Case $\tilde{w}_{f-1-i} = t_1$. From [LLHLM18, Table 5], the matrix of the partial Frobenius $A^{(f-1-i)}$ has the following form:

$$ \begin{pmatrix} c_{11} + v c_{11}^* & c_{12} & c_{13} \\ v c_{21} & c_{22} + v c_{22}^* & c_{23} \\ v c_{31} & v c_{32} & c_{33} + v c_{33}^* \end{pmatrix}. $$

Recall from item (3) above that $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}\llbracket c_{ij}, c_{kk}^* - [\bar{c}_{kk}^*] \rrbracket$, $1 \le i, j, k \le 3\rrbracket$ by the relations

$$ \begin{array}{l@{\quad}l@{\quad}l} c_{ii}c_{jj} = 0 & \text{for } i \neq j, & c_{11}c_{23} = 0, \qquad c_{31}c_{22} = 0, \\ c_{33}c_{12} = 0, & & c_{12}c_{23} = c_{22}c_{13}, \qquad c_{11}c_{32} = c_{12}c_{31}, \\ c_{21}c_{33} = c_{31}c_{23}, & & \end{array} $$

and

$$ \begin{align*} & (-1 - a + c)c_{22}^*c_{33} + (-1 - a + b)c_{22}c_{33}^* - (-1 - a + c)c_{23}c_{32} = 0 \\ & (a - b)c_{33}^*c_{11} + (-1 - b + c)c_{33}c_{11}^* - (a - b)c_{13}c_{31} = 0 \\ & (b - c)c_{11}^*c_{22} + (a - c)c_{11}c_{22}^* - (b - c)c_{12}c_{21} = 0 \\ & c_{11}c_{22}^*c_{33}^* + c_{22}c_{11}^*c_{33} + c_{33}c_{11}^*c_{22}^* - c_{11}^*c_{23}c_{32} - c_{22}^*c_{13}c_{31} \\ & \qquad - c_{33}^*c_{12}c_{21} + c_{13}c_{32}c_{21} = 0. \end{align*} $$

(Note that $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ is formally smooth of relative dimension three over the ring $\tilde{R}$ defined in [LLHLM18, Corollary 8.4].)

We provide details for the matching of $c_{(0,0)} = (c_{11}, c_{22}, c_{33}, c_{13}, c_{23}, c_{12})$. We have

$$ \bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / c_{(0,0)} = \mathbb{F}\llbracket c_{ij}, c_{kk}^* - [\bar{c}_{kk}^*] \rrbracket, \quad 1 \le j < i \le 3, 1 \le k \le 3\rrbracket $$

and

$$ A^{(f-1-i)} \mod c_{(0,0)} = \begin{pmatrix} v c_{11}^* & 0 & 0 \\ v c_{21} & v c_{22}^* & 0 \\ v c_{31} & v c_{32} & v c_{33}^* \end{pmatrix}. $$

On the other hand, $\tilde{w}'_{f-1-i} = \alpha\beta\gamma t_1$ and $A^{(f-1-i)}$ has the form

$$ \begin{pmatrix} v^2 c_{11}^* & 0 & 0 \\ v^2 c_{21}^* & v c_{22}^* & 0 \\ v^2 c_{31}^* & v c_{32}^* & c_{33}^* \end{pmatrix}. $$
We note that $\tilde{z}_{f-1-i} = t_{(1,0,-1)}$ and

$$ \begin{pmatrix} vc_{11}^* & 0 & 0 \\ vc_{21} & vc_{22}^* & 0 \\ vc_{31} & vc_{32} & vc_{33}^* \end{pmatrix} = \begin{pmatrix} v^2 c_{11}^{*'} & 0 & 0 \\ v^2 c_{21}^{*'} & vc_{22}^{*'} & 0 \\ v^2 c_{31}^{*'} & vc_{32}^{*'} & c_{33}^{*'} \end{pmatrix} \begin{pmatrix} v^{-1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & v \end{pmatrix} $$

under the isomorphism

$$ \overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / c_{(0,0)} \cong \overline{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla} $$

given by the change of variables

$$ c_{ij} = c'_{ij} \quad \text{for } 1 \le j < i \le 3, \quad c_{kk}^* = c_{kk'}^* \quad \text{for } 1 \le k \le 3. $$

We now explain the matching for the ideal $w_0 \stackrel{\text{def}}{=} c_{(0,0)} \cap c_{(0,1)} = (c_{22}, c_{33}, c_{23}, c_{33}^*c_{12} - c_{13}c_{32})$. We have

$$ \overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / w_0 = \frac{\mathbb{F}\llbracket c_{13}, c_{21}, c_{31}, c_{32}, c_{11}^* - [\tilde{c}_{11}^*], c_{22}^* - [\tilde{c}_{22}^*], c_{33}^* - [\tilde{c}_{33}^*] \rrbracket}{(c_{13}((a-c)c_{22}^*c_{31} - (b-c)c_{32}c_{21}))} $$

(we have used the relations

$$ c_{33}^*c_{12} - c_{13}c_{32} = 0, \quad c_{33}^*c_{11} - c_{13}c_{31} = 0, \quad (a-c)c_{22}^*c_{11} - (b-c)c_{12}c_{21} = 0 $$

holding in $\overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / w_0$. Moreover,

$$ A^{(f-1-i)} \mod w_0 = \begin{pmatrix} c_{13}c_{31}(c_{33}^*)^{-1} + vc_{11}^* & c_{13}c_{32}(c_{33}^*)^{-1} & c_{13} \\ vc_{21} & vc_{22}^* & 0 \\ vc_{31} & vc_{32} & vc_{33}^* \end{pmatrix} $$

On the other hand, $\tilde{w}_{f-1-i} = \alpha\beta\alpha = \tilde{z}_i^*$, $(a', b', c') = (c, b, a)$ and $A^{(f-1-i)}$ has the form

$$ \begin{pmatrix} c'_{11} & c'_{11}c'_{32}(c'_{31}^*)^{-1} & c'_{11}d'_{33}(c'_{31}^*)^{-1} + vc'_{13}^* \\ 0 & vc'_{22} & vc'_{23} \\ vc'_{31} & vc'_{32} & vd'_{33} \end{pmatrix} $$

The ring $\overline{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}\llbracket c'_{11}, c'_{23}, c'_{32}, d'_{33}, c'_{13} - [\tilde{c}'_{13}], c'_{22} - [\tilde{c}'_{22}], c'_{31} - [\tilde{c}'_{31}] \rrbracket$ by the relation

$$ c'_{11}((a' - b')c'_{23}c'_{32} - (a' - c')c'_{22}d'_{33}) = 0. $$

We now note that

$$ \begin{pmatrix} c_{13}c_{31}(c_{33}^*)^{-1} + vc_{11}^* & c_{13}c_{32}(c_{33}^*)^{-1} & c_{13} \\ vc_{21} & vc_{22}^* & 0 \\ vc_{31} & vc_{32} & vc_{33}^* \end{pmatrix} $$
$$ = \begin{pmatrix} c'_{11} & c'_{11}c'_{32}(c'_{31})^{-1} & c'_{11}d'_{33}(c'_{31})^{-1} + vc'_{13} \\ 0 & vc'_{22} & vc'_{23} \\ vc'_{31} & vc'_{32} & vd'_{33} \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} $$

under the isomorphism

$$ \bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / \mathfrak{w}_0 \cong \bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla} $$

given by the change of variables

$$ \begin{aligned} c_{13} &= c'_{11}, & c_{21} &= c'_{23}, & c_{31} &= d'_{33}, \\ c_{32} &= c'_{32}, & c_{11} &= c'_{13}, & c_{22} &= c'_{22}, & c_{33} &= c'_{31}. \end{aligned} $$

Case $\tilde{w}_{f-1-i} = \beta\alpha t_\perp$ and $\alpha\beta\alpha t_\perp$: The computations of these two cases are very similar to those we already performed and are left to the reader.

