the point $y_{pdR} = (\underline{B}, 0) = ([1:0:0], [0:0:1])$. On $\bar{Y}^0$ we can thus remove $y_0$ from our equations. Let us write

$$N = \begin{pmatrix} 0 & u_{12} & u_{13} \\ 0 & u_{23} & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

for the universal matrix over $\bar{Y}^0$. The ideal defining

$$\bar{Y}_{w_0}^0 \subset Z := \mathrm{Spec}(k[x_1, x_2, y_1, u_{12}, u_{23}, u_{13}])$$

is then given by

$$I_{w_0} = (u_{23}x_2, u_{12}(x_2 + x_1y_1), u_{12}x_1 + u_{13}x_2, u_{23}y_1 - u_{13}(x_2 + x_1y_1)).$$

We remark that we can replace $u_{12}(x_2+x_1y_1)$ by $u_{12}x_2-x_{13}x_2y_1$ using the third equation, and that automatically $y_1u_{12}u_{23} = 0$ using our new equation and $u_{23}y_1-u_{13}(x_2+x_1y_1) = 0$. We then check (e.g. using Macaulay2) that

$$0 \to \mathcal{O}_Z^2 \xrightarrow{A'} \mathcal{O}_Z^6 \xrightarrow{A} \mathcal{O}_Z^5 \xrightarrow{A''} \mathcal{O}_Z$$

is a resolution of $\mathcal{O}_Z/I_{w_0}$, where

$$A' = \begin{pmatrix} y_1 & y_1 u_{13} - u_{12} \\ -x_2 & 0 \\ x_1 & u_{23} \\ 0 & -u_{12} u_{23} \\ 0 & -x_2 u_{23} \\ 0 & x_1 u_{12} + x_2 u_{12} \end{pmatrix}, \quad A'' = \begin{pmatrix} x_1 u_{12} + x_2 u_{13} \\ x_2 u_{23} \\ y_1 u_{12} u_{23} \\ x_1 y_1 u_{13} - y_1 u_{23} + x_2 u_{13} \\ x_2 y_1 u_{13} - x_2 u_{12} \end{pmatrix}^t$$

$$A = \begin{pmatrix} -x_2 u_{23} & -y_1 u_{23} & 0 & x_2 & -y_1 u_{13} & 0 \\ x_1 u_{12} + x_2 u_{13} & y_1 u_{13} & -y_1 u_{12} & -y_1 & 0 & -y_1 u_{13} + u_{12} \\ 0 & x_1 & x_2 & 0 & 1 & 0 \\ 0 & 0 & 0 & -x_2 & u_{12} & 0 \\ 0 & 0 & 0 & x_1 & u_{13} & u_{23} \end{pmatrix}.$$

Let $i: \bar{Y}_{w_0}^0 \hookrightarrow Z$ denote the canonical closed embedding. Then the dualizing sheaf can be computed as $\omega_{\bar{Y}_{w_0}^0} = i^*\mathrm{Ext}_{\mathcal{O}_Z}^3(\mathcal{O}_{\bar{Y}_{w_0}^0}, \mathcal{O}_Z)$ which is given by

$$\omega_{\bar{Y}_{w_0}^0} \simeq \mathcal{O}_Z^2 / \langle (y_1, y_1 u_{13} - u_{12}), (x_2, 0), (x_1, u_{12}), (0, u_{12} u_{23}) \rangle,$$

as $x_2 u_{23} = x_2 u_{12} + x_2 u_{12} = 0$ on $\bar{Y}_{w_0}^0$. It follows that the fiber of $\omega_{\bar{Y}_{w_0}^0}$ at $y_{pdR}$ is 2-dimensional. $\square$

**Lemma 7.7.** Let $J \subset \Delta_{GL_3}$.
the point $y_{pdR} = (\underline{B}, 0) = ([1:0:0], [0:0:1])$. On $\bar{Y}^0$ we can thus remove $y_0$ from our equations. Let us write

$$N = \begin{pmatrix} 0 & u_{12} & u_{13} \\ 0 & u_{23} & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

for the universal matrix over $\bar{Y}^0$. The ideal defining

$$\bar{Y}_{w_0}^0 \subset Z := \mathrm{Spec}(k[x_1, x_2, y_1, u_{12}, u_{23}, u_{13}])$$

is then given by

$$I_{w_0} = (u_{23}x_2, u_{12}(x_2 + x_1y_1), u_{12}x_1 + u_{13}x_2, u_{23}y_1 - u_{13}(x_2 + x_1y_1)).$$

We remark that we can replace $u_{12}(x_2+x_1y_1)$ by $u_{12}x_2-x_{13}x_2y_1$ using the third equation, and that automatically $y_1u_{12}u_{23} = 0$ using our new equation and $u_{23}y_1-u_{13}(x_2+x_1y_1) = 0$. We then check (e.g. using Macaulay2) that

$$0 \to \mathcal{O}_Z^2 \xrightarrow{A'} \mathcal{O}_Z^6 \xrightarrow{A} \mathcal{O}_Z^5 \xrightarrow{A''} \mathcal{O}_Z$$

is a resolution of $\mathcal{O}_Z/I_{w_0}$, where

$$A' = \begin{pmatrix} y_1 & y_1 u_{13} - u_{12} \\ -x_2 & 0 \\ x_1 & u_{23} \\ 0 & -u_{12} u_{23} \\ 0 & -x_2 u_{23} \\ 0 & x_1 u_{12} + x_2 u_{12} \end{pmatrix}, \quad A'' = \begin{pmatrix} x_1 u_{12} + x_2 u_{13} \\ x_2 u_{23} \\ y_1 u_{12} u_{23} \\ x_1 y_1 u_{13} - y_1 u_{23} + x_2 u_{13} \\ x_2 y_1 u_{13} - x_2 u_{12} \end{pmatrix}^t$$

$$A = \begin{pmatrix} -x_2 u_{23} & -y_1 u_{23} & 0 & x_2 & -y_1 u_{13} & 0 \\ x_1 u_{12} + x_2 u_{13} & y_1 u_{13} & -y_1 u_{12} & -y_1 & 0 & -y_1 u_{13} + u_{12} \\ 0 & x_1 & x_2 & 0 & 1 & 0 \\ 0 & 0 & 0 & -x_2 & u_{12} & 0 \\ 0 & 0 & 0 & x_1 & u_{13} & u_{23} \end{pmatrix}.$$

Let $i: \bar{Y}_{w_0}^0 \hookrightarrow Z$ denote the canonical closed embedding. Then the dualizing sheaf can be computed as $\omega_{\bar{Y}_{w_0}^0} = i^*\mathrm{Ext}_{\mathcal{O}_Z}^3(\mathcal{O}_{\bar{Y}_{w_0}^0}, \mathcal{O}_Z)$ which is given by

$$\omega_{\bar{Y}_{w_0}^0} \simeq \mathcal{O}_Z^2 / \langle (y_1, y_1 u_{13} - u_{12}), (x_2, 0), (x_1, u_{12}), (0, u_{12} u_{23}) \rangle,$$

as $x_2 u_{23} = x_2 u_{12} + x_2 u_{12} = 0$ on $\bar{Y}_{w_0}^0$. It follows that the fiber of $\omega_{\bar{Y}_{w_0}^0}$ at $y_{pdR}$ is 2-dimensional. $\square$

**Lemma 7.7.** Let $J \subset \Delta_{GL_3}$.
1. For $w \in W(\text{GL}_3) \simeq \mathfrak{S}_3$ the component $X_{3,w}$ is smooth if $w \neq w_0$.

2. If $x_{\text{pdR}} = (g\underline{B}_3, h\underline{B}_3, 0) \in X_{3,w_0}(L)$, with $g\underline{B}_3 \neq h\underline{B}_3$, then $x_{\text{pdR}}$ is a smooth point of $X_{3,w_0}$.

3. For $\emptyset \neq J \subset \Delta_{\text{GL}_3} = \{s_1, s_2\}$ the component $X_{3,J,\overline{w}}$ is smooth for any $\overline{w} \in W_J \setminus W_{\text{GL}_3}$.

**Proof.** Point 1 is Proposition 7.6. For the point 2, denote $w'$ the index of the Schubert stratum in which $x_{\text{pdR}}$ lies. By [BHS19, Proposition 2.5.3(ii)] it is thus enough (as $\overline{U}_{w_0} = \text{GL}_3 / \underline{B}_3 \times \text{GL}_3 / \underline{B}_3$ is smooth) to prove that $\text{codim}_t(t^{w_0 w'^{-1}}) = \lg(w_0) - \lg(w')$. But this codimension is what we have denoted $\ell(w_0 w'^{-1})$ in the proof of Proposition 4.1. As $w' \neq 1$ and $n = 3$, $w_0 w'^{-1}$ is a product of distinct simple reflections thus $\ell(w_0 w'^{-1}) = \lg(w_0 w'^{-1}) = \lg(w_0) - \lg(w')$. For point 3, as $n = 3$ we have that $J = \{s_1\}, \{s_2\}$ or $J = \{s_1, s_2\}$. Denote $\underline{P} = \underline{P}_J$. In the case $J = \{s_1, s_2\}$, then $\underline{P}_J = \text{GL}_3$ and $X_{3,J} = \tilde{\mathfrak{g}}$ is smooth. It is sufficient to prove the case of $J = \{s_1\}$ (the other case is exactly the same), where an explicite computation gives the smoothness (alternatively, when $w^{\min}$ has length $\le 1$, [BD, Corollary 5.3.4] also implies smoothness). $\square$

**Corollary 7.8.** Let $w = (w_\tau)_\tau \in W$ and let $I = \coprod_\tau I_\tau \subset \Delta$. Let $x_{\text{pdR}} = (x_{\text{pdR},\tau})_\tau = (g_\tau \underline{B}_\tau, h_\tau \underline{B}_\tau, N_\tau)$ be a point such that $N_\tau = 0$ whenever $I_\tau = \emptyset, w_\tau = 1$. If

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

is not a finite free over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-q\text{tri},w^{\min}w_0}$-module, then there exists an embedding $\tau$ such that $I_\tau = \emptyset, w_\tau = 1$ and $w_{x,\mathcal{R},\tau} = 1$.

**Proof.** Assume that there is no $\tau$ such that $I_\tau = \emptyset$ and $w_\tau = w_{x,\mathcal{R},\tau} = 1$. Lemma 7.7, then shows that the local model $X_I$ is smooth at $x_{\text{pdR}}$. By 7.2 the support

$$ \mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri},ww_0} = \text{supp} \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) $$

is smooth. Thus $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda))$ is a free of rank $m_x$ over $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\text{tri},ww_0}$. By Remark 6.3 it follows that $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda))$ is a free of rank $m_x$ over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-q\text{tri},ww_0}$.

The same argument also applies to $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda))$. $\square$

**Proposition 7.9.** For all $w \in W$ the sheaf $\mathcal{B}_x(L(w \cdot \lambda))$ is cyclic. Moreover, for all $w \in W$ such that $ww_0 \ge w_{x,\mathcal{R}}$ the sheaf $\mathcal{M}_\infty(L(w \cdot \lambda))$ is free of rank $m_x$ over its support.

**Proof.** Recall that, for $w \in W$, $Z_w$ is the closure in $\widetilde{\mathcal{N}} \times_{\widetilde{\mathcal{N}}} \widetilde{\mathcal{N}}$ of the preimage $V_w$ of the Bruhat Cell $U_w = \underline{G}_L(1, w) \subset \underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$. By [CG10, Prop. 3.3.4], $V_w$ can be identified with the conormal bundle of $U_w$ in $\widetilde{\mathcal{N}} \times \widetilde{\mathcal{N}} \simeq T^*(\underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B})$. As $\mathfrak{g}$ is isomorphic to direct sum of copies of $\mathfrak{gl}_3$, the closure $\overline{U}_w$ of $U_w$ in $\underline{G}_L/\underline{B} \times \underline{G}_L/\underline{B}$ is smooth, hence a local complete intersection. This proves that the conormal bundle of
1. For $w \in W(\mathrm{GL}_3) \simeq \mathfrak{S}_3$ the component $X_{3,w}$ is smooth if $w \neq w_0$.

2. If $x_{\mathrm{pdR}} = (g\underline{B}_3, h\underline{B}_3, 0) \in X_{3,w_0}(L)$, with $g\underline{B}_3 \neq h\underline{B}_3$, then $x_{\mathrm{pdR}}$ is a smooth point of $X_{3,w_0}$.

3. For $\emptyset \neq J \subset \Delta_{\mathrm{GL}_3} = \{s_1, s_2\}$ the component $X_{3,J,\bar{w}}$ is smooth for any $\bar{w} \in W_J \setminus W_{\mathrm{GL}_3}$.

**Proof.** Point 1 is Proposition 7.6. For the point 2, denote $w'$ the index of the Schubert stratum in which $x_{\mathrm{pdR}}$ lies. By [BHS19, Proposition 2.5.3(ii)] it is thus enough (as $\overline{U}_{w_0} = \mathrm{GL}_3 / \underline{B}_3 \times \mathrm{GL}_3 / \underline{B}_3$ is smooth) to prove that $\mathrm{codim}_t(t^{w_0 w'^{-1}}) = \lg(w_0) - \lg(w')$. But this codimension is what we have denoted $\ell(w_0 w'^{-1})$ in the proof of Proposition 4.1. As $w' \neq 1$ and $n = 3$, $w_0 w'^{-1}$ is a product of distinct simple reflections thus $\ell(w_0 w'^{-1}) = \lg(w_0 w'^{-1}) = \lg(w_0) - \lg(w')$. For point 3, as $n = 3$ we have that $J = \{s_1\}, \{s_2\}$ or $J = \{s_1, s_2\}$. Denote $\underline{P} = \underline{P}_J$. In the case $J = \{s_1, s_2\}$, then $\underline{P}_J = \mathrm{GL}_3$ and $X_{3,J} = \tilde{\mathfrak{g}}$ is smooth. It is sufficient to prove the case of $J = \{s_1\}$ (the other case is exactly the same), where an explicite computation gives the smoothness (alternatively, when $w^{\min}$ has length $\le 1$, [BD, Corollary 5.3.4] also implies smoothness). $\square$

**Corollary 7.8.** Let $w = (w_\tau)_\tau \in W$ and let $I = \coprod_\tau I_\tau \subset \Delta$. Let $x_{\mathrm{pdR}} = (x_{\mathrm{pdR},\tau})_\tau = (g_\tau \underline{B}_\tau, h_\tau \underline{B}_\tau, N_\tau)$ be a point such that $N_\tau = 0$ whenever $I_\tau = \emptyset, w_\tau = 1$. If

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

is not a finite free over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri},w^{\min}w_0}$-module, then there exists an embedding $\tau$ such that $I_\tau = \emptyset, w_\tau = 1$ and $w_{x,\mathcal{R},\tau} = 1$.

**Proof.** Assume that there is no $\tau$ such that $I_\tau = \emptyset$ and $w_\tau = w_{x,\mathcal{R},\tau} = 1$. Lemma 7.7, then shows that the local model $X_I$ is smooth at $x_{\mathrm{pdR}}$. By 7.2 the support

$$ \mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri},ww_0} = \mathrm{supp} \, \mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda)) $$

is smooth. Thus $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda))$ is a free of rank $m_x$ over $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri},ww_0}$. By Remark 6.3 it follows that $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda))$ is a free of rank $m_x$ over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-q\mathrm{tri},ww_0}$.

The same argument also applies to $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_I(w \cdot \lambda))$. $\square$

**Proposition 7.9.** For all $w \in W$ the sheaf $\mathcal{B}_x(L(w \cdot \lambda))$ is cyclic. Moreover, for all $w \in W$ such that $ww_0 \ge w_{x,\mathcal{R}}$ the sheaf $\mathcal{M}_\infty(L(w \cdot \lambda))$ is free of rank $m_x$ over its support.

**Proof.** Recall that, for $w \in W$, $Z_w$ is the closure in $\tilde{\mathcal{N}} \times_{\mathcal{N}} \tilde{\mathcal{N}}$ of the preimage $V_w$ of the Bruhat Cell $U_w = \underline{G}_L(1, w) \subset \underline{G}_L / \underline{B} \times \underline{G}_L / \underline{B}$. By [CG10, Prop. 3.3.4], $V_w$ can be identified with the conormal bundle of $U_w$ in $\tilde{\mathcal{N}} \times \tilde{\mathcal{N}} \simeq T^*(\underline{G}_L / \underline{B} \times \underline{G}_L / \underline{B})$. As $\mathfrak{g}$ is isomorphic to direct sum of copies of $\mathfrak{gl}_3$, the closure $\overline{U}_w$ of $U_w$ in $\underline{G}_L / \underline{B} \times \underline{G}_L / \underline{B}$ is smooth, hence a local complete intersection. This proves that the conormal bundle of
$\overline{U}_w$ is a closed smooth subscheme of $\tilde{\mathcal{N}} \times \tilde{\mathcal{N}}$ containing $V_w$ as an open dense subset so that it coincides with $Z_w$ and $Z_w$ is smooth. This implies that $\mathfrak{z}_w$ is a smooth. As $\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))$ is Cohen-Macaulay, it follows from Proposition 7.2 3) and from the fact that $a_{w,w'} = 0$ for $w \neq w'$ (see [BHS19, Rk. 2.4.5]) that the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))$ is locally free over its support. $\square$

## 7.4 The case of dual Vermas

For later use, let us recall the following Lemma.

**Lemma 7.10.** Let $R$ be a commutative local ring and let $I \subset J$ two ideals of $R$. Let $m \ge 1$ and $\pi : (R/I)^m \to (R/J)^m$ a surjective $R$-linear map. Then there exist isomorphisms

$$ \varphi : (R/J)^m \to (R/J)^m, \quad \psi : (R/I)^m \to (R/I)^m $$

such that $\varphi \circ \pi = \pi \circ \psi = \mathrm{can}^{\oplus m}$ where can : $R/I \to R/J$ is the quotient map.

*Proof.* Let $(e_1, \dots, e_m)$ be the standard basis of $(R/I)^m$ as an $(R/I)$-module and $(f_1, \dots, f_m)$ the standard basis of $(R/J)^m$. Then $(\pi(e_1), \dots, \pi(e_m))$ is a generating family of $(R/J)^m$. As a surjective endomorphism of a module is bijective, we see that $(\pi(e_1), \dots, \pi(e_m))$ is also a basis of $(R/I)^m$. Therefore we can define $\varphi$ by the formula $\varphi(\pi(e_i)) = f_i$. Now, for any $1 \le i \le m$, let $f'_i \in (R/I)^m$ such that $\pi(f'_i) = f_i$. By Nakayama Lemma the family $(f'_1, \dots, f'_m)$ generates $(R/I)^m$ and so is a basis of $(R/I)^m$. We can therefore define $\psi$ by the formula $\psi(e_i) = f'_i$. $\square$

We will use the previous Corollary 7.8 to start a devissage which will be assured by the following two Lemmas.

**Lemma 7.11.** Let $M$ be an object of $\mathcal{O}_{\chi_\lambda}$ and let $Q_1, \dots, Q_r$ be quotients of $M$. Let $Q$ be the smallest quotient of $M$ dominating all the $Q_i$, i.e. $Q = M/(M_1 \cap \dots \cap M_r)$ where $M_i = \mathrm{Ker}(M \to Q_i)$ for $1 \le i \le r$. We assume that

(i) for any $1 \le i \le r$, the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(Q_i)$ is free of rank $m_x$ over it support;

(ii) for any $1 \le i \le r$, the sheaf $\mathcal{B}_x(Q_i)$ is cyclic (generated by one element);

(iii) for any $1 \le i \le r$, $\mathrm{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q_i) = \mathrm{Supp} \mathcal{B}_x(Q_i)$;

(iv) the sheaf $\mathcal{B}_x(Q)$ is cyclic.

Then the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_x$ over its support and

$$ \mathrm{Supp}(\mathcal{M}_{\infty,x,\mathcal{R}}(Q)) = \mathrm{Supp}(\mathcal{B}_x(Q)). $$
$\overline{U}_w$ is a closed smooth subscheme of $\tilde{\mathcal{N}} \times \tilde{\mathcal{N}}$ containing $V_w$ as an open dense subset so that it coincides with $Z_w$ and $Z_w$ is smooth. This implies that $\mathfrak{z}_w$ is a smooth. As $\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))$ is Cohen-Macaulay, it follows from Proposition 7.2 3) and from the fact that $a_{w,w'} = 0$ for $w \neq w'$ (see [BHS19, Rk. 2.4.5]) that the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_0 \cdot \lambda))$ is locally free over its support. $\square$

## 7.4 The case of dual Vermas

For later use, let us recall the following Lemma.

**Lemma 7.10.** Let $R$ be a commutative local ring and let $I \subset J$ two ideals of $R$. Let $m \ge 1$ and $\pi : (R/I)^m \to (R/J)^m$ a surjective $R$-linear map. Then there exist isomorphisms

$$ \varphi : (R/J)^m \to (R/J)^m, \quad \psi : (R/I)^m \to (R/I)^m $$

such that $\varphi \circ \pi = \pi \circ \psi = \text{can}^{\oplus m}$ where can : $R/I \to R/J$ is the quotient map.

*Proof.* Let $(e_1, \dots, e_m)$ be the standard basis of $(R/I)^m$ as an $(R/I)$-module and $(f_1, \dots, f_m)$ the standard basis of $(R/J)^m$. Then $(\pi(e_1), \dots, \pi(e_m))$ is a generating family of $(R/J)^m$. As a surjective endomorphism of a module is bijective, we see that $(\pi(e_1), \dots, \pi(e_m))$ is also a basis of $(R/I)^m$. Therefore we can define $\varphi$ by the formula $\varphi(\pi(e_i)) = f_i$. Now, for any $1 \le i \le m$, let $f'_i \in (R/I)^m$ such that $\pi(f'_i) = f_i$. By Nakayama Lemma the family $(f'_1, \dots, f'_m)$ generates $(R/I)^m$ and so is a basis of $(R/I)^m$. We can therefore define $\psi$ by the formula $\psi(e_i) = f'_i$. $\square$

We will use the previous Corollary 7.8 to start a devissage which will be assured by the following two Lemmas.

**Lemma 7.11.** Let $M$ be an object of $\mathcal{O}_{\chi_\lambda}$ and let $Q_1, \dots, Q_r$ be quotients of $M$. Let $Q$ be the smallest quotient of $M$ dominating all the $Q_i$, i.e. $Q = M/(M_1 \cap \dots \cap M_r)$ where $M_i = \text{Ker}(M \to Q_i)$ for $1 \le i \le r$. We assume that

(i) for any $1 \le i \le r$, the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(Q_i)$ is free of rank $m_x$ over it support;

(ii) for any $1 \le i \le r$, the sheaf $\mathcal{B}_x(Q_i)$ is cyclic (generated by one element);

(iii) for any $1 \le i \le r$, $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q_i) = \text{Supp} \mathcal{B}_x(Q_i)$;

(iv) the sheaf $\mathcal{B}_x(Q)$ is cyclic.

Then the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_x$ over its support and

$$ \text{Supp}(\mathcal{M}_{\infty,x,\mathcal{R}}(Q)) = \text{Supp}(\mathcal{B}_x(Q)). $$
Proof. To ease notation we note $m = m_x$. Let's prove the result when $r = 2$. Let $A = \overline{R}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ be the ring of global sections of $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ and let $I_i = \text{Ann}(\mathcal{B}_x(Q_i))$ for $i \in \{1,2\}$. Define $Q_0$ the largest common quotient of $Q_1$ and $Q_2$, i.e. $Q_0 = M/(M_1+M_2)$. Then we have a short exact sequence

$$0 \to Q \to Q_1 \oplus Q_2 \to Q_0 \to 0.$$

By exactness of $M_{\infty,x,\mathcal{R}}$, we have a short exact sequence

$$0 \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q) \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_1) \oplus \mathcal{M}_{\infty,x,\mathcal{R}}(Q_2) \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0) \to 0$$

where the map $Q_1 \oplus Q_2 \to Q_0$ is given by $(x, y) \mapsto x - y$.

We fix isomorphisms $(A/I_i)^m \xrightarrow{\sim} \mathcal{M}_{\infty,x,\mathcal{R}}(Q_i)$ for $i \in \{1,2\}$. As $Q_0$ is a quotient of both $Q_1$ and $Q_2$, we have surjective maps

$$(A/I_i)^m \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_i) \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0),$$

which factor through $(A/(I_1 + I_2))^m$. Using Lemma 7.10 we can choose the previous isomorphisms such that the following diagram commutes

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    (A/I_1)^m \oplus (A/I_2)^m \arrow[r, "(x,y)\mapsto x-y"] \arrow[d, "\simeq"'] & A/(I_1 + I_2)^m \arrow[d, "\simeq"'] & 0 \\
    \mathcal{M}_{\infty,x,\mathcal{R}}(Q_1) \oplus \mathcal{M}_{\infty,x,\mathcal{R}}(Q_2) & \mathcal{M}_{\infty,x,\mathcal{R}}(Q_1) \oplus \mathcal{M}_{\infty,x,\mathcal{R}}(Q_2) & \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0) \arrow[l, "\simeq"'] \arrow[d, "\simeq"']
\end{tikzcd}
\qquad (9)$$

As the kernel of the upper horizontal map is isomorphic to $(A/(I_1 \cap I_2))^m$, we obtain a commutative diagram

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    0 \arrow[r] & (A/(I_1 \cap I_2))^m \arrow[r] \arrow[d, "\simeq"'] & (A/I_1)^m \oplus (A/I_2)^m \arrow[r] \arrow[d, "\simeq"'] & A/(I_1 + I_2)^m \arrow[d, "\simeq"'] & 0 \\
    0 \arrow[r] & \mathcal{M}_{\infty,x,\mathcal{R}}(Q) \arrow[r] \arrow[d, "\simeq"'] & \mathcal{M}_{\infty,x,\mathcal{R}}(Q_1) \oplus \mathcal{M}_{\infty,x,\mathcal{R}}(Q_2) \arrow[r] \arrow[d, "\simeq"'] & \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0) \arrow[d, "\simeq"'] & 0.
\end{tikzcd}
\qquad (10)$$

As $\text{Ann}(\mathcal{B}_x(Q)) = I_1 \cap I_2$ and $\mathcal{B}_x(Q)$ is cyclic, there exists an isomorphism $\mathcal{B}_x(Q) \simeq A/(I_1 \cap I_2)$. Moreover, by hypothesis, we have $\text{Supp}(\mathcal{B}_x(Q_i)) = \text{Spec}(A/I_i)$ so that the maps $A/(I_1 \cap I_2) \simeq \mathcal{B}_x(Q) \to \mathcal{B}_x(Q_i)$ factors through isomorphisms $A/I_i \simeq \mathcal{B}_x(Q_i)$. Therefore, by exactness of $\mathcal{B}_x$, we also have a commutative diagram

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    0 \arrow[r] & (A/(I_1 \cap I_2))^m \arrow[r, "x \mapsto (x,x)"] \arrow[d, "\simeq"'] & (A/I_1) \oplus (A/I_2) \arrow[d, "\simeq"'] & 0 \\
    0 \arrow[r] & \mathcal{B}_x(Q) \arrow[r, "x \mapsto (x,x)"] \arrow[d, "\simeq"'] & \mathcal{B}_x(Q_1) \oplus \mathcal{B}_x(Q_2) \arrow[r, "x \mapsto (x,x)"] \arrow[d, "\simeq"'] & \mathcal{B}_x(Q_0) \arrow[d, "\simeq"'] & 0.
\end{tikzcd}$$

This implies that we have an isomorphism $A/(I_1 + I_2) \simeq \mathcal{B}_x(Q_0)$. As $\mathcal{B}_x(Q_0)$ is Cohen-Macaulay, so is $A/(I_1 + I_2)$. As the ring $A/(I_1 + I_2)$ is Cohen-Macaulay, the vertical
right arrow of diagram (9) is a surjective map $(A/(I_1 + I_2))^m \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0)$ between two Cohen-Macaulay modules with the same cycle by Corollary 7.5. It is therefore an isomorphism and the Snake Lemma allows us to conclude that the left vertical arrow in (10) is an isomorphism.

Assume that the result is proved for some integer $r \ge 2$. Let $Q_1, \dots, Q_{r+1}$ be quotients of $M$ satisfying the hypotheses of the Lemma. Let $Q'$ be the smallest quotient of $M$ dominating all the $Q_i$ for $1 \le i \le r$. Note that $\mathcal{B}_x(Q')$ is a quotient of $\mathcal{B}_x(Q)$ and is therefore cyclic. By induction, $\mathcal{M}_{\infty,x,\mathcal{R}}(Q')$ is free of rank $m$ over its support and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q') = \text{Supp} \mathcal{B}_x(Q')$. The quotient $Q$ is now the smallest quotient of $M$ dominating $Q'$ and $Q_{r+1}$. Therefore the case $r=2$ implies that $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m$ over its support and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q) = \text{Supp} \mathcal{B}_x(Q)$, which concludes the induction. $\square$

**Lemma 7.12.** Let $M$ be an object of the category $\mathcal{O}_{\chi_\lambda}$. Assume that $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_x$ elements and $\mathcal{B}_x(M)$ is cyclic. Then $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is locally free of rank $m_x$ over its support, its support is Cohen-Macaulay and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(M) = \text{Supp} \mathcal{B}_x(M)$.

*Proof.* We prove the result by induction on the length of $M$. If $M$ is simple this is done in Proposition 7.9. Thus we can assume that we have a short exact sequence

$$0 \to L \to M \to Q \to 0$$

with $L$ simple such that $\mathcal{M}_{\infty,x,\mathcal{R}}(L) \ne 0$ and that the result is true for $Q$. Let $I = \text{Ann}(\mathcal{M}_{\infty,x,\mathcal{R}}(M))$, $I_B = \text{Ann}(\mathcal{B}_x(M))$, $J = \text{Ann}(\mathcal{B}_x(Q))$ and $K = \text{Ann}(\mathcal{B}_x(L))$. Then we have two short exact sequences

$$0 \to \mathcal{B}_x(L) \to \mathcal{B}_x(M) \to \mathcal{B}_x(Q) \to 0 \\ 0 \to \mathcal{M}_{\infty,x,\mathcal{R}}(L) \to \mathcal{M}_{\infty,x,\mathcal{R}}(M) \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q) \to 0.$$

The first exact sequence shows that $\hat{R}_{\infty,x}/K \simeq J/I_B$ so that $I_B = JK$. The second exact sequence shows that $I_B \subset I$. Therefore, as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_x$ elements, we have a surjective map

$$\mathcal{B}_x(M)^{m_x} \simeq (\hat{R}_{\infty,x}/I_B)^{m_x} \to \mathcal{M}_{\infty,x,\mathcal{R}}(M).$$

These modules are both Cohen-Macaulay of the same dimension with identical associated maximal cycle by Corollary 7.5, therefore this map is an isomorphism and $I_B = I$. Moreover as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is Cohen-Macaulay, so is its support. $\square$

**Theorem 7.13.** For any $w \in W$ such that $ww_0 \ge w_{x,\mathcal{R}}$, the coherent sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w.\lambda)^\vee)$ is locally free of rank $m_x$ over its support.

*Proof.* As $M(w \cdot \lambda)^\vee$ is a quotient of $M(\lambda)^\vee$ for any $w \in W$, Lemma 7.12 implies that it is sufficient to prove the result for $w=1$.
right arrow of diagram (9) is a surjective map $(A/(I_1 + I_2))^m \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q_0)$ between two Cohen-Macaulay modules with the same cycle by Corollary 7.5. It is therefore an isomorphism and the Snake Lemma allows us to conclude that the left vertical arrow in (10) is an isomorphism.

Assume that the result is proved for some integer $r \ge 2$. Let $Q_1, \dots, Q_{r+1}$ be quotients of $M$ satisfying the hypotheses of the Lemma. Let $Q'$ be the smallest quotient of $M$ dominating all the $Q_i$ for $1 \le i \le r$. Note that $\mathcal{B}_x(Q')$ is a quotient of $\mathcal{B}_x(Q)$ and is therefore cyclic. By induction, $\mathcal{M}_{\infty,x,\mathcal{R}}(Q')$ is free of rank $m$ over its support and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q') = \text{Supp} \mathcal{B}_x(Q')$. The quotient $Q$ is now the smallest quotient of $M$ dominating $Q'$ and $Q_{r+1}$. Therefore the case $r=2$ implies that $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m$ over its support and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(Q) = \text{Supp} \mathcal{B}_x(Q)$, which concludes the induction. $\square$

**Lemma 7.12.** Let $M$ be an object of the category $\mathcal{O}_{\chi_\lambda}$. Assume that $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_x$ elements and $\mathcal{B}_x(M)$ is cyclic. Then $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is locally free of rank $m_x$ over its support, its support is Cohen-Macaulay and $\text{Supp} \mathcal{M}_{\infty,x,\mathcal{R}}(M) = \text{Supp} \mathcal{B}_x(M)$.

*Proof.* We prove the result by induction on the length of $M$. If $M$ is simple this is done in Proposition 7.9. Thus we can assume that we have a short exact sequence

$$0 \to L \to M \to Q \to 0$$

with $L$ simple such that $\mathcal{M}_{\infty,x,\mathcal{R}}(L) \ne 0$ and that the result is true for $Q$. Let $I = \text{Ann}(\mathcal{M}_{\infty,x,\mathcal{R}}(M))$, $I_B = \text{Ann}(\mathcal{B}_x(M))$, $J = \text{Ann}(\mathcal{B}_x(Q))$ and $K = \text{Ann}(\mathcal{B}_x(L))$. Then we have two short exact sequences

$$0 \to \mathcal{B}_x(L) \to \mathcal{B}_x(M) \to \mathcal{B}_x(Q) \to 0 \\ 0 \to \mathcal{M}_{\infty,x,\mathcal{R}}(L) \to \mathcal{M}_{\infty,x,\mathcal{R}}(M) \to \mathcal{M}_{\infty,x,\mathcal{R}}(Q) \to 0.$$

The first exact sequence shows that $\hat{R}_{\infty,x}/K \simeq J/I_B$ so that $I_B = JK$. The second exact sequence shows that $I_B \subset I$. Therefore, as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_x$ elements, we have a surjective map

$$\mathcal{B}_x(M)^{m_x} \simeq (\hat{R}_{\infty,x}/I_B)^{m_x} \to \mathcal{M}_{\infty,x,\mathcal{R}}(M).$$

These modules are both Cohen-Macaulay of the same dimension with identical associated maximal cycle by Corollary 7.5, therefore this map is an isomorphism and $I_B = I$. Moreover as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is Cohen-Macaulay, so is its support. $\square$

**Theorem 7.13.** For any $w \in W$ such that $ww_0 \ge w_{x,\mathcal{R}}$, the coherent sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w.\lambda)^\vee)$ is locally free of rank $m_x$ over its support.

*Proof.* As $M(w \cdot \lambda)^\vee$ is a quotient of $M(\lambda)^\vee$ for any $w \in W$, Lemma 7.12 implies that it is sufficient to prove the result for $w=1$.
Recall that $W = \prod_{\tau:F \hookrightarrow L} W_{\tau}$ and write $w_{x,\mathcal{R}} = (w_{x,\tau})$. Let $J \subset \operatorname{Hom}(F, L)$ be the set embeddings such that $w_{x,\tau} = 1$. Let $E$ be the set of elements $w = (w_v) \in W$ such that $w_{\tau} \in \{s_1, s_2\}$ if $\tau \in J$ and $w_{\tau} = 1$ if $\tau \notin J$. By Corollary 7.8 and Lemma 7.12, for $w \in E$, the module $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)^{\vee})$ is free of rank $m_x$ over its support and $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)^{\vee}) = \mathcal{B}_x(M(w \cdot \lambda)^{\vee})^{m_x}$. Let $Q$ be the smallest quotient of $M(\lambda)^{\vee}$ dominating all the $M(w \cdot \lambda)^{\vee}$ for $w \in E$. Lemma 7.11 implies that $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_x$ over its support and $\mathcal{M}_{\infty,x,\mathcal{R}}(Q) = \mathcal{B}_x(Q)^{m_x}$. Let $N$ be the kernel of the map $M \to Q$.

Let $I$ of the form $\coprod_{\tau \in J} \{s_{i_{\tau}}\}$ where $i_{\tau} \in \{1, 2\}$. Then the image of the map $M_I(\lambda)^{\vee} \hookrightarrow M(\lambda)^{\vee} \to Q$ is $Q_I := \coprod_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \coprod_{\tau \notin J} M(\lambda_{\tau})^{\vee}$. By Corollary 7.8, the module $\mathcal{M}_{\infty,x,\mathcal{R}}(M_I(\lambda)^{\vee})$ is free of rank $m_x$ over its support. Thus $\mathcal{M}_{\infty,x,\mathcal{R}}(Q_I)$ is generated by $m_x$ elements, and its quotient

$$L_I := \coprod_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \coprod_{\tau \notin J} M(w_{x,\tau}w_0 \cdot \lambda_{\tau})^{\vee},$$

satisfies

$$\mathcal{M}_{\infty,x,\mathcal{R}}(L_I) = \mathcal{M}_{\infty,x,\mathcal{R}}(\coprod_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \coprod_{\tau \notin J} L(w_{x,\tau}w_0 \cdot \lambda_{\tau})).$$

by Proposition 7.2. Moreover, by Proposition 7.9, this module is free of rank $m_x$ over its support so that its fiber at $x$ has dimension $m_x$. This implies that the following surjective maps are all isomorphisms

$$k(x)^{m_x} \simeq \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(\lambda)^{\vee}) \otimes k(x) \xrightarrow{\sim} \mathcal{M}_{\infty,x,\mathcal{R}}(Q_I) \otimes k(x) \\ \xrightarrow{\sim} \mathcal{M}_{\infty,x,\mathcal{R}}(L_I) \otimes k(x) \simeq k(x)^{m_x}.$$

As moreover $\operatorname{Ker}(M_I(\lambda)^{\vee} \to Q_I) = N \cap M_I(\lambda)^{\vee}$, we see that the map

$$\mathcal{M}_{\infty,x,\mathcal{R}}(N \cap M_I(\lambda)^{\vee}) \otimes k(x) \to \mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee}) \otimes k(x)$$

is zero. As $M(\lambda)^{\vee}$ is multiplicity-free, we have $N = \sum_I (N \cap M_I(\lambda)^{\vee})$ and we conclude that the map

$$\mathcal{M}_{\infty,x,\mathcal{R}}(N) \otimes k(x) \to \mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee}) \otimes k(x)$$

is zero. Therefore $\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee}) \otimes k(x) \simeq \mathcal{M}_{\infty,x,\mathcal{R}}(Q) \otimes k(x) \simeq k(x)^{m_x}$ and we conclude with Lemma 7.12 since $\mathcal{B}_x(M(\lambda)^{\vee})$ is cyclic. $\square$

## 7.5 The case of the antidominant projective

**Theorem 7.14.** The coherent sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_0 \cdot \lambda))$ is free of rank $m_x$ over its support.

*Proof.* Recall that $A = U(t)_m$ and set $D := L \otimes_{A^W} A$. By Proposition 2.17, the action of $Z(\mathfrak{g})$ on $P(w_0 \cdot \lambda)$ induces a structure of $D$-module on $P(w_0 \cdot \lambda)$. As $M(\lambda)^{\vee}$ is an injective
Recall that $W = \coprod_{\tau:F \hookrightarrow L} W_{\tau}$ and write $w_{x,R} = (w_{x,\tau})$. Let $J \subset \mathrm{Hom}(F, L)$ be the set embeddings such that $w_{x,\tau} = 1$. Let $E$ be the set of elements $w = (w_v) \in W$ such that $w_{\tau} \in \{s_1, s_2\}$ if $\tau \in J$ and $w_{\tau} = 1$ if $\tau \notin J$. By Corollary 7.8 and Lemma 7.12, for $w \in E$, the module $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda)^{\vee})$ is free of rank $m_x$ over its support and $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda)^{\vee}) = \mathcal{B}_x(M(w \cdot \lambda)^{\vee})^{m_x}$. Let $Q$ be the smallest quotient of $M(\lambda)^{\vee}$ dominating all the $M(w \cdot \lambda)^{\vee}$ for $w \in E$. Lemma 7.11 implies that $\mathcal{M}_{\infty,x,R}(Q)$ is free of rank $m_x$ over its support and $\mathcal{M}_{\infty,x,R}(Q) = \mathcal{B}_x(Q)^{m_x}$. Let $N$ be the kernel of the map $M \to Q$.

Let $I$ of the form $\coprod_{\tau \in J} \{s_{i_{\tau}}\}$ where $i_{\tau} \in \{1, 2\}$. Then the image of the map $M_I(\lambda)^{\vee} \hookrightarrow M(\lambda)^{\vee} \to Q$ is $Q_I := \boxtimes_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \boxtimes_{\tau \notin J} M(\lambda_{\tau})^{\vee}$. By Corollary 7.8, the module $\mathcal{M}_{\infty,x,R}(M_I(\lambda)^{\vee})$ is free of rank $m_x$ over its support. Thus $\mathcal{M}_{\infty,x,R}(Q_I)$ is generated by $m_x$ elements, and its quotient

$$L_I := \boxtimes_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \boxtimes_{\tau \notin J} M(w_{x,\tau}w_0 \cdot \lambda_{\tau})^{\vee},$$

satisfies

$$\mathcal{M}_{\infty,x,R}(L_I) = \mathcal{M}_{\infty,x,R}(\boxtimes_{\tau \in J} L(s_{3-i_{\tau}} \cdot \lambda_{\tau}) \boxtimes_{\tau \notin J} L(w_{x,\tau}w_0 \cdot \lambda_{\tau})).$$

by Proposition 7.2. Moreover, by Proposition 7.9, this module is free of rank $m_x$ over its support so that its fiber at $x$ has dimension $m_x$. This implies that the following surjective maps are all isomorphisms

$$k(x)^{m_x} \simeq \mathcal{M}_{\infty,x,R}(M_I(\lambda)^{\vee}) \otimes k(x) \xrightarrow{\sim} \mathcal{M}_{\infty,x,R}(Q_I) \otimes k(x) \\ \xrightarrow{\sim} \mathcal{M}_{\infty,x,R}(L_I) \otimes k(x) \simeq k(x)^{m_x}.$$

As moreover $\mathrm{Ker}(M_I(\lambda)^{\vee} \to Q_I) = N \cap M_I(\lambda)^{\vee}$, we see that the map

$$\mathcal{M}_{\infty,x,R}(N \cap M_I(\lambda)^{\vee}) \otimes k(x) \to \mathcal{M}_{\infty,x,R}(M(\lambda)^{\vee}) \otimes k(x)$$

is zero. As $M(\lambda)^{\vee}$ is multiplicity-free, we have $N = \sum_I (N \cap M_I(\lambda)^{\vee})$ and we conclude that the map

$$\mathcal{M}_{\infty,x,R}(N) \otimes k(x) \to \mathcal{M}_{\infty,x,R}(M(\lambda)^{\vee}) \otimes k(x)$$

is zero. Therefore $\mathcal{M}_{\infty,x,R}(M(\lambda)^{\vee}) \otimes k(x) \simeq \mathcal{M}_{\infty,x,R}(Q) \otimes k(x) \simeq k(x)^{m_x}$ and we conclude with Lemma 7.12 since $\mathcal{B}_x(M(\lambda)^{\vee})$ is cyclic. $\square$

## 7.5 The case of the antidominant projective

**Theorem 7.14.** The coherent sheaf $\mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda))$ is free of rank $m_x$ over its support.

*Proof.* Recall that $A = U(t)_m$ and set $D := L \otimes_{Aw} A$. By Proposition 2.17, the action of $Z(g)$ on $P(w_0 \cdot \lambda)$ induces a structure of $D$-module on $P(w_0 \cdot \lambda)$. As $M(\lambda)^{\vee}$ is an injective
object, it follows from [Soe90, Prop. 6], that $M(\lambda)^\vee \simeq P(w_0 \cdot \lambda) \otimes_D (D/\mathfrak{m}_D)$, where $\mathfrak{m}_D$ is the maximal ideal of $D$. We have also a local map of local algebras $\alpha : D \to \mathcal{O}_{\bar{X}_{\infty,x,R}^{\text{qtri}}}$ defined in section 6.3. It follows from Corollary 6.14 that these define the same action of $D$ on $\mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda))$. As moreover the functor $\mathcal{M}_{\infty,x,R}$ is exact, we have an isomorphism $\mathcal{M}_{\infty,x,R}(M(\lambda)^\vee) \simeq \mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes_D (D/\mathfrak{m}_D)$. As moreover the map $A \otimes_{AW} A \to \mathcal{O}_{\bar{X}_{\infty,x,R}^{\text{qtri}}}$ is a local map of local rings, we have an isomorphism $\mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes k(x) \xrightarrow{\sim} \mathcal{M}_{\infty,x,R}(M(\lambda)^\vee) \otimes k(x)$ and thus $\dim_L \mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes k(x) = m_x$ by Theorem 7.13. We conclude by Lemma 7.12. $\square$

**Corollary 7.15.** Let $Q$ be a quotient of the anti-dominant projective $P(w_0 \cdot \lambda)$ in the category $\mathcal{O}_{\chi\chi}$. If $\mathcal{M}_{\infty,x,R}(Q) \neq 0$, then it is finite free of rank $m_x$ over its support and its support is Cohen-Macaulay.

*Proof.* As $\mathcal{M}_{\infty,x,R}(P(w_0 \cdot \lambda))$ (resp. $\mathcal{B}_x(P(w_0 \cdot \lambda)$)) is free of rank $m_x$ (resp. 1) over its support by Theorems 7.14 and 7.4, we have that $\mathcal{M}_{\infty,x,R}(Q)$ (resp. $\mathcal{B}_x(Q)$) is generated by at most $m_x$ elements (resp. cyclic). It follows from Lemma 7.12, $\mathcal{M}_{\infty,x,R}(Q)$ is free of rank $m_x$ over its support and that its support is Cohen-Macaulay. $\square$

**Corollary 7.16.** For all $w \in W$, the coherent sheaf

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

is free of rank $m_x$ over its support.

*Proof.* By Corollary 7.15, it is sufficient to prove that $\mathcal{M}_{\infty,x,R}(P(w \cdot \lambda)^\vee)$ is non zero and that there exists a surjective map

$$ P(w_0 \cdot \lambda) \to P(w \cdot \lambda)^\vee. $$

As $P(w_0 \cdot \lambda)$ is the projective envelope of $L(w_0 \cdot \lambda)$, this is equivalent to showing that the socle of $P(w \cdot \lambda)$ is isomorphic to $L(w_0 \cdot \lambda)$. By [Str03, Thm. 8.1], the socle of $P(w \cdot \lambda)$ is isomorphic to $L(w_0 \cdot \lambda)^m$ with $m = [P(w \cdot \lambda) : M(\lambda)] = [M(\lambda) : L(w \cdot \lambda)]$ by [Hum08, Thm. 3.9]. As $\mathfrak{g}$ is isomorphic to a direct sum of copies of $\mathfrak{gl}_{3,L}$, we have $[M(\lambda) : L(w \cdot \lambda)] = 1$ for any $w \in W$.

Moreover, as $[M(\lambda) : L(\lambda)] = 1$, we have

$$ [P(w \cdot \lambda)^\vee : L(\lambda)] = [P(w \cdot \lambda) : L(\lambda)] = 1. $$

As $\mathcal{M}_{\infty,x,R}(L(\lambda)) \neq 0$, we have $\mathcal{M}_{\infty,x,R}(P(w \cdot \lambda)^\vee) \neq 0$. $\square$
object, it follows from [Soe90, Prop. 6], that $M(\lambda)^{\vee} \simeq P(w_0 \cdot \lambda) \otimes_D (D/\mathfrak{m}_D)$, where $\mathfrak{m}_D$ is the maximal ideal of $D$. We have also a local map of local algebras $\alpha : D \to \mathcal{O}_{\bar{X}_{\infty,x,R}}^{\text{qtri}}$ defined in section 6.3. It follows from Corollary 6.14 that these define the same action of $D$ on $M_{\infty,x,R}(P(w_0 \cdot \lambda))$. As moreover the functor $M_{\infty,x,R}$ is exact, we have an isomorphism $M_{\infty,x,R}(M(\lambda)^{\vee}) \simeq M_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes_D (D/\mathfrak{m}_D)$. As moreover the map $A \otimes_{AW} A \to \mathcal{O}_{\bar{X}_{\infty,x,R}}^{\text{qtri}}$ is a local map of local rings, we have an isomorphism $M_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes k(x) \xrightarrow{\sim} M_{\infty,x,R}(M(\lambda)^{\vee}) \otimes k(x)$ and thus $\dim_L M_{\infty,x,R}(P(w_0 \cdot \lambda)) \otimes k(x) = m_x$ by Theorem 7.13. We conclude by Lemma 7.12. $\square$

**Corollary 7.15.** Let $Q$ be a quotient of the anti-dominant projective $P(w_0 \cdot \lambda)$ in the category $\mathcal{O}_{\chi\chi}$. If $M_{\infty,x,R}(Q) \neq 0$, then it is finite free of rank $m_x$ over its support and its support is Cohen-Macaulay.

*Proof.* As $M_{\infty,x,R}(P(w_0 \cdot \lambda))$ (resp. $\mathcal{B}_x(P(w_0 \cdot \lambda)$)) is free of rank $m_x$ (resp. 1) over its support by Theorems 7.14 and 7.4, we have that $M_{\infty,x,R}(Q)$ (resp. $\mathcal{B}_x(Q)$) is generated by at most $m_x$ elements (resp. cyclic). It follows from Lemma 7.12, $M_{\infty,x,R}(Q)$ is free of rank $m_x$ over its support and that its support is Cohen-Macaulay. $\square$

**Corollary 7.16.** For all $w \in W$, the coherent sheaf

$$M_{\infty,x,R}(P(w \cdot \lambda)^{\vee}),$$

is free of rank $m_x$ over its support.

*Proof.* By Corollary 7.15, it is sufficient to prove that $M_{\infty,x,R}(P(w \cdot \lambda)^{\vee})$ is non zero and that there exists a surjective map

$$P(w_0 \cdot \lambda) \to P(w \cdot \lambda)^{\vee}.$$

As $P(w_0 \cdot \lambda)$ is the projective envelope of $L(w_0 \cdot \lambda)$, this is equivalent to showing that the socle of $P(w \cdot \lambda)$ is isomorphic to $L(w_0 \cdot \lambda)$. By [Str03, Thm. 8.1], the socle of $P(w \cdot \lambda)$ is isomorphic to $L(w_0 \cdot \lambda)^m$ with $m = [P(w \cdot \lambda) : M(\lambda)] = [M(\lambda) : L(w \cdot \lambda)]$ by [Hum08, Thm. 3.9]. As $\mathfrak{g}$ is isomorphic to a direct sum of copies of $\mathfrak{gl}_{3,L}$, we have $[M(\lambda) : L(w \cdot \lambda)] = 1$ for any $w \in W$.

Moreover, as $[M(\lambda) : L(\lambda)] = 1$, we have

$$[P(w \cdot \lambda)^{\vee} : L(\lambda)] = [P(w \cdot \lambda) : L(\lambda)] = 1.$$

As $M_{\infty,x,R}(L(\lambda)) \neq 0$, we have $M_{\infty,x,R}(P(w \cdot \lambda)^{\vee}) \neq 0$. $\square$
## 7.6 Duality

For a Cohen-Macaulay sheaf $\mathcal{F}$ on $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ of dimension $\dim \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$, we write $\omega_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{\bullet}$ for the dualizing complex and set

$$ \mathcal{F}^{\vee} := R \operatorname{Hom}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}} (\mathcal{F}, \omega_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{\bullet}) [-\dim \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}]. $$

This complex $\mathcal{F}^{\vee}$ is a coherent sheaf concentrated in degree 0 to which we refer to $\mathcal{F}^{\vee}$ as the shifted Serre dual of $\mathcal{F}$.

**Lemma 7.17.** Let $\mathcal{F}$ be a maximal Cohen-Macaulay coherent sheaf over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$. Then $[\mathcal{F}^{\vee}] = [\mathcal{F}]$. As a consequence if $\mathcal{Y} \subset \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ is a maximal Cohen-Macaulay closed subscheme, we have $[\omega_{\mathcal{Y}}] = [\mathcal{Y}]$.

*Proof.* Let $R$ be a local complete regular ring such that $O_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}$ is isomorphic to a quotient of $R$. Then we can compute $\mathcal{F}^{\vee}$ by the formula $\mathcal{F}^{\vee} = \operatorname{Ext}_R^d(\mathcal{F}, R)$ where $d$ is the codimension of $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ in $\operatorname{Spec}(R)$. By definition, we have $[\mathcal{F}] = \sum_z a(z)z$ where the sum is over all maximal points in $\operatorname{Supp}(\mathcal{F})$ and $a(z)$ is the length of the finite length $R_z$-module $\mathcal{F}_z$. Let $z \in \operatorname{Spec}(R)$ be a maximal point of the support of $\mathcal{F}$. The localization $R_z$ of $R$ at $z$ is a local regular ring and we have $\mathcal{F}_z^{\vee} \simeq \operatorname{Ext}_{R_z}^d(\mathcal{F}_z, R_z)$. As $\operatorname{Ext}_{R_z}^d(-, R_z)$ is an exact functor on the subcategory of finite length $R_z$-modules and $\dim_{k(z)} \operatorname{Ext}_{R_z}^d(k(z), R_z) = 1$, the length of the $R_z$-module $\operatorname{Ext}_{R_z}^d(\mathcal{F}_z, R_z)$ is $a(z)$. So we have the proved the claim. $\square$

**Proposition 7.18.** Let $M$ be a subobject of the anti-dominant projective $P(w_0 \cdot \lambda)$. Assume that $M_{\infty,x,\mathcal{R}}(M) \neq 0$ and let $\mathcal{Y}$ be the support of $M_{\infty,x,\mathcal{R}}(M)$. Then $M_{\infty,x,\mathcal{R}}(M)$ is isomorphic to $\omega_{\mathcal{Y}}^{m_x}$ and $\mathcal{Y}$ is Cohen-Macaulay.

*Proof.* Let $Q$ be the quotient of $P(w_0 \cdot \lambda)$ by $M$. If $M_{\infty,x,\mathcal{R}}(Q) = 0$, then Theorem 7.14 implies the result. So we can assume that $M_{\infty,x,\mathcal{R}}(M) \neq 0$ and $M_{\infty,x,\mathcal{R}}(Q) \neq 0$. By Corollary 7.15, $M_{\infty,x,\mathcal{R}}(Q)$ is isomorphic to $O_{\mathcal{Z}}^{m_x}$ for $\mathcal{Z} \subset \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ maximal Cohen-Macaulay. Using Lemma 7.10, we can construct a commutative diagram

$$ \begin{array}{ccc} 0 & \to & M_{\infty,x,\mathcal{R}}(M) \to M_{\infty,x,\mathcal{R}}(P(w_0 \cdot \lambda)) \to M_{\infty,x,\mathcal{R}}(Q) \to 0 \\ \Big\downarrow & & \Big\downarrow \\ 0 & \to & \operatorname{Ker} \quad \Big\downarrow \quad \operatorname{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{m_x} \quad \Big\downarrow \quad \operatorname{O}_{\mathcal{Z}}^{m_x} \quad \to 0. \end{array} $$

Let $I$ be the ideal defining $\mathcal{Z}$. As $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ and $\mathcal{Z}$ are Cohen-Macaulay of the same dimension, the involutivity of the duality implies that we have $I \simeq \omega_{\mathcal{O}_{\mathcal{Y}}}$ where $\mathcal{Y} = \operatorname{Supp}(I)$ so that we have the result. $\square$
## 7.6 Duality

For a Cohen-Macaulay sheaf $\mathcal{F}$ on $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ of dimension $\dim \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$, we write $\omega_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{\bullet}$ for the dualizing complex and set

$$ \mathcal{F}^{\vee} := R \operatorname{Hom}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}} (\mathcal{F}, \omega_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{\bullet}) [-\dim \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}]. $$

This complex $\mathcal{F}^{\vee}$ is a coherent sheaf concentrated in degree 0 to which we refer to $\mathcal{F}^{\vee}$ as the *shifted* Serre dual of $\mathcal{F}$.

**Lemma 7.17.** Let $\mathcal{F}$ be a maximal Cohen-Macaulay coherent sheaf over $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$. Then $[\mathcal{F}^{\vee}] = [\mathcal{F}]$. As a consequence if $\mathcal{Y} \subset \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ is a maximal Cohen-Macaulay closed subscheme, we have $[\omega_{\mathcal{Y}}] = [\mathcal{Y}]$.

*Proof.* Let $R$ be a local complete regular ring such that $\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}$ is isomorphic to a quotient of $R$. Then we can compute $\mathcal{F}^{\vee}$ by the formula $\mathcal{F}^{\vee} = \operatorname{Ext}_R^d(\mathcal{F}, R)$ where $d$ is the codimension of $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ in $\operatorname{Spec}(R)$. By definition, we have $[\mathcal{F}] = \sum_z a(z)z$ where the sum is over all maximal points in $\operatorname{Supp}(\mathcal{F})$ and $a(z)$ is the length of the finite length $R_z$-module $\mathcal{F}_z$. Let $z \in \operatorname{Spec}(R)$ be a maximal point of the support of $\mathcal{F}$. The localization $R_z$ of $R$ at $z$ is a local regular ring and we have $\mathcal{F}_z^{\vee} \simeq \operatorname{Ext}_{R_z}^d(\mathcal{F}_z, R_z)$. As $\operatorname{Ext}_{R_z}^d(-, R_z)$ is an exact functor on the subcategory of finite length $R_z$-modules and $\dim_{k(z)} \operatorname{Ext}_{R_z}^d(k(z), R_z) = 1$, the length of the $R_z$-module $\operatorname{Ext}_{R_z}^d(\mathcal{F}_z, R_z)$ is $a(z)$. So we have the proved the claim. $\square$

**Proposition 7.18.** Let $M$ be a subobject of the anti-dominant projective $P(w_0 \cdot \lambda)$. Assume that $M_{\infty,x,\mathcal{R}}(M) \neq 0$ and let $\mathcal{Y}$ be the support of $M_{\infty,x,\mathcal{R}}(M)$. Then $M_{\infty,x,\mathcal{R}}(M)$ is isomorphic to $\omega_{\mathcal{Y}}^{m_x}$ and $\mathcal{Y}$ is Cohen-Macaulay.

*Proof.* Let $Q$ be the quotient of $P(w_0 \cdot \lambda)$ by $M$. If $M_{\infty,x,\mathcal{R}}(Q) = 0$, then Theorem 7.14 implies the result. So we can assume that $M_{\infty,x,\mathcal{R}}(M) \neq 0$ and $M_{\infty,x,\mathcal{R}}(Q) \neq 0$. By Corollary 7.15, $M_{\infty,x,\mathcal{R}}(Q)$ is isomorphic to $\mathcal{O}_{\mathcal{Z}}^{m_x}$ for $\mathcal{Z} \subset \overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ maximal Cohen-Macaulay. Using Lemma 7.10, we can construct a commutative diagram

$$ \begin{array}{ccc} 0 & \to & M_{\infty,x,\mathcal{R}}(M) \to M_{\infty,x,\mathcal{R}}(P(w_0 \cdot \lambda)) \to M_{\infty,x,\mathcal{R}}(Q) \to 0 \\ \Big\downarrow & & \Big\downarrow \\ 0 & \to & \operatorname{Ker} \quad \Big\downarrow \\ & & \mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}}^{m_x} \quad \Big\downarrow \\ & & \mathcal{O}_{\mathcal{Z}}^{m_x} \quad \Big\downarrow \\ & & \end{array} $$

Let $I$ be the ideal defining $\mathcal{Z}$. As $\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\text{qtri}}$ and $\mathcal{Z}$ are Cohen-Macaulay of the same dimension, the involutivity of the duality implies that we have $I \simeq \omega_{\mathcal{O}_{\mathcal{Y}}}$ where $\mathcal{Y} = \operatorname{Supp}(I)$ so that we have the result. $\square$
**Theorem 7.19.** For all $w \in W$, with $ww_0 \ge w_{x,R}$, the sheaf $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))$ is isomorphic to

$$ (\omega_{\bar{\chi}_{\infty,x,R}^{\text{qtri},ww_0}})^{\oplus m_x} . $$

*Proof.* It follows from Propositions 7.18 and 7.3 that $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda)) \simeq \omega_{\mathcal{Y}}^{m_x}$ where $\mathcal{Y} = \text{Supp} \mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))$ is Cohen-Macaulay. However it follows from Theorem 6.16 that $\mathcal{Y} \subset \bar{\chi}_{\infty,x,R}^{\text{qtri},ww_0}$. By Lemma 7.17, we have an equality $[\omega_{\mathcal{Y}}] = [\mathcal{Y}]$ and it follows from Corollary 7.5 that $[\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))] = m_x[\bar{\chi}_{\infty,x,R}^{\text{qtri},ww_0}]$. Therefore we have $[\mathcal{Y}] = [\bar{\chi}_{\infty,x,R}^{\text{qtri},ww_0}]$ and thus $\mathcal{Y} = \bar{\chi}_{\infty,x,R}^{\text{qtri},ww_0}$. $\square$

We choose for all $\lambda$ dominant weight, and all $w \in W$ a surjective map $\pi_w : P(w_0 \cdot \lambda) \to P(w \cdot \lambda)^{\vee}$ (see proof of Corollary 7.16).

**Lemma 7.20.** For all map $f_{w,w'} : P(w \cdot \lambda)^{\vee} \to P(w' \cdot \lambda)^{\vee}$ there exists a map $\tilde{f}_{w,w'} : P(w_0 \cdot \lambda) \to P(w_0 \cdot \lambda)$ such that the following diagram commutes

$$ \begin{tikzcd} P(w_0 \cdot \lambda) \arrow[r, "\tilde{f}_{w,w'}"] \arrow[d, "\pi_w"] & P(w_0 \cdot \lambda) \arrow[d, "\pi_{w'}"] \\ P(w \cdot \lambda)^{\vee} \arrow[r, "f_{w,w'}"] \arrow[d, "\pi_w"'] & P(w \cdot \lambda)^{\vee} \end{tikzcd} \qquad (11) $$

*Proof.* As $\pi_{w'} : P(w_0 \cdot \lambda) \to P(w' \cdot \lambda)^{\vee}$ is surjective and $P(w_0 \cdot \lambda)$ is projective, the map $\text{Hom}(P(w_0 \cdot \lambda, P(w_0 \cdot \lambda)) \to \text{Hom}(P(w_0 \cdot \lambda), P(w' \cdot \lambda)^{\vee})$ is surjective, thus there exists $\tilde{f}_{w,w'}$ mapping to $f_{w,w'} \circ \pi_w$. This proves the claim. $\square$

**Lemma 7.21.** Let $\mathcal{F}$ be either $\mathcal{B}$, $\mathcal{B}_x$ or $\mathcal{M}_{\infty,x,R}$. There exists a family of isomorphisms indexed by $w \in W$

$$ \Psi_w : \mathcal{F}(P(w \cdot \lambda)^{\vee}) \xrightarrow{\sim} \mathcal{F}(P(w \cdot \lambda))^{\vee}. $$

such that for any $w, w' \in W$ and any if $f_{w,w'} : P(w \cdot \lambda)^{\vee} \to P(w' \cdot \lambda)^{\vee}$, the following diagram commutes

$$ \begin{tikzcd} \mathcal{F}(P(w \cdot \lambda)^{\vee}) \arrow[r, "\mathcal{F}(f_{w,w'})"] \arrow[d, "\Psi_w"'] & \mathcal{F}(P(w' \cdot \lambda)^{\vee}) \arrow[d, "\Psi_{w'}"'] \\ \mathcal{F}(P(w \cdot \lambda))^{\vee} \arrow[r, "\mathcal{F}(f_{w,w'}^{\vee})^{\vee}"] \arrow[d, "\Psi_{w'}"'] & \mathcal{F}(P(w' \cdot \lambda))^{\vee} \end{tikzcd} \qquad (12) $$

where we denote by the same symbol $(\cdot)^{\vee}$ the duality in $\mathcal{O}$ and Serre duality on coherent sheaves.

*Proof.* Let $w \in W$. The sheaves $\mathcal{F}(P(w \cdot \lambda)^{\vee})$ and $\mathcal{F}(P(w \cdot \lambda))^{\vee}$ are isomorphic to the same quotient of $\mathcal{F}(P(w_0 \cdot \lambda))$ by Theorem 7.4 for $\mathcal{B}$, $\mathcal{B}_x$ and Corollary 7.16 and Proposition 7.18 for $\mathcal{M}_{\infty,x,R}$. This implies that there exists an isomorphism $\Psi_w : \mathcal{F}(P(w \cdot \lambda)^{\vee}) \to \mathcal{F}(P(w \cdot \lambda))^{\vee}$ such that the following diagram commutes
**Theorem 7.19.** For all $w \in W$, with $ww_0 \ge w_{x,R}$, the sheaf $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))$ is isomorphic to

$$ (\omega_{\bar{\mathcal{X}}_{\infty,x,R}^{\text{qtri},ww_0}})^{\oplus m_x} . $$

*Proof.* It follows from Propositions 7.18 and 7.3 that $\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda)) \simeq \omega_{\mathcal{Y}}^{m_x}$ where $\mathcal{Y} = \text{Supp} \mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))$ is Cohen-Macaulay. However it follows from Theorem 6.16 that $\mathcal{Y} \subset \bar{\mathcal{X}}_{\infty,x,R}^{\text{qtri},ww_0}$. By Lemma 7.17, we have an equality $[\omega_{\mathcal{Y}}] = [\mathcal{Y}]$ and it follows from Corollary 7.5 that $[\mathcal{M}_{\infty,x,R}(M(w \cdot \lambda))] = m_x[\bar{\mathcal{X}}_{\infty,x,R}^{\text{qtri},ww_0}]$. Therefore we have $[\mathcal{Y}] = [\bar{\mathcal{X}}_{\infty,x,R}^{\text{qtri},ww_0}]$ and thus $\mathcal{Y} = \bar{\mathcal{X}}_{\infty,x,R}^{\text{qtri},ww_0}$. $\square$

We choose for all $\lambda$ dominant weight, and all $w \in W$ a surjective map $\pi_w : P(w_0 \cdot \lambda) \to P(w \cdot \lambda)^{\vee}$ (see proof of Corollary 7.16).

**Lemma 7.20.** For all map $f_{w,w'} : P(w \cdot \lambda)^{\vee} \to P(w' \cdot \lambda)^{\vee}$ there exists a map $\tilde{f}_{w,w'} : P(w_0 \cdot \lambda) \to P(w_0 \cdot \lambda)$ such that the following diagram commutes

$$ \begin{tikzcd} P(w_0 \cdot \lambda) \arrow[r, "\tilde{f}_{w,w'}"] \arrow[d, "\pi_w"] & P(w_0 \cdot \lambda) \arrow[d, "\pi_{w'}"'] \\ P(w \cdot \lambda)^{\vee} \arrow[r, "f_{w,w'}"] \arrow[d, "\pi_w"'] & P(w \cdot \lambda)^{\vee} \end{tikzcd} \qquad (11) $$

*Proof.* As $\pi_{w'} : P(w_0 \cdot \lambda) \to P(w' \cdot \lambda)^{\vee}$ is surjective and $P(w_0 \cdot \lambda)$ is projective, the map $\text{Hom}(P(w_0 \cdot \lambda, P(w_0 \cdot \lambda)) \to \text{Hom}(P(w_0 \cdot \lambda), P(w' \cdot \lambda)^{\vee})$ is surjective, thus there exists $\tilde{f}_{w,w'}$ mapping to $f_{w,w'} \circ \pi_w$. This proves the claim. $\square$

**Lemma 7.21.** Let $\mathcal{F}$ be either $\mathcal{B}$, $\mathcal{B}_x$ or $\mathcal{M}_{\infty,x,R}$. There exists a family of isomorphisms indexed by $w \in W$

$$ \Psi_w : \mathcal{F}(P(w \cdot \lambda)^{\vee}) \xrightarrow{\sim} \mathcal{F}(P(w \cdot \lambda))^{\vee}. $$

such that for any $w, w' \in W$ and any if $f_{w,w'} : P(w \cdot \lambda)^{\vee} \to P(w' \cdot \lambda)^{\vee}$, the following diagram commutes

$$ \begin{tikzcd} \mathcal{F}(P(w \cdot \lambda)^{\vee}) \arrow[r, "\mathcal{F}(f_{w,w'})"] \arrow[d, "\Psi_w"'] & \mathcal{F}(P(w' \cdot \lambda)^{\vee}) \arrow[d, "\Psi_{w'}"'] \\ \mathcal{F}(P(w \cdot \lambda))^{\vee} \arrow[r, "\mathcal{F}(f_{w,w'}^{\vee})^{\vee}"] \arrow[d, "\mathcal{F}(f_{w,w'}^{\vee})^{\vee}"'] & \mathcal{F}(P(w' \cdot \lambda))^{\vee} \end{tikzcd} \qquad (12) $$

where we denote by the same symbol $(\cdot)^{\vee}$ the duality in $\mathcal{O}$ and Serre duality on coherent sheaves.

*Proof.* Let $w \in W$. The sheaves $\mathcal{F}(P(w \cdot \lambda)^{\vee})$ and $\mathcal{F}(P(w \cdot \lambda))^{\vee}$ are isomorphic to the same quotient of $\mathcal{F}(P(w_0 \cdot \lambda))$ by Theorem 7.4 for $\mathcal{B}$, $\mathcal{B}_x$ and Corollary 7.16 and Proposition 7.18 for $\mathcal{M}_{\infty,x,R}$. This implies that there exists an isomorphism $\Psi_w : \mathcal{F}(P(w \cdot \lambda)^{\vee}) \to \mathcal{F}(P(w \cdot \lambda))^{\vee}$ such that the following diagram commutes
$$
\begin{equation} \tag{13}
\begin{aligned}
& \mathcal{F}(P(w_0 \cdot \lambda)) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee \\
& \xleftarrow{\mathcal{F}(\pi_w)} \xrightarrow{\mathcal{F}(\pi_w)^\vee} \\
& \mathcal{F}(P(w \cdot \lambda)^\vee) \xrightarrow{\Psi_w} \mathcal{F}(P(w \cdot \lambda))^\vee
\end{aligned}
\end{equation}
$$

Fix $w, w'$ and let's show that the diagram (12) is commutative. Let $f_{w,w'} \in \operatorname{Hom}(P(w \cdot \lambda)^\vee, P(w' \cdot \lambda)^\vee)$. By Lemma 7.20, there exists a map $\tilde{f}_{w,w'} \in \operatorname{End}(P(w_0 \cdot \lambda))$ such that the diagram (11) is commutative. We first consider the following diagram

$$
\begin{equation} \tag{14}
\begin{aligned}
& \mathcal{F}(P(w_0 \cdot \lambda)) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee \\
& \xleftarrow{\mathcal{F}(\tilde{f}_{w,w'})} \xrightarrow{\mathcal{F}(\tilde{f}_{w,w'}^\vee)} \\
& \mathcal{F}(P(w_0 \cdot \lambda)^\vee) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee
\end{aligned}
\end{equation}
$$

But as $\tilde{f}_{w,w'} \in \operatorname{End}_{\mathcal{O}}(P(w_0 \cdot \lambda), P(w_0 \cdot \lambda)) \simeq D = L \otimes_{A_W} A$, it follows from Corollary 6.14 for $\mathcal{F} = \mathcal{M}_{\infty,x,\mathcal{R}}$ and [Bez16, Prop. 23] for $\mathcal{F} = \mathcal{B}_x$, and the fact that $\Psi_{w_0}$ is $\mathcal{O}_{\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{tri}}}$-linear, that this diagram commutes. Now consider the diagram

<figure><img src="image_6.png" /></figure>

All faces, except maybe the right hand one (which is the one of the statement), of
this cube are commutative diagrams by functoriality and diagrams (11), (13), (14).
Moreover $\mathcal{F}(\pi_w)$, $\mathcal{F}(\pi_w^\vee)^\vee$, $\mathcal{F}(\pi_{w'})$, $\mathcal{F}(\pi_{w'}^\vee)^\vee$ are surjective, thus the last right hand face
also commutes.
$\square$

**Corollary 7.22.** For any $M \in \mathcal{O}_{\text{alg}}$, there is a compatible choice of isomorphisms

$$
\Psi_M : \mathcal{F}(M^{\vee}) \xrightarrow{\sim} \mathcal{F}(M)^{\vee},
$$

where $\mathcal{F}$ is either the functor $\mathcal{B}, \mathcal{B}_x$ or $\mathcal{M}_{\infty,x,\mathcal{R}}$. In particular, $\mathcal{F}$ is compatible with duality.

*Proof.* Choose a resolution

$$
\bigoplus_i P(\mu_i) \to \bigoplus_j P(\lambda_j) \to M \to 0. \quad (15)
$$
$$
\begin{equation} \tag{13}
\begin{aligned}
& \mathcal{F}(P(w_0 \cdot \lambda)) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee \\
& \xleftarrow{\mathcal{F}(\pi_w)} \xrightarrow{\mathcal{F}(\pi_w)^\vee} \\
& \mathcal{F}(P(w \cdot \lambda)^\vee) \xrightarrow{\Psi_w} \mathcal{F}(P(w \cdot \lambda))^\vee
\end{aligned}
\end{equation}
$$

Fix $w, w'$ and let's show that the diagram (12) is commutative. Let $f_{w,w'} \in \operatorname{Hom}(P(w \cdot \lambda)^\vee, P(w' \cdot \lambda)^\vee)$. By Lemma 7.20, there exists a map $\tilde{f}_{w,w'} \in \operatorname{End}(P(w_0 \cdot \lambda))$ such that the diagram (11) is commutative. We first consider the following diagram

$$
\begin{equation} \tag{14}
\begin{aligned}
& \mathcal{F}(P(w_0 \cdot \lambda)) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee \\
& \xleftarrow{\mathcal{F}(\tilde{f}_{w,w'})} \xleftarrow{\mathcal{F}(\tilde{f}_{w,w'}^\vee)} \\
& \mathcal{F}(P(w_0 \cdot \lambda)^\vee) \xrightarrow{\Psi_{w_0}} \mathcal{F}(P(w_0 \cdot \lambda))^\vee
\end{aligned}
\end{equation}
$$

But as $\tilde{f}_{w,w'} \in \operatorname{End}_{\mathcal{O}}(P(w_0 \cdot \lambda), P(w_0 \cdot \lambda)) \simeq D = L \otimes_{A_W} A$, it follows from Corollary 6.14 for $\mathcal{F} = \mathcal{M}_{\infty,x,\mathcal{R}}$ and [Bez16, Prop. 23] for $\mathcal{F} = \mathcal{B}_x$, and the fact that $\Psi_{w_0}$ is $\mathcal{O}_{\mathcal{X}_{\infty,x,\mathcal{R}}^{\text{tri}}}$-linear, that this diagram commutes. Now consider the diagram

<figure><img src="image_6.png" /></figure>

All faces, except maybe the right hand one (which is the one of the statement), of
this cube are commutative diagrams by functoriality and diagrams (11), (13), (14).
Moreover $\mathcal{F}(\pi_w)$, $\mathcal{F}(\pi_w^\vee)^\vee$, $\mathcal{F}(\pi_{w'})$, $\mathcal{F}(\pi_{w'}^\vee)^\vee$ are surjective, thus the last right hand face
also commutes.
$\square$

**Corollary 7.22.** For any $M \in \mathcal{O}_{\text{alg}}$, there is a compatible choice of isomorphisms

$$
\Psi_M : \mathcal{F}(M^{\vee}) \xrightarrow{\sim} \mathcal{F}(M)^{\vee},
$$

where $\mathcal{F}$ is either the functor $\mathcal{B}, \mathcal{B}_x$ or $\mathcal{M}_{\infty,x,\mathcal{R}}$. In particular, $\mathcal{F}$ is compatible with duality.

*Proof.* Choose a resolution

$$
\bigoplus_i P(\mu_i) \to \bigoplus_j P(\lambda_j) \to M \to 0. \quad (15)
$$
Then we have two exact sequences

$$0 \to \mathcal{F}(M^{\vee}) \to \bigoplus_j \mathcal{F}(P(\lambda_j)^{\vee}) \to \bigoplus_i \mathcal{F}(P(\mu_i)^{\vee}),$$

and

$$0 \to \mathcal{F}(M)^{\vee} \to \bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee} \to \bigoplus_i \mathcal{F}(P(\mu_i))^{\vee}.$$

For the second one, recall that if $K$ denote the Kernel in equation (15) so that

$$0 \to K \to \bigoplus_j P(\lambda_j) \to M \to 0,$$

then, as $\mathcal{F}(K)$ is CM of the same dimension as the other modules, we have

$$0 \to \mathcal{F}(M)^{\vee} \to \bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee} \to \mathcal{F}(K)^{\vee} \to 0,$$

which is exact. Moreover, by the previous Lemma 7.21 we have a commutative diagram with vertical isomorphisms

$$\begin{tikzcd} 0 \arrow[r] \arrow[d, "\bigoplus_j \mathcal{F}(P(\lambda_j)^{\vee})"] & \mathcal{F}(M^{\vee}) \arrow[r] \arrow[d, "\bigoplus_j \Psi_{\lambda_j}"] & \bigoplus_i \mathcal{F}(P(\mu_i)^{\vee}) \arrow[d, "\bigoplus_i \Psi_{\mu_i}"] \\ 0 \arrow[r] \arrow[d, "\bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee}"] & \mathcal{F}(M)^{\vee} \arrow[d, "\bigoplus_i \mathcal{F}(P(\mu_i))^{\vee}"] & 0 \arrow[r] \arrow[d, "\bigoplus_i \mathcal{F}(P(\mu_i)^{\vee})"] \arrow[d, "\bigoplus_j \mathcal{F}(P(\lambda_j)^{\vee})"] \end{tikzcd}$$

which induces an isomorphism $\Psi_M : \mathcal{F}(M^{\vee}) \to \mathcal{F}(M)^{\vee}$. $\square$

<figure><img src="image_7.png" /></figure>

**Corollary 7.23.** There exists an isomorphism of functors $\mathcal{B}_x^{m_x} \simeq \mathcal{M}_{\infty,x,\mathcal{R}}$.

*Proof.* By a similar argument to the proof of Lemma 7.21, we can construct a family indexed by $w \in W$ of isomorphisms

$$\Phi_w : \mathcal{B}_x(P(w \cdot \lambda)^{\vee})^{m_x} \xrightarrow{\sim} \mathcal{M}_{\infty,x,\mathcal{R}}(P(w \cdot \lambda)^{\vee})$$

such that, for any $w, w' \in W$ and any $f_{w,w'} \in \operatorname{Hom}(P(w \cdot \lambda)^{\vee}, P(w' \cdot \lambda)^{\vee})$, the following diagram commutes

$$\begin{tikzcd} \mathcal{B}_x(P(w \cdot \lambda)^{\vee})^{m_x} \arrow[r, "\mathcal{B}_x(f_{w,w'})"'] & \mathcal{B}_x(P(w' \cdot \lambda)^{\vee}) \arrow[d, "\Phi_w"] & \mathcal{M}_{\infty,x,\mathcal{R}}(P(w \cdot \lambda)^{\vee}) \arrow[r, "\mathcal{M}_{\infty,x,\mathcal{R}}(f_{w,w'})"'] & \mathcal{M}_{\infty,x,\mathcal{R}}(P(w' \cdot \lambda)^{\vee}). \arrow[d, "\Phi_{w'}"] \end{tikzcd}$$

Such a family of isomorphisms provides an isomorphism of functors between $\mathcal{B}_x^{m_x}$ and $\mathcal{M}_{\infty,x,\mathcal{R}}$ restricted to the full subcategory of $\mathcal{O}_\lambda$ of injective objects. As the category $\mathcal{O}_\lambda$ has enough injectives, this isomorphism extends to all of $\mathcal{O}_\lambda$. $\square$

<figure><img src="image_8.png" /></figure>
Then we have two exact sequences

$$0 \to \mathcal{F}(M^{\vee}) \to \bigoplus_j \mathcal{F}(P(\lambda_j)^{\vee}) \to \bigoplus_i \mathcal{F}(P(\mu_i)^{\vee}),$$

and

$$0 \to \mathcal{F}(M)^{\vee} \to \bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee} \to \bigoplus_i \mathcal{F}(P(\mu_i))^{\vee}.$$

For the second one, recall that if $K$ denote the Kernel in equation (15) so that

$$0 \to K \to \bigoplus_j P(\lambda_j) \to M \to 0,$$

then, as $\mathcal{F}(K)$ is CM of the same dimension as the other modules, we have

$$0 \to \mathcal{F}(M)^{\vee} \to \bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee} \to \mathcal{F}(K)^{\vee} \to 0,$$

which is exact. Moreover, by the previous Lemma 7.21 we have a commutative diagram with vertical isomorphisms

$$0 \to \mathcal{F}(M^{\vee}) \to \bigoplus_j \mathcal{F}(P(\lambda_j)^{\vee}) \to \bigoplus_i \mathcal{F}(P(\mu_i)^{\vee}) \\ \qquad \bigoplus_j \Psi_{\lambda_j} \qquad \bigoplus_i \Psi_{\mu_i} \\ 0 \to \mathcal{F}(M)^{\vee} \to \bigoplus_j \mathcal{F}(P(\lambda_j))^{\vee} \to \bigoplus_i \mathcal{F}(P(\mu_i))^{\vee}$$

which induces an isomorphism $\Psi_M : \mathcal{F}(M^{\vee}) \to \mathcal{F}(M)^{\vee}$. $\square$

<figure><img src="image_7.png" /></figure>

**Corollary 7.23.** There exists an isomorphism of functors $\mathcal{B}_x^{m_x} \simeq \mathcal{M}_{\infty,x,\mathcal{R}}$.

*Proof.* By a similar argument to the proof of Lemma 7.21, we can construct a family indexed by $w \in W$ of isomorphisms

$$\Phi_w : \mathcal{B}_x(P(w \cdot \lambda)^{\vee})^{m_x} \xrightarrow{\sim} \mathcal{M}_{\infty,x,\mathcal{R}}(P(w \cdot \lambda)^{\vee})$$

such that, for any $w, w' \in W$ and any $f_{w,w'} \in \operatorname{Hom}(P(w \cdot \lambda)^{\vee}, P(w' \cdot \lambda)^{\vee})$, the following diagram commutes

$$\begin{tikzcd} \mathcal{B}_x(P(w \cdot \lambda)^{\vee})^{m_x} \arrow[r, "\mathcal{B}_x(f_{w,w'})"'] & \mathcal{B}_x(P(w' \cdot \lambda)^{\vee}) \\ \mathcal{M}_{\infty,x,\mathcal{R}}(P(w \cdot \lambda)^{\vee}) \arrow[u, "\Phi_w"'] \arrow[r, "\mathcal{M}_{\infty,x,\mathcal{R}}(f_{w,w'})"'] & \mathcal{M}_{\infty,x,\mathcal{R}}(P(w' \cdot \lambda)^{\vee}). \end{tikzcd}$$

Such a family of isomorphisms provides an isomorphism of functors between $\mathcal{B}_x^{m_x}$ and $\mathcal{M}_{\infty,x,\mathcal{R}}$ restricted to the full subcategory of $\mathcal{O}_{\lambda}$ of injective objects. As the category $\mathcal{O}_{\lambda}$ has enough injectives, this isomorphism extends to all of $\mathcal{O}_{\lambda}$. $\square$

<figure><img src="image_8.png" /></figure>
## 7.7 Consequences

In this section we keep the setting introduced in subsection 7.3. In particular $n = 3$.

**Lemma 7.24.** Let $\rho, \lambda, \mathcal{R}$ be as above and let $x \in \mathcal{X}_{\infty}(L)$ the point corresponding to $\rho$. Then for all $M \in \mathcal{O}_{\chi_{\lambda}}$,

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x) \simeq (\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}])' $$

*Proof.* By construction (see Remark 6.2), we have

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

By Corollary 6.11, the $\mathcal{X}_{\infty} \times \hat{T}$-structure on the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ factors through $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}} \to \mathcal{X}_{\infty} \times \hat{T}$. Thus,

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x) \simeq (\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_{\infty}^{\mathrm{la}}[\mathfrak{m}_x])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}])' \quad \square $$

**Corollary 7.25.** Let $\delta : T \to L^\times$ be a continuous character and let $\chi^S : T^S \to L$ be a character such that there exists $f \in S^\dagger(K^p)[\chi^S \otimes \delta]$ an overconvergent p-adic eigenform on the group $U(3)$. Assume that the Galois representation $\rho$ associated to $f$ is crystalline strictly dominant and $\varphi$-generic at $p$ satisfying (1). Let $r = |\{\tau \in \Sigma_F | \omega_{x,\mathcal{R},\tau} = 1\}|$. Then

$$ \dim S^\dagger(K^p)[\chi^S \otimes \delta] = 2^r \dim S^{\mathrm{cl}}(K^p)[\chi^S \otimes \delta] \neq 0. $$

*Proof.* The assumptions imply that the character $\delta$ is locally algebraic and that it factors as $\delta = \delta_{\lambda}\delta_{\mathcal{R}}$ for some $\lambda \in X^*(T)^+$ and some unramified character $\delta_{\mathcal{R}}$. By Breuil's adjunction formula [Bre15, Théorème 4.3] (see also [BHS19, eq. (5.5)]) and [BHS19, Lemma 5.2.3] we have

$$ \begin{aligned} S^\dagger(K^p)[\chi^S \otimes \delta] &= \mathrm{Hom}_{U(\mathfrak{g})}(M(\lambda), \Pi^{\mathrm{la}}[\chi^S])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], \\ S^{\mathrm{cl}}(K^p)[\chi^S \otimes \delta] &= \mathrm{Hom}_{U(\mathfrak{g})}(L(\lambda), \Pi^{\mathrm{la}}[\chi^S])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}]. \end{aligned} $$

In particular, by Lemma 7.24, these spaces are identified with the dual vector spaces of the fiber of $\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda))$ resp. of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ at $k(x)$. Thus, as $m_x = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda)) \otimes k(x)$, the result is a direct corollary of Theorem 7.19 (and Proposition 7.6). $\square$

We can also deduce the following corollary on the structure of the completed cohomology $\Pi$ (see Definition 5.2), which is a representation of $G := U(\mathbb{Q}_p)$. Let $\mathfrak{gl}_3$ be the Lie algebra (over $L$) of the group $\mathrm{GL}_3$ and for a dominant $\lambda$ we consider the extension

$$ N(\lambda) = [L(s_1 \cdot \lambda) \oplus L(s_2 \cdot \lambda) - L(\lambda)] \in \mathrm{Ext}^1_{\mathcal{O}}(L(\lambda), L(s_1 \cdot \lambda) \oplus L(s_2 \cdot \lambda)), $$

which is non trivial when mapped in each of $\mathrm{Ext}^1_{\mathcal{O}}(L(\lambda), L(s_i \cdot \lambda))$, for $i = 1, 2$. This extension is the quotient of the Verma module $M(\lambda)$ by $M(s_1 s_2 \cdot \lambda) + M(s_2 s_1 \cdot \lambda)$.
## 7.7 Consequences

In this section we keep the setting introduced in subsection 7.3. In particular $n = 3$.

**Lemma 7.24.** Let $\rho, \lambda, \mathcal{R}$ be as above and let $x \in \mathcal{X}_{\infty}(L)$ the point corresponding to $\rho$. Then for all $M \in \mathcal{O}_{\chi_{\lambda}}$,

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x) \simeq (\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}])' $$

*Proof.* By construction (see Remark 6.2), we have

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

By Corollary 6.11, the $\mathcal{X}_{\infty} \times \hat{T}$-structure on the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ factors through $\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}} \to \mathcal{X}_{\infty} \times \hat{T}$. Thus,

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x) \simeq (\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_{\infty}^{\mathrm{la}}[\mathfrak{m}_x])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}])' \quad \square $$

**Corollary 7.25.** Let $\delta : T \to L^\times$ be a continuous character and let $\chi^S : T^S \to L$ be a character such that there exists $f \in S^\dagger(K^p)[\chi^S \otimes \delta]$ an overconvergent p-adic eigenform on the group $U(3)$. Assume that the Galois representation $\rho$ associated to $f$ is crystalline strictly dominant and $\varphi$-generic at $p$ satisfying (1). Let $r = |\{\tau \in \Sigma_F | \omega_{x,\mathcal{R},\tau} = 1\}|$. Then

$$ \dim S^\dagger(K^p)[\chi^S \otimes \delta] = 2^r \dim S^{\mathrm{cl}}(K^p)[\chi^S \otimes \delta] \neq 0. $$

*Proof.* The assumptions imply that the character $\delta$ is locally algebraic and that it factors as $\delta = \delta_{\lambda}\delta_{\mathcal{R}}$ for some $\lambda \in X^*(T)^+$ and some unramified character $\delta_{\mathcal{R}}$. By Breuil's adjunction formula [Bre15, Théorème 4.3] (see also [BHS19, eq. (5.5)]) and [BHS19, Lemma 5.2.3] we have

$$ \begin{aligned} S^\dagger(K^p)[\chi^S \otimes \delta] &= \mathrm{Hom}_{U(\mathfrak{g})}(M(\lambda), \Pi^{\mathrm{la}}[\chi^S])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], \\ S^{\mathrm{cl}}(K^p)[\chi^S \otimes \delta] &= \mathrm{Hom}_{U(\mathfrak{g})}(L(\lambda), \Pi^{\mathrm{la}}[\chi^S])^{N_0}[\mathfrak{m}_{\delta_{\mathcal{R}}}]. \end{aligned} $$

In particular, by Lemma 7.24, these spaces are identified with the dual vector spaces of the fiber of $\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda))$ resp. of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ at $k(x)$. Thus, as $m_x = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda)) \otimes k(x)$, the result is a direct corollary of Theorem 7.19 (and Proposition 7.6). $\square$

We can also deduce the following corollary on the structure of the completed cohomology $\Pi$ (see Definition 5.2), which is a representation of $G := U(\mathbb{Q}_p)$. Let $\mathfrak{gl}_3$ be the Lie algebra (over $L$) of the group $\mathrm{GL}_3$ and for a dominant $\lambda$ we consider the extension

$$ N(\lambda) = [L(s_1 \cdot \lambda) \oplus L(s_2 \cdot \lambda) - L(\lambda)] \in \mathrm{Ext}^1_{\mathcal{O}}(L(\lambda), L(s_1 \cdot \lambda) \oplus L(s_2 \cdot \lambda)), $$

which is non trivial when mapped in each of $\mathrm{Ext}^1_{\mathcal{O}}(L(\lambda), L(s_i \cdot \lambda))$, for $i = 1, 2$. This extension is the quotient of the Verma module $M(\lambda)$ by $M(s_1 s_2 \cdot \lambda) + M(s_2 s_1 \cdot \lambda)$.
As before we consider the Lie algebra

$$ \mathfrak{g} = \mathrm{Lie}(\underline{G}_L \simeq (\mathrm{Res}_{F \otimes \mathbb{Q}_{\mathbb{Q}_p}/\mathbb{Q}_p} \mathrm{GL}_3) \times_{\mathbb{Q}_p} L) \simeq \prod_{\tau \in \Sigma_F} \mathfrak{g} \mathfrak{l}_3 $$

with Borel $\mathfrak{b} \simeq \prod_{\tau} \mathfrak{b}_{\tau}$. Associated to a dominant weight $\lambda = (\lambda_{\tau})_{\tau} \in X^{*}(\underline{T})^{+}$ and $w_{\mathcal{R}} = (w_{\mathcal{R},\tau})_{\tau \in \Sigma_F} \in W$ we define the object

$$ N(\lambda, w_{\mathcal{R}}) = \left( \bigotimes_{\tau: w_{\mathcal{R},\tau} \neq 1} L(\lambda_{\tau}) \otimes \bigotimes_{\tau: w_{\mathcal{R},\tau} = 1} N(\lambda_{\tau}) \right) $$

of the category $\mathcal{O}_{\chi_{\lambda}} = \bigotimes_{\tau} \mathcal{O}_{\chi_{\lambda_{\tau}}}^{\mathfrak{g}l_3, \mathfrak{b}_{\tau}}$. We also define

$$ S(\lambda, w_{\mathcal{R}}) = \bigotimes_{\tau} S(\lambda_{\tau}, w_{\mathcal{R},\tau}) \in \mathcal{O}_{\chi_{\lambda}}, $$

where

$$ S(\lambda_{\tau}, w_{\mathcal{R},\tau}) = \left\{ \begin{array}{ll} \bigoplus_{w \le w_{\mathcal{R},\tau}, w_0} L(w \cdot \lambda_{\tau}) & \text{if } w_{\mathcal{R},\tau} \ne 1 \\ \bigoplus_{\ell(w) \ne 1} L(w \cdot \lambda_{\tau}) \bigoplus N(\lambda_{\tau}) & \text{if } w_{\mathcal{R},\tau} = 1 \end{array} \right. , $$

so that $S(\lambda, w_{\mathcal{R}}) = \bigoplus_{w \le w_{\mathcal{R}}, w_0} L(w \cdot \lambda)$ if $w_{\mathcal{R},\tau} \ne 1$ for all $\tau$, and

$$ N(\lambda, w_{\mathcal{R}}) \subset S(\lambda, w_{\mathcal{R}}), $$

otherwise.

If $M$ is a $U(\mathfrak{g})$-module, we denote $\mathrm{Hom}_E(M, E)$ the $U(\mathfrak{g})$-module with underlying vector space $\mathrm{Hom}_E(M, E)$ and action of $\mathfrak{r} \in U(\mathfrak{g})$ given by

$$ (\mathfrak{r} \cdot \phi)(m) := \phi(\dot{\mathfrak{r}}m), \quad \phi \in \mathrm{Hom}_E(M, E), m \in M, $$

where $\mathfrak{r} \mapsto \dot{\mathfrak{r}}$ is the anti-involution of $U(\mathfrak{g})$ extending $-1$ on $\mathfrak{g}$. We denote $\bar{B}$ the Borel opposite to $\underline{B}$, whose Lie algebra is $\bar{\mathfrak{b}}$ with $\bar{\mathfrak{n}}$ its nilpotent radical. We then denote $B = \underline{B}(\mathbb{Q}_p), \bar{B} = \bar{B}(\mathbb{Q}_p)$ and $\delta_B$ the modulus character of $B$. We then denote $M' := \mathrm{Hom}_E(M, E)^{\bar{\mathfrak{n}}^{\infty}}$ the vectors which are killed by a finite power of $\bar{\mathfrak{n}}$. If $M = \bigoplus_{\lambda \in X^{*}(\underline{T})_L} M_{\lambda} \in \mathcal{O}^{\mathfrak{g}, \mathfrak{b}}$, then $M' \in \mathcal{O}^{\mathfrak{g}, \bar{\mathfrak{b}}}$. Finally recall that if $M \in \mathcal{O}^{\mathfrak{g}, \bar{\mathfrak{b}}}$ and $\delta$ is a smooth character of $\underline{T}(\mathbb{Q}_p)$, then Orlik-Strauch constructed (see [OS10] or also [Bre16])

$$ \mathcal{F}_{\bar{B}}^{G}(M, \delta), $$

which is a locally analytic representation of $G$. In particular, locally analytic principal series are of this form : if $M = M(\lambda)^{\vee} \in \mathcal{O}^{\mathfrak{g}, \mathfrak{b}}$, then

$$ \mathcal{F}_{\bar{B}}^{G}((M(\lambda)^{\vee})', \delta) = \mathrm{ind}_{\bar{B}}^{G}(\delta_{\lambda}\delta)^{\mathrm{la}}. \qquad (16) $$

Let $\rho : \mathrm{Gal}_E \to \mathrm{GL}_n(L)$ be a crystalline, Hodge-Tate regular and $\varphi$-generic autodual representation satisfying Hypothesis 5.9 such that $\Pi[\mathfrak{m}_{\rho}] \ne 0$ where $\mathfrak{m}_{\rho}$ is the ideal of
As before we consider the Lie algebra

$$
\mathfrak{g} = \mathrm{Lie}(\underline{G}_L \simeq (\mathrm{Res}_{F \otimes \mathbb{Q}_{\mathbb{Q}_p}/\mathbb{Q}_p} \mathrm{GL}_3) \times_{\mathbb{Q}_p} L) \simeq \prod_{\tau \in \Sigma_F} \mathfrak{g} \mathfrak{l}_3
$$

with Borel **b** ≃ ∏<sub>τ</sub> **b**<sub>τ</sub>. Associated to a dominant weight λ = (λ<sub>τ</sub>)<sub>τ</sub> ∈ X<sup>*</sup>(T)<sup>+</sup> and
w<sub>**R**</sub> = (w<sub><sup>**R**</sup>&#x20D7;<sub>τ</sub></sub>)<sub>τ∈Σ<sub>F</sub></sub> ∈ W we define the object

$$
N(\lambda, w_{\mathcal{R}}) = \left( \bigotimes_{\tau: w_{\mathcal{R},\tau} \neq 1} L(\lambda_{\tau}) \bigotimes \bigotimes_{\tau: w_{\mathcal{R},\tau} = 1} N(\lambda_{\tau}) \right)
$$

of the category $\mathcal{O}_{\chi_{\lambda}} = \boxtimes_{\tau} \mathcal{O}_{\chi_{\lambda_{\tau}}}^{\mathfrak{g}_{3,\mathfrak{b}_{\tau}}}$. We also define

$$
S(\lambda, w_{\mathcal{R}}) = \boxtimes_{\tau} S(\lambda_{\tau}, w_{\mathcal{R},\tau}) \in \mathcal{O}_{\lambda_{\chi}},
$$

where

$$
S(\lambda_{\tau}, w_{\mathcal{R},\tau}) = \begin{cases} \bigoplus_{w \le w_{\mathcal{R},\tau} w_0} L(w \cdot \lambda_{\tau}) & \text{if } w_{\mathcal{R},\tau} \ne 1 \\ \bigoplus_{\ell(w) \ne 1} L(w \cdot \lambda_{\tau}) \bigoplus N(\lambda_{\tau}) & \text{if } w_{\mathcal{R},\tau} = 1 \end{cases},
$$

so that $S(\lambda, w_{\mathcal{R}}) = \bigoplus_{w \le w_{\mathcal{R}} w_0} L(w \cdot \lambda)$ if $w_{\mathcal{R},\tau} \ne 1$ for all $\tau$, and

$$
N(\lambda, w_{\mathcal{R}}) \subset S(\lambda, w_{\mathcal{R}}),
$$

otherwise.

If $M$ is a $U(\mathfrak{g})$-module, we denote $\mathrm{Hom}_E(M, E)$ the $U(\mathfrak{g})$-module with underlying vector space $\mathrm{Hom}_E(M, E)$ and action of $\mathfrak{r} \in U(\mathfrak{g})$ given by

$$
(\mathfrak{r} \cdot \phi)(m) := \phi(\dot{\mathfrak{r}}m), \quad \phi \in \mathrm{Hom}_E(M, E), m \in M,
$$

where $\mathfrak{r} \mapsto \dot{\mathfrak{r}}$ is the anti-involution of $U(\mathfrak{g})$ extending $-1$ on $\mathfrak{g}$. We denote $\bar{B}$ the Borel opposite to $\underline{B}$, whose Lie algebra is $\bar{\mathfrak{b}}$ with $\bar{\mathfrak{n}}$ its nilpotent radical. We then denote $B = \underline{B}(\mathbb{Q}_p), \bar{B} = \bar{B}(\mathbb{Q}_p)$ and $\delta_B$ the modulus character of $B$. We then denote $M' := \mathrm{Hom}_E(M, E)^{\bar{\mathfrak{n}}^\infty}$ the vectors which are killed by a finite power of $\bar{\mathfrak{n}}$. If $M = \bigoplus_{\lambda \in X^*(T)_L} M_\lambda \in \mathcal{O}^{\mathfrak{g}, \mathfrak{b}}$, then $M' \in \mathcal{O}^{\mathfrak{g}, \bar{\mathfrak{b}}}$. Finally recall that if $M \in \mathcal{O}^{\mathfrak{g}, \bar{\mathfrak{b}}}$ and $\delta$ is a smooth character of $\underline{T}(\mathbb{Q}_p)$, then Orlik-Strauch constructed (see [OS10] or also [Bre16])

$$
\mathcal{F}_{\bar{B}}^{G}(M, \delta),
$$

which is a locally analytic representation of $G$. In particular, locally analytic principal
series are of this form : if $M = M(\lambda)^{\vee} \in \mathcal{O}^{g,b}$, then

$$
\mathcal{F}_{\bar{B}}^{G}((M(\lambda)^{\vee})', \delta) = \mathrm{ind}_{\bar{B}}^{G}(\delta_{\lambda}\delta)^{\mathrm{la}}. \qquad (16)
$$

Let $\rho: \mathrm{Gal}_E \to \mathrm{GL}_n(L)$ be a crystalline, Hodge-Tate regular and $\varphi$-generic autod-
ual representation satisfying Hypothesis 5.9 such that $\Pi[\mathfrak{m}_\rho] \neq 0$ where $\mathfrak{m}_\rho$ is the ideal of
$\mathbb{T}^S \otimes L$ associated to $\rho$. Let $\mathcal{R}$ a choice of refinement and $\delta_{\mathcal{R}}$ the associated unramified character. Denote $\lambda = (\lambda_{\tau})_{\tau} := \text{HT}(\rho) - \delta_G \in X^*(T)^+$ the (dominant) algebraic character associated to $\rho$ as before, where $\text{HT}(\rho) = (h_{1,\tau} > \dots > h_{n,\tau})_{\tau \in \Sigma_F} \in X^*(\mathbb{T})$ gives the Hodge-Tate weights of $\rho$. As $\Pi[m_{\rho}] \neq 0$ and $\rho$ satisfies Hypothesis 5.9, it corresponds to a point $x \in \mathcal{X}_{\infty}(L)$. Denote $w_{\rho,\mathcal{R}} = (w_{\rho,\mathcal{R},\tau})_{\tau \in \Sigma_F}$ and $m_{\rho} := m_x \geq 1$ as in Section 7.1.

**Corollary 7.26.** For $\rho, \lambda, \mathcal{R}$ as above and all $w \le w_{\mathcal{R}}w_0$, we have

$$ \dim \mathrm{Hom}_{\mathcal{G}}(\mathrm{ind}_{\mathcal{B}}^{\mathcal{G}}(\delta_{w \cdot \lambda} \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})^{\mathrm{la}}, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]) = m_{\rho}. $$

*Proof.* By [Bre15, Proposition 4.2] and [BHS19, Lemma 5.2.3], we have, for all $M \in \mathcal{O}$

$$ \begin{aligned} \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] &\simeq \mathrm{Hom}_{G(\mathbb{Q}_{\rho})}(\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1}), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]) \\ &\simeq \mathrm{Hom}_{(\mathfrak{g}, \bar{B}_{\rho})}(M \otimes_{\mathcal{L}} C_c^{\infty}(N_{\mathcal{B}}(L), \delta_{\mathcal{R}}), \Pi[\mathfrak{m}_{\rho}]). \end{aligned} $$

Thus, using equation (16) and Lemma 7.24 we deduce that the statement is equivalent to

$$ \dim \mathrm{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda)^{\vee}, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}][\mathfrak{m}_{\delta}]) = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)^{\vee}) \otimes k(x) = m_x, $$

which is Theorem 7.13. $\square$

**Corollary 7.27.** For $\rho, \lambda, \mathcal{R}$ as before, we have an injection of $(\mathfrak{g}, B(L))$-modules

$$ (S(\lambda, w_{\rho, \mathcal{R}}) \otimes_{\mathcal{L}} C_c^{\infty}(N_{\mathcal{B}}(L), \delta_{\mathcal{R}}))^{\oplus m_{\rho}} \hookrightarrow \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}], $$

or, equivalently, an injection of $\mathcal{G}$-representations

$$ \mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})^{\oplus m_{\rho}} \subset \Pi[\mathfrak{m}_{\rho}]. $$

Moreover each map from $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M(w \cdot \lambda)', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$ to $\Pi[\mathfrak{m}_{\rho}]$ factors through the previous representation $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$.

*Proof.* The two statements about injections are equivalent and each of the $m_{\rho}$ asserted maps comes from a section of

$$ \mathrm{Hom}_{U(\mathfrak{g})}(S(\lambda, w_{\rho, \mathcal{R}}), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], $$

by the adjunction recalled in the proof of the previous corollary.

We already know, by [BHS19], that for all $w \le w_{\rho,\mathcal{R}}w_0$ we have, in previously used notations

$$ \dim \mathrm{Hom}_{U(\mathfrak{g})}(L(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] = m_x = m_{\rho}. $$

Moreover, for each $w_{\rho,\mathcal{R}}w_0 \ge w$ with $w_{\rho,\mathcal{R},\tau} \ne 1$ if $w_{\tau} = 1$, we have

$$ \begin{aligned} m_{\rho} &= \dim \mathrm{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] \\ &= \dim \mathrm{Hom}_{U(\mathfrak{g})}(L(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], \end{aligned} $$
$\mathbb{T}^S \otimes L$ associated to $\rho$. Let $\mathcal{R}$ a choice of refinement and $\delta_{\mathcal{R}}$ the associated unramified character. Denote $\lambda = (\lambda_{\tau})_{\tau} := \mathrm{HT}(\rho) - \delta_G \in X^*(T)^+$ the (dominant) algebraic character associated to $\rho$ as before, where $\mathrm{HT}(\rho) = (h_{1,\tau} > \cdots > h_{n,\tau})_{\tau \in \Sigma_F} \in X^*(\mathbb{T})$ gives the Hodge-Tate weights of $\rho$. As $\Pi[m_{\rho}] \neq 0$ and $\rho$ satisfies Hypothesis 5.9, it corresponds to a point $x \in \mathcal{X}_{\infty}(L)$. Denote $w_{\rho,\mathcal{R}} = (w_{\rho,\mathcal{R},\tau})_{\tau \in \Sigma_F}$ and $m_{\rho} := m_x \geq 1$ as in Section 7.1.

**Corollary 7.26.** For $\rho, \lambda, \mathcal{R}$ as above and all $w \le w_{\mathcal{R}}w_0$, we have

$$ \dim \mathrm{Hom}_{\mathcal{G}}(\mathrm{ind}_{\mathcal{B}}^{\mathcal{G}}(\delta_{w \cdot \lambda} \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})^{\mathrm{la}}, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]) = m_{\rho}. $$

*Proof.* By [Bre15, Proposition 4.2] and [BHS19, Lemma 5.2.3], we have, for all $M \in \mathcal{O}$

$$ \begin{aligned} \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{\mathrm{N}^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] &\simeq \mathrm{Hom}_{G(\mathbb{Q}_{\rho})}(\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1}), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]) \\ &\simeq \mathrm{Hom}_{(\mathfrak{g}, \bar{B}_{\rho})}(M \otimes_{\mathcal{L}} C_{\mathrm{c}}^{\infty}(N_{\mathcal{B}}(L), \delta_{\mathcal{R}}), \Pi[\mathfrak{m}_{\rho}]). \end{aligned} $$

Thus, using equation (16) and Lemma 7.24 we deduce that the statement is equivalent to

$$ \dim \mathrm{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda)^{\vee}, \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}][\mathfrak{m}_{\delta}]) = \dim \mathcal{M}_{\infty,x,\mathcal{R}}(M(w \cdot \lambda)^{\vee}) \otimes k(x) = m_x, $$

which is Theorem 7.13. $\square$

**Corollary 7.27.** For $\rho, \lambda, \mathcal{R}$ as before, we have an injection of $(\mathfrak{g}, B(L))$-modules

$$ (S(\lambda, w_{\rho, \mathcal{R}}) \otimes_{\mathcal{L}} C_{\mathrm{c}}^{\infty}(N_{\mathcal{B}}(L), \delta_{\mathcal{R}}))^{\oplus m_{\rho}} \hookrightarrow \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}], $$

or, equivalently, an injection of $\mathcal{G}$-representations

$$ \mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})^{\oplus m_{\rho}} \subset \Pi[\mathfrak{m}_{\rho}]. $$

Moreover each map from $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M(w \cdot \lambda)', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$ to $\Pi[\mathfrak{m}_{\rho}]$ factors through the previous representation $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$.

*Proof.* The two statements about injections are equivalent and each of the $m_{\rho}$ asserted maps comes from a section of

$$ \mathrm{Hom}_{U(\mathfrak{g})}(S(\lambda, w_{\rho, \mathcal{R}}), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{\mathrm{N}^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], $$

by the adjunction recalled in the proof of the previous corollary.

We already know, by [BHS19], that for all $w \le w_{\rho,\mathcal{R}}w_0$ we have, in previously used notations

$$ \dim \mathrm{Hom}_{U(\mathfrak{g})}(L(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{\mathrm{N}^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] = m_x = m_{\rho}. $$

Moreover, for each $w_{\rho,\mathcal{R}}w_0 \ge w$ with $w_{\rho,\mathcal{R},\tau} \ne 1$ if $w_{\tau} = 1$, we have

$$ \begin{aligned} m_{\rho} &= \dim \mathrm{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{\mathrm{N}^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}] \\ &= \dim \mathrm{Hom}_{U(\mathfrak{g})}(L(w \cdot \lambda), \Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{\mathrm{N}^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}], \end{aligned} $$
by Corollary 7.8. Thus, for those w, all maps from M(w · λ) factors through L(w · λ).

So we really need to take care of the direct factors of S(λ, w<sub>ρ,R</sub>) where a factor N(λ<sub>τ</sub>) appears. Such a factors is of the form

$$
\boxtimes_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \boxtimes_{\tau \in I_2} N(\lambda_{\tau}), \quad \Sigma = I_1 \sqcup I_2,
$$

and is a quotient of M(w · λ) where w = (w<sub>τ</sub>) with w<sub>τ</sub> = 1 if τ ∈ I<sub>2</sub>, and even of M<sub>I</sub>(w · λ)
where I = { s<sub>1,τ</sub> | τ ∈ I<sub>1</sub> such that w<sub>τ</sub> = 1 = w<sub>ρ,R,τ</sub> }.

We first prove that any map from $M_I(w \cdot \lambda)$ has to factor through $S(\lambda, w_{\rho,\mathcal{R}})$ and
more precisely through the previous factor.

Choose $\tau_0 \in I_2$ so that $w_{\rho,\mathcal{R},\tau_0} = w_{\tau_0} = 1$ and for $i = 1,2$ let $s_i^{\tau_0} \in W$ with
$(s_i^{\tau_0})_{\tau} = w_{\tau} = \text{if } \tau \neq \tau_0$, and $w_{\tau_0} = s_i$. Then

$$
M_I(s_i^{\tau_0} \cdot \lambda) \subset M_I(w \cdot \lambda).
$$

Moreover, $M_I(s_i^{\tau_0} \cdot \lambda)$ has a quotient

$$
Q_i^{\tau_0} := \boxtimes_{\tau \in I_1} L(\lambda_{\tau}) \boxtimes L(s_i \cdot \lambda_{\tau_0}) \boxtimes \boxtimes_{\tau \in I_2 : \tau \neq \tau_0} M(w_{\tau} \cdot \lambda_{\tau}),
$$

and we first prove that maps from $M_I(s_i^{\tau_0} \cdot \lambda)$ into $\Pi^{\text{la}}[\mathfrak{m}_{\rho}]$ factors through $Q_i^{\tau_0}$. This is equivalent to proving that $M_{\infty,x,\mathcal{R}}(M(s_i^{\tau_0})) \otimes k(x) \to M_{\infty,x,\mathcal{R}}(M(\lambda)) \otimes k(x)$ factors through $M_{\infty,x,\mathcal{R}}(Q_i^{\tau_0}) \otimes k(x)$. By Corollary 7.23, this is equivalent to the same question for $\mathcal{B}_x$.

Claim 7.28. If $G = G_1 \times G_2$, $\lambda = (\lambda_1, \lambda_2)$ is an algebraic weight and $\mathcal{O}_{\chi_{\lambda}} = \mathcal{O}_{\chi_{\lambda_1}} \boxtimes \mathcal{O}_{\chi_{\lambda_2}}$,
then

$$
\mathcal{B}_G(M_1 \boxtimes M_2) = \mathcal{B}_{G_1}(M_1) \boxtimes \mathcal{B}_{G_2}(M_2),
$$

where $\mathcal{B}_H$ is Bezrukavnikov's functor of Theorem 7.4 for the group $H$, under the obvious isomorphism of Steinberg varieties

$X_G = X_{G_1} \times X_{G_2}.$

Proof. This follows from the very construction of Bezrukvanikov's functor. The functors $\mathcal{B}_G$ is even defined on the larger category $D^b(\text{Perv}_{\underline{N}}(\underline{G}/\underline{N}))$ and compatible with its monoidal structure. Then, by Theorem 7.4, we know that dual Vermas are sent to the structure sheaves of the respective components by $\mathcal{B}_G$, and similarly for $\mathcal{B}_{G_i}$. In particular we have an isomorphism

$$
\mathcal{B}_G(M((w_1, w_2) \cdot (\lambda_1, \lambda_2))^\vee) \simeq \mathcal{B}_{G_1}(M(w_1 \cdot \lambda_1)^\vee) \boxtimes \mathcal{B}_{G_2}(M(w_2 \cdot \lambda_2)^\vee).
$$

As the functors $\mathcal{B}_G$ and $\mathcal{B}_{G_1} \boxtimes \mathcal{B}_{G_2}$ are both monoïdal and triangulated, using translation functors we deduce that $\mathcal{B}_G$ and $\mathcal{B}_{G_1} \boxtimes \mathcal{B}_{G_2}$ are isomorphic on projective objects. Thus by the same proof of Lemma 7.21 and Lemma 7.23, we deduce the isomorphism of functors on $\mathcal{O}_{\chi_{\lambda}}$. $\square$
by Corollary 7.8. Thus, for those w, all maps from M(w · λ) factors through L(w · λ).

So we really need to take care of the direct factors of S(λ, w<sub>ρ,R</sub>) where a factor N(λ<sub>τ</sub>) appears. Such a factors is of the form

$$
\boxtimes_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \boxtimes_{\tau \in I_2} N(\lambda_{\tau}), \quad \Sigma = I_1 \sqcup I_2,
$$

and is a quotient of M(w · λ) where w = (w<sub>τ</sub>) with w<sub>τ</sub> = 1 if τ ∈ I<sub>2</sub>, and even of M<sub>I</sub>(w · λ)
where I = { s<sub>1,τ</sub> | τ ∈ I<sub>1</sub> such that w<sub>τ</sub> = 1 = w<sub>ρ,R,τ</sub> }.

We first prove that any map from $M_I(w \cdot \lambda)$ has to factor through $S(\lambda, w_{\rho,\mathcal{R}})$ and
more precisely through the previous factor.

Choose $\tau_0 \in I_2$ so that $w_{\rho,\mathcal{R},\tau_0} = w_{\tau_0} = 1$ and for $i = 1,2$ let $s_i^{\tau_0} \in W$ with
$(s_i^{\tau_0})_{\tau} = w_{\tau} = \text{if } \tau \neq \tau_0$, and $w_{\tau_0} = s_i$. Then

$$
M_I(s_i^{\tau_0} \cdot \lambda) \subset M_I(w \cdot \lambda).
$$

Moreover, $M_I(s_i^{\tau_0} \cdot \lambda)$ has a quotient

$$
Q_i^{\tau_0} := \boxtimes_{\tau \in I_1} L(\lambda_\tau) \boxtimes L(s_i \cdot \lambda_{\tau_0}) \boxtimes \boxtimes_{\tau \in I_2 : \tau \neq \tau_0} M(w_\tau \cdot \lambda_\tau),
$$

and we first prove that maps from $M_I(s_i^{\tau_0} \cdot \lambda)$ into $\Pi^{\text{la}}[\mathfrak{m}_{\rho}]$ factors through $Q_i^{\tau_0}$. This is equivalent to proving that $M_{\infty,x,\mathcal{R}}(M(s_i^{\tau_0})) \otimes k(x) \to M_{\infty,x,\mathcal{R}}(M(\lambda)) \otimes k(x)$ factors through $M_{\infty,x,\mathcal{R}}(Q_i^{\tau_0}) \otimes k(x)$. By Corollary 7.23, this is equivalent to the same question for $\mathcal{B}_x$.

Claim 7.28. If $G = G_1 \times G_2$, $\lambda = (\lambda_1, \lambda_2)$ is an algebraic weight and $\mathcal{O}_{\chi_\lambda} = \mathcal{O}_{\chi_{\lambda_1}} \boxtimes \mathcal{O}_{\chi_{\lambda_2}}$, then

$$
\mathcal{B}_G(M_1 \boxtimes M_2) = \mathcal{B}_{G_1}(M_1) \boxtimes \mathcal{B}_{G_2}(M_2),
$$

where $\mathcal{B}_H$ is Bezrukavnikov's functor of Theorem 7.4 for the group $H$, under the obvious isomorphism of Steinberg varieties

$X_G = X_{G_1} \times X_{G_2}.$

Proof. This follows from the very construction of Bezrukvanikov's functor. The functors $\mathcal{B}_G$ is even defined on the larger category $D^b(\text{Perv}_{\underline{N}}(\underline{G}/\underline{N}))$ and compatible with its monoidal structure. Then, by Theorem 7.4, we know that dual Vermas are sent to the structure sheaves of the respective components by $\mathcal{B}_G$, and similarly for $\mathcal{B}_{G_i}$. In particular we have an isomorphism

$$
\mathcal{B}_G(M((w_1, w_2) \cdot (\lambda_1, \lambda_2))^{\vee}) \simeq \mathcal{B}_{G_1}(M(w_1 \cdot \lambda_1)^{\vee}) \boxtimes \mathcal{B}_{G_2}(M(w_2 \cdot \lambda_2)^{\vee}).
$$

As the functors $\mathcal{B}_G$ and $\mathcal{B}_{G_1} \boxtimes \mathcal{B}_{G_2}$ are both monoïdal and triangulated, using translation functors we deduce that $\mathcal{B}_G$ and $\mathcal{B}_{G_1} \boxtimes \mathcal{B}_{G_2}$ are isomorphic on projective objects. Thus by the same proof of Lemma 7.21 and Lemma 7.23, we deduce the isomorphism of functors on $\mathcal{O}_{\chi_{\lambda}}$. $\square$
So we want to prove that $\mathcal{B}_x(M_I(s_i^{\tau_0} \cdot \lambda)) \otimes k(x) \to \mathcal{B}_x(M_I(w \cdot \lambda)) \otimes k(x)$ factors through $Q_i^{\tau_0}$. By the previous claim, it suffices to show one $\tau$ at a time using $M(s_i^{\tau_0} \cdot \lambda) = M(s_i \cdot \lambda_{\tau_0}) \boxtimes M_I(w^{\tau_0} \cdot \lambda^{\tau_0})$. Thus when $\tau \neq \tau_0$ this is obvious for $\tau \in I_2$ and reduces to freeness of $X_{3,I,w_\tau}$ when $\tau \in I_1$ as before. For $\tau = \tau_0$ this amount to show that, for $k \neq \ell \in \{1,2\}$ the map

$$ \mathcal{B}_x(M(s_k s_\ell \cdot \lambda_{\tau_0})) \otimes k(x) \to \mathcal{B}_x(M(s_i \cdot \lambda_{\tau_0})) \otimes k(x), $$

vanishes. But as $\mathcal{B}_x(M(s_i \cdot \lambda_{\tau_0}))$ is free of rank 1, this is obvious. Thus, we have a factorisation through $Q_i^{\tau_0}$ for all $\tau_0, i = 1, 2$, thus

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda)) \otimes k(x) = \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x), $$

where

$$ M = \bigbox_{\tau: w_{\rho, \mathcal{R}, \tau} \neq 1} L(\lambda_{\tau}) \boxtimes \bigbox_{\tau: w_{\rho, \mathcal{R}, \tau} = 1} \underbrace{M(\lambda_{\tau}) / (M(s_1 s_2 \cdot \lambda_{\tau}) + M(s_2 s_1 \cdot \lambda_{\tau}))}_{N(\lambda_{\tau})}. $$

Now we prove the last part of the statement, i.e. any map from (the Orlik-Strauch induction of) a Verma $M(w \cdot \lambda)$ to $\Pi^{\text{la}}[\mathfrak{m}_{\rho}]$ will factor through $S(\lambda, w_{\mathcal{R}})$. Assume given a map in $\text{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda), \Pi^{\text{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}]$, and let $I_1 = \{\tau \in \Sigma | w_{\rho, \mathcal{R}, \tau} = w_{\tau} = 1\}$ and $I_2$ its complement, then by the previous argument the map factors through

$$ \bigbox_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2} N(\lambda_{\tau}). $$

As any quotient of this module is a sub-representation of $S(\lambda, w_{\rho, \mathcal{R}})$, we have that any map

$$ \mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M(w \cdot \lambda)^*, \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1}) \to \Pi[\mathfrak{m}_{\rho}] $$

factors through $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho, \mathcal{R}})^*, \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$.

We now prove the injective part. As $S(\lambda, w_{\rho, \mathcal{R}})$ is the direct sum of terms of the form, for $w \in W$,

$$ M_{w, I_1, I_2} = \bigbox_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2} N(\lambda_{\tau}), \quad \Sigma = I_1 \cup I_2, $$

where $I_2 \subset \{ \tau \in \Sigma | w_{\rho, \mathcal{R}, \tau} = w_{\tau} = 1 \}$, we first prove that the direct sum of $m_{\rho}$ copies of (the Orlik-Strauch induction of) each term $M_{w, I_1, I_2}$ injects in $\Pi^{\text{la}}[m_{\rho}]$. First remark

$$ \dim \text{Hom}_{U(\mathfrak{g})}(M_{w, I_1, I_2}, \Pi^{\text{la}}[\mathfrak{m}_{\rho}])[\mathfrak{m}_{\delta_{\mathcal{R}}}] = 2^{|I_2|} m_{\rho}, $$

by the previous factorisation of $M_I(w \cdot \rho)$ for some $I \subset I_1$ and the computation using $\mathcal{B}_x$ (and using Proposition 7.6). Now each quotient of $M_{w, I_1, I_2}$ is of the form

$$ M_{w, I_1 \cup J, I_2 \setminus J} = \bigbox_{\tau \in I_1 \cup J} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2 \setminus J} N(\lambda_{\tau}), $$
So we want to prove that $\mathcal{B}_x(M_I(s_i^{\tau_0} \cdot \lambda)) \otimes k(x) \to \mathcal{B}_x(M_I(w \cdot \lambda)) \otimes k(x)$ factors through $Q_i^{\tau_0}$. By the previous claim, it suffices to show one $\tau$ at a time using $M(s_i^{\tau_0} \cdot \lambda) = M(s_i \cdot \lambda_{\tau_0}) \boxtimes M_I(w^{\tau_0} \cdot \lambda^{\tau_0})$. Thus when $\tau \neq \tau_0$ this is obvious for $\tau \in I_2$ and reduces to freeness of $X_{3,I,w_\tau}$ when $\tau \in I_1$ as before. For $\tau = \tau_0$ this amount to show that, for $k \neq \ell \in \{1, 2\}$ the map

$$ \mathcal{B}_x(M(s_ks_\ell \cdot \lambda_{\tau_0})) \otimes k(x) \to \mathcal{B}_x(M(s_i \cdot \lambda_{\tau_0})) \otimes k(x), $$

vanishes. But as $\mathcal{B}_x(M(s_i \cdot \lambda_{\tau_0}))$ is free of rank 1, this is obvious. Thus, we have a factorisation through $Q_i^{\tau_0}$ for all $\tau_0, i = 1, 2$, thus

$$ \mathcal{M}_{\infty,x,\mathcal{R}}(M_I(w \cdot \lambda)) \otimes k(x) = \mathcal{M}_{\infty,x,\mathcal{R}}(M) \otimes k(x), $$

where

$$ M = \bigbox_{\tau:w_{\rho,\mathcal{R},\tau} \neq 1} L(\lambda_{\tau}) \boxtimes \bigbox_{\tau:w_{\rho,\mathcal{R},\tau}=1} \frac{M(\lambda_{\tau})/(M(s_1s_2 \cdot \lambda_{\tau}) + M(s_2s_1 \cdot \lambda_{\tau}))}{N(\lambda_{\tau})}. $$

Now we prove the last part of the statement, i.e. any map from (the Orlik-Strauch induction of) a Verma $M(w \cdot \lambda)$ to $\Pi^{\text{la}}[\mathfrak{m}_{\rho}]$ will factor through $S(\lambda, w_{\mathcal{R}})$. Assume given a map in $\text{Hom}_{U(\mathfrak{g})}(M(w \cdot \lambda), \Pi^{\text{la}}[\mathfrak{m}_{\rho}])^{N^0}[\mathfrak{m}_{\delta_{\mathcal{R}}}]$, and let $I_1 = \{\tau \in \Sigma | w_{\rho,\mathcal{R},\tau} = w_{\tau} = 1\}$ and $I_2$ its complement, then by the previous argument the map factors through

$$ \bigbox_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2} N(\lambda_{\tau}). $$

As any quotient of this module is a sub-representation of $S(\lambda, w_{\rho,\mathcal{R}})$, we have that any map

$$ \mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(M(w \cdot \lambda)^*, \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1}) \to \Pi[\mathfrak{m}_{\rho}] $$

factors through $\mathcal{F}_{\mathcal{B}}^{\mathcal{G}}(S(\lambda, w_{\rho,\mathcal{R}})^*, \delta_{\mathcal{R}} \delta_{\mathcal{B}}^{-1})$.

We now prove the injective part. As $S(\lambda, w_{\rho,\mathcal{R}})$ is the direct sum of terms of the form, for $w \in W$,

$$ M_{w,I_1,I_2} = \bigbox_{\tau \in I_1} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2} N(\lambda_{\tau}), \quad \Sigma = I_1 \cup I_2, $$

where $I_2 \subset \{\tau \in \Sigma | w_{\rho,\mathcal{R},\tau} = w_{\tau} = 1\}$, we first prove that the direct sum of $m_{\rho}$ copies of (the Orlik-Strauch induction of) each term $M_{w,I_1,I_2}$ injects in $\Pi^{\text{la}}[m_{\rho}]$. First remark

$$ \dim \text{Hom}_{U(\mathfrak{g})}(M_{w,I_1,I_2}, \Pi^{\text{la}}[\mathfrak{m}_{\rho}])[\mathfrak{m}_{\delta_{\mathcal{R}}}] = 2^{|I_2|} m_{\rho}, $$

by the previous factorisation of $M_I(w \cdot \rho)$ for some $I \subset I_1$ and the computation using $\mathcal{B}_x$ (and using Proposition 7.6). Now each quotient of $M_{w,I_1,I_2}$ is of the form

$$ M_{w,I_1 \cup J, I_2 \setminus J} = \bigbox_{\tau \in I_1 \cup J} L(w_{\tau} \cdot \lambda_{\tau}) \boxtimes \bigbox_{\tau \in I_2 \setminus J} N(\lambda_{\tau}), $$
for some $J \subset I_2$ (remark that if $\tau \in I_2, w_\tau = 1$), and thus

$$ \dim \operatorname{Hom}_{U(\mathfrak{g})}(M_{w, I_1 \cup J, I_2 \setminus J}, \Pi^{\text{la}}[\mathfrak{m}_{\rho}]) [\mathfrak{m}_{\delta_{\mathcal{R}}}] = 2^{|I_2|-|J|} m_{\rho}. $$

We deduce that the dimension of homomorphisms modulo those which factors through a strict quotient is

$$ 2^{|I_2|} m_{\rho} - \sum_{\emptyset \neq J \subset I_2} (-1)^{|J|+1} 2^{|I_2|-|J|} m_{\rho} = m_{\rho}. $$

In particular there are $m_\rho$ independent injective maps from $\mathcal{F}_{\bar{B}}^{G}(M'_{w,I_1,I_2}, \delta_{\mathcal{R}})$ to $\Pi^{\text{la}}$. Now when $w$ and $I_1, I_2$ varies, these objects have distincts irreducible in their socle. Thus the direct sum of all those maps

$$ \bigoplus_{w, I_1, I_2} \mathcal{F}_{\bar{B}}^{G}(M'_{w, I_1, I_2}, \delta_{\mathcal{R}})^{\oplus m_{\rho}} = \mathcal{F}_{\bar{B}}^{G}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\bar{B}}^{-1})^{\oplus m_{\rho}}, $$

injects into $\Pi[\mathfrak{m}_{\rho}]$. $\square$

*Remark 7.29.* In particular, for each $\tau$ such that $w_{\mathcal{R},\tau} = 1$ we deduce the injection of the locally analytic representation

$$ \mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}) = [\mathrm{LA}_{s_1} \oplus \mathrm{LA}_{s_2} - \mathrm{LALG}], $$

as representation of $\mathrm{GL}_3(F_\tau)$ (acting through $\tau$), where

$$ \mathrm{LALG} := L(\lambda_{\tau}) \otimes_L \mathrm{ind}_{\bar{B}_{\tau}}^{G_{\tau}} (\delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}), $$

is an irreducible locally algebraic representation which appears in cosocle, where

$$ \mathrm{LA}_s = \mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(L(s \cdot \lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}) $$

is the irreducible, non-locally algebraic, socle of the locally analytic principal series

$$ \mathrm{LA}_s \subset \mathrm{ind}_{\bar{B}_{\tau}}^{G_{\tau}} (\delta_{s \cdot \lambda_{\tau}} \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1})^{\mathrm{la}}. $$

In this case, the locally algebraic representation LALG appears with multiplicity $m_\rho$ in the socle by the main result of [BHS19], but also with multiplicity $m_\rho$ as an higher order Jordan-Hölder factor, namely, in the cosocle of the previous $\mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1})$.

# 8 Existence of very critical classical modular forms

In this section we show the existence of a classical form $f$ satisfying the hypothesis of Theorem 1.2. The main difficulty is to find a form satisfying the Taylor-Wiles hypothesis, which is moreover completely critical at $p$ (i.e. $w_{\rho f, \mathcal{R}} = 1$).
for some $J \subset I_2$ (remark that if $\tau \in I_2, w_\tau = 1$), and thus

$$ \dim \operatorname{Hom}_{U(\mathfrak{g})}(M_{w, I_1 \cup J, I_2 \setminus J}, \Pi^{\text{la}}[\mathfrak{m}_{\rho}]) [\mathfrak{m}_{\delta_{\mathcal{R}}}] = 2^{|I_2| - |J|} m_{\rho}. $$

We deduce that the dimension of homomorphisms modulo those which factors through a strict quotient is

$$ 2^{|I_2|} m_{\rho} - \sum_{\emptyset \neq J \subset I_2} (-1)^{|J|+1} 2^{|I_2|-|J|} m_{\rho} = m_{\rho}. $$

In particular there are $m_\rho$ independent injective maps from $\mathcal{F}_{\bar{B}}^{G}(M'_{w, I_1, I_2}, \delta_{\mathcal{R}})$ to $\Pi^{\text{la}}$. Now when $w$ and $I_1, I_2$ varies, these objects have distincts irreducible in their socle. Thus the direct sum of all those maps

$$ \bigoplus_{w, I_1, I_2} \mathcal{F}_{\bar{B}}^{G}(M'_{w, I_1, I_2}, \delta_{\mathcal{R}})^{\oplus m_{\rho}} = \mathcal{F}_{\bar{B}}^{G}(S(\lambda, w_{\rho, \mathcal{R}})', \delta_{\mathcal{R}} \delta_{\bar{B}}^{-1})^{\oplus m_{\rho}}, $$

injects into $\Pi[\mathfrak{m}_{\rho}]$. $\square$

*Remark 7.29.* In particular, for each $\tau$ such that $w_{\mathcal{R},\tau} = 1$ we deduce the injection of the locally analytic representation

$$ \mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}) = [\mathrm{LA}_{s_1} \oplus \mathrm{LA}_{s_2} - \mathrm{LALG}], $$

as representation of $\mathrm{GL}_3(F_\tau)$ (acting through $\tau$), where

$$ \mathrm{LALG} := L(\lambda_{\tau}) \otimes_L \mathrm{ind}_{\bar{B}_{\tau}}^{G_{\tau}} (\delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}), $$

is an irreducible locally algebraic representation which appears in cosocle, where

$$ \mathrm{LA}_s = \mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(L(s \cdot \lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1}) $$

is the irreducible, non-locally algebraic, socle of the locally analytic principal series

$$ \mathrm{LA}_s \subset \mathrm{ind}_{\bar{B}_{\tau}}^{G_{\tau}} (\delta_{s \cdot \lambda_{\tau}} \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1})^{\mathrm{la}}. $$

In this case, the locally algebraic representation LALG appears with multiplicity $m_\rho$ in the socle by the main result of [BHS19], but also with multiplicity $m_\rho$ as an higher order Jordan-Hölder factor, namely, in the cosocle of the previous $\mathcal{F}_{\bar{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})', \delta_{\mathcal{R},\tau} \delta_{\bar{B}_{\tau}}^{-1})$.

# 8 Existence of very critical classical modular forms

In this section we show the existence of a classical form $f$ satisfying the hypothesis of Theorem 1.2. The main difficulty is to find a form satisfying the Taylor-Wiles hypothesis, which is moreover completely critical at $p$ (i.e. $w_{\rho f, \mathcal{R}} = 1$).
For a finite extension $F$ of $\mathbb{Q}_p$, we denote by $\mathrm{rec}_F : F^\times \to \mathrm{Gal}_F^{\mathrm{ab}}$ the local reciprocity map sending a uniformizer of $F$ on a geometric Frobenius. If $K$ is a number field we denote by $\mathrm{Art}_K$ the Artin reciprocity map $\mathrm{Art}_K^\times/K^\times \to \mathrm{Gal}_K^{\mathrm{ab}}$ such that, for any finite place $v$ of $K$ the precomposition of $\mathrm{Art}_K$ with the inclusion $K_v^\times \hookrightarrow \mathrm{Art}_K^\times$ is $\mathrm{rec}_{K_v}$. If $\Psi$ is a character of $\mathrm{Art}_K^\times/K^\times$ and $v$ is a finite place of $K$ such that $\Psi_v$ is unramified, we write $\Psi(v)$ for the evaluation of $\Psi_v$ at a uniformizer of $F_v^\times$. First, we remark the following,

**Lemma 8.1.** Let $K/\mathbb{Q}_p$ be a finite extension and let $\rho_p : \mathrm{Gal}_K \to \mathrm{GL}_n(\overline{\mathbb{Q}}_p)$ be a crystalline representation with regular Hodge-Tate weights such that there exists a refinement $F_\bullet \subset D_{\mathrm{cris}}(\rho_p)$ which contains the Hodge filtration. We moreover assume that the eigenvalues of the linearization of the crystalline Frobenius on $D_{\mathrm{cris}}(\rho_p)$ are pairwise distinct. Then $\rho_p$ is a split sum of characters.

*Proof.* This is a simple application of weak admissibility. Up to extending scalars, we can assume that $D = D_{\mathrm{cris}}(\rho_p) = \bigoplus_\tau D_\tau$ is split, and is a filtered $\varphi$-module. We consider the linearization $\varphi_\tau^f$ of the Frobenius on $D_\tau$, where $f = [K_0 : \mathbb{Q}_p]$. We write $\mathrm{Fil}^\bullet D_\tau$ for the filtration on $D_\tau$ induced by the Hodge-filtration on $D$. The assumption is that the Hodge filtration on $D$ is $\varphi$-stable i.e. there is a full flag of $K \otimes \overline{\mathbb{Q}}_p$-modules $F_\bullet$, stable under $\varphi$, such that, for all $\tau$, if $k_1^\tau \le \dots \le k_n^\tau$ are the (opposite) $\tau$-Hodge-Tate weights (with multiplicities) then $F_{i,\tau} \subset \mathrm{Fil}^{k_{n-i+1}} D_\tau$. Denote the eigenvalues of $\varphi^f$ on $F_{i,\tau}$ by $(\varphi_1, \dots, \varphi_i)$. Thus by weak admissibility,

$$ \frac{1}{f}(v(\varphi_1) + \dots + v(\varphi_i)) \ge \sum_{\tau} \sum_{k=1}^{i} k_{n+1-k}^{\tau} $$

Now, if $G_i$ is a complementary $\varphi$-stable subspace of $F_i$ in $D$ (which exists due to the assumptions on the eigenvalues of $\varphi^f$), then we see directly that the $\tau$-Hodge-Tate weights of $G_i$ are $k_1^\tau, \dots, k_{n-i}^\tau$. Thus by weak admissibility again,

$$ \frac{1}{f}(v(\varphi_{i+1}) + \dots + v(\varphi_n)) \ge \sum_{\tau} \sum_{k=1}^{n-i} k_k^\tau. $$

But by weak admissibility of $D$, the endpoints of both polygons gives

$$ \frac{1}{f}(v(\varphi_1) + \dots + v(\varphi_n)) = \sum_{\tau} \sum_{i} k_i^\tau. $$

Thus both $G_i$ and $F_i$ are weakly admissible, thus admissible, thus $\rho_p$ splits accordingly. As this is true for all $i$, we get the Lemma. $\square$

It follows that, when $n=3$, an eigenform $f$ as in Theorem 1.2 has a split representation at $p$. In the case of modular forms, it was asked by Greenberg (see the work of Ghate and Vatsal [Gha04], [GV04]) if a cuspform whose representation is split at $p$ is necessarily a CM form. The natural generalization of this question to $\mathrm{GL}_3$ would suggest that we cannot find a form $f$ to apply Theorem 1.2 with very large image. Fortunately, we can construct an analog of a CM form for $\mathrm{GL}_3$ (more precisely for $U(3)$) which still has adequate image modulo $p$.
For a finite extension $F$ of $\mathbb{Q}_p$, we denote by $\text{rec}_F : F^\times \to \text{Gal}_F^{ab}$ the local reciprocity map sending a uniformizer of $F$ on a geometric Frobenius. If $K$ is a number field we denote by $\text{Art}_K$ the Artin reciprocity map $\text{A}_K^\times/K^\times \to \text{Gal}_K^{ab}$ such that, for any finite place $v$ of $K$ the precomposition of $\text{Art}_K$ with the inclusion $K_v^\times \hookrightarrow \text{A}_K^\times$ is $\text{rec}_{K_v}$. If $\Psi$ is a character of $\text{A}_K^\times/K^\times$ and $v$ is a finite place of $K$ such that $\Psi_v$ is unramified, we write $\Psi(v)$ for the evaluation of $\Psi_v$ at a uniformizer of $F_v^\times$. First, we remark the following,

**Lemma 8.1.** Let $K/\mathbb{Q}_p$ be a finite extension and let $\rho_p : \text{Gal}_K \to \text{GL}_n(\overline{\mathbb{Q}}_p)$ be a crystalline representation with regular Hodge-Tate weights such that there exists a refinement $F_\bullet \subset D_{\text{cris}}(\rho_p)$ which contains the Hodge filtration. We moreover assume that the eigenvalues of the linearization of the crystalline Frobenius on $D_{\text{cris}}(\rho_p)$ are pairwise distinct. Then $\rho_p$ is a split sum of characters.

*Proof.* This is a simple application of weak admissibility. Up to extending scalars, we can assume that $D = D_{\text{cris}}(\rho_p) = \bigoplus_{\tau} D_{\tau}$ is split, and is a filtered $\varphi$-module. We consider the linearization $\varphi_{\tau}^f$ of the Frobenius on $D_{\tau}$, where $f = [K_0 : \mathbb{Q}_p]$. We write $\text{Fil} \cdot D_{\tau}$ for the filtration on $D_{\tau}$ induced by the Hodge-filtration on $D$. The assumption is that the Hodge filtration on $D$ is $\varphi$-stable i.e. there is a full flag of $K \otimes \overline{\mathbb{Q}}_p$-modules $F_\bullet$, stable under $\varphi$, such that, for all $\tau$, if $k_1^\tau \le \dots \le k_n^\tau$ are the (opposite) $\tau$-Hodge-Tate weights (with multiplicities) then $F_{i,\tau} \subset \text{Fil}^{k_{n-i+1}} D_{\tau}$. Denote the eigenvalues of $\varphi^f$ on $F_{i,\tau}$ by $(\varphi_1, \dots, \varphi_i)$. Thus by weak admissibility,

$$ \frac{1}{f}(v(\varphi_1) + \dots + v(\varphi_i)) \ge \sum_{\tau} \sum_{k=1}^{i} k_{n+1-k}^{\tau} $$

Now, if $G_i$ is a complementary $\varphi$-stable subspace of $F_i$ in $D$ (which exists due to the assumptions on the eigenvalues of $\varphi^f$), then we see directly that the $\tau$-Hodge-Tate weights of $G_i$ are $k_1^\tau, \dots, k_{n-i}^\tau$. Thus by weak admissibility again,

$$ \frac{1}{f}(v(\varphi_{i+1}) + \dots + v(\varphi_n)) \ge \sum_{\tau} \sum_{k=1}^{n-i} k_k^{\tau}. $$

But by weak admissibility of $D$, the endpoints of both polygons gives

$$ \frac{1}{f}(v(\varphi_1) + \dots + v(\varphi_n)) = \sum_{\tau} \sum_{i} k_i^{\tau}. $$

Thus both $G_i$ and $F_i$ are weakly admissible, thus admissible, thus $\rho_p$ splits accordingly. As this is true for all $i$, we get the Lemma. $\square$

It follows that, when $n = 3$, an eigenform $f$ as in Theorem 1.2 has a split representation at $p$. In the case of modular forms, it was asked by Greenberg (see the work of Ghate and Vatsal [Gha04], [GV04]) if a cuspform whose representation is split at $p$ is necessarily a CM form. The natural generalization of this question to $\text{GL}_3$ would suggest that we cannot find a form $f$ to apply Theorem 1.2 with very large image. Fortunately, we can construct an analog of a CM form for $\text{GL}_3$ (more precisely for $U(3)$) which still has adequate image modulo $p$.
## 8.1 Choosing a Hecke character

Let $E$ be a CM field with totally real subfield $E^+ = F$ and let $F'$ be a totally real field disjoint from $E$, such that $F'/\mathbb{Q}$ is Galois and such that $[F' : \mathbb{Q}] = 3$. Set $K = EF'$. This is a CM field. We moreover assume that all the ramified primes of $K/E$ lie above split primes in $E/E^+$. Choose two distinct primes $p$ and $\ell$ such that $\ell$ is totally split in $K = EF'$ and primes above $p$ in $E^+ = F$ are totally split in $K$. Moreover assume $p > 8(= 2(n+1))$ when $n=3$) and $\zeta_p \notin E$.

*Example 8.2.* 1. The easiest choice is $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i\sqrt{3})$ so that $7$ is split in $E$. For this $F'$, we can also choose $E = \mathbb{Q}(i, \sqrt{3})$, with maximal totally real subfield $E^+ = \mathbb{Q}(\sqrt{3})$ so that $E/E^+$ is unramified everywhere.

2. The second easiest choice for $F'$ is $F' = \mathbb{Q}(\zeta_9)^+$. In this case we can choose $E = \mathbb{Q}(i\sqrt{5})$.

3. If $E = \mathbb{Q}(i)$, we can choose $F' = \mathbb{Q}(\alpha)$ with $\alpha$ a root of $X^3 - X^2 - 4X - 1$, which has discriminant $13^2$.

4. If $F' = \mathbb{Q}(\alpha)$ and $E = \mathbb{Q}(i)$, we can choose any prime $p > 8$, $\ell$ congruent to $1, 5, 21, 25 \pmod{52}$, like $5, 53, 73, \dots$. In particular in that case we better should exclude $p = 13$ as in the early version [Bel10] (who knows?).

5. If $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i\sqrt{3})$, we can choose any prime congruent to $1, 13 \pmod{21}$ like $13, 43, 97, \dots$.

6. If $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i, \sqrt{3})$, we can take any prime $\ell \equiv 1, 13 \pmod{84}$ like $13, 97, 169, \dots$ and $p \equiv 1, 13 \pmod{21}$ like $13, 43, 97, \dots$.

7. If we really want to use $p = 13$ and that $p = 13$ is inert in $F = E^+$, and if we want moreover $E/E^+$ to be unramified everywhere, we can choose $E = \mathbb{Q}(i, \sqrt{7})$ with $F' = \mathbb{Q}(\beta) \subset \mathbb{Q}(\zeta_{43})$ as $43$ is split in $\mathbb{Q}(i, \sqrt{7})/\mathbb{Q}(\sqrt{7})$, with $\beta$ a root of $X^3 - X^2 - 14X - 8$.

In the following we say that a weight $\underline{k} \in \mathbb{Z}^{\text{Hom}(K,\mathbb{C})}$ is *very regular* if, for $\tau_1 \neq \tau_2$ in $\text{Hom}(K, \mathbb{C})$, we have $|k_{\tau_1} - k_{\tau_2}| \ge 2$.

Let $\Psi$ be an algebraic Hecke character of $\mathbb{A}_K^\times$ with algebraic very regular weight $\underline{k} = (k_v)_{v|\infty}$, such that $\Psi^c = \Psi^\vee$ and such that $\Psi$ is unramified both at $p$ and $\ell$. Choose an isomorphism $\iota : \mathbb{C} \simeq \overline{\mathbb{Q}}_p$. We moreover assume that

($\Psi, p$) if $\mathfrak{p}|p$ in $E$, we have $\Psi(v)\Psi(v')^{-1} \notin \{1, p\}$ for $v \neq v'$ places of $K$ dividing $\mathfrak{p}$.

($\Psi, \ell$) There exists $\lambda|\ell$ in $E$, and $\lambda'|\lambda$ in $E(\zeta_p)$, such that for all $v_1 \neq v_2$ places of $K$ dividing $\lambda$, if $v'_1, v'_2$ are the corresponding places above $\lambda'$ in $K(\zeta_p)$, $\iota(\Psi(v'_1)) \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}} \neq \iota(\Psi(v'_2)) \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}}$.

Consider moreover the following hypothesis on $\Psi$:
## 8.1 Choosing a Hecke character

Let $E$ be a CM field with totally real subfield $E^+ = F$ and let $F'$ be a totally real field disjoint from $E$, such that $F'/\mathbb{Q}$ is Galois and such that $[F' : \mathbb{Q}] = 3$. Set $K = EF'$. This is a CM field. We moreover assume that all the ramified primes of $K/E$ lie above split primes in $E/E^+$. Choose two distinct primes $p$ and $\ell$ such that $\ell$ is totally split in $K = EF'$ and primes above $p$ in $E^+ = F$ are totally split in $K$. Moreover assume $p > 8(= 2(n+1))$ when $n=3$) and $\zeta_p \notin E$.

*Example 8.2.* 1. The easiest choice is $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i\sqrt{3})$ so that $7$ is split in $E$. For this $F'$, we can also choose $E = \mathbb{Q}(i, \sqrt{3})$, with maximal totally real subfield $E^+ = \mathbb{Q}(\sqrt{3})$ so that $E/E^+$ is unramified everywhere.

2. The second easiest choice for $F'$ is $F' = \mathbb{Q}(\zeta_9)^+$. In this case we can choose $E = \mathbb{Q}(i\sqrt{5})$.

3. If $E = \mathbb{Q}(i)$, we can choose $F' = \mathbb{Q}(\alpha)$ with $\alpha$ a root of $X^3 - X^2 - 4X - 1$, which has discriminant $13^2$.

4. If $F' = \mathbb{Q}(\alpha)$ and $E = \mathbb{Q}(i)$, we can choose any prime $p > 8$, $\ell$ congruent to $1, 5, 21, 25 \pmod{52}$, like $5, 53, 73, \dots$. In particular in that case we better should exclude $p = 13$ as in the early version [Bel10] (who knows?).

5. If $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i\sqrt{3})$, we can choose any prime congruent to $1, 13 \pmod{21}$ like $13, 43, 97, \dots$.

6. If $F' = \mathbb{Q}(\zeta_7)^+$ and $E = \mathbb{Q}(i, \sqrt{3})$, we can take any prime $\ell \equiv 1, 13 \pmod{84}$ like $13, 97, 169, \dots$ and $p \equiv 1, 13 \pmod{21}$ like $13, 43, 97, \dots$.

7. If we really want to use $p = 13$ and that $p = 13$ is inert in $F = E^+$, and if we want moreover $E/E^+$ to be unramified everywhere, we can choose $E = \mathbb{Q}(i, \sqrt{7})$ with $F' = \mathbb{Q}(\beta) \subset \mathbb{Q}(\zeta_{43})$ as $43$ is split in $\mathbb{Q}(i, \sqrt{7})/\mathbb{Q}(\sqrt{7})$, with $\beta$ a root of $X^3 - X^2 - 14X - 8$.

In the following we say that a weight $\underline{k} \in \mathbb{Z}^{\text{Hom}(K,\mathbb{C})}$ is *very regular* if, for $\tau_1 \neq \tau_2$ in $\text{Hom}(K, \mathbb{C})$, we have $|k_{\tau_1} - k_{\tau_2}| \ge 2$.

Let $\Psi$ be an algebraic Hecke character of $\mathbb{A}_K^\times$ with algebraic very regular weight $\underline{k} = (k_v)_{v|\infty}$, such that $\Psi^c = \Psi^\vee$ and such that $\Psi$ is unramified both at $p$ and $\ell$. Choose an isomorphism $\iota : \mathbb{C} \simeq \overline{\mathbb{Q}}_p$. We moreover assume that

($\Psi, p$) if $\mathfrak{p}|p$ in $E$, we have $\Psi(v)\Psi(v')^{-1} \notin \{1, p\}$ for $v \neq v'$ places of $K$ dividing $\mathfrak{p}$.

($\Psi, \ell$) There exists $\lambda|\ell$ in $E$, and $\lambda'|\lambda$ in $E(\zeta_p)$, such that for all $v_1 \neq v_2$ places of $K$ dividing $\lambda$, if $v_1', v_2'$ are the corresponding places above $\lambda'$ in $K(\zeta_p)$, $\iota(\Psi(v_1')) \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}} \neq \iota(\Psi(v_2')) \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}}$.

Consider moreover the following hypothesis on $\Psi$:
(Ψ, Ram) If $v$ is a place of $K$ such that $\Psi$ is ramified at $v$, then $v$ divides a prime which is totally split in $K/\mathbb{Q}$.

Let $\Psi_p : \mathbb{A}_K^\times \to \overline{\mathbb{Q}}_p^\times$ be the $p$-adic realization of $\Psi$ and $\iota$, and $\psi_p : \mathrm{Gal}_K \to \overline{\mathbb{Q}}_p^\times$ such that $\psi_p = \Psi_p \circ \mathrm{Art}_K$. It is a Galois representation satisfying $\psi_p^\vee = \psi_p^c$.

## 8.2 Galois induction

**Definition 8.3.** We denote by $\rho$ the induced Galois representation

$$ \rho = \mathrm{ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p = \{f : \mathrm{Gal}_E \to \overline{\mathbb{Z}}_p^{-\times} | f(gk) = \psi_p^{-1}(k)f(g) \forall g \in \mathrm{Gal}_E, k \in \mathrm{Gal}_K\}, $$

where the action of $g \in \mathrm{Gal}_E$ is given by $(g \cdot f)(x) = f(g^{-1}x)$.

Then $\rho$ is a three dimensional Galois representation since $[K : E]$ is Galois of degree 3. We claim the following

**Lemma 8.4.** 1. The representation $\bar{\rho} := \rho \otimes \overline{\mathbb{F}}_p$ is absolutely irreducible, in particular $\rho$ is absolutely irreducible.

2. The representation $\bar{\rho}(\mathrm{Gal}_{E(\zeta_p)})$ is adequate.

3. The representation $\rho$ is polarized, i.e. $\rho^c \simeq \rho^\vee$.

4. The representation $\rho_{\mathrm{Gal}_{E_v}}$ is split, $\varphi$-generic, Hodge-Tate regular for any $v|p$ in $E$,

5. If $v$ is a place of $E$ such that $\rho$ is ramified at $v$, then $\mathrm{Hom}_{\mathrm{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$.

*Proof.* We will actually prove that $\bar{\rho}(\mathrm{Gal}_{E(\zeta_p)})$ acts absolutely irreducibly, which will imply point 1 and point 2 will follow by [Tho12] Lemma 2.4. To prove point 1, remark that if we denote by $\sigma \in \mathrm{Gal}_E$ a lift of a generator of the Galois group $\mathrm{Gal}(K/E) = < \bar{\sigma} > = \mathbb{Z}/3\mathbb{Z}$, then $\rho$ has a basis given by $f, \sigma \cdot f, \sigma^2 \cdot f$, where $f$ is the function

$$ f : \mathrm{Gal}_E = \mathrm{Gal}_K \coprod \sigma \mathrm{Gal}_K \coprod \sigma^2 \mathrm{Gal}_K \to \overline{\mathbb{Z}}_p^\times, k \in \mathrm{Gal}_K \mapsto \psi_p^{-1}(k), \sigma k, \sigma^2 k \mapsto 0. $$

Then $\sigma^3 \cdot f = \psi_p(\sigma^3)f$. Thus, after restricting to $\mathrm{Gal}_K$, there is an isomorphism $\rho|_{\mathrm{Gal}_K} \simeq \psi_p \oplus \psi_p^\sigma \oplus \psi_p^{\sigma^2}$, where $\psi_p^\sigma = \psi_p(\sigma^{-1} \cdot \sigma)$. We reduce mod $p$, where we have a similar reduction after restricting to $\mathrm{Gal}_K$. Because of the hypothesis $(\Psi, \ell)$ away from $p$, we have that $\bar{\rho}_{\mathrm{Gal}_{E(\zeta_p)\lambda'}}$, for $\lambda' | \ell$, is the sum of three distinct characters. Moreover the group $\mathrm{Gal}_E$ acts transitively on these three eigenspaces. Therefore this representation is absolutely irreducible. To prove point 3, we compute $\rho^\vee$. By [CR81, Prop. 10.28], we have an isomorphism

$$ \rho^\vee \simeq \mathrm{Ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p^{-1} = \mathrm{Ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p^c \simeq \rho^c. $$
(Ψ, Ram) If $v$ is a place of $K$ such that $\Psi$ is ramified at $v$, then $v$ divides a prime which is totally split in $K/\mathbb{Q}$.

Let $\Psi_p : \mathbb{A}_K^\times \to \overline{\mathbb{Q}}_p^\times$ be the $p$-adic realization of $\Psi$ and $\iota$, and $\psi_p : \mathrm{Gal}_K \to \overline{\mathbb{Q}}_p^\times$ such that $\psi_p = \Psi_p \circ \mathrm{Art}_K$. It is a Galois representation satisfying $\psi_p^\vee = \psi_p^c$.

## 8.2 Galois induction

**Definition 8.3.** We denote by $\rho$ the induced Galois representation

$$ \rho = \mathrm{ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p = \{f : \mathrm{Gal}_E \to \overline{\mathbb{Z}}_p^{-\times} | f(gk) = \psi_p^{-1}(k)f(g) \forall g \in \mathrm{Gal}_E, k \in \mathrm{Gal}_K\}, $$

where the action of $g \in \mathrm{Gal}_E$ is given by $(g \cdot f)(x) = f(g^{-1}x)$.

Then $\rho$ is a three dimensional Galois representation since $[K : E]$ is Galois of degree 3. We claim the following

**Lemma 8.4.** 1. The representation $\bar{\rho} := \rho \otimes \overline{\mathbb{F}}_p$ is absolutely irreducible, in particular $\rho$ is absolutely irreducible.

2. The representation $\bar{\rho}(\mathrm{Gal}_{E(\zeta_p)})$ is adequate.

3. The representation $\rho$ is polarized, i.e. $\rho^c \simeq \rho^\vee$.

4. The representation $\rho_{\mathrm{Gal}_{E_v}}$ is split, $\varphi$-generic, Hodge-Tate regular for any $v|p$ in $E$,

5. If $v$ is a place of $E$ such that $\rho$ is ramified at $v$, then $\mathrm{Hom}_{\mathrm{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$.

*Proof.* We will actually prove that $\bar{\rho}(\mathrm{Gal}_{E(\zeta_p)})$ acts absolutely irreducibly, which will imply point 1 and point 2 will follow by [Tho12] Lemma 2.4. To prove point 1, remark that if we denote by $\sigma \in \mathrm{Gal}_E$ a lift of a generator of the Galois group $\mathrm{Gal}(K/E) = < \bar{\sigma} > = \mathbb{Z}/3\mathbb{Z}$, then $\rho$ has a basis given by $f, \sigma \cdot f, \sigma^2 \cdot f$, where $f$ is the function

$$ f : \mathrm{Gal}_E = \mathrm{Gal}_K \coprod \sigma \mathrm{Gal}_K \coprod \sigma^2 \mathrm{Gal}_K \to \overline{\mathbb{Z}}_p^\times, k \in \mathrm{Gal}_K \mapsto \psi_p^{-1}(k), \sigma k, \sigma^2 k \mapsto 0. $$

Then $\sigma^3 \cdot f = \psi_p(\sigma^3)f$. Thus, after restricting to $\mathrm{Gal}_K$, there is an isomorphism $\rho|_{\mathrm{Gal}_K} \simeq \psi_p \oplus \psi_p^\sigma \oplus \psi_p^{\sigma^2}$, where $\psi_p^\sigma = \psi_p(\sigma^{-1} \cdot \sigma)$. We reduce mod $p$, where we have a similar reduction after restricting to $\mathrm{Gal}_K$. Because of the hypothesis $(\Psi, \ell)$ away from $p$, we have that $\bar{\rho}_{\mathrm{Gal}_{E(\zeta_p)\lambda'}}$, for $\lambda' | \ell$, is the sum of three distinct characters. Moreover the group $\mathrm{Gal}_E$ acts transitively on these three eigenspaces. Therefore this representation is absolutely irreducible. To prove point 3, we compute $\rho^\vee$. By [CR81, Prop. 10.28], we have an isomorphism

$$ \rho^\vee \simeq \mathrm{Ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p^{-1} = \mathrm{Ind}_{\mathrm{Gal}_K}^{\mathrm{Gal}_E} \psi_p^c \simeq \rho^c. $$
Let us prove 4. As $p$ is totally split in $K/F$, we have for $v|p$ in $E$, $\text{Gal}_{E_v} \subset \text{Gal}_K$ so that $\rho_{|\text{Gal}_{E_v}} \simeq \psi_{p,v} \oplus \psi_{p,v}^{\sigma} \oplus \psi_{p,v}^{\sigma^2}$. As the group $\text{Gal}_E$ acts transitively on the three places of $K$ over $v$, we have $\rho_{|\text{Gal}_{E_v}} \simeq \bigoplus_{v'|v} \psi_{p,v'}$. Therefore $\rho_{|\text{Gal}_{E_v}}$ is crystalline and the eigenvalues of the Frobenius endomorphism of $D_{\text{cris}}(\rho_{|\text{Gal}_{E_v}})$ are the $\Psi(v')$ for $v'|v$ in $K$. It follows from hypothesis $(\Psi, p)$ that $\rho_{|\text{Gal}_{E_v}}$ is $\varphi$-generic. Moreover the Hodge–Tate weights of $\rho_{|\text{Gal}_{E_v}}$ corresponds to the algebraic (infinitesimal) weight of $\Psi$, which was assumed regular so that $\rho_{\text{Gal}_{E_v}}$ is Hodge–Tate regular.

Finally we prove 5. Let $v$ be a place of $E$ such that $\rho_v$ is ramified. Then either $v$ is ramified in $K/E$ or $\Psi_v$ is ramified. Assume in a first time that $\Psi_v$ is ramified. Then $(\Psi, \text{Ram})$ implies that $v$ divides a prime of $\mathbb{Q}$ which is totally split in $K$. In particular, $v$ is split in $K/E$. As above, we have $\rho_v \simeq \bigoplus_{v'|v} \psi_{p,v'}$ with $\psi_{p,v'} = \Psi_{v'} \circ \text{rec}_{K_{v'}}^{-1}$ as $v' \nmid p$. Therefore it follows from Lemma 8.5 below that $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$.

Now assume that $v$ is non split in $K$. As $K/E$ is Galois there is a unique place $w$ of $K$ over $v$ and $\rho_v \simeq \text{Ind}_{\text{Gal}_{K_w}}^{\text{Gal}_{E_v}} \psi_{p,w}$. By Frobenius reciprocity, we have

$$ \text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) \simeq \text{Hom}_{\text{Gal}_{K_w}}(\psi_{p,w} \oplus \psi_{p,w}^{\sigma} \oplus \psi_{p,w}^{\sigma^2}, \psi_{p,w} \chi_{\text{cyc}|K_w}). $$

Assume that $\psi_{p,w} = \psi_{p,w}^{\sigma} \chi_{\text{cyc}|K_w}$. As $\chi_{\text{cyc}|K_w} = \chi_{\text{cyc}|K_w}^{\sigma}$, we deduce $\psi_{p,w}^{\sigma} = \psi_{p,w}^{\sigma^2} \chi_{\text{cyc}|K_w}$ and $\psi_{p,w}^{\sigma^2} = \psi_{p,w}^{\sigma^3} \chi_{\text{cyc}|K_w} = \psi_{p,w} \chi_{\text{cyc}|K_w}$ so that $\psi_{p,w} = \psi_{p,w} \chi_{\text{cyc}|K_w}^3$ which is false. We prove similarly than $\psi_{p,w} \neq \psi_{p,w}^{\sigma^2} \chi_{\text{cyc}|K_w}$ and deduce $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. If $\Psi_w \neq \Psi_w \circ \sigma$, then the characters $\Psi_w, \Psi_w \circ \sigma, \Psi \circ \sigma^2$ are pairwise distinct and $\rho_v$ is irreducible so that $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. If $\Psi_w = \Psi_w \circ \sigma$, then $\rho_v$ is not irreducible, but clearly $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$ (as $\psi_{p|\text{Gal}_{E_v}} \neq \psi_{p|\text{Gal}_{E_v}}(1)$). $\square$

**Lemma 8.5.** Let $\Psi : \mathbb{A}_K^\times/K^\times$ be an algebraic Hecke character of very regular weight $k$. Then, if $\ell$ is a prime number which is totally split in $K$, then $\Psi_v \neq \Psi_w|_w$ for all places $v, w$ of $K$ dividing $\ell$.

*Proof.* Let $\Psi$ and $\ell$ be as in the statement. Fix $\iota : \mathbb{C} \simeq \bar{\mathbb{Q}}_\ell$ and let $|\cdot|_\ell$ be the unique absolute value on $\bar{\mathbb{Q}}_\ell$ extending the one on $\mathbb{Q}_\ell$. Let $\Psi_\iota$ be the continuous character $\mathbb{A}_K^\times/K^\times K_\infty^\times \to \bar{\mathbb{Q}}_\ell^\times$ defined by

$$ \Psi_{\iota,w}(x_w) = \begin{cases} \Psi_w(x_w) & \text{if } w \nmid \ell, w \nmid \infty \\ 1 & \text{if } w \nmid \infty \\ \iota(\Psi_w(w_w)) \prod_{\tau \in \text{Hom}(K_w, \bar{\mathbb{Q}}_\ell), \tau|w} \tau(x_w)^{k_\iota - 1} & \text{if } w|\ell, \end{cases} $$

where $\tau|w$ means that $|\cdot|_\ell \circ \tau$ extends the absolute value given by $w$ on $K$, and $(k_\sigma)_{\sigma \in \text{Hom}(K, \mathbb{C})}$ is the weight of $\Psi$. As the group $\mathbb{A}_K^\times/K^\times K_\infty^\times$ is compact, we have $\text{Im}(\Psi_\iota) \subset \bar{\mathbb{Z}}_\ell^\times$. As $\ell$ is totally split $\iota$ induces a bijection between $\{v|\ell\}$ and $\text{Hom}(K, \mathbb{C})$. Let $v$ be a place of $K$ dividing $\ell$ corresponding to $\tau$ (i.e. $|\cdot|_\ell \circ \iota^{-1}\tau$ extends $|\cdot|_v$) and denote $k_v := k_{\iota^{-1}\tau}$. We have

$$ |\iota \Psi_v(\ell) \tau(\ell)^{k_v}|_\ell = 1 $$

so that $|\iota(\Psi_v(\ell))| = l^{k_v}$. As $\ell$ is a uniformizer of $K_v$, for any $v|\ell$, the result follows. $\square$
Let us prove 4. As $p$ is totally split in $K/F$, we have for $v|p$ in $E$, $\text{Gal}_{E_v} \subset \text{Gal}_K$ so that $\rho_{|\text{Gal}_{E_v}} \simeq \psi_{p,v} \oplus \psi_{p,v}^{\sigma} \oplus \psi_{p,v}^{\sigma^2}$. As the group $\text{Gal}_E$ acts transitively on the three places of $K$ over $v$, we have $\rho_{|\text{Gal}_{E_v}} \simeq \bigoplus_{v'|v} \psi_{p,v'}$. Therefore $\rho_{|\text{Gal}_{E_v}}$ is crystalline and the eigenvalues of the Frobenius endomorphism of $D_{\text{cris}}(\rho_{|\text{Gal}_{E_v}})$ are the $\Psi(v')$ for $v'|v$ in $K$. It follows from hypothesis $(\Psi, p)$ that $\rho_{|\text{Gal}_{E_v}}$ is $\varphi$-generic. Moreover the Hodge-Tate weights of $\rho_{|\text{Gal}_{E_v}}$ corresponds to the algebraic (infinitesimal) weight of $\Psi$, which was assumed regular so that $\rho_{\text{Gal}_{E_v}}$ is Hodge-Tate regular.

Finally we prove 5. Let $v$ be a place of $E$ such that $\rho_v$ is ramified. Then either $v$ is ramified in $K/E$ or $\Psi_v$ is ramified. Assume in a first time that $\Psi_v$ is ramified. Then $(\Psi, \text{Ram})$ implies that $v$ divides a prime of $\mathbb{Q}$ which is totally split in $K$. In particular, $v$ is split in $K/E$. As above, we have $\rho_v \simeq \bigoplus_{v'|v} \psi_{p,v'}$ with $\psi_{p,v'} = \Psi_{v'} \circ \text{rec}_{K_{v'}}^{-1}$ as $v' \nmid p$. Therefore it follows from Lemma 8.5 below that $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$.

Now assume that $v$ is non split in $K$. As $K/E$ is Galois there is a unique place $w$ of $K$ over $v$ and $\rho_v \simeq \text{Ind}_{\text{Gal}_{K_w}}^{\text{Gal}_{E_v}} \psi_{p,w}$. By Frobenius reciprocity, we have

$$ \text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) \simeq \text{Hom}_{\text{Gal}_{K_w}}(\psi_{p,w} \oplus \psi_{p,w}^{\sigma} \oplus \psi_{p,w}^{\sigma^2}, \psi_{p,w} \chi_{\text{cyc}|K_w}). $$

Assume that $\psi_{p,w} = \psi_{p,w}^{\sigma} \chi_{\text{cyc}|K_w}$. As $\chi_{\text{cyc}|K_w} = \chi_{\text{cyc}|K_w}^{\sigma}$, we deduce $\psi_{p,w}^{\sigma} = \psi_{p,w}^{\sigma^2} \chi_{\text{cyc}|K_w}$ and $\psi_{p,w}^{\sigma^2} = \psi_{p,w}^{\sigma^3} \chi_{\text{cyc}|K_w} = \psi_{p,w} \chi_{\text{cyc}|K_w}$ so that $\psi_{p,w} = \psi_{p,w} \chi_{\text{cyc}|K_w}^3$ which is false. We prove similarly than $\psi_{p,w} \neq \psi_{p,w}^{\sigma^2} \chi_{\text{cyc}|K_w}$ and deduce $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. If $\Psi_w \neq \Psi_w \circ \sigma$, then the characters $\Psi_w, \Psi_w \circ \sigma, \Psi \circ \sigma^2$ are pairwise distinct and $\rho_v$ is irreducible so that $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. If $\Psi_w = \Psi_w \circ \sigma$, then $\rho_v$ is not irreducible, but clearly $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$ (as $\psi_{p|\text{Gal}_{E_v}} \neq \psi_{p|\text{Gal}_{E_v}}(1)$). $\square$

**Lemma 8.5.** Let $\Psi : \mathbb{A}_K^\times/K^\times$ be an algebraic Hecke character of very regular weight $k$. Then, if $\ell$ is a prime number which is totally split in $K$, then $\Psi_v \neq \Psi_w|_w$ for all places $v, w$ of $K$ dividing $\ell$.

*Proof.* Let $\Psi$ and $\ell$ be as in the statement. Fix $\iota : \mathbb{C} \simeq \bar{\mathbb{Q}}_\ell$ and let $|\cdot|_\ell$ be the unique absolute value on $\bar{\mathbb{Q}}_\ell$ extending the one on $\mathbb{Q}_\ell$. Let $\Psi_\iota$ be the continuous character $\mathbb{A}_K^\times/K^\times K_\infty^\times \to \bar{\mathbb{Q}}_\ell^\times$ defined by

$$ \Psi_{\iota, w}(x_w) = \begin{cases} \Psi_w(x_w) & \text{if } w \nmid \ell, w \nmid \infty \\ 1 & \text{if } w \nmid \infty \\ \iota(\Psi_w(w_w)) \prod_{\tau \in \text{Hom}(K_w, \bar{\mathbb{Q}}_\ell), \tau|w} \tau(x_w)^{k_\iota - 1} & \text{if } w|\ell, \end{cases} $$

where $\tau|w$ means that $|\cdot|_\ell \circ \tau$ extends the absolute value given by $w$ on $K$, and $(k_\sigma)_{\sigma \in \text{Hom}(K, \mathbb{C})}$ is the weight of $\Psi$. As the group $\mathbb{A}_K^\times/K^\times K_\infty^\times$ is compact, we have $\text{Im}(\Psi_\iota) \subset \bar{\mathbb{Z}}_\ell^\times$. As $\ell$ is totally split $\iota$ induces a bijection between $\{v|\ell\}$ and $\text{Hom}(K, \mathbb{C})$. Let $v$ be a place of $K$ dividing $\ell$ corresponding to $\tau$ (i.e. $|\cdot|_\ell \circ \iota^{-1}\tau$ extends $|\cdot|_v$) and denote $k_v := k_{\iota^{-1}\tau}$. We have

$$ |\iota \Psi_v(\ell) \tau(\ell)^{k_v}|_\ell = 1 $$

so that $|\iota(\Psi_v(\ell))| = l^{k_v}$. As $\ell$ is a uniformizer of $K_v$, for any $v|\ell$, the result follows. $\square$
## 8.3 Construction of an explicit set of Hecke characters

In this subsection we explain one way to find a $\Psi$ as before, satisfying hypothesis $(\Psi, p)$, $(\Psi, \ell)$, $(\Psi, \text{Ram})$. Fix $E$ a CM extension, with $E^+ = F$ its maximal totally real subfield, so that $[E : E^+] = 2$. Fix also $F'$ disjoint from $E$, a totally real degree 3 Galois extension of $\mathbb{Q}$. Choose $p, \ell$ two primes with are totally split in $K := EF'$ such that $p > 8$. The following Lemma is a more precise version of [CHT08, Lem. 4.1.1].

**Lemma 8.6.** Let $F$ be a number field. Let $S$ be a finite set of places of $F$. Let $\chi_S$ be an unramified continuous character $F_S^\times := \prod_{v \in S} F_v^\times \to \mathbb{C}^\times$ of finite order. Let $T$ be a set of finite places of $F$, disjoint from $S$ and of Dirichlet density 1. Then there exists a continuous character $\chi : \mathbb{A}_F^\times / F^\times \to \mathbb{C}^\times$ of finite order such that $\chi|_{F_S^\times} = \chi_S$ and the ramification places of $\chi$ are in $T$.

*Proof.* Let $U^S$ be the product of the $O_{F_v}^\times$ for $v \notin S$. Then $F^\times \cap U^S$ is a finitely generated subgroup of $F^\times$. Let us write $m$ for the order of the finite cyclic group $\chi_S(F^\times \cap U^S)$. It follows from the proof of Theorem 1 in [Che51] that we can find finitely many places $w_1, \dots, w_r$ in $T$ such that the subgroup of $F^\times \cap U^S$ congruent to 1 modulo $\mathfrak{p}_{w_1}, \dots, \mathfrak{p}_{w_r}$ is contained in $(F^\times \cap U^S)^m$. We conclude as in the proof of [CHT08, Lem. 4.1.1] choosing for $U$ the product of the $U_v$ for $v$ not in $S$ nor $\{w_1, \dots, w_r\}$ and a small enough subgroup at $w_1, \dots, w_r$. $\square$

**Lemma 8.7.** Let $K$ be an (imaginary) CM field with totally real subfield $K^+$ and complex conjugacy $c$. Denote $\psi : \mathbb{A}_K^\times / K^\times \to \mathbb{C}^\times$ be a continuous character. Assume that there exists a finite set $S$ of places of $K$ which are split in $K/K^+$ and such that $\psi_v^{-1} = \psi_{cv}$ for $v \in S$. Moreover, assume that $S$ contains the Archimedean places. Let $T$ be a finite set of places of $K$ that contains $S$ and is stable under $c$, such that $\psi$ is unramified outside of $T$. Then there exists a Hecke character $\tilde{\psi} : \mathbb{A}_K^\times / K^\times \to \mathbb{C}^\times$ such that $\tilde{\psi}^{-1} = \tilde{\psi}^c$ and $\tilde{\psi}_v = \psi_v$ for $v \in S$ and such that $\tilde{\psi}_v$ is unramified outside of $T$.

*Proof.* Let $\theta = \psi \circ N_{K/K^+}$. As $S$ contains the Archimedean places, the character $\theta$ is trivial at Archimedean places and is therefore a character of finite order. Let $U_T \subset \prod_{v \in T \setminus S} K_v^\times$ be a compact open subgroup such that $\theta|_{U_T}$ is trivial and such that $c(U_T) = U_T$. Let

$$U = \left( \prod_{v \notin T} \mathcal{O}_{K_v^\times} \right) \cdot U_T \cdot \left( \prod_{v \in S} K_v^\times \right).$$

We have an injection of compact groups

$$N_{K/K^+}(\mathbb{A}_K^\times) / (N_{K/K^+}(\mathbb{A}_K^\times) \cap K^\times U) \hookrightarrow \mathbb{A}_K^\times / K^\times U.$$

Under our hypothesis, the character $\psi|_{N_{K/K^+}(\mathbb{A}_K^\times)}$ is trivial on $(N_{K/K^+}(\mathbb{A}_K^\times) \cap K^\times U)$. Therefore it extends to a character $\alpha$ of finite order of $\mathbb{A}_K^\times$ trivial on $K^\times U$. We thus have $\psi \circ N_{K/K^+} = \alpha \circ N_{K/K^+}$. It is easy to check that the character $\tilde{\psi} = \psi \alpha^{-1}$ satisfies our requirements. $\square$
## 8.3 Construction of an explicit set of Hecke characters

In this subsection we explain one way to find a $\Psi$ as before, satisfying hypothesis $(\Psi, p)$, $(\Psi, \ell)$, $(\Psi, \text{Ram})$. Fix $E$ a CM extension, with $E^+ = F$ its maximal totally real subfield, so that $[E : E^+] = 2$. Fix also $F'$ disjoint from $E$, a totally real degree 3 Galois extension of $\mathbb{Q}$. Choose $p, \ell$ two primes with are totally split in $K := EF'$ such that $p > 8$. The following Lemma is a more precise version of [CHT08, Lem. 4.1.1].

**Lemma 8.6.** Let $F$ be a number field. Let $S$ be a finite set of places of $F$. Let $\chi_S$ be an unramified continuous character $F_S^\times := \prod_{v \in S} F_v^\times \to \mathbb{C}^\times$ of finite order. Let $T$ be a set of finite places of $F$, disjoint from $S$ and of Dirichlet density 1. Then there exists a continuous character $\chi : \mathbb{A}_F^\times / F^\times \to \mathbb{C}^\times$ of finite order such that $\chi|_{F_S^\times} = \chi_S$ and the ramification places of $\chi$ are in $T$.

*Proof.* Let $U^S$ be the product of the $O_{F_v}^\times$ for $v \notin S$. Then $F^\times \cap U^S$ is a finitely generated subgroup of $F^\times$. Let us write $m$ for the order of the finite cyclic group $\chi_S(F^\times \cap U^S)$. It follows from the proof of Theorem 1 in [Che51] that we can find finitely many places $w_1, \dots, w_r$ in $T$ such that the subgroup of $F^\times \cap U^S$ congruent to 1 modulo $\mathfrak{p}_{w_1}, \dots, \mathfrak{p}_{w_r}$ is contained in $(F^\times \cap U^S)^m$. We conclude as in the proof of [CHT08, Lem. 4.1.1] choosing for $U$ the product of the $U_v$ for $v$ not in $S$ nor $\{w_1, \dots, w_r\}$ and a small enough subgroup at $w_1, \dots, w_r$. $\square$

**Lemma 8.7.** Let $K$ be an (imaginary) CM field with totally real subfield $K^+$ and complex conjugacy $c$. Denote $\psi : \mathbb{A}_K^\times / K^\times \to \mathbb{C}^\times$ be a continuous character. Assume that there exists a finite set $S$ of places of $K$ which are split in $K/K^+$ and such that $\psi_v^{-1} = \psi_{cv}$ for $v \in S$. Moreover, assume that $S$ contains the Archimedean places. Let $T$ be a finite set of places of $K$ that contains $S$ and is stable under $c$, such that $\psi$ is unramified outside of $T$. Then there exists a Hecke character $\tilde{\psi} : \mathbb{A}_K^\times / K^\times \to \mathbb{C}^\times$ such that $\tilde{\psi}^{-1} = \tilde{\psi}^c$ and $\tilde{\psi}_v = \psi_v$ for $v \in S$ and such that $\tilde{\psi}_v$ is unramified outside of $T$.

*Proof.* Let $\theta = \psi \circ N_{K/K^+}$. As $S$ contains the Archimedean places, the character $\theta$ is trivial at Archimedean places and is therefore a character of finite order. Let $U_T \subset \prod_{v \in T \setminus S} K_v^\times$ be a compact open subgroup such that $\theta|_{U_T}$ is trivial and such that $c(U_T) = U_T$. Let

$$U = \left( \prod_{v \notin T} \mathcal{O}_{K_v^\times} \right) \cdot U_T \cdot \left( \prod_{v \in S} K_v^\times \right).$$

We have an injection of compact groups

$$N_{K/K^+}(\mathbb{A}_K^\times) / (N_{K/K^+}(\mathbb{A}_K^\times) \cap K^\times U) \hookrightarrow \mathbb{A}_K^\times / K^\times U.$$

Under our hypothesis, the character $\psi|_{N_{K/K^+}(\mathbb{A}_K^\times)}$ is trivial on $(N_{K/K^+}(\mathbb{A}_K^\times) \cap K^\times U)$. Therefore it extends to a character $\alpha$ of finite order of $\mathbb{A}_K^\times$ trivial on $K^\times U$. We thus have $\psi \circ N_{K/K^+} = \alpha \circ N_{K/K^+}$. It is easy to check that the character $\tilde{\psi} = \psi \alpha^{-1}$ satisfies our requirements. $\square$
**Proposition 8.8.** For each choice of fields $E$ and $F'$ and places $p$ and $\ell$ and very regular weight $\underline{k}$ as above there exists a Hecke character $\Psi : \text{A}_K^\times/K^\times \to \text{C}^\times$ satisfying $(\Psi, p)$, $(\Psi, \ell)$ and $(\Psi, \text{Ram})$ and such that $\Psi^{-1} = \Psi^c$.

*Proof.* Let $\underline{k}$ be a very regular weight. It follows from [Sch88], Section 0.3, that there exists a Hecke character $\Psi_0$ of $\text{A}_K^\times/K^\times$ with weight $\underline{k}$. Using Lemma 8.6, we can construct a Hecke character $\theta$ of finite order such that, setting $\Psi_1 = \Psi_0\theta$, we have

• the character $\Psi_1$ satisfies $(\Psi_1, p)$ and $(\Psi_1, \ell)$;

• there exists finitely many primes $\ell_1, \dots, \ell_r$, different from $p$ and $\ell$, which are totally split in $K$ and such that $\Psi_1$ is only ramified at places dividing $\ell_1, \dots, \ell_r$;

• we have $\Psi_{1,w}^{-1} = \Psi_{1,cw}$ for any place $w$ of $K$ dividing $\ell$ or $p$.

Now it follows from Lemma 8.7 that there exists a Hecke character $\Psi$ of $\text{A}_K^\times/K^\times$ such that

• $\Psi^{-1} = \Psi^c$;

• $\Psi_v = \Psi_{1,v}$ if $v$ is a place of $K$ dividing $p$ or $\ell$;

• $\Psi$ is ramified only at places dividing $\ell_1, \dots, \ell_r$.

<figure><img src="image_9.png" /></figure>

## 8.4 Automorphic Induction and base change

Let $\Psi$ and $\rho$ as in subsection 8.1 and let $U$ denote the unitary group in three variables for $E/E^+$ that is compact at infinity and quasi-split at all finite places. We need to find an automorphic form for $U$ whose associated Galois representation is induced representation $\rho$ from 8.3.

**Proposition 8.9.** There exists an automorphic representation $\Pi$ of $\text{GL}_{3,E}$, cuspidal, cohomological at infinity, unramified at $\ell$ and $p$, polarized, whose associated Galois representation is given by $\rho$.

*Proof.* This is the content of [Hen12] Théorème 3 (as $K/E$ cyclic of degree 3) for the existence of the automorphic representation, Théorème 5 for the compatibility with the local correspondence at $\ell$ and $p$ and at infinity (cf. the following remark of [Hen12]). Polarization can be checked after base change of the automorphic induction to $K$, where it follows as $\Psi^c = \Psi^\vee$, and as $\Psi \neq \Psi^\sigma$ for $\sigma \in \text{Gal}(K/E)$ such that $\sigma \neq 1$. Moreover, the automorphic induction is also cuspidal (Theorem 2 of [Hen12]). $\square$

**Conjecture 8.10.** There exists a cohomological, cuspidal, automorphic representation $\pi$ of $U$ whose base change to $\text{GL}_{3,E}$ is $\Pi$.
**Proposition 8.8.** For each choice of fields $E$ and $F'$ and places $p$ and $\ell$ and very regular weight $\underline{k}$ as above there exists a Hecke character $\Psi : \text{A}_K^\times/K^\times \to \text{C}^\times$ satisfying $(\Psi, p)$, $(\Psi, \ell)$ and $(\Psi, \text{Ram})$ and such that $\Psi^{-1} = \Psi^c$.

*Proof.* Let $\underline{k}$ be a very regular weight. It follows from [Sch88], Section 0.3, that there exists a Hecke character $\Psi_0$ of $\text{A}_K^\times/K^\times$ with weight $\underline{k}$. Using Lemma 8.6, we can construct a Hecke character $\theta$ of finite order such that, setting $\Psi_1 = \Psi_0\theta$, we have

• the character $\Psi_1$ satisfies $(\Psi_1, p)$ and $(\Psi_1, \ell)$;

• there exists finitely many primes $\ell_1, \dots, \ell_r$, different from $p$ and $\ell$, which are totally split in $K$ and such that $\Psi_1$ is only ramified at places dividing $\ell_1, \dots, \ell_r$;

• we have $\Psi_{1,w}^{-1} = \Psi_{1,cw}$ for any place $w$ of $K$ dividing $\ell$ or $p$.

Now it follows from Lemma 8.7 that there exists a Hecke character $\Psi$ of $\text{A}_K^\times/K^\times$ such that

• $\Psi^{-1} = \Psi^c$;

• $\Psi_v = \Psi_{1,v}$ if $v$ is a place of $K$ dividing $p$ or $\ell$;

• $\Psi$ is ramified only at places dividing $\ell_1, \dots, \ell_r$.

<figure><img src="image_9.png" /></figure>

## 8.4 Automorphic Induction and base change

Let $\Psi$ and $\rho$ as in subsection 8.1 and let $U$ denote the unitary group in three variables for $E/E^+$ that is compact at infinity and quasi-split at all finite places. We need to find an automorphic form for $U$ whose associated Galois representation is induced representation $\rho$ from 8.3.

**Proposition 8.9.** There exists an automorphic representation $\Pi$ of $\text{GL}_{3,E}$, cuspidal, cohomological at infinity, unramified at $\ell$ and $p$, polarized, whose associated Galois representation is given by $\rho$.

*Proof.* This is the content of [Hen12] Théorème 3 (as $K/E$ cyclic of degree 3) for the existence of the automorphic representation, Théorème 5 for the compatibility with the local correspondence at $\ell$ and $p$ and at infinity (cf. the following remark of [Hen12]). Polarization can be checked after base change of the automorphic induction to $K$, where it follows as $\Psi^c = \Psi^\vee$, and as $\Psi \neq \Psi^\sigma$ for $\sigma \in \text{Gal}(K/E)$ such that $\sigma \neq 1$. Moreover, the automorphic induction is also cuspidal (Theorem 2 of [Hen12]). $\square$

**Conjecture 8.10.** There exists a cohomological, cuspidal, automorphic representation $\pi$ of $U$ whose base change to $\text{GL}_{3,E}$ is $\Pi$.
**Proposition 8.11.** If $E/E^+$ is everywhere unramified (e.g. for $E = \mathbb{Q}(i, \sqrt{3})$ or $\mathbb{Q}(i, \sqrt{7}))$, then the previous conjecture is true.

*Proof.* This is [Lab11] Theorem 5.4. □

<figure><img src="image_10.png" /></figure>

**Proposition 8.12.** If $E$ is quadratic imaginary, then the previous conjecture is true.

*Proof.* By [Mor10] Corollary 8.5.3 (ii), there exists $\pi'$ an automorphic representation for $GU(3)$ associated to $\Pi \times 1$, which is automorphic for $GL_3 \times GL_1$. By [HS22] Lemma A.7 (based on [HT01]), there exists $\pi$, an automorphic representation of $U(3)$ associated to $\pi'$. □

<figure><img src="image_11.png" /></figure>

**Corollary 8.13.** If $E$ is quadratic imaginary or if $E/E^+$ is everywhere unramified, then there exists a classical form on $U(3)$ satisfying the hypothesis of Theorem 1.2.

*Proof.* Let $\pi$ be the automorphic representation of $U$ considered above, and let $f \in \pi$ be an eigenform for the Hecke operators away from a set $S$ of bad places of $\pi$. Then $\rho_f = \rho_\pi = \rho$ is crystalline at $p$ and $\varphi$-generic. In particular it has $3! = 6$ refinements which are automorphic and split at $p$. Hence there exists an automorphic refinement $\mathcal{R}$ of $f$ with relative position $w_\mathcal{R} = 1$ with respect to the Hodge filtration. In particular, for this choice of a refinement, there exists a refined classical modular form $f'$ satisfying all hypothesis of Theorem 1.2. But, by Lemma 8.4(5) we know that $f$ gives, for all $v \in S \backslash S_p$, a point of $\mathcal{X}_{\rho_v}^\square$ which satisfies $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. When $v$ splits in $E/E^+$, such a $v$ is a smooth point by [All16] Prop 1.2.2. □

<figure><img src="image_12.png" /></figure>

## References

[All16] P. B. ALLEN – “Deformations of polarized automorphic Galois representations and adjoint Selmer groups”, Duke Math. J. 165 (2016), no. 13, p. 2407–2460 (English).

[AS22] S. AGRAWAL & M. STRAUCH – “From category $\mathcal{O}^\infty$ to locally analytic representations”, J. Algebra 592 (2022), p. 169–232.

[BC06] J. BELLAÏCHE & G. CHENEVIER – “Smoothness of the Hecke curve of $GL_2$ at critical Eisenstein points.”, J. Inst. Math. Jussieu 5 (2006), no. 2, p. 333–349 (English).

[BD] C. BREUIL & Y. DING – “Bernstein eigenvarieties”, arXiv:2109.06696.

[BD16] J. BELLAÏCHE & M. DIMITROV – “On the eigencurve at classical weight 1 points”, Duke Math. J. 165 (2016), no. 2, p. 245–266 (English).
**Proposition 8.11.** If $E/E^+$ is everywhere unramified (e.g. for $E = \mathbb{Q}(i, \sqrt{3})$ or $\mathbb{Q}(i, \sqrt{7}))$, then the previous conjecture is true.

*Proof.* This is [Lab11] Theorem 5.4. □

<figure><img src="image_10.png" /></figure>

**Proposition 8.12.** If $E$ is quadratic imaginary, then the previous conjecture is true.

*Proof.* By [Mor10] Corollary 8.5.3 (ii), there exists $\pi'$ an automorphic representation for $GU(3)$ associated to $\Pi \times 1$, which is automorphic for $GL_3 \times GL_1$. By [HS22] Lemma A.7 (based on [HT01]), there exists $\pi$, an automorphic representation of $U(3)$ associated to $\pi'$. □

<figure><img src="image_11.png" /></figure>

**Corollary 8.13.** If $E$ is quadratic imaginary or if $E/E^+$ is everywhere unramified, then there exists a classical form on $U(3)$ satisfying the hypothesis of Theorem 1.2.

*Proof.* Let $\pi$ be the automorphic representation of $U$ considered above, and let $f \in \pi$ be an eigenform for the Hecke operators away from a set $S$ of bad places of $\pi$. Then $\rho_f = \rho_\pi = \rho$ is crystalline at $p$ and $\varphi$-generic. In particular it has $3! = 6$ refinements which are automorphic and split at $p$. Hence there exists an automorphic refinement $\mathcal{R}$ of $f$ with relative position $w_\mathcal{R} = 1$ with respect to the Hodge filtration. In particular, for this choice of a refinement, there exists a refined classical modular form $f'$ satisfying all hypothesis of Theorem 1.2. But, by Lemma 8.4(5) we know that $f$ gives, for all $v \in S\backslash S_p$, a point of $\mathcal{X}_{\rho_v}^\square$ which satisfies $\text{Hom}_{\text{Gal}_{E_v}}(\rho_v, \rho_v(1)) = 0$. When $v$ splits in $E/E^+$, such a $v$ is a smooth point by [All16] Prop 1.2.2. □

<figure><img src="image_12.png" /></figure>

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[HMS14] Y. HU, S. MORRA & B. SCHRAEN – “Sur la fidélité de certaines représentations de $GL_2(F)$ sous une algèbre d’Iwasawa”, *Rend. Semin. Mat. Univ. Padova* **131** (2014), p. 49–65.

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[Koh07] J. KOHLHAASE – “Invariant distributions on $p$-adic analytic groups.”, Duke Math. J. **137** (2007), no. 1, p. 19–62.

[Lab11] J.-P. LABESSE – “Changement de base CM et séries discrètes”, in *On the stabilization of the trace formula*, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, 2011, p. 429–470.

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[Mor10] S. MOREL – *On the cohomology of certain noncompact Shimura varieties. With an appendix by Robert Kottwitz.*, Ann. Math. Stud., vol. 173, Princeton, NJ: Princeton University Press, 2010 (English).

[Mow23] S. MOWLAVI – “The trianguline variety, tangent spaces and the grothendieck-springer resolution”, arXiv preprint arXiv:2303.06022 (2023).

[OS10] S. ORLIK & M. STRAUCH – “On the irreducibility of locally analytic principal series representations”, Represent. Theory **14** (2010), p. 713–746.

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[Wu] Z. WU – “Local models for the trianguline variety and partially classical families”, arXiv:2103.03823.
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[Tho12] J. THORNE – “On the automorphy of *l*-adic Galois representations with small residual image”, *J. Inst. Math. Jussieu* **11** (2012), no. 4, p. 855–920, With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne.

[Wu] Z. WU – “Local models for the trianguline variety and partially classical families”, arXiv:2103.03823.