < \dots < h_{n,\tau})_{\tau:F \hookrightarrow L} \in X^*(\mathcal{T})$ be the character given by the Hodge–Tate weights of $\rho_x$. Let $\delta'_G = \det \frac{1-n}{2} \delta_G = (0, -1, \dots, 1-n)_{\tau:F \hookrightarrow L} \in X^*(\mathcal{T})$, where $\delta_G$ is the half sum of the positive roots, and define $\lambda := w_0(h) - \delta'_G \in X^*(\mathcal{T})^+$. Then, for $I \subset \Delta$ and $w \in W$, the schematic supports of $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda))$ and $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee})$ are either $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri},w^{\min}w_0}$ or empty.

*Proof.* Let $M$ be $\tilde{M}_I(w^{\min} \cdot \lambda)$ or $\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee}$. As $R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$ is generically reduced and equi-dimensional by Lemma 4.4 and as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is Cohen–Macaulay of dimension $\dim R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$, its schematic support is reduced and is a union of irreducible components of $\text{Spec } R_{\infty,x,\mathcal{R}}^{I-q\text{tri}}$, i.e. it is a union of $\text{Spec } R_{\infty,x,\mathcal{R}}^{I-q\text{tri},w'}$ for some $w' \in W$.

By Proposition 2.15, the module $M$ is annihilated by $I_w \subset A_I \otimes_L Z(\mathfrak{g})$. This implies in particular that the action of $A_I \otimes_L Z(\mathfrak{g})$ on $M$ factors through $h_{\lambda}$. By functoriality, this gives rise to a structure of an $A_I \otimes_{Aw} A$-module on $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. Note that the
map $(\kappa_1, \kappa_2)$ of section 4.1 provides a morphism of L-algebras $A_I \otimes_{A^W} A \to R_{\infty,x,\mathcal{R}}^{I-qtri}$ and, using Theorem 6.6, a second structure of an $A_I \otimes_{A^W} A$-module on $M_{\infty,x,\mathcal{R}}(M)$. It follows from Lemma 6.12, Corollary 6.14 and Remark 6.15 that this two actions coincide up to composition with $\mathrm{Id} \otimes \mathrm{Ad}(w_0)$. We deduce that $M_{\infty,x,\mathcal{R}}(M)$ is killed by the ideal of $R_{\infty,x,\mathcal{R}}^{I-qtri}$ defining the inverse image of $T_{I,ww_0} \subset \mathfrak{z}_I \times_{\mathrm{t}/W} \mathrm{t}$. Therefore Lemma 4.5 (see also Remark 2.16) implies that the action of $R_{\infty,x,\mathcal{R}}^{I-qtri}$ factors through $R_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ so that the schematic support of $M_{\infty,x,\mathcal{R}}(M)$ is $\mathrm{Spec} R_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$. $\square$

# 7 Main results

Let $x = (\rho_p, \rho^p, z) \in \mathcal{X}_{\infty}(L)$ corresponding to a classical automorphic form of tame level $K^p$. Moreover, we assume that (the Galois representation defined by) $x$ is crystalline, Hodge–Tate regular and $\varphi$-generic (see section 6.2) at $p$. This means that $x \in \mathcal{X}_{\bar{\rho},S}(L) \subset \mathcal{X}_{\infty}(L)$ and that there exists an automorphic representation $\pi = \pi_{\infty} \otimes_{\mathbb{C}} \pi_f$ of $U(\mathbb{A}_{\mathbb{Q}})$ whose associated Galois representation $\rho$ is the representation defined by $x$ and such that $\pi_f \otimes W$ occurs in the locally algebraic vectors of $\Pi_m$ for some algebraic representation $W$ depending on $\rho$. In particular, the automorphic representation $\pi$ is finite slope at $p$. It follows from the proof of [BHS17a, Cor. 3.12] that the image $\rho^p$ of $x$ in $\mathrm{Spf}(\otimes_{v \in S, p|v} \bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}}$ lies in the smooth locus.

We fix a refinement $\mathcal{R} = (\varphi_{1,v}, \dots, \varphi_{n,v})_v$ of $x$. Let us denote the $\tau$-Hodge–Tate weights of $\rho_{x,v}$ for $v|p$ in $F$ and $\tau : F_v \hookrightarrow L$ by $h_{v,\tau} := (h_{1,v,\tau} < \dots < h_{1,v,\tau})$. Given this collection of Hodge–Tate weights we write $h = (h_{v,\tau})_{v,\tau}$ and $h_v = (h_{v,\tau})_{\tau}$. We then define $R_{\rho_v}^{\mathrm{cris},h_v}$ to be the crystalline deformation ring of $\rho_v$ of labelled Hodge–Tate weight $h_v$ and set

$$R_{\rho_p}^{\mathrm{cris},h} = \widehat{\otimes}_{v|p} R_{\rho_v}^{\mathrm{cris},h_v}.$$

We further define

$$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h} = \widehat{\mathcal{X}}^{p}_{\rho^p} \times (\mathrm{Spf} R_{\rho_v}^{\mathrm{cris},h_v}) \times \widehat{\mathbb{U}}^g.$$

Note that it follows from the definitions that $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}$ embeds into $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},w_0}$ for any choice of a refinement $\mathcal{R}$.

We set

$$\mu_{v,\tau} = (h_{1,v,\tau}, h_{2,v,\tau} + 1, \dots, h_{n,v,\tau} + (n-1)) = h_{v,\tau} - \delta'_{G,v,\tau},$$

and $\mu = (\mu_{v,\tau})_{v,\tau}$, which is thus antidominant (for the upper Borel), and $\lambda = w_0(h) - \delta'_G = w_0 \cdot \mu \in X^*(\mathbb{T})^+$. For all $v|p$ in $F$, we denote by $W_v$ the Weyl group of $\mathrm{GL}_n(F_v)$, which we identify with $\mathfrak{S}_n$ and denote by $s_{1,v}, \dots, s_{n-1,v}$ the simple reflections with respect to the choice of the upper Borel $B_v \subset \mathrm{GL}_{n,F_v}$. Moreover, $w_{0,v} = s_{n-1,v} \dots s_{2,v} s_{1,v} s_{2,v} \dots s_{n-1,v}$ will denote the longest element of $W_v$. We then write $W = \prod_v W_v$ the Weyl group of $\mathbb{G}_{\mathbb{Q}_p} \simeq \prod_{v|p} \mathrm{GL}_{n,F_v}$ with respect to the Borel $B =$
map $(\kappa_1, \kappa_2)$ of section 4.1 provides a morphism of L-algebras $A_I \otimes_{A^W} A \to R_{\infty,x,\mathcal{R}}^{I-qtri}$ and, using Theorem 6.6, a second structure of an $A_I \otimes_{A^W} A$-module on $M_{\infty,x,\mathcal{R}}(M)$. It follows from Lemma 6.12, Corollary 6.14 and Remark 6.15 that this two actions coincide up to composition with $\mathrm{Id} \otimes \mathrm{Ad}(w_0)$. We deduce that $M_{\infty,x,\mathcal{R}}(M)$ is killed by the ideal of $R_{\infty,x,\mathcal{R}}^{I-qtri}$ defining the inverse image of $T_{I,ww_0} \subset \mathfrak{z}_I \times_{\mathrm{t}/W} \mathrm{t}$. Therefore Lemma 4.5 (see also Remark 2.16) implies that the action of $R_{\infty,x,\mathcal{R}}^{I-qtri}$ factors through $R_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ so that the schematic support of $M_{\infty,x,\mathcal{R}}(M)$ is $\mathrm{Spec} R_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$. $\square$

# 7 Main results

Let $x = (\rho_p, \rho^p, z) \in \mathcal{X}_{\infty}(L)$ corresponding to a classical automorphic form of tame level $K^p$. Moreover, we assume that (the Galois representation defined by) $x$ is crystalline, Hodge–Tate regular and $\varphi$-generic (see section 6.2) at $p$. This means that $x \in \mathcal{X}_{\bar{\rho},S}(L) \subset \mathcal{X}_{\infty}(L)$ and that there exists an automorphic representation $\pi = \pi_{\infty} \otimes_{\mathbb{C}} \pi_f$ of $U(\mathbb{A}_{\mathbb{Q}})$ whose associated Galois representation $\rho$ is the representation defined by $x$ and such that $\pi_f \otimes W$ occurs in the locally algebraic vectors of $\Pi_m$ for some algebraic representation $W$ depending on $\rho$. In particular, the automorphic representation $\pi$ is finite slope at $p$. It follows from the proof of [BHS17a, Cor. 3.12] that the image $\rho^p$ of $x$ in $\mathrm{Spf}(\otimes_{v \in S, p \nmid v} \bar{R}_{\bar{\rho}_v}^{\square})^{\mathrm{rig}}$ lies in the smooth locus.

We fix a refinement $\mathcal{R} = (\varphi_{1,v}, \dots, \varphi_{n,v})_v$ of $x$. Let us denote the $\tau$-Hodge–Tate weights of $\rho_{x,v}$ for $v \nmid p$ in $F$ and $\tau : F_v \hookrightarrow L$ by $h_{v,\tau} := (h_{1,v,\tau} < \dots < h_{1,v,\tau})$. Given this collection of Hodge–Tate weights we write $h = (h_{v,\tau})_{v,\tau}$ and $h_v = (h_{v,\tau})_{\tau}$. We then define $R_{\rho_v}^{\mathrm{cris},h_v}$ to be the crystalline deformation ring of $\rho_v$ of labelled Hodge–Tate weight $h_v$ and set

$$R_{\rho_p}^{\mathrm{cris},h} = \widehat{\otimes}_{v|p} R_{\rho_v}^{\mathrm{cris},h_v}.$$

We further define

$$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h} = \widehat{\mathcal{X}}^{p}_{\rho^p} \times (\mathrm{Spf} R_{\rho_v}^{\mathrm{cris},h_v}) \times \widehat{\mathbb{U}}^g.$$

Note that it follows from the definitions that $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}$ embeds into $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},w_0}$ for any choice of a refinement $\mathcal{R}$.

We set

$$\mu_{v,\tau} = (h_{1,v,\tau}, h_{2,v,\tau} + 1, \dots, h_{n,v,\tau} + (n-1)) = h_{v,\tau} - \delta'_{G,v,\tau},$$

and $\mu = (\mu_{v,\tau})_{v,\tau}$, which is thus antidominant (for the upper Borel), and $\lambda = w_0(h) - \delta'_G = w_0 \cdot \mu \in X^*(\mathbb{T})^+$. For all $v \nmid p$ in $F$, we denote by $W_v$ the Weyl group of $\mathrm{GL}_n(F_v)$, which we identify with $\mathfrak{S}_n$ and denote by $s_{1,v}, \dots, s_{n-1,v}$ the simple reflections with respect to the choice of the upper Borel $B_v \subset \mathrm{GL}_{n,F_v}$. Moreover, $w_{0,v} = s_{n-1,v} \dots s_{2,v} s_{1,v} s_{2,v} \dots s_{n-1,v}$ will denote the longest element of $W_v$. We then write $W = \prod_v W_v$ the Weyl group of $\mathbb{G}_{\mathbb{Q}_p} \simeq \prod_{v|p} \mathrm{GL}_{n,F_v}$ with respect to the Borel $B =$
$\prod_{v|p} B_v$. Because of the product structure, we will sometimes abuse notations and simply write $s_i$ for the simple reflections and $w_0$ for the longest element.

For a scheme $X$ of dimension $d$ we write $Z^0(X) = Z_d(X)$ for the free abelian group on the irreducible components of $X$. Moreover, for $d' \le d$ we write $Z_{d'}(X)$ for the free abelian group on the irreducible and reduced closed subschemes of dimension $d'$. We recall that a coherent sheaf $\mathcal{F}$ on $X$ with $d'$-dimensional support defines a class $[\mathcal{F}] \in Z_{d'}(X)$, see e.g. [BHS19, Equation (2.13)].

## 7.1 Sheaves and supports.

Let $\lambda = w_0 \cdot \mu \in X^*(\underline{T})^+$ dominant, integral.

We moreover write

$$m_x = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda)) \otimes k(x). \qquad (8)$$

It follows from [BHS19, Thm. 5.1.3] that $m_x \ge 1$ and the proof of [BHS19, Thm. 5.3.3] implies that $m_x$ does not depend on the choice of a refinement $\mathcal{R}$. To $x$ and $\mathcal{R}$ we associate a permutation

$$w_{x,\mathcal{R}} = (w_{x,\mathcal{R}_v})_{v \in \Sigma} = (w_{x,\mathcal{R}_v,\tau})_{v,\tau} \in W$$

defined as in [HMS14, § 3.7]. We recall that these permutations encode the relative position of the Hodge–Tate flags with respect to the full flag corresponding to the refinement $\mathcal{R}$. We recall that, for any object $M$ of $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is zero or Cohen–Macaulay of dimension $d$.

**Lemma 7.1.** Let $R$ be a Cohen–Macaulay noetherian local ring of dimension $d'$ and let $M$ and $M'$ be two finitely generated Cohen–Macaulay modules. Let $(t_1, \dots, t_m)$ be a regular sequence of elements of the maximal ideal of $R$ which is also $M$ and $M'$-regular. Assume that $[M] = [M']$ in $Z_{d'}(\text{Spec } R)$. Then

$$[M/(t_1, \dots, t_m)M] = [M'/(t_1, \dots, t_m)M'] \in Z_{d'-m}(R).$$

*Proof.* By induction it is sufficient to prove the result when $m = 1$. Set $t = t_1$. Let $\mathfrak{p}$ be a prime ideal of $R$ which is a generic point of $\text{Supp}(M)$ or $\text{Supp}(M')$. It is sufficient to prove that $[M_{\mathfrak{p}}/tM_{\mathfrak{p}}] = [M'_{\mathfrak{p}}/tM'_{\mathfrak{p}}]$ in $Z_{d'-1}(\text{Spec } R_{\mathfrak{p}}/(t))$, i.e. that $M_{\mathfrak{p}}/tM_{\mathfrak{p}}$ and $M'_{\mathfrak{p}}/tM'_{\mathfrak{p}}$ are two $R_{\mathfrak{p}}/(t)$-modules of the same length. This is a consequence of [Sta24, Lemma 02QG]. $\square$

Let $\mathcal{N} \subset \mathfrak{g}$ be the nilpotent cone and let $\tilde{\mathcal{N}} \to \mathcal{N}$ be the Springer resolution. Similarly to the definition of the closed subschemes $X_w \subset X$ in 4.1 we define

$$Z_w \subset \tilde{\mathcal{N}} \times_{\mathcal{N}} \tilde{\mathcal{N}} \subset X$$
$\prod_{v|p} B_v$. Because of the product structure, we will sometimes abuse notations and simply write $s_i$ for the simple reflections and $w_0$ for the longest element.

For a scheme $X$ of dimension $d$ we write $Z^0(X) = Z_d(X)$ for the free abelian group on the irreducible components of $X$. Moreover, for $d' \le d$ we write $Z_{d'}(X)$ for the free abelian group on the irreducible and reduced closed subschemes of dimension $d'$. We recall that a coherent sheaf $\mathcal{F}$ on $X$ with $d'$-dimensional support defines a class $[\mathcal{F}] \in Z_{d'}(X)$, see e.g. [BHS19, Equation (2.13)].

## 7.1 Sheaves and supports.

Let $\lambda = w_0 \cdot \mu \in X^*(\underline{T})^+$ dominant, integral.

We moreover write

$$m_x = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda)) \otimes k(x). \qquad (8)$$

It follows from [BHS19, Thm. 5.1.3] that $m_x \ge 1$ and the proof of [BHS19, Thm. 5.3.3] implies that $m_x$ does not depend on the choice of a refinement $\mathcal{R}$. To $x$ and $\mathcal{R}$ we associate a permutation

$$w_{x,\mathcal{R}} = (w_{x,\mathcal{R}_v})_{v \in \Sigma} = (w_{x,\mathcal{R}_v,\tau})_{v,\tau} \in W$$

defined as in [HMS14, § 3.7]. We recall that these permutations encode the relative position of the Hodge–Tate flags with respect to the full flag corresponding to the refinement $\mathcal{R}$. We recall that, for any object $M$ of $\mathcal{O}_{\text{alg}}^{\infty}$ or $\tilde{\mathcal{O}}_{\text{alg}}$, the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is zero or Cohen–Macaulay of dimension $d$.

**Lemma 7.1.** Let $R$ be a Cohen–Macaulay noetherian local ring of dimension $d'$ and let $M$ and $M'$ be two finitely generated Cohen–Macaulay modules. Let $(t_1, \dots, t_m)$ be a regular sequence of elements of the maximal ideal of $R$ which is also $M$ and $M'$-regular. Assume that $[M] = [M']$ in $Z_{d'}(\text{Spec } R)$. Then

$$[M/(t_1, \dots, t_m)M] = [M'/(t_1, \dots, t_m)M'] \in Z_{d'-m}(R).$$

*Proof.* By induction it is sufficient to prove the result when $m = 1$. Set $t = t_1$. Let $\mathfrak{p}$ be a prime ideal of $R$ which is a generic point of $\text{Supp}(M)$ or $\text{Supp}(M')$. It is sufficient to prove that $[M_{\mathfrak{p}}/tM_{\mathfrak{p}}] = [M'_{\mathfrak{p}}/tM'_{\mathfrak{p}}]$ in $Z_{d'-1}(\text{Spec } R_{\mathfrak{p}}/(t))$, i.e. that $M_{\mathfrak{p}}/tM_{\mathfrak{p}}$ and $M'_{\mathfrak{p}}/tM'_{\mathfrak{p}}$ are two $R_{\mathfrak{p}}/(t)$-modules of the same length. This is a consequence of [Sta24, Lemma 02QG]. $\square$

Let $\mathcal{N} \subset \mathfrak{g}$ be the nilpotent cone and let $\tilde{\mathcal{N}} \to \mathcal{N}$ be the Springer resolution. Similarly to the definition of the closed subschemes $X_w \subset X$ in 4.1 we define

$$Z_w \subset \tilde{\mathcal{N}} \times_{\mathcal{N}} \tilde{\mathcal{N}} \subset X$$
to be the Zariski closure of preimage under $\tilde{N} \times_N \tilde{N} \to \underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$ of the orbit $\underline{G}_L(1, w) \subset \underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$. Set

$$Z_w = g(f^{-1}(Z_w \cap \hat{X}_{I,w,x_{dR}})) \times \hat{X}_{\rho p}^p \times \hat{U}^g \subset \bar{X}_{\infty,x,\mathcal{R}}^{\text{qtri}}$$

where $f$ and $g$ are the maps from Theorem 4.7.

In the following we will make use of the following abusive notation for (local) formal schemes: Let Spf $R$ be a (local) affine formal scheme. Then we will say that Spf $R$ is reduced, if $R$ is reduced. Moreover, we will say that Spf $R$ is irreducible if Spec $R$ is irreducible. More generally, for a given irreducible component Spec $R/\mathfrak{a} \subset \text{Spec } R$, we will refer to the formal subscheme Spf $R/\mathfrak{a} \subset \text{Spf } R$ as an irreducible component of Spf $R$. Similarly, we will write $Z^0(\text{Spf } R) = Z^0(\text{Spec } R)$ for the free abelian group on the irreducible components of Spf $R$ to which we also refer as the irreducible components of Spf $R$, etc.

**Proposition 7.2.** Let $w \in W$. Then the following properties hold:

1) For all $I \subset \Delta$ and all $\bar{w} \in W_I \setminus W$ satisfying $w^{\min}w_0 \ge w_{x,\mathcal{R}}$, the formal subscheme $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ is reduced and irreducible and coincides with an irreducible component of $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri}$.

2) The schematic support of $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$, for $w \in W$, is contained in $\bar{X}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ if $ww_0 \ge w_{x,\mathcal{R}}$, and this sheaf is zero otherwise. Moreover,

$$[\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))] = m_x[\bar{X}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}] \in Z^0(\bar{X}_{\infty,x,\mathcal{R}}^{I-qtri})$$

for $ww_0 \ge w_{x,\mathcal{R}}$, where $m_x$ is the integer defined by (8).

3) There is an equality

$$[\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))] = m_x \sum_{w' \le w} a_{w,w'}[Z_w] \in Z^0(\bar{X}_{\infty,x,\mathcal{R}}^{I-qtri})$$

where the $a_{w,w'} \in \mathbb{N}$ are the integers defined in [BHS19, Thm. 2.4.7]. In particular $a_{w,w} = 1$.

4) For all $I \subset \Delta$, the sheaves

$$\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda)^{\vee})$$

are non zero if and only if $w^{\min}w_0 \ge w_{x,\mathcal{R}}$.

5) For all $I \subset \Delta$, the support of

$$\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee}),$$

for $\bar{w} \in W_I \setminus W$, is $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w^{\min}w_0}$ if $w^{\min}w_0 \ge w_{x,\mathcal{R}}$ and these sheaves are zero otherwise.
to be the Zariski closure of preimage under $\tilde{N} \times_N \tilde{N} \to \underline{G_L}/\underline{B} \times \underline{G_L}/\underline{B}$ of the orbit $\underline{G_L}(1, w) \subset \underline{G_L}/\underline{B} \times \underline{G_L}/\underline{B}$. Set

$$ \mathcal{Z}_w = g(f^{-1}(Z_w \cap \hat{X}_{I,w,x_{dR}})) \times \hat{\chi}_{\rho p}^p \times \hat{\mathbb{U}}^g \subset \bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}} $$

where $f$ and $g$ are the maps from Theorem 4.7.

In the following we will make use of the following abusive notation for (local) formal schemes: Let Spf $R$ be a (local) affine formal scheme. Then we will say that Spf $R$ is reduced, if $R$ is reduced. Moreover, we will say that Spf $R$ is irreducible if Spec $R$ is irreducible. More generally, for a given irreducible component Spec $R/\mathfrak{a} \subset \text{Spec } R$, we will refer to the formal subscheme Spf $R/\mathfrak{a} \subset \text{Spf } R$ as an irreducible component of Spf $R$. Similarly, we will write $Z^0(\text{Spf } R) = Z^0(\text{Spec } R)$ for the free abelian group on the irreducible components of Spf $R$ to which we also refer as the irreducible components of Spf $R$, etc.

**Proposition 7.2.** Let $w \in W$. Then the following properties hold:

1) For all $I \subset \Delta$ and all $\bar{w} \in W_I \setminus W$ satisfying $w^{\min}w_0 \ge w_{x,\mathcal{R}}$, the formal subscheme $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ is reduced and irreducible and coincides with an irreducible component of $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri}$.

2) The schematic support of $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$, for $w \in W$, is contained in $\bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}$ if $ww_0 \ge w_{x,\mathcal{R}}$, and this sheaf is zero otherwise. Moreover,

$$ [\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))] = m_x[\bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri,ww_0}] \in Z^0(\bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri}) $$

for $ww_0 \ge w_{x,\mathcal{R}}$, where $m_x$ is the integer defined by (8).

3) There is an equality

$$ [\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))] = m_x \sum_{w' \le w} a_{w,w'}[\mathcal{Z}_w] \in Z^0(\bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri}) $$

where the $a_{w,w'} \in \mathbb{N}$ are the integers defined in [BHS19, Thm. 2.4.7]. In particular $a_{w,w} = 1$.

4) For all $I \subset \Delta$, the sheaves

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda)^{\vee}) $$

are non zero if and only if $w^{\min}w_0 \ge w_{x,\mathcal{R}}$.

5) For all $I \subset \Delta$, the support of

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee}), $$

for $\bar{w} \in W_I \setminus W$, is $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w^{\min}w_0}$ if $w^{\min}w_0 \ge w_{x,\mathcal{R}}$ and these sheaves are zero otherwise.
6) The module $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank $m_x$ over $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{cris},h} \subset \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},w_0}$.

7) For any $I \subset \Delta$ and any $w \in W$, the sheaves

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w^{\min} \cdot \lambda)^{\vee}) $$

are generically free of rank $m_x$ over their support.

*Proof.* We first prove point 1)). As $\mathcal{X}^p$ is smooth at $\rho^p$ (as recalled in the beginning of this section), the formal completion $\widetilde{\mathcal{X}}^{\rho^p}$ is formally smooth. As $\widetilde{\mathcal{U}}^g$ is also formally smooth, the claim follows from the fact that

$$ \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}} \text{ and } \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to \widehat{\mathcal{X}}_{I, x_{pdR}} $$

are formally smooth and that $X_{I,w,x_{pdR}}$ is an irreducible component of $\widehat{\mathcal{X}}_{I, x_{pdR}}$.

By Theorem 6.16, the schematic support of the Cohen-Macaulay sheaves

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)^{\vee}) $$

is contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w}$ which is irreducible. By Proposition 6.5, as the sheaves are Cohen-Macaulay of dimension $t + \dim \beta_I = \dim \mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri}$ (e.g. [BHS19, equation (5.8)] and Proposition 3.20), we deduce that, if non empty, their schematic support is all $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w}$.

By Remark 6.3 we deduce also that

$$ \text{supp}(\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda))) \subset \bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri, w^{\min} w_0} $$

for $w \in {}^I W$. Note that the Jordan-Hölder factors of $M_I(w \cdot \lambda)$ are among the $L(w' \cdot \lambda)$ with $w' \ge w$ and that $L(w \cdot \lambda)$ is the cosople of $M_I(w \cdot \lambda)$. Therefore $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) \ne 0$ if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda)) \ne 0$ if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w \cdot \lambda)) \ne 0$. Therefore the non nullity assertions in 4) and 5) follow from the exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (Proposition 6.5) and from [BHS19, Thm. 5.3.3]. This proves 4) and 5).

We prove point 6). By [BHS19, Remark 4.3.1 and Proof of Theorem 5.3.3, Step 7], the schematic support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is contained in the crystalline locus $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{cris},h} \subset \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri}}$, which is smooth and irreducible of the same dimension as the support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$. Thus these coincide and $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank $m_x$ over the crystalline locus.

No we prove point 2). The first assertion has already been proved with 4) and 5), therefore it remains to prove the assertion on the cycle. It follows from the proof [BHS19, Thm. 5.3.3] that $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$ is generically free of rank $m_x$ for $ww_0 \ge w_{x,\mathcal{R}}$. As $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},ww_0}$ is Cohen-Macaulay, the result is a consequence of point 5) and of Lemma 7.1 applied with

$$ M = \mathcal{O}_{\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},ww_0}}^{m_x} \text{ and } M' = \mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)) $$
6) The module $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank $m_x$ over $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{cris},h} \subset \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},w_0}$.

7) For any $I \subset \Delta$ and any $w \in W$, the sheaves

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w^{\min} \cdot \lambda)^{\vee}) $$

are generically free of rank $m_x$ over their support.

*Proof.* We first prove point 1)). As $\mathcal{X}^p$ is smooth at $\rho^p$ (as recalled in the beginning of this section), the formal completion $\widetilde{\mathcal{X}}^{\rho^p}$ is formally smooth. As $\widetilde{\mathcal{U}}^g$ is also formally smooth, the claim follows from the fact that

$$ \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}} \text{ and } \mathcal{X}_{r_v, \mathcal{R}_v}^{I_v - \text{qtri}, \square} \to \widehat{\mathcal{X}}_{I, x_{\text{pdR}}} $$

are formally smooth and that $X_{I,w,x_{\text{pdR}}}$ is an irreducible component of $\widehat{\mathcal{X}}_{I, x_{\text{pdR}}}$.

By Theorem 6.16, the schematic support of the Cohen-Macaulay sheaves

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)^{\vee}) $$

is contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w}$ which is irreducible. By Proposition 6.5, as the sheaves are Cohen-Macaulay of dimension $t + \dim \beta_I = \dim \mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri}$ (e.g. [BHS19, equation (5.8)] and Proposition 3.20), we deduce that, if non empty, their schematic support is all $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-qtri,w}$.

By Remark 6.3 we deduce also that

$$ \text{supp}(\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda))) \subset \bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-qtri, w^{\min} w_0} $$

for $w \in {}^I W$. Note that the Jordan-Hölder factors of $M_I(w \cdot \lambda)$ are among the $L(w' \cdot \lambda)$ with $w' \ge w$ and that $L(w \cdot \lambda)$ is the cosople of $M_I(w \cdot \lambda)$. Therefore $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) \ne 0$ if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda)) \ne 0$ if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w \cdot \lambda)) \ne 0$. Therefore the non nullity assertions in 4) and 5) follow from the exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (Proposition 6.5) and from [BHS19, Thm. 5.3.3]. This proves 4) and 5).

We prove point 6). By [BHS19, Remark 4.3.1 and Proof of Theorem 5.3.3, Step 7], the schematic support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is contained in the crystalline locus $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{cris},h} \subset \mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri}}$, which is smooth and irreducible of the same dimension as the support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$. Thus these coincide and $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank $m_x$ over the crystalline locus.

No we prove point 2). The first assertion has already been proved with 4) and 5), therefore it remains to prove the assertion on the cycle. It follows from the proof [BHS19, Thm. 5.3.3] that $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$ is generically free of rank $m_x$ for $ww_0 \ge w_{x,\mathcal{R}}$. As $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},ww_0}$ is Cohen-Macaulay, the result is a consequence of point 5) and of Lemma 7.1 applied with

$$ M = \mathcal{O}_{\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},ww_0}}^{m_x} \text{ and } M' = \mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)) $$
and to a regular sequence generating the maximal ideal of $U(t)_m$. This sequence is $M'$-regular by Proposition 6.5.

We deduce 3) from 2) together with formulas (5.23) and (5.24) of [BHS19] and the fact that the Verma modules form a basis of the Grothendieck group of the category $\mathcal{O}_{\chi\lambda}$.

We prove point 7). As $\chi_{\infty,x,\mathcal{R}}^{I-qtri,w'}$ is generically smooth for any $w'$, the module $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generically free, say of rank $r$, over its support where

$$M \in \{\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)), \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee})\}.$$

Now we claim that there exists an open and subset $U$ in the regular locus of $\text{Spec}(R_{\infty,x,\mathcal{R}}^{I-qtri,w^{\min}w_0})$ such that $U$ intersects the support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\min} \cdot \lambda))$. The claim then implies $r = m_x$. Indeed, the restriction of $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda))$ to $U$ is locally free since $U$ is regular. Therefore $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda))$ is locally free of rank $r$ over its support intersected with $U$. It follows from the point 3) that $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w' \cdot \lambda))$ is not supported at the generic point of $\mathcal{Z}_{w^{\min}w_0}$ for $w' > w^{\min}$ and that $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\min} \cdot \lambda))$ has length $m_x$ at the generic point of $\mathcal{Z}_{w^{\min}w_0}$. As $L(w^{\min} \cdot \lambda)$ appears with multiplicity one in $M_I(w^{\min} \cdot \lambda)$ and all other subquotient are of the form $L(w' \cdot \lambda)$ with $w' > w^{\min}$, we have $r = m_x$. We now construct an open subset $U$ with the claimed properties. We set

$$U = g(f^{-1}(V_{w^{\min}w_0} \cap \tilde{X}_{I,w^{\min}w_0,x_{dR}})) \times \tilde{X}_{\rho p}^p \times \mathbb{U}^g,$$

where $f$ and $g$ are the maps of Theorem 4.7 and $V_{w^{\min}w_0}$ is the preimage of the Schubert cell $\underline{G}_L(1, w^{\min}w_0) \subset \underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$ in $X_{I,w^{\min}w_0}$. This is an open and smooth subset of $X_{I,w^{\min}w_0}$: indeed, the maps $f$ and $g$ are formally smooth, the formal scheme $\chi_{\infty,x,\mathcal{R}}^{qtri} \to \chi_{\rho p,\mathcal{R}}^{qtri}$ is formally smooth and the point as $\rho^p$ lies in the smooth locus of $\mathcal{X}^p$. $\square$

**Proposition 7.3.** Assume that $x_{\text{pdR}}$ is a smooth point of $X_{w^{\min}w_0}$. Then

$$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\min} \cdot \lambda)^{\vee})$$

are finite free $\mathcal{O}_{\tilde{X}_{\infty,x,\mathcal{R}}^{qtri,w^{\min}w_0}}$-modules.

*Proof.* We write $w^{\min} = w$ to simplify the notations. By Remark 6.3, the two $U(t)$-module structures on $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda))$ coming from the $U(t)$-action on $\tilde{M}(w \cdot \lambda)$ and the one coming from the derivative of the locally analytic action, coincide. Thus we have the equality between $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$ and the localisation

$$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)) \simeq i_*i^*\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda)),$$

where $i: \hat{T}^{\text{sm}} \to \hat{T}$ denotes the inclusion of the closed subspace of smooth characters. A similar remark applies to the dual Verma module. In particular, it is enough to show that the $\mathcal{O}_{\chi_{\infty,x,\mathcal{R}}^{qtri,ww_0}}$-modules

$$\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}^{\vee}(w \cdot \lambda))$$
and to a regular sequence generating the maximal ideal of $U(t)_m$. This sequence is $M'$-regular by Proposition 6.5.

We deduce 3) from 2) together with formulas (5.23) and (5.24) of [BHS19] and the fact that the Verma modules form a basis of the Grothendieck group of the category $\mathcal{O}_{\chi\lambda}$.

We prove point 7). As $\chi_{\infty,x,\mathcal{R}}^{I-qtri,w'}$ is generically smooth for any $w'$, the module $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generically free, say of rank $r$, over its support where

$$M \in \{\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)), \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda)^{\vee})\}.$$

Now we claim that there exists an open and subset $U$ in the regular locus of $\text{Spec}(R_{\infty,x,\mathcal{R}}^{I-qtri,w^{\min}w_0})$ such that $U$ intersects the support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\min} \cdot \lambda))$. The claim then implies $r = m_x$. Indeed, the restriction of $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}_I(w^{\min} \cdot \lambda))$ to $U$ is locally free since $U$ is regular. Therefore $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w^{\min} \cdot \lambda))$ is locally free of rank $r$ over its support intersected with $U$. It follows from the point 3) that $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w' \cdot \lambda))$ is not supported at the generic point of $\mathcal{Z}_{w^{\min}w_0}$ for $w' > w^{\min}$ and that $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\min} \cdot \lambda))$ has length $m_x$ at the generic point of $\mathcal{Z}_{w^{\min}w_0}$. As $L(w^{\min} \cdot \lambda)$ appears with multiplicity one in $M_I(w^{\min} \cdot \lambda)$ and all other subquotient are of the form $L(w' \cdot \lambda)$ with $w' > w^{\min}$, we have $r = m_x$. We now construct an open subset $U$ with the claimed properties. We set

$$U = g(f^{-1}(V_{w^{\min}w_0} \cap \tilde{X}_{I,w^{\min}w_0,x_{dR}})) \times \tilde{X}_{\rho p}^p \times \mathbb{U}^g,$$

where $f$ and $g$ are the maps of Theorem 4.7 and $V_{w^{\min}w_0}$ is the preimage of the Schubert cell $\underline{G}_L(1, w^{\min}w_0) \subset \underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$ in $X_{I,w^{\min}w_0}$. This is an open and smooth subset of $X_{I,w^{\min}w_0}$: indeed, the maps $f$ and $g$ are formally smooth, the formal scheme $\tilde{X}_{\infty,x,\mathcal{R}}^{qtri} \to \tilde{X}_{\rho p,\mathcal{R}}^{qtri}$ is formally smooth and the point as $\rho^p$ lies in the smooth locus of $\mathcal{X}^p$. $\square$

**Proposition 7.3.** Assume that $x_{pdR}$ is a smooth point of $X_{w^{\min}w_0}$. Then

$$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\min} \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\min} \cdot \lambda)^{\vee})$$

are finite free $\mathcal{O}_{\tilde{X}_{\infty,x,\mathcal{R}}^{qtri,w^{\min}w_0}}$-modules.

*Proof.* We write $w^{\min} = w$ to simplify the notations. By Remark 6.3, the two $U(t)$-module structures on $\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda))$ coming from the $U(t)$-action on $\tilde{M}(w \cdot \lambda)$ and the one coming from the derivative of the locally analytic action, coincide. Thus we have the equality between $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda))$ and the localisation

$$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)) \simeq i_*i^*\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda)),$$

where $i: \hat{T}^{\text{sm}} \to \hat{T}$ denotes the inclusion of the closed subspace of smooth characters. A similar remark applies to the dual Verma module. In particular, it is enough to show that the $\mathcal{O}_{\chi_{\infty,x,\mathcal{R}}^{qtri,ww_0}}$-modules

$$\mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}(w \cdot \lambda)) \text{ and } \mathcal{M}_{\infty,x,\mathcal{R}}(\tilde{M}^{\vee}(w \cdot \lambda))$$
are finite free. But these modules are Cohen-Macaulay with support the localization at $x$ of $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri},ww_0}$, which is smooth. $\square$

## 7.2 Recollection on Bezrukavnikov's functor

The aim if this section (or even of the paper) is to identify the patching functor that takes objects in $\mathcal{O}_{\text{alg}}$ (or more generally in $\mathcal{O}_{\text{alg}}^{\infty}$) to Cohen-Macaulay modules on certain Galois deformation rings with a functor constructed by Bezrukavnikov in geometric representation theory (more precisely: with the pullback from our local models to the Galois deformation rings). Before doing so, we will need to recall the result of Bezrukavnikov.

Recall that $X = \tilde{g} \times_g \tilde{g}$ where $g$ is the Lie algebra of $\mathcal{G}_L = \prod_{v \in \Sigma} (L \times_{\mathbb{Q}_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_n)$ as in section 4.1 and denote by $X^\wedge$ the completion of $X$ along the preimage of $\{(0,0)\} \in t \times_{t/W} t$ in $X$. Moreover, we write $\bar{X} = X \times_t \{0\}$, where the fiber product is taken with respect to the map $\kappa_1: X \to t$ of 4.1 that maps $(g\bar{B}, h\bar{B}, N)$ to $\text{ad}(g^{-1})(N)$ (mod $n) \in t$. As in the preceding sections we fix the shift

$$ \delta'_{\bar{G}} = \det^{\frac{1-n}{2}} \delta_{\bar{G}} \in X^*(\bar{T}) $$

of the half sum of the positive roots $\delta_{\bar{G}}$.

**Theorem 7.4 (Bezrukavnikov).** Let $\lambda \in X^*(\bar{T})$ be a dominant character. There exists an exact functor

$$ \mathcal{B} : \mathcal{O}_{\chi_{\lambda}} \to \text{Coh}_{\mathcal{G}_L}(X^{\wedge}), $$

such that

1) for all $M \in \mathcal{O}_{\chi_{\lambda}}$ the sheaf $\mathcal{B}(M)$ is a Cohen-Macaulay sheaf,

2) for all $w \in W$ there is an isomorphism $\mathcal{B}(M(ww_0 \cdot \lambda)^{\vee}) \simeq \mathcal{O}_{\bar{X}_w}$,

3) for all $w \in W$ there is an isomorphism $\mathcal{B}(M(ww_0 \cdot \lambda)) \simeq \omega_{\bar{X}_w}$,

4) the image $\mathcal{B}(P(w_0 \cdot \lambda))$ of the anti-dominant projective $P(w_0 \cdot \lambda)$ is the structure sheaf $\mathcal{O}_{\bar{X}}$,

5) the image $\mathcal{B}(L(\lambda))$ of the algebraic representation $L(\lambda)$ is the line bundle $\mathcal{O}(-\delta'_{\bar{G}}) \boxtimes \mathcal{O}(-\delta'_{\bar{G}})$ on $\mathcal{G}_L/\bar{B} \times \mathcal{G}_L/\bar{B}$ which is viewed as a closed subscheme of $X^\wedge$ via

$$ (g\bar{B}, h\bar{B}) \mapsto (g\bar{B}, h\bar{B}, 0). $$

This result is (a small part of a result) due to Bezrukavnikov and his collaborators whose proof is spread out through the papers [Bez16, BR12, BL23, BR22]). For the convenience of the reader, we explain how to get the result in the previous form.
are finite free. But these modules are Cohen-Macaulay with support the localization at $x$ of $\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{qtri,ww}_0}$, which is smooth. $\square$

## 7.2 Recollection on Bezrukavnikov's functor

The aim if this section (or even of the paper) is to identify the patching functor that takes objects in $\mathcal{O}_{\text{alg}}$ (or more generally in $\mathcal{O}_{\text{alg}}^{\infty}$) to Cohen-Macaulay modules on certain Galois deformation rings with a functor constructed by Bezrukavnikov in geometric representation theory (more precisely: with the pullback from our local models to the Galois deformation rings). Before doing so, we will need to recall the result of Bezrukavnikov.

Recall that $X = \tilde{g} \times_g \tilde{g}$ where $g$ is the Lie algebra of $\mathcal{G}_L = \prod_{v \in \Sigma} (L \times_{\mathbb{Q}_p} \text{Res}_{F_v/\mathbb{Q}_p} \text{GL}_n)$ as in section 4.1 and denote by $X^\wedge$ the completion of $X$ along the preimage of $\{(0,0)\} \in t \times_{t/W} t$ in $X$. Moreover, we write $\bar{X} = X \times_t \{0\}$, where the fiber product is taken with respect to the map $\kappa_1: X \to t$ of 4.1 that maps $(g\bar{B}, h\bar{B}, N)$ to $\text{ad}(g^{-1})(N)$ (mod $n) \in t$. As in the preceding sections we fix the shift

$$ \delta'_{\bar{G}} = \det^{\frac{1-n}{2}} \delta_{\bar{G}} \in X^*(\bar{T}) $$

of the half sum of the positive roots $\delta_{\bar{G}}$.

**Theorem 7.4 (Bezrukavnikov).** Let $\lambda \in X^*(\bar{T})$ be a dominant character. There exists an exact functor

$$ \mathcal{B} : \mathcal{O}_{\chi_{\lambda}} \to \text{Coh}_{\mathcal{G}_L}(X^{\wedge}), $$

such that

1) for all $M \in \mathcal{O}_{\chi_{\lambda}}$ the sheaf $\mathcal{B}(M)$ is a Cohen-Macaulay sheaf,

2) for all $w \in W$ there is an isomorphism $\mathcal{B}(M(ww_0 \cdot \lambda)^{\vee}) \simeq \mathcal{O}_{\bar{X}_w}$,

3) for all $w \in W$ there is an isomorphism $\mathcal{B}(M(ww_0 \cdot \lambda)) \simeq \omega_{\bar{X}_w}$,

4) the image $\mathcal{B}(P(w_0 \cdot \lambda))$ of the anti-dominant projective $P(w_0 \cdot \lambda)$ is the structure sheaf $\mathcal{O}_{\bar{X}}$,

5) the image $\mathcal{B}(L(\lambda))$ of the algebraic representation $L(\lambda)$ is the line bundle $\mathcal{O}(-\delta'_{\bar{G}}) \boxtimes \mathcal{O}(-\delta'_{\bar{G}})$ on $\mathcal{G}_L/\bar{B} \times \mathcal{G}_L/\bar{B}$ which is viewed as a closed subscheme of $X^\wedge$ via

$$ (g\bar{B}, h\bar{B}) \mapsto (g\bar{B}, h\bar{B}, 0). $$

This result is (a small part of a result) due to Bezrukavnikov and his collaborators whose proof is spread out through the papers [Bez16, BR12, BL23, BR22]). For the convenience of the reader, we explain how to get the result in the previous form.
Proof. By the main result of [Bez16], there are reverse equivalence of categories

$$ \Psi : D_{I^0, I^0} \leftrightarrow D^b(\mathrm{Coh}(\tilde{\mathfrak{g}} \times_{\mathfrak{g}} \tilde{\mathfrak{g}})) : \Phi_{I^0, I^0}, $$

which we can then localize on $X^\wedge \subset X$. Up to use translation functors, we can focus on the case $\lambda = 0$. By [Bez16, Corollary 42] the functor $\Psi$ in fact takes values in $(\mathcal{G}$-equivariant) coherent sheaves on $X$, when restricted to perverse sheaves $F \in \mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{B})$. Moreover, the Beillinson-Bernstein localization theorem, more precisely by [BG99] Localization Theorem 2.2, provides an exact fully faithfull embedding of categories

$$ \mathcal{O}_{\chi_0} \to \mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{N}). $$

Composing the Beillinson-Bernstein equivalence with Bezrukavnikov's functor (noting that the blocks $\mathcal{O}_{\chi_0}$ and $\mathcal{O}_{\chi_\lambda}$ are equivalent) we get the exact functor $\mathcal{B}$.

Denote $\mu = w_0 \cdot \lambda$ denote the antidominant weight in the dot-orbit of $\lambda$. Now the proof of [BL23, Proposition 5.8] implies that $\mathcal{B}(M(s \cdot \mu)^\vee) = \mathcal{O}_{\bar{X}_s}$ for all simple reflection $s$ and $\mathcal{B}(P(\mu)) = \mathcal{O}_{\bar{X}}$. Bezrukavnikov's main result [Bez16, Theorem 1] implies that $\Psi$ (hence $\mathcal{B}$) intertwines the convolutions on both sides. Here the convolution on the category $\mathcal{O}_{\chi_\lambda} \simeq \mathcal{O}_{\chi_0}$ is inherited from the convolution in $\mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{B})$ defined as in [BR22, 7.]. We write $w = s_1 \dots s_r$ and compute convolutions on both sides. By [BR12, Theorem 2.2.1] we have

$$ \mathcal{O}_{\bar{X}_w} = \mathcal{O}_{\bar{X}_{s_1}} \star \dots \star \mathcal{O}_{\bar{X}_{s_r}}. $$

By [BR22, Lemma 7.7] we have $M(w \cdot \mu)^\vee = M(s_1 \cdot \mu)^\vee \star \dots \star M(s_r \cdot \mu)^\vee$ and hence $\mathcal{B}(M(w \cdot \mu)^\vee) = \mathcal{O}_{\bar{X}_w}$. Moreover, by [BR12, Theorem 2.2.1] again, the dualizing sheaf of $\bar{X}_w$ is given by the convolution

$$ \omega_{\bar{X}_w} = \omega_{\bar{X}_{s_1}} \star \dots \star \omega_{\bar{X}_{s_r}}. $$

But [BR12, Proposition 1.10.3] implies that the inverse of $\mathcal{O}_{\bar{X}_s}$ for the convolution is $\omega_{\bar{X}_s}$, and as $\mathcal{B}$ is compatible with convolution, and as the inverse of $M(s \cdot \mu)^\vee$ is $M(s \cdot \mu)$ (again using [BR22, Lemma 7.7] for example), we deduce $\omega_{\bar{X}_s} = \mathcal{B}(M(s \cdot \mu))$. Finally 5. is a consequence of [BL23, Lemma 6.7] (with $P = \mathcal{G}$). $\square$

Recall that we have fixed a point $x \in \mathcal{X}_\infty$ associated which we have defined the positive integer $m_x$ in (8).

**Corollary 7.5.** The functor $\mathcal{B}$ induces an exact functor

$$ \mathcal{B}_x : \mathcal{O}_{\chi_\lambda} \to \mathrm{Coh}(\mathcal{X}_{\infty, x, \mathcal{R}}^{\mathrm{qtri}}) $$

such that, for all $M \in \mathcal{O}_{\chi_\lambda}$ the sheaf $\mathcal{B}_x(M)$ is a Cohen-Macaulay sheaf and such that

$$ [\mathcal{M}_{\infty, x, \mathcal{R}}(M)] = m_x[\mathcal{B}_x(M)] \in \mathbb{Z}^0(\bar{\mathcal{X}}_{\infty, x, \mathcal{R}}^{I-q\mathrm{tri}}). $$
Proof. By the main result of [Bez16], there are reverse equivalence of categories

$$ \Psi : D_{I^0, I^0} \leftrightarrow D^b(\mathrm{Coh}(\tilde{\mathfrak{g}} \times_{\mathfrak{g}} \tilde{\mathfrak{g}})) : \Phi_{I^0, I^0}, $$

which we can then localize on $X^\wedge \subset X$. Up to use translation functors, we can focus on the case $\lambda = 0$. By [Bez16, Corollary 42] the functor $\Psi$ in fact takes values in $(\mathcal{G}$-equivariant) coherent sheaves on $X$, when restricted to perverse sheaves $F \in \mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{B})$. Moreover, the Beillinson-Bernstein localization theorem, more precisely by [BG99] Localization Theorem 2.2, provides an exact fully faithfull embedding of categories

$$ \mathcal{O}_{\chi_0} \to \mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{N}). $$

Composing the Beillinson-Bernstein equivalence with Bezrukavnikov's functor (noting that the blocks $\mathcal{O}_{\chi_0}$ and $\mathcal{O}_{\chi_\lambda}$ are equivalent) we get the exact functor $\mathcal{B}$.

Denote $\mu = w_0 \cdot \lambda$ denote the antidominant weight in the dot-orbit of $\lambda$. Now the proof of [BL23, Proposition 5.8] implies that $\mathcal{B}(M(s \cdot \mu)^\vee) = \mathcal{O}_{\bar{X}_s}$ for all simple reflection $s$ and $\mathcal{B}(P(\mu)) = \mathcal{O}_{\bar{X}}$. Bezrukavnikov's main result [Bez16, Theorem 1] implies that $\Psi$ (hence $\mathcal{B}$) intertwines the convolutions on both sides. Here the convolution on the category $\mathcal{O}_{\chi_\lambda} \simeq \mathcal{O}_{\chi_0}$ is inherited from the convolution in $\mathrm{Perv}_{\mathcal{N}}(\mathcal{G}/\mathcal{B})$ defined as in [BR22, 7.]. We write $w = s_1 \dots s_r$ and compute convolutions on both sides. By [BR12, Theorem 2.2.1] we have

$$ \mathcal{O}_{\bar{X}_w} = \mathcal{O}_{\bar{X}_{s_1}} \star \cdots \star \mathcal{O}_{\bar{X}_{s_r}}. $$

By [BR22, Lemma 7.7] we have $M(w \cdot \mu)^\vee = M(s_1 \cdot \mu)^\vee \star \cdots \star M(s_r \cdot \mu)^\vee$ and hence $\mathcal{B}(M(w \cdot \mu)^\vee) = \mathcal{O}_{\bar{X}_w}$. Moreover, by [BR12, Theorem 2.2.1] again, the dualizing sheaf of $\bar{X}_w$ is given by the convolution

$$ \omega_{\bar{X}_w} = \omega_{\bar{X}_{s_1}} \star \cdots \star \omega_{\bar{X}_{s_r}}. $$

But [BR12, Proposition 1.10.3] implies that the inverse of $\mathcal{O}_{\bar{X}_s}$ for the convolution is $\omega_{\bar{X}_s}$, and as $\mathcal{B}$ is compatible with convolution, and as the inverse of $M(s \cdot \mu)^\vee$ is $M(s \cdot \mu)$ (again using [BR22, Lemma 7.7] for example), we deduce $\omega_{\bar{X}_s} = \mathcal{B}(M(s \cdot \mu))$. Finally 5. is a consequence of [BL23, Lemma 6.7] (with $P = \mathcal{G}$). $\square$

Recall that we have fixed a point $x \in \mathcal{X}_\infty$ associated which we have defined the positive integer $m_x$ in (8).

**Corollary 7.5.** The functor $\mathcal{B}$ induces an exact functor

$$ \mathcal{B}_x : \mathcal{O}_{\chi_\lambda} \to \mathrm{Coh}(\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}) $$

such that, for all $M \in \mathcal{O}_{\chi_\lambda}$ the sheaf $\mathcal{B}_x(M)$ is a Cohen-Macaulay sheaf and such that

$$ [\mathcal{M}_{\infty,x,\mathcal{R}}(M)] = m_x[\mathcal{B}_x(M)] \in \mathbb{Z}^0(\bar{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri}}). $$
Proof. Let $\underline{G}_1$ be the completion of $\underline{G}$ at the unit element. As the representations $(\rho_v)_{v|p}$ defined by the point $x$ are crystalline and hence de Rham we may choose a basis $\alpha$ of $W(x) = \prod_{v \in \Sigma} W_{\mathrm{dR}}(D_{\mathrm{rig}}(\rho_{x,v})[1/t])$ and define a point $x_{\mathrm{pdR}}$ associated to $x$ (or rather to the representations $(\rho_v)_{v|p}$) as in (6). For all $M \in \mathcal{O}_{\chi_\lambda}$, the sheaf $\mathcal{B}(M)$ is a $\underline{G}_L$-equivariant sheaf on $X^\wedge$ and hence gives rise to a $\underline{G}_1$-equivariant sheaf on $\hat{X}_{x_{\mathrm{pdR}}}$. Now by [BHS19, Theorem 3.4.4. and Corollary 3.5.8], see also Theorem 4.7 above, we have a diagram

<figure><img src="image_5.png" /></figure>

More precisely, the map $\pi$ forgets the deformation of the fixed basis $\alpha$, and hence it is a $\underline{G}_1$-torsor. Moreover, $W$ formally smooth and $\underline{G}_1$-equivariant for the natural left actions $g \cdot \tilde{\alpha} := \tilde{\alpha} \circ g^{-1}$ on the source (acting only on the deformation of the isomorphisms $\alpha_v : L \otimes_{\mathbb{Q}_p} F_v \xrightarrow{\sim} W_v$) and $g \cdot (kB, hB, N) = (gkB, ghB, g^{-1}Ng)$ on the target of $W$.

It follows that the pullback of $\mathcal{B}(M)^{\wedge}_{x_{\mathrm{pdR}}}$ at $\hat{X}_{x_{\mathrm{pdR}}}$ along $W$ is a $\underline{G}_1$-equivariant sheaf and hence descends to a coherent sheaf

$$ \mathcal{B}_x(M) \in \mathrm{Coh}(\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}). $$

It follows from the construction that $M \mapsto \mathcal{B}_x(M)$ and that $\mathcal{B}_x(M)$ is Cohen-Macaulay, as $\mathcal{B}(M)$ is. Moreover, $\mathcal{B}_x$ is exact, as $W$ is formally smooth and hence flat.

It remains to check the assertion on cycles. But as taking cycles is additive and $\mathcal{B}_x$ is exact, we only need to check this equality on a generating set of the Grothendieck group of $\mathcal{O}_{\chi_\lambda}$, such as the Verma modules $M(w \cdot \mu)$. Hence the desired equality follows from the previous result on Bezrukavnikov's functor together with Proposition 7.2. $\square$

## 7.3 A detail study of local models when $n = 3$

From now on we assume $n = 3$, so that the group $\underline{G}_L$ is

$$ \underline{G}_L \simeq (\mathrm{Res}_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p / \mathbb{Q}_p} \mathrm{GL}_3) \times_{\mathbb{Q}_p} L \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \mathrm{Res}_{F_v / \mathbb{Q}_p} \mathrm{GL}_{3,F_v}) \simeq \prod_{\tau \in \Sigma_F} \mathrm{GL}_{3,L}. $$

We identify the previous local Weyl group $W$ with $\prod_{\tau} W_{\tau}$ and each $W_{\tau}$ with $W_{\mathrm{GL}_3} \simeq \mathfrak{S}_3$ and denote $s_{1,\tau}$, $s_{2,\tau}$ the two simple reflection corresponding to the choice of the upper Borel, and $w_{0,\tau} = s_{1,\tau}s_{2,\tau}s_{1,\tau}$ the longest element in $W_{\tau}$. If $\tau$ is understood, we often omit it from the notation.

As in section 4.1 we denote by $X$ the Steinberg variety for the group

$$ \underline{G} = \mathrm{Res}_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p / \mathbb{Q}_p} \mathrm{GL}_3, $$
Proof. Let $\underline{G}_1$ be the completion of $\underline{G}$ at the unit element. As the representations $(\rho_v)_{v|p}$ defined by the point $x$ are crystalline and hence de Rham we may choose a basis $\alpha$ of $W(x) = \prod_{v \in \Sigma} W_{\mathrm{dR}}(D_{\mathrm{rig}}(\rho_{x,v})[1/t])$ and define a point $x_{\mathrm{pdR}}$ associated to $x$ (or rather to the representations $(\rho_v)_{v|p}$) as in (6). For all $M \in \mathcal{O}_{\chi_\lambda}$, the sheaf $\mathcal{B}(M)$ is a $\underline{G}_L$-equivariant sheaf on $X^\wedge$ and hence gives rise to a $\underline{G}_1$-equivariant sheaf on $\hat{X}_{x_{\mathrm{pdR}}}$. Now by [BHS19, Theorem 3.4.4. and Corollary 3.5.8], see also Theorem 4.7 above, we have a diagram

<figure><img src="image_5.png" /></figure>

More precisely, the map $\pi$ forgets the deformation of the fixed basis $\alpha$, and hence it is a $\underline{G}_1$-torsor. Moreover, $W$ formally smooth and $\underline{G}_1$-equivariant for the natural left actions $g \cdot \tilde{\alpha} := \tilde{\alpha} \circ g^{-1}$ on the source (acting only on the deformation of the isomorphisms $\alpha_v : L \otimes_{\mathbb{Q}_p} F_v \xrightarrow{\sim} W_v$) and $g \cdot (kB, hB, N) = (gkB, ghB, g^{-1}Ng)$ on the target of $W$.

It follows that the pullback of $\mathcal{B}(M)^{\wedge}_{x_{\mathrm{pdR}}}$ at $\hat{X}_{x_{\mathrm{pdR}}}$ along $W$ is a $\underline{G}_1$-equivariant sheaf and hence descends to a coherent sheaf

$$ \mathcal{B}_x(M) \in \mathrm{Coh}(\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}). $$

It follows from the construction that $M \mapsto \mathcal{B}_x(M)$ and that $\mathcal{B}_x(M)$ is Cohen-Macaulay, as $\mathcal{B}(M)$ is. Moreover, $\mathcal{B}_x$ is exact, as $W$ is formally smooth and hence flat.

It remains to check the assertion on cycles. But as taking cycles is additive and $\mathcal{B}_x$ is exact, we only need to check this equality on a generating set of the Grothendieck group of $\mathcal{O}_{\chi_\lambda}$, such as the Verma modules $M(w \cdot \mu)$. Hence the desired equality follows from the previous result on Bezrukavnikov's functor together with Proposition 7.2. $\square$

## 7.3 A detail study of local models when $n = 3$

From now on we assume $n = 3$, so that the group $\underline{G}_L$ is

$$ \underline{G}_L \simeq (\mathrm{Res}_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p / \mathbb{Q}_p} \mathrm{GL}_3) \times_{\mathbb{Q}_p} L \simeq \prod_{v \in S_p} (L \times_{\mathbb{Q}_p} \mathrm{Res}_{F_v / \mathbb{Q}_p} \mathrm{GL}_{3,F_v}) \simeq \prod_{\tau \in \Sigma_F} \mathrm{GL}_{3,L}. $$

We identify the previous local Weyl group $W$ with $\prod_{\tau} W_{\tau}$ and each $W_{\tau}$ with $W_{\mathrm{GL}_3} \simeq \mathfrak{S}_3$ and denote $s_{1,\tau}$, $s_{2,\tau}$ the two simple reflection corresponding to the choice of the upper Borel, and $w_{0,\tau} = s_{1,\tau}s_{2,\tau}s_{1,\tau}$ the longest element in $W_{\tau}$. If $\tau$ is understood, we often omit it from the notation.

As in section 4.1 we denote by $X$ the Steinberg variety for the group

$$ \underline{G} = \mathrm{Res}_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p / \mathbb{Q}_p} \mathrm{GL}_3, $$
over L. As L is assumed to contain all Galois conjugates of F we have $X \simeq \prod_{\tau \in \Sigma_F} X_3$ (see Remark 4.6 for the notation $X_3$). The Steinberg variety $X$ (resp. $X_3$) has dimension $9^{|\Sigma_F|}$ (resp. 9) and $6^{|\Sigma_F|}$ (resp. 6) irreducible components $X_w, w \in W$ (resp. $X_{3,w}, w \in \mathfrak{S}_3$), see e.g. [BHS19, Proposition 2.2.5].

**Proposition 7.6.** For $w = (w_\tau)_{\tau \in \Sigma_F}$, let $s = |\{\tau \in \Sigma_F | w_\tau = w_0\}|$. Then the component $X_w$ is smooth if and only if $s = 0$. Moreover, if $s \neq 0$, then the component $X_w$ is Cohen-Macaulay but not Gorenstein. More precisely, let

$$x_{\text{pdR}} = (g\underline{B}, h\underline{B}, N) = (g_{\tau}\underline{B}_{\tau}, N_{\tau}, h_{\tau}\underline{B}_{\tau}) \in X_w(L) = \prod_{\tau \in \Sigma_F} X_{3,w_{\tau}}(L),$$

and assume that $N_\tau = 0$ when $w_\tau = w_0$. Then

$$\dim_L \omega_{X_w} \otimes k(x_{\text{pdR}}) = 2^r,$$

where $r := |\{\tau | w_\tau = w_0, \text{ and } g_\tau \underline{B}_\tau = h_\tau \underline{B}_\tau\}|$.

*Proof.* The smoothness is a consequence of Proposition 4.1. As $X = \prod_{\tau \in \Sigma_F} X_3$, it is enough to prove the analogous result for $X_3$ only. Indeed, by base change and composition of upper shriek functors, the dualizing sheaf of $X$ is a derived tensor product $\otimes_{\tau}^{\mathbb{L}} p_{\tau}^* \omega_{X_3}$, where $p_{\tau}: X \to X_3$ is projection to the $\tau$-component. But as the product $X = \prod_{\tau} X_3$ is a product over a field, we find

$$\omega_X = \bigotimes_{\tau} p_{\tau}^* \omega_{X_3}.$$

Thus from now on we denote $X_3$ simply by $X$.

It is thus enough to prove that the fiber of $\omega_{X_{w_0}}$, is 2-dimensional at a point of the form $(g\underline{B}, 0, g\underline{B})$. Let $q: \tilde{\mathfrak{g}} \to \mathfrak{g}$ denote the Grothendieck resolution, then $X \simeq \underline{G}_L \times_{\underline{B}} q^{-1}(\mathfrak{b})$. Moreover, $Y := q^{-1}(\mathfrak{b})$ decomposes into irreducible components $Y = \bigcup_{w \in W} Y_w$ such that $X_w \simeq \underline{G}_L \times_{\underline{B}} Y_w$. Hence it is enough to prove that $\omega_{Y_{w_0}}$ has fiber dimension 2 at the point $y_{\text{pdR}} = (\underline{B}, 0)$. As $X_{w_0}$ is Cohen-Macaulay and flat over $\mathfrak{t}$ (cf [BHS19, Proposition 2.2.3]), we have the base change formula $\omega_{X_{w_0}} \otimes_X \bar{X} \simeq \omega_{\bar{X}_{w_0}}$. We are thus reduced to compute the dualizing sheaf $\omega_{\bar{Y}_{w_0}}$ of the irreducible component

$$\bar{Y}_{w_0} = Y_{w_0} \times_{\mathfrak{t}} \{0\}$$

of $\bar{Y} = q^{-1}(\mathfrak{n})$. This scheme now has dimension 3 and we can use explicit computations.

A point of $\bar{Y}(L)$ is of the form $(g\underline{B}, N) \in (\underline{G}/\underline{B} \times \mathfrak{g})(L)$. We use the embedding $\underline{G}/\underline{B} \hookrightarrow \mathbb{P}_L^2 \times (\mathbb{P}_L^2)^\vee$ that sends a full flag $(0 \subset \mathcal{L} \subset \mathcal{P} \subset k^3)$ to $(\mathcal{L} \subset k^3, \mathcal{P} \subset k^3)$. In homogeneous coordinates $([x_0 : x_1 : x_2], [y_0 : y_1 : y_2])$ the condition $\mathcal{L} \subset \mathcal{P}$ is given by $x_0 y_0 + x_1 y_1 + x_2 y_2 = 0$. Let $\bar{Y}^0 \subset \bar{Y}$ denote the open subset defined by the condition $x_0 = y_2 = 1$. It is enough to compute on this open subset, as this is a neighborhood of
over L. As L is assumed to contain all Galois conjugates of F we have $X \simeq \prod_{\tau \in \Sigma_F} X_3$ (see Remark 4.6 for the notation $X_3$). The Steinberg variety X (resp. $X_3$) has dimension $9^{|\Sigma_F|}$ (resp. 9) and $6^{|\Sigma_F|}$ (resp. 6) irreducible components $X_w, w \in W$ (resp. $X_{3,w}, w \in \mathfrak{S}_3$), see e.g. [BHS19, Proposition 2.2.5].

**Proposition 7.6.** For $w = (w_\tau)_{\tau \in \Sigma_F}$, let $s = |\{\tau \in \Sigma_F | w_\tau = w_0\}|$. Then the component $X_w$ is smooth if and only if $s = 0$. Moreover, if $s \neq 0$, then the component $X_w$ is Cohen-Macaulay but not Gorenstein. More precisely, let

$$x_{\text{pdR}} = (g\underline{B}, h\underline{B}, N) = (g_{\tau}\underline{B}_{\tau}, N_{\tau}, h_{\tau}\underline{B}_{\tau}) \in X_{\underline{w}}(L) = \prod_{\tau \in \Sigma_F} X_{3,w_{\tau}}(L),$$

and assume that $N_\tau = 0$ when $w_\tau = w_0$. Then

$$\dim_L \omega_{X_w} \otimes k(x_{\text{pdR}}) = 2^r,$$

where $r := |\{\tau | w_\tau = w_0, \text{ and } g_\tau \underline{B}_\tau = h_\tau \underline{B}_\tau\}|$.

*Proof.* The smoothness is a consequence of Proposition 4.1. As $X = \prod_{\tau \in \Sigma_F} X_3$, it is enough to prove the analogous result for $X_3$ only. Indeed, by base change and composition of upper shriek functors, the dualizing sheaf of $X$ is a derived tensor product $\otimes_{\tau}^{\mathbb{L}} p_{\tau}^* \omega_{X_3}$, where $p_{\tau}: X \to X_3$ is projection to the $\tau$-component. But as the product $X = \prod_{\tau} X_3$ is a product over a field, we find

$$\omega_X = \bigotimes_{\tau} p_{\tau}^* \omega_{X_3}.$$

Thus from now on we denote $X_3$ simply by $X$.

It is thus enough to prove that the fiber of $\omega_{X_{w_0}}$, is 2-dimensional at a point of the form $(g\underline{B}, 0, g\underline{B})$. Let $q: \tilde{\mathfrak{g}} \to \mathfrak{g}$ denote the Grothendieck resolution, then $X \simeq \underline{G}_L \times_{\underline{B}} q^{-1}(\mathfrak{b})$. Moreover, $Y := q^{-1}(\mathfrak{b})$ decomposes into irreducible components $Y = \bigcup_{w \in W} Y_w$ such that $X_w \simeq \underline{G}_L \times_{\underline{B}} Y_w$. Hence it is enough to prove that $\omega_{Y_{w_0}}$ has fiber dimension 2 at the point $y_{\text{pdR}} = (\underline{B}, 0)$. As $X_{w_0}$ is Cohen-Macaulay and flat over $\mathfrak{t}$ (cf [BHS19, Proposition 2.2.3]), we have the base change formula $\omega_{X_{w_0}} \otimes_X \bar{X} \simeq \omega_{\bar{X}_{w_0}}$. We are thus reduced to compute the dualizing sheaf $\omega_{\bar{Y}_{w_0}}$ of the irreducible component

$$\bar{Y}_{w_0} = Y_{w_0} \times_{\mathfrak{t}} \{0\}$$

of $\bar{Y} = q^{-1}(\mathfrak{n})$. This scheme now has dimension 3 and we can use explicit computations.

A point of $\bar{Y}(L)$ is of the form $(g\underline{B}, N) \in (\underline{G}/\underline{B} \times \mathfrak{g})(L)$. We use the embedding $\underline{G}/\underline{B} \hookrightarrow \mathbb{P}_L^2 \times (\mathbb{P}_L^2)^\vee$ that sends a full flag $(0 \subset \mathcal{L} \subset \mathcal{P} \subset k^3)$ to $(\mathcal{L} \subset k^3, \mathcal{P} \subset k^3)$. In homogeneous coordinates $([x_0 : x_1 : x_2], [y_0 : y_1 : y_2])$ the condition $\mathcal{L} \subset \mathcal{P}$ is given by $x_0 y_0 + x_1 y_1 + x_2 y_2 = 0$. Let $\bar{Y}^0 \subset \bar{Y}$ denote the open subset defined by the condition $x_0 = y_2 = 1$. It is enough to compute on this open subset, as this is a neighborhood of