**3.6.3. Ideal relations in deformation rings.** In this subsection, we collect some results about sums of intersections of minimal primes in the potentially crystalline deformation rings $R_{\bar{\mathfrak{p}}}^\tau$. These computations play a crucial role in Section 5.1, where they are used to compute the value of a patching functor on certain representations as the limit of the value of the patching functor on simpler pieces of the representation.

Thanks to Theorem 3.6.4, all computations that we need to perform can be done on the rings $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ given in Table 3. We continue to adopt the notation and setting of Theorem 3.6.4.

We will frequently make use of the following.

**LEMMA 3.6.11.** Suppose we have a surjection of rings $g : S \to R$. Assume that $R$ and $S$ are equidimensional of dimension $d$, have the same number of minimal primes, and $S$ is reduced. Then $g$ is an isomorphism.

*Proof.* The first two hypotheses imply that the kernel of $g$ is nilpotent since $g$ induces an isomorphism between the underlying topological spaces of $\text{Spec}(R)$ and $\text{Spec}(S)$. But since $S$ is reduced, this kernel must in fact be $0$. $\square$

Ideal relations in $\bar{R}_{\mathfrak{M}, \text{id}}^{\text{expl}, \nabla}$

**LEMMA 3.6.12.** In the ring $\bar{R}_{\mathfrak{M}, \text{id}}^{\text{expl}, \nabla}$, we have

$$ (\mathfrak{w}_0 \cap \mathfrak{c}_{(\varepsilon_1, 0)}) + (\mathfrak{w}_0 \cap \mathfrak{c}_{(\varepsilon_2, 0)}) = \mathfrak{w}_0. $$
*Proof.* An elementary check on the list of generators of the ideals $w_0$, $c_{(\varepsilon_1,0)}$ and $c_{(\varepsilon_2,0)}$ (cf. Table 3) shows that

$$w_0 \cap c_{(\varepsilon_1,0)} \supseteq (c_{33}, c_{23}, c_{22}), \quad w_0 \cap c_{(\varepsilon_2,0)} \supseteq (c_{33}, c_{22}, c_{13}c_{32} - c_{12}c_{33}^*).$$

In particular (again by looking at the list of generators of $w_0$), we have ($w_0 \cap c_{(\varepsilon_1,0)} + (w_0 \cap c_{(\varepsilon_2,0)}) \supseteq w_0$. The reverse inclusion is obvious. $\square$

LEMMA 3.6.13. In the ring $\overline{R}_{\overline{\mathbb{M}}_{\mathrm{id}}}$, we have

$$w_0 \cap c_{(\varepsilon_1,0)} \cap c_{(\varepsilon_2,0)} = (c_{22}, c_{33}).$$

*Proof.* We let $\tilde{R}$ be the ring with the same presentation as $\overline{R}_{\overline{\mathbb{M}}_{\mathrm{id}}}$ except that all $c_{ij}^*$ are set to 1. As explained in [LLHLM18, Corollary 8.4], there is a natural identification of $\overline{R}_{\overline{\mathbb{M}}_{\mathrm{id}}}$ with the power series over $\tilde{R}$ with three variables, and we can work with the ring $\tilde{R}$ instead of $\overline{R}_{\overline{\mathbb{M}}_{\mathrm{id}}}$. All the ideals that we consider come from $\tilde{R}$ and are given by generators with the same name.

From Table 3, we immediately obtain ($c_{22}, c_{33}) \subseteq w_0 \cap c_{(\varepsilon_1,0)} \cap c_{(\varepsilon_2,0)}$, and, hence, we need to prove that the surjection

$$\tilde{R}/(c_{22}, c_{33}) \to \tilde{R}/(w_0 \cap c_{(\varepsilon_1,0)} \cap c_{(\varepsilon_2,0)}) \quad (3.15)$$

is an isomorphism. The ring on the right-hand side is equidimensional of dimension three and has four minimal primes. By Lemma 3.6.11, it suffices to show that $\tilde{R}/(c_{22}, c_{33})$ is reduced, is equidimensional of dimension three, and has four minimal primes.

Now $\tilde{R}/(c_{22}, c_{33})$ is the quotient of the power series ring $\mathbb{F}\llbracket c_{11}, c_{ij}, 1 \le i, j, \le 3, i \ne j \rrbracket$ by the ideal generated by the following elements:

$$\begin{array}{l@{\qquad}l@{\qquad}l}
c_{11}c_{23}, & c_{12}c_{23}, & c_{11}c_{32} - c_{12}c_{31}, \\
c_{31}c_{23}, & c_{23}c_{32}, & \kappa c_{11} = c_{21}c_{12}, \\
c_{11} - c_{13}c_{31}, & c_{11} - c_{12}c_{21} - c_{13}c_{31} - c_{23}c_{32} + c_{21}c_{13}c_{32} &
\end{array}$$

for some $\kappa \in \mathbb{F}^*$ depending on $(a, b, c)$. By standard manipulations, we conclude that $\tilde{R}/(c_{22}, c_{33})$ is isomorphic to the quotient of the power series ring $\mathbb{F}\llbracket c_{ij}, 1 \le i, j, \le 3, i \ne j \rrbracket$ by the ideal generated by the following elements:

$$\begin{array}{l@{\qquad}l@{\qquad}l}
c_{31}c_{23}, & c_{23}c_{32}, & c_{12}c_{23}, \\
c_{31}(c_{12} - c_{13}c_{32}), & c_{13}(\kappa c_{31} - c_{21}c_{32}), & c_{21}(c_{12} - c_{13}c_{32});
\end{array}$$
hence, by writing $\tilde{c}_{12} \stackrel{\text{def}}{=} c_{12} - c_{13}c_{32}$, $\tilde{c}_{31} \stackrel{\text{def}}{=} \kappa c_{31} - c_{21}c_{32}$, we obtain

$$ \tilde{R}/(c_{22}, c_{33}) \cong \frac{\mathbb{F}[\langle \tilde{c}_{12}, \tilde{c}_{31}, c_{23}, c_{13}, c_{21}, c_{32} \rangle]}{(\tilde{c}_{12}\tilde{c}_{31}, \tilde{c}_{12}c_{23}, \tilde{c}_{12}c_{21}, \tilde{c}_{31}c_{23}, \tilde{c}_{31}c_{13}, c_{23}c_{32})}. $$

This latter ring is easily seen to be equidimensional of dimension three and has four minimal primes. It is furthermore reduced since it is the quotient of a power series ring by an ideal generated by square-free monomials. $\square$

From Lemma 3.6.13, the same argument used in the proof of Lemma 3.6.12 gives the following.

**LEMMA 3.6.14.** In the ring $\bar{R}_{\mathfrak{M},id}^{\text{expl},\nabla}$, we have

$$ (\mathfrak{w}_0 \cap \mathfrak{w}_{\varepsilon_1} \cap \mathfrak{c}_{(\varepsilon_2,0)}) + (\mathfrak{w}_0 \cap \mathfrak{w}_{\varepsilon_2} \cap \mathfrak{c}_{(\varepsilon_1,0)}) = \mathfrak{w}_0 \cap \mathfrak{c}_{(\varepsilon_1,0)} \cap \mathfrak{c}_{(\varepsilon_2,0)}. $$

*Proof.* From Table 3, we immediately deduce that $(\mathfrak{w}_0 \cap \mathfrak{w}_{\varepsilon_1} \cap \mathfrak{c}_{(\varepsilon_2,0)}) \supseteq (c_{33})$ and $(\mathfrak{w}_0 \cap \mathfrak{w}_{\varepsilon_2} \cap \mathfrak{c}_{(\varepsilon_1,0)}) \supseteq (c_{22})$. The conclusion now follows as in Lemma 3.6.12 by noting that $(c_{22}, c_{33}) = \mathfrak{w}_0 \cap \mathfrak{c}_{(\varepsilon_1,0)} \cap \mathfrak{c}_{(\varepsilon_2,0)}$. $\square$

Ideal relations in $\bar{R}_{\mathfrak{M},\alpha}^{\text{expl},\nabla}$.

**LEMMA 3.6.15.** Consider the ring $\bar{R}_{\mathfrak{M},\alpha}^{\text{expl},\nabla}$ (cf. Table 3) and let $I_{\Lambda_0} \stackrel{\text{def}}{=} \mathfrak{c}_{(0,0)} \cap \mathfrak{c}_{(0,1)} \cap \mathfrak{c}_{(\varepsilon_2,0)}$.

Then

$$ (1) \quad I_{\Lambda_0} = (c_{11}c_{33}^* - c_{13}c_{31}, c_{23}c_{31}, c_{23}(c_{32}c_{21}^* - d_{22}c_{31})), $$

$$ (2) \quad \mathfrak{c}_{(\varepsilon_1,1)} \cap \mathfrak{c}_{(0,0)} \cap \mathfrak{c}_{(\varepsilon_2,0)} = (c_{11}, c_{13}c_{31}), $$

$$ (3) \quad \mathfrak{c}_{(0,1)} \cap \mathfrak{c}_{(0,0)} = (c_{23}, c_{11}c_{33}^* - c_{13}c_{31}). $$

*Proof.* As in the proof of Lemma 3.6.13, it suffices to work in the ring $\tilde{R}$ which has the same presentation as $\bar{R}_{\mathfrak{M},\alpha}^{\text{expl},\nabla}$ except that all $c_{ij}^*$ are set to 1. Recall that $\tilde{c}_{32} \stackrel{\text{def}}{=} c_{32} - d_{22}c_{31}$ in $\tilde{R}$.

We start with item (1). From Table (3), we easily deduce

$$ I_{\Lambda_0} \supseteq (c_{11} - c_{13}c_{31}, c_{23}c_{31}, c_{23}\tilde{c}_{32}), $$

that is, a surjection

$$ \tilde{R}/(c_{11} - c_{13}c_{31}, c_{23}c_{31}, c_{23}\tilde{c}_{32}) \to \tilde{R}/I_{\Lambda_0} \quad (3.16) $$
which we claim is an isomorphism. Note that, by construction, the ring $\tilde{R}/I_{\Lambda_0}$ has three minimal primes, and it is equidimensional of dimension three.

From [LLHLM18, Proposition 8.11], we immediately deduce that the ring $\tilde{R}/(c_{11} - c_{13}c_{31}, c_{23}c_{31}, c_{23}\tilde{c}_{32})$ is isomorphic to the quotient of the power series ring $\mathbb{F}\llbracket c_{31}, c_{13}, d_{22}, \tilde{c}_{32}, c_{23}\rrbracket$ by the ideal generated by the following elements:

$$c_{13}c_{31}c_{23}, \quad c_{13}c_{31}d_{22} - \frac{b-c}{a-b}c_{13}\tilde{c}_{32}, \quad c_{23}c_{31}, \\ c_{23}\tilde{c}_{32}, \quad (a-b)c_{13}c_{31}d_{22} + (c-b)c_{13}\tilde{c}_{32} + (-1-a+c)c_{23}c_{31}$$

that is, by the ideal generated by

$$c_{13}\left(c_{31}d_{22} - \frac{b-c}{a-b}\tilde{c}_{32}\right), \quad c_{23}c_{31}, \quad c_{23}\tilde{c}_{32},$$

or, equivalently, the ideal generated by

$$c_{13}\left(c_{31}d_{22} - \frac{b-c}{a-b}\tilde{c}_{32}\right), \quad c_{23}c_{31}, \quad c_{23}\left(c_{31}d_{22} - \frac{b-c}{a-b}\tilde{c}_{32}\right).$$

In other words, by an evident change of variables, we have

$$\tilde{R}/(c_{11} - c_{13}c_{31}, c_{23}c_{31}, c_{23}\tilde{c}_{32}) \cong \mathbb{F}\llbracket X, Y, Z, W, c'_{22}\rrbracket/(XY, YZ, WZ)$$

and the latter ring is equidimensional of dimension three and has three irreducible components. It is, moreover, reduced since the ideal of relations is generated by square-free monomials.

The argument to prove (2) is completely analogous and we only sketch it. It is immediate to obtain from Table 3 the inclusion $c_{(\varepsilon_{1,1})} \cap c_{(0,0)} \cap c_{(\varepsilon_{2,0})} \supseteq (c_{11}, c_{13}c_{31})$. From [LLHLM18, Proposition 8.11], we see that $\tilde{R}/(c_{11}, c_{13}c_{31})$ is isomorphic to

$$\mathbb{F}\llbracket c_{13}, c_{23}, c_{31}, \tilde{c}_{32}, d_{22} \rrbracket/(c_{13}c_{31}, c_{13}\tilde{c}_{32}, c_{31}c_{23}\tilde{c}_{32}, c_{31}(\kappa c_{23} - c_{13}d_{22})) = \\ = \mathbb{F}\llbracket c_{13}, c_{23}, c_{31}, \tilde{c}_{32}, d_{22} \rrbracket/(c_{13}c_{31}, c_{13}\tilde{c}_{32}, c_{31}c_{23}),$$

for some $\kappa \in \mathbb{F}^\times$ depending on $(a, b, c)$. The conclusion is obtained by an analogous argument to the previous case.

The proof of item (3) is similar and left to the reader. (Note that $c_{(0,1)} \cap c_{(0,0)} \supseteq (c_{23}, c_{11} - c_{13}c_{31})$ and use the explicit presentation of $\tilde{R}$ given by [LLHLM18, Proposition 8.11] to check that $\tilde{R}/(c_{23}, c_{11} - c_{13}c_{31})$ is reduced with two irreducible components of dimension three.) $\square$
LEMMA 3.6.16. In $\overline{R}_{\mathfrak{M},\alpha}^{\text{expl.}^{\vee}}$, we have

$$ (I_{\Lambda_0} \cap c_{(\varepsilon_1, 1)}) + (I_{\Lambda_0} \cap c_{(\varepsilon_2, 1)} \cap c_{(\varepsilon_2 - \varepsilon_1, 0)}) = I_{\Lambda_0}, \quad (3.17) $$

$$ \begin{aligned} & (c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(0,1)}) + (c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(\varepsilon_2, 1)}) \\ & = c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)}, \end{aligned} \quad (3.18) $$

$$ (c_{(0,1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)}) + (c_{(0,1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2 - \varepsilon_1, 0)}) = c_{(0,1)} \cap c_{(0,0)}. \quad (3.19) $$

Proof. We work in $\tilde{R}$ and set $\tilde{c}_{32} \stackrel{\text{def}}{=} c_{32} - d_{22}c_{31}$. An elementary check on the list of generators of the ideals $I_{\Lambda_0}$, $c_{(\varepsilon_2, 1)}$, $c_{(\varepsilon_2 - \varepsilon_1, 0)}$, and $c_{(\varepsilon_1, 1)}$ shows that

$$ (I_{\Lambda_0} \cap c_{(\varepsilon_1, 1)}) \supseteq (c_{11} - c_{13}c_{31}, c_{23}c_{31}), \quad (I_{\Lambda_0} \cap c_{(\varepsilon_2, 1)} \cap c_{(\varepsilon_2 - \varepsilon_1, 0)}) \supseteq (c_{23}\tilde{c}_{32}), $$

and we deduce (again by looking at the list of generators of $I_{\Lambda_0}$) that

$$ (I_{\Lambda_0} \cap c_{(\varepsilon_1, 1)}) + (I_{\Lambda_0} \cap c_{(\varepsilon_2, 1)} \cap c_{(\varepsilon_2 - \varepsilon_1, 0)}) \supseteq I_{\Lambda_0}. $$

The reverse inclusion is obvious.

Similarly, $(c_{11}) \subseteq c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(\varepsilon_2, 1)}$ and $(c_{11} - c_{13}c_{31}) \subseteq c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(0,1)}$. Hence, $(c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(0,1)}) + (c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)} \cap c_{(\varepsilon_2, 1)}) \supseteq c_{(\varepsilon_1, 1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)}$.

The proof of item (3.19) is similar and left to the reader (note that $c_{11} - c_{13}c_{31} \in (c_{(0,1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2, 0)})$ and $c_{23} \in (c_{(0,1)} \cap c_{(0,0)} \cap c_{(\varepsilon_2 - \varepsilon_1, 0)})$). $\square$

REMARK 3.6.17. The ideal relation appearing in Lemmas 3.6.15 and 3.6.16 are compatible with the outer automorphism of $\tilde{W}^{\vee}$. Explicitly, define $\tilde{\delta} \stackrel{\text{def}}{=} (123)t_{(0,0,-1)}$ and let $\tilde{w} \mapsto \tilde{\delta}\tilde{w}\tilde{\delta}^{-1}$ be the corresponding outer automorphism on $\tilde{W}^{\vee}$. Then $\delta$ acts on $\Sigma_0$ and the ideal relations of Lemmas 3.6.15 and 3.6.16 hold for shape $\beta$ with $c_{(\omega_i, a_i)}$ replaced by $c_{((\tilde{\delta}^*)^{-1}(\omega_i), a_i)}$. As an example, for shape $\beta$, the relation (3.19) becomes $(c_{(\varepsilon_1, 1)} \cap c_{(\varepsilon_1, 0)} \cap c_{(0,0)}) + (c_{(\varepsilon_1, 1)} \cap c_{(\varepsilon_1, 0)} \cap c_{(\varepsilon_1 - \varepsilon_2, 0)}) = c_{(\varepsilon_1, 1)} \cap c_{(\varepsilon_1, 0)}$.

## 4. Lattices in generic Deligne–Lusztig representations

The aim of this section is to classify lattices with irreducible cosocle in generic $\text{GL}_3(\mathbb{F}_q)$ Deligne–Lusztig representations, providing the crucial representation-theoretic input to deduce Breuil’s lattice conjecture from the weight part of Serre conjecture. The main result (Theorem 4.1.9) states that the submodule structure of lattices with irreducible cosocle can be predicted using the extension graph introduced in Section 2.
*Outline of the proof.* Let $R$ be a generic Deligne–Lusztig representation. We have two main steps in the proof of the classification theorem: first, by local algebraic methods, we describe the reduction of lattices with irreducible cosocles isomorphic to a lower alcove weight with *defect zero* (Theorem 4.2.16). Second, in Section 4.3, we introduce another notion of distance related to the second part of Theorem 4.1.9, which we call *saturation distance*. It turns out that this notion is closely related to the submodule structure of the reduction of lattices.

Using a crucial global input coming from the geometry of Galois deformation
rings, we show that

(1) if measured from $\sigma$ the graph and saturation distances coincide, then the reduction of the lattice with cosocle isomorphic to $\sigma$ is as predicted in Theorem 4.1.9 (Proposition 4.3.16);

(2) measured from a lower alcove weight of defect zero, the saturation distance and the graph distance coincide (Proposition 4.3.17, which uses Theorem 4.2.16).

In contrast to the other notions of distance that we introduce, the saturation
distance involves lattices in characteristic zero, making it far more flexible.
Taking advantage of this flexibility, we finally show by an induction on defect
that the saturation distance and the graph distance coincide, completing the proof
of Theorem 4.1.9.

*Structure of Section 4.* For $R$ as above and $\sigma \in \text{JH}(\bar{R})$, let $R^\sigma$ denote the unique (up to homothety) $O$-lattice in $R$ with irreducible cosocle $\sigma$ and write $\bar{R}^\sigma$ to denote its reduction modulo $\sigma$. The first main step is in Section 4.2. The argument uses the modular representation theory of algebraic groups to embed $\bar{R}^\sigma$ into the $G \stackrel{\text{def}}{=} \text{G}_0(\text{F}_p)$-restriction of a tensor product $V_\mu$ of algebraic Weyl modules with non-$p$-restricted highest weight (see Section 4.2.2 for the definition of $V_\mu$). The content of Section 4.2.3 is the description of $V_\mu|_G$ provided by Theorem 4.2.7. This theorem describes the Jordan–Hölder constituents of $V_\mu|_G$ and the existence of nontrivial extensions between constituents at graph distance one (cf. the key technical result Proposition 4.2.10). The embedding of $\bar{R}^\sigma$ in $V_\mu$ is constructed in Section 4.2.4. One first proves the existence of a nonzero (and unique up to scalar) morphism $\bar{R}^\sigma \to V_\mu$ (Proposition 4.2.15); an inductive argument, using the description of the submodule structure of $V_\mu|_G$, then shows that the image of this morphism contains all the constituents of $\bar{R}^\sigma$. The submodule structure of $\bar{R}^\sigma$ is then obtained from that of $V_\mu$.

The second part of the proof of Theorem 4.1.9 is the content of Section 4.3.
The key insight is the introduction of the auxiliary notion of saturation distance
on JH($\bar{R}$) in Section 4.3.1 which relates the position of saturated lattices in $R^\sigma$.
Using a global input, we give in Section 4.3.1 a first coarse relation between the saturation and the graph distance (cf. Corollary 4.3.8). Subsequently, we prove in Section 4.3.2 that the three distances are actually equal provided that $\sigma$ verifies an appropriate condition relating its defect, its graph distance, and its saturated distance (we say that $\sigma$ is *maximally saturated* in $R$). In particular, if $\sigma$ is maximally saturated in $R$, the structure of $\bar{R}^\sigma$ is predicted by the extension graph (Proposition 4.3.16).

We are hence left to prove that all constituents of $R$ are maximally saturated. This is shown by an induction argument in Section 4.3.3. The proof of Theorem 4.1.9 concludes this section.

**4.1. The classification statement.**

**4.1.1. Some generalities.** Let $\mathcal{C}$ be a nonzero finite abelian category over $\mathbb{F}$. Let $M$ be a nonzero object of $\mathcal{C}$. A decreasing (respectively increasing) filtration $\mathcal{F}$ on $M$ is a collection of subobjects $\mathcal{F}^n(M) \subset M$ (respectively $\mathcal{F}_n(M) \subset M$) for $n \in \mathbb{Z}$ such that $\mathcal{F}^{n+1}(M) \subset \mathcal{F}^n(M)$ (respectively $\mathcal{F}_n(M) \subset \mathcal{F}_{n+1}(M)$) for all $i$. A filtration $\mathcal{F}$ is exhaustive and separated if $\mathcal{F}^n(M) = 0$ for $i$ sufficiently large (respectively small) and $\mathcal{F}^n(M) = M$ for $n$ sufficiently small (respectively large), and this property will always be assumed to hold. We write $\mathrm{gr}_{\mathcal{F}}^n(M) \stackrel{\mathrm{def}}{=} \frac{\mathcal{F}^n(M)}{\mathcal{F}^{n+1}(M)}$ (respectively $\mathrm{gr}_{n}^{\mathcal{F}}(M) \stackrel{\mathrm{def}}{=} \frac{\mathcal{F}_{n}(M)}{\mathcal{F}_{n-1}(M)}$) and omit $\mathcal{F}$ from the notation if it is clear from context. A filtration is *semisimple* if $\mathrm{gr}_{\mathcal{F}}^n(M)$ (respectively $\mathrm{gr}_{n}^{\mathcal{F}}(M)$) is semisimple for all $n \in \mathbb{Z}$. By shifting the filtration, we will assume that $\mathrm{gr}_{\mathcal{F}}^n(M) = 0$ (respectively $\mathrm{gr}_{n}^{\mathcal{F}}(M) = 0$) for $n < 0$ and that $\mathrm{gr}_{\mathcal{F}}^0(M) \neq 0$ (respectively $\mathrm{gr}_{0}^{\mathcal{F}}(M) \neq 0$). The length of a filtration is the maximal $\ell \in \mathbb{Z}$ such that $\mathrm{gr}_{\mathcal{F}}^{\ell-1}(M) \neq 0$ (respectively $\mathrm{gr}_{\ell-1}^{\mathcal{F}}(M) \neq 0$).

The *socle* of $M$, denoted as $\mathrm{soc}(M)$, is defined to be the maximal (with respect to inclusion) semisimple subobject of $M$. The *radical* of $M$, denoted as $\mathrm{rad}(M)$, is the minimal (with respect to inclusion) subobject of $M$ whose corresponding quotient is semisimple. The *cosocle* of $M$ is $\mathrm{cosoc}(M) \stackrel{\mathrm{def}}{=} M/\mathrm{rad}(M)$. We inductively define the *radical* and *socle filtration* on $M$: we set $\mathrm{rad}^0(M) = M$ and let $\mathrm{rad}^n(M) \stackrel{\mathrm{def}}{=} \mathrm{rad}(\mathrm{rad}^{n-1}(M))$, and set $\mathrm{soc}_{-1}(M) = 0$ and let $\mathrm{soc}_n(M)$ be the inverse image, via the canonical projection $M \to M/\mathrm{soc}_{n-1}(M)$, of $\mathrm{soc}(M/\mathrm{soc}_{n-1}(M)) \subseteq M/\mathrm{soc}_{n-1}(M)$. Then the radical (respectively socle) filtration is a decreasing (respectively increasing) semisimple filtration. Moreover, $\mathrm{gr}_{\mathrm{rad}}^0(M) = \mathrm{cosoc}(M)$ and $\mathrm{gr}_{0}^{\mathrm{soc}}(M) = \mathrm{soc}(M)$. Since formation of cosocle (respectively socle) is right (respectively left) exact, the filtration induced from the radical (respectively socle) filtration on a quotient object (respectively subobject) is the radical (respectively socle) filtration. The lengths
of the radical and socle filtrations coincide, and we call this value the Loewy length of M and denote it by ℓℓ(M). Any semisimple filtration has length at least ℓℓ(M), and we say that it is a Loewy series if its length equals ℓℓ(M). If ℘ is a decreasing Loewy series, then we necessarily have

$$ \mathrm{rad}^n(M) \subseteq \mathcal{F}^n(M) \subseteq \mathrm{soc}_{\ell\ell(M)-n-1}(M) \quad (4.1) $$

for all $n \in \{0, \dots, \ell\ell(M)\}$. We say that M is rigid if $\mathrm{rad}^n(M) = \mathrm{soc}_{\ell\ell(M)-n-1}(M)$ for all $n$.

We say that M is multiplicity free if every Jordan–Hölder factor of M appears with multiplicity one. We now suppose that M is multiplicity free. We say that $\sigma \in \mathrm{JH}(M)$ points to $\sigma' \in \mathrm{JH}(M)$ if there exists a subquotient of M which is isomorphic to a nontrivial extension of $\sigma$ by $\sigma'$. We say that a subset $S \subset \mathrm{JH}(M)$ is closed if $\sigma \in S$ and $\sigma$ points to $\sigma'$ imply that $\sigma' \in S$.

**PROPOSITION 4.1.1.** *The assignment of JH(N) to a subobject N ⊂ M gives a bijection between subobjects of M and closed subsets of JH(M).*

*Proof.* It is easy to see that $\mathrm{JH}(N)$ is a closed subset of $\mathrm{JH}(M)$. Suppose that $S \subset \mathrm{JH}(M)$ is a closed subset. Let $N$ be the minimal subobject of $M$ with $S \subset \mathrm{JH}(N)$. Suppose that $\mathrm{JH}(N) \setminus S$ is nonempty and contains $\sigma$. If $\sigma' \in \mathrm{JH}(N)$ points to $\sigma$, then $\sigma'$ is not in $S$ since $S$ is closed. By replacing $\sigma$ by $\sigma'$ repeatedly, we can assume, without loss of generality, that $\sigma'$ does not point to $\sigma$ for all $\sigma' \in \mathrm{JH}(N)$. Let $N'$ be the maximal subobject of $N$ such that $\sigma \notin \mathrm{JH}(N')$. This maximality implies that the socle of $N/N'$ must be isomorphic to $\sigma$. By the assumption above, there is no subobject of $N/N'$ which is an extension by $\sigma$, and, therefore, $N/N'$ is isomorphic to $\sigma$. Then the existence of $N'$ contradicts the minimality of $N$. $\square$

Suppose that M is multiplicity free and that ℘ is a decreasing (respectively increasing) filtration on M. For $\sigma \in \mathrm{JH}(M)$, we define $d_M^\mathcal{F}(\sigma)$ (respectively $d_M^\mathcal{F}(\sigma)$) to be the unique value n such that $\mathrm{Hom}_C(\sigma, \mathrm{gr}_n^\mathcal{F}(M))$ (respectively $\mathrm{Hom}_C(\sigma, \mathrm{gr}_n^\mathcal{F}(M)))$ is nonzero. For ℘ semisimple, we say that $\sigma \in \mathrm{JH}(M)$ ℘-points to $\sigma' \in \mathrm{JH}(M)$ if $\sigma$ points to $\sigma'$ and the (shifted) induced filtration on the subquotient which is isomorphic to a nontrivial extension of $\sigma$ by $\sigma'$ has length 2.

If M is multiplicity free, we can attach a directed acyclic graph $\Gamma(M)$ and a subgraph $\Gamma_\mathcal{F}(M)$ of $\Gamma(M)$ (respectively $\Gamma_\mathcal{F}(M)$) of $\Gamma(M)$) where the vertices are in bijection with the Jordan–Hölder factors of M and there is an arrow $\sigma \to \sigma'$ in $\Gamma(M)$ if $\sigma$ points to $\sigma'$ and an arrow $\sigma \to \sigma'$ in $\Gamma_\mathcal{F}(M)$ (respectively in $\Gamma_\mathcal{F}(M)$) if $\sigma$ $\mathcal{F}$-points to $\sigma'$. An extension path (in M) is a directed path in
$\Gamma(M)$ and an extension path in $\mathcal{F}$ is a directed path in $\Gamma_{\mathcal{F}}(M)$ (respectively $\Gamma_{\mathcal{F}}^*(M)$). The following proposition is immediate from the definitions.

**PROPOSITION 4.1.2.** Let $N$ be a subquotient of $M$. If $\mathcal{F}$ is a semisimple filtration on $M$, then it naturally induces a semisimple filtration on $N$. Moreover, $\Gamma(N)$ (respectively $\Gamma_{\mathcal{F}}(N)$ and $\Gamma_{\mathcal{F}}^*(N)$) is the maximal subgraph of $\Gamma(M)$ (respectively $\Gamma_{\mathcal{F}}^*(M)$ and $\Gamma_{\mathcal{F}}(M)$) with vertices corresponding to $\text{JH}(N)$.

**LEMMA 4.1.3.** Suppose that $M$ is multiplicity free and $\sigma \in \text{JH}(M)$. If $d_{\text{rad}}^{M}(\sigma) > 0$ (respectively $d_{\text{rad}}^{\text{soc}}(\sigma) > 0$), there is a $\sigma'$ such that $d_{\text{rad}}^{M}(\sigma') = d_{\text{rad}}^{M}(\sigma) - 1$ (respectively $d_{\text{rad}}^{\text{soc}}(\sigma') = d_{\text{rad}}^{\text{soc}}(\sigma) - 1$) and $\sigma' \text{ rad-points to } \sigma$ (respectively $\sigma$ soc-points to $\sigma'$). $\square$

*Proof.* We consider the radical filtration; the socle filtration is analyzed similarly. Let $d = d_{\text{rad}}^{M}(\sigma)$. Then the radical filtration on $\text{rad}^{d-1}(M)$ is a shift of the radical filtration on $M$ by definition and induces the radical filtration on $\text{rad}^{d-1}(M)/\text{rad}^{d+1}(M)$. Then there is a $\sigma' \in \text{JH}(\text{gr}_{\text{rad}}^{d-1}(M))$ which rad-points to $\sigma$, otherwise $\sigma$ would not be in the radical of $\text{rad}^{d-1}(M)/\text{rad}^{d+1}(M)$ which is $\text{gr}_{\text{rad}}^{d}(M)$. $\square$

**COROLLARY 4.1.4.** Suppose that $M$ is multiplicity free and $\sigma \in \text{JH}(M)$. Then there is an extension path in the radical (respectively socle) filtration of length $d_{\text{rad}}^{M}(\sigma)$ (respectively $d_{\text{rad}}^{\text{soc}}(\sigma)$) ending (respectively beginning) with $\sigma$.

*Proof.* The case $d_{\text{rad}}^{M}(\sigma) = 0$ (respectively $d_{\text{rad}}^{\text{soc}}(\sigma) = 0$) is trivial. The induction step follows from Lemma 4.1.3. $\square$

**LEMMA 4.1.5.** Suppose that $M$ is multiplicity free and $\mathcal{F}$ is an increasing semisimple filtration on $M$. If $\sigma$ points to $\sigma'$, then $d_M^{\mathcal{F}}(\sigma) > d_M^{\mathcal{F}}(\sigma')$.

*Proof.* Let $d$ be $d_M^{\mathcal{F}}(\sigma)$. By Proposition 4.1.1, $\mathcal{F}_d(M)$ contains $\sigma'$ as a Jordan-Hölder factor. Thus, $d_M^{\mathcal{F}}(\sigma) \ge d_M^{\mathcal{F}}(\sigma')$. If $d_M^{\mathcal{F}}(\sigma) = d_M^{\mathcal{F}}(\sigma')$, then there is a subquotient of $M$ which is isomorphic to a direct sum of $\sigma$ and $\sigma'$. This contradicts the fact that $\sigma$ points to $\sigma'$. $\square$

**PROPOSITION 4.1.6.** Suppose that $M \in \mathcal{C}$ is multiplicity free. Then $M$ is rigid if and only if for every $\sigma \in \text{JH}(M)$, there is an extension path in the radical (respectively socle) filtration of length $\ell\ell(M) - 1 - d_{\text{rad}}^{M}(\sigma)$ (respectively $\ell\ell(M) - 1 - d_{\text{rad}}^{\text{soc}}(\sigma)$) beginning (respectively ending) with $\sigma$.
*Proof.* First, suppose that $M$ is rigid. There is an extension path $P_{\text{rad}}$ in the radical filtration of length $d_{\text{rad}}^{M}(\sigma)$ ending at $\sigma$ by Corollary 4.1.4. By rigidity, this is an extension path in the socle filtration of length $d_{\text{rad}}^{M}(\sigma) = \ell\ell(M) - 1 - d_{M}^{\text{soc}}(\sigma)$.

Now suppose that there is an extension path in the radical filtration of length $\ell\ell(M) - 1 - d_{\text{rad}}^{M}(\sigma)$ starting with $\sigma$. Then

$$d_{M}^{\text{soc}}(\sigma) \geq \ell\ell(M) - 1 - d_{\text{rad}}^{M}(\sigma)$$

by Lemma 4.1.5. The reverse inequality is implied by (4.1), and we conclude that $\text{gr}_{\text{rad}}^{n}(M)$ and $\text{gr}_{\ell\ell(M)-1-n}^{\text{soc}}(M)$ are isomorphic and, thus, that $M$ is rigid. $\square$

The following is self-evident.

**PROPOSITION 4.1.7.** Let $M \in \mathcal{C}$ be multiplicity free. Then the dual object $M^*$ in the dual abelian category $\mathcal{C}^*$ is also multiplicity free. A decreasing filtration $\mathcal{F}$ on $M$ gives rise to an increasing filtration $\mathcal{F}^*$ on $M^*$. Then the map sending $\sigma \in \text{JH}(M)$ to $\sigma^* \in \text{JH}(M^*)$ extends to isomorphisms of directed graphs $\Gamma(M) \to \Gamma(M^*)$ and $\Gamma_{\mathcal{F}}(M) \to \Gamma_{\mathcal{F}^*}(M)^*$ where $-^*$ denotes the transpose of a directed graph. In particular, if $\sigma_0 \to \sigma_1 \to \dots \to \sigma_n$ is an extension path in $\mathcal{F}$, then $\sigma_n^* \to \sigma_{n-1}^* \to \dots \to \sigma_0^*$ is an extension path in $\mathcal{F}^*$.

**4.1.2. The main result.** We now use the notation from Section 4.1.1 with $\mathcal{C}$ the category of finite $\mathbb{F}[G]$-modules. Let $R$ be a 2-generic Deligne–Lusztig representation of $G$ over $E$. By [Her09, Appendix, Theorem 3.4] (see the proof of Proposition 2.3.5), $R$ is residually multiplicity free. We will show that the elements in $\text{JH}(\bar{R})$ are $p$-regular in Lemma 4.2.13. If $\sigma \in \text{JH}(\bar{R})$, then there exists a unique (up to homothety) $\mathcal{O}$-lattice $R^\sigma \subseteq R$ with irreducible cosocle isomorphic to $\sigma$ by [EGS15, Lemma 4.1.1], and we write $\bar{R}^\sigma$ to denote its reduction modulo $\mathcal{W}$. In Section 2, we defined a distance function $d_{\text{gph}}$ on $p$-regular Serre weights. We now want to relate the graph distance to the submodule structure of $\bar{R}^\sigma$. To simplify notation, we fix $R$ and write $d_{\text{rad}}^\sigma(\sigma')$ for $d_{\text{rad}}^{\bar{R}^\sigma}(\sigma')$.

**DEFINITION 4.1.8.** Let $V$ be a set of (isomorphism classes of) weights $\sigma = F(\mu)$ with $\mu$ $p$-regular. Let $\sigma \in V$. Let $\Gamma$ be a directed graph with vertex set $V$. Then we say that $\Gamma$ is predicted by the extension graph with respect to $\sigma$ if there is an edge from $\kappa_1$ to $\kappa_2$ if and only if $d_{\text{gph}}(\kappa_1, \kappa_2) = 1$ and $d_{\text{gph}}(\sigma, \kappa_1) \le d_{\text{gph}}(\sigma, \kappa_2)$ (see Definition 2.1.8).

Our main result on the representation theory side is the following.

**THEOREM 4.1.9.** Let $R$ be as above and 13-generic. Let $\sigma \in \text{JH}(\bar{R})$. Then
(1) $d_{\text{gph}}(\sigma, \kappa) = d_{\text{rad}}^{\sigma}(\kappa)$ for all $\kappa \in \text{JH}(\overline{R})$, in particular,

$$ \ell \ell(\overline{R}^{\sigma}) = 2\text{Def}_{R}(\sigma) + 3(f - \text{Def}_{R}(\sigma)) + 1; $$

(2) $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$ is predicted by the extension graph with respect to $\sigma$;

(3) $\Gamma(\overline{R}^{\sigma})$ is predicted by the extension graph with respect to $\sigma$;

(4) $\overline{R}^{\sigma}$ is rigid; and

(5) if $\kappa \in \text{JH}(\overline{R})$ and $R^{\kappa} \hookrightarrow R^{\sigma}$ is a saturated inclusion, then $p^{d_{\text{gph}}(\kappa, \sigma)} R^{\sigma} \hookrightarrow R^{\kappa}$ is a saturated inclusion.

**REMARK 4.1.10.** By Proposition 2.3.5, every maximal geodesic in $\text{JH}(\overline{R})$ starting from $\sigma$ has the same length. Then item (4) follows from items (1) and (2) and Proposition 4.1.6. Furthermore, (3) and Proposition 4.1.1 give a classification of submodules of $\overline{R}^{\sigma}$, from which one can easily deduce items (1) and (2).

The proof of Theorem 4.1.9 will be carried out in the following subsections.

**4.2. Injective envelopes.** We now relax our hypotheses on $G_0$ but keep much of the related notation. Let $G_0$ be a connected reductive group over $\mathbb{F}_p$ and let $G$ be the base change $G_0 \times_{\mathbb{F}_p} \mathbb{F}$. Assume that $G$ is split and isomorphic to $G_s \times_{\mathbb{F}_p} \mathbb{F}$, where $G_s$ is a connected split reductive group over $\mathbb{F}_p$. Let $G_0^{\text{der}}$ be the derived subgroup of $G_0$ and let $G^{\text{der}}$ be the base change $G_0^{\text{der}} \times_{\mathbb{F}_p} \mathbb{F}$. Assume that $G_0^{\text{der}}$ is simply connected. Let $G$ (respectively $G^{\text{der}}$) be the finite group $G_0(\mathbb{F}_p)$ (respectively $G_0^{\text{der}}(\mathbb{F}_p)$). Let $F : G \to G$ denote the relative Frobenius with respect to $G_s$. There is an automorphism $\pi$ of $G_s$, and hence its based root datum, so that $F \circ \pi : G \to G$ is the relative Frobenius with respect to $G_0$. This definition of $\pi$ is consistent with the special case introduced in Section 1.4. Let $h$ be the Coxeter number of $G$. We will eventually specialize to the case where $G$ is a product of copies of $\text{GL}_3$ so that $h = 3$.

We define $\text{Proj}(\sigma)$ to be the projective hull of $\sigma$ in the category of $\mathbb{F}[G]$-modules. As $\mathbb{F}[G]$ is a Frobenius algebra, we have an isomorphism $\text{Proj}(\sigma) \cong \text{Inj}(\sigma)$ where $\text{Inj}(\sigma)$ denotes the injective envelope of $\sigma$ (again in the category of $\mathbb{F}[G]$-modules; cf. [Alp86, Section 6, Theorems 4 and 6]).

**4.2.1. Algebraic groups, Frobenius kernels, and finite groups.** In this section, we compare injective envelopes of weights for representations of Frobenius kernel and of finite groups. One goal is to prove that, under genericity conditions,
the graph distance introduced in Section 2.1 can be characterized in terms of nonvanishing of Ext<sup>1</sup><sub>G</sub>-groups (Lemma 4.2.6). We, moreover, introduce and describe a non-p-restricted Weyl module $V_μ$ which will play a key role in the proof of a particular case of Theorem 4.1.9.

Let $G_1$ denote the Frobenius kernel of $G$ and $G_1T$ denote the product of $T$ and $G_1$ in $G$. We similarly define $G_1^{\text{der}}$ and $G_1^{\text{der}}T^{\text{der}}$. The $G$-representation $L(\mu)$ remains irreducible when restricted to $G_1T$ and $G_1$ (cf. [Jan03, II.3.10 and II.9.6]) and we will use the standard notations $L_1(\mu) = L(\mu)|_{G_1}$, $\hat{L}_1(\mu) = L(\mu)|_{G_1T}$ (cf. loc. cit.). We write $\hat{Q}_1(\mu)$ to denote the injective envelope of the irreducible representation $\hat{L}_1(\mu)$ in the category of $G_1T$-modules. It restricts to an injective envelope of $L_1(\mu)$ in the category of $G_1$-modules. As in the case of the finite group $G$, it is isomorphic to a projective cover of $L_1(\mu)$ as well. We recall the following important result.

**THEOREM 4.2.1.** Assume that $p > 2(h-1)$ and that $G$ has no factors of type $A_1$. Then $\hat{Q}_1(\mu)$ has a unique $G$-module structure which will be denoted by $Q_1(\mu)$ in what follows. In particular, $\text{soc}_G Q_1(\mu)$ is isomorphic to $L(\mu)$ since $L(\mu)$ is the unique extension of $\hat{L}_1(\mu)$. Assume that $\mu \in X_1(T)$ is $h-2$-deep. Then

$$Q_1(\mu)|_G \cong \text{Proj}(F(\mu)) \cong \text{Inj}(F(\mu)). \qquad (4.2)$$

*Proof.* The first part of the theorem is well known; cf. [Jan03, II.11.11]. In what follows, we deduce the isomorphism (4.2) from [Pil93, Lemma 6.1] (where it is stated when $G$ is semisimple). It suffices to show that $Q_1(\mu)|_G$ is injective and its socle is $F(\mu)$.

**Claim 1.** Let $M$ be a $G$-module. Then $\text{soc}_{G^{\text{der}}}(M|_{G^{\text{der}}}) = \text{soc}_G(M)|_{G^{\text{der}}}$. The analogous statement holds true for $G_1T$ and $G_1^{\text{der}}T^{\text{der}}$ and for the finite groups $G$ and $G^{\text{der}}$.

*Proof of Claim 1.* Let $Z \stackrel{\text{def}}{=} G/G^{\text{der}}$ (respectively $Z_1 \stackrel{\text{def}}{=} G_1/G_1^{\text{der}}$). For $\bullet \in \{\emptyset, 1\}$, the group $Z_\bullet$ is diagonalizable, and, hence, by the Hochschild–Serre spectral sequence [Jan03, I.6.9(3)], the restriction functor $\text{Res}_{G_\bullet}^{G_\bullet}$ (which is exact) induces a canonical isomorphism

$$\text{Ext}_{G_\bullet}^i(M, N) \xrightarrow{\sim} H^0(Z_\bullet, \text{Ext}_{G_\bullet}^i(M|_{G_\bullet^{\text{der}}}, N|_{G_\bullet^{\text{der}}})) \qquad (4.3)$$

for all $i \in \mathbb{N}$ and all $G_\bullet$-modules $M, N$. Since the $G_\bullet$-restriction of an irreducible $G_\bullet$-module remains irreducible, we conclude that $\text{Res}_{G_\bullet}^{G_\bullet}$ commutes with the formation of socles and cosocles. By [Jan03, II.9.6 (11)], this implies the required statement for the groups $G_1T$, $G_1^{\text{der}}T^{\text{der}}$.
Since $Z \stackrel{\text{def}}{=} G/G^{\text{der}}$ has order prime to $p$, we also have a canonical isomorphism

$$ \mathrm{Ext}_{G}^{i}(M, N) \xrightarrow{\sim} \mathrm{H}^{0}(Z, \mathrm{Ext}_{G^{\text{der}}}^{i}(M|_{G^{\text{der}}}, N|_{G^{\text{der}}})) \quad (4.4) $$

for all $i \in \mathbb{N}$ and all $G$-modules $M, N$ which implies the statement in the case of finite groups. $\square$

**Claim 2.** Let $\nu$ be a $p$-restricted weight. Then $\hat{Q}_1(\nu)|_{G_1^{\text{der}}\underline{T}^{\text{der}}}$ is the injective envelope of $\hat{L}_1(\nu)|_{G_1^{\text{der}}\underline{T}^{\text{der}}}$ as a $G_1^{\text{der}}\underline{T}^{\text{der}}$-module.

**Proof of Claim 2.** By Claim 1, the socle of $\hat{Q}_1(\nu)|_{G_1^{\text{der}}\underline{T}^{\text{der}}}$ is isomorphic to $\hat{L}_1(\nu)|_{G_1^{\text{der}}\underline{T}^{\text{der}}}$. It suffices to prove injectivity. By [Jan03, II.9.4], it is enough to prove injectivity for the restriction to $G_1^{\text{der}}$. As $G_1^{\text{der}}$ and $G_1$ are both finite and $G_1^{\text{der}}$ is closed in $G_1$, we deduce that $\mathrm{Res}_{G_1^{\text{der}}}$ maps injectives to injectives (since it has an exact left adjoint, cf. [Jan03, I.3.5, I.8.16]). $\square$

We are now ready to prove (4.2). We show that $Q_1(\mu)|_G$ is an injective $G$-module with socle isomorphic to $F(\mu)$. By [Pil93, Lemma 6.1] (which also holds in the nonsplit case; cf. the final remark of [Pil93, Section 11]) and Claim 2, we have

$$ Q_1(\mu)|_{G^{\text{der}}} \cong \mathrm{Inj}(F(\mu)|_{G^{\text{der}}}) \quad (4.5) $$

so that $Q_1(\mu)|_{G^{\text{der}}}$ is an injective $G^{\text{der}}$-module with socle isomorphic to $F(\mu)|_{G^{\text{der}}}$.

We first show injectivity. The functor of $G/G^{\text{der}}$-invariants is exact on the category of $G/G^{\text{der}}$-representations. We hence obtain a canonical isomorphism

$$ \mathrm{Hom}_G(\bullet, Q_1(\mu)|_G) \cong (\mathrm{Hom}_{G'}(\bullet, Q_1(\mu)|_{G^{\text{der}}}))^{G/G^{\text{der}}}, $$

and $\mathrm{Hom}_G(\bullet, Q_1(\mu)|_G)$, being the composite of two exact functors, is therefore exact.

The socle of $Q_1(\mu)|_G$ contains a submodule isomorphic to $F(\mu)$ and its restriction to $G^{\text{der}}$ is isomorphic to $F(\mu)|_{G^{\text{der}}}$. Thus, the socle of $Q_1(\mu)|_G$ is isomorphic to $F(\mu)$. (We are grateful to the referee for simplifying the argument in our first version.) $\square$

Recall that an irreducible $G$-module $L(\kappa)$, with $\kappa \in X_+^*(T)$, is said to be $p$-bounded if $\langle \kappa, \alpha^\vee \rangle < 2(h-1)p$ for all coroots $\alpha \in \underline{R}^\vee$; a $G$-module is $p$-bounded if all its Jordan–Hölder factors are $p$-bounded. Similarly, a $G$-module is defined to be $m$-deep if the highest weights of all its Jordan–Hölder factors are $m$-deep. The following lemmas will be used several times in the rest of this section.
LEMMA 4.2.2 [Pil97, Lemma 3.1]. Let M be a $\mathcal{G}$-module. If M is $3(h-1)$-deep with p-bounded highest weight, then

$$ \mathrm{soc}_i^G(M)|_G = \mathrm{soc}_i^G(M)|_G, \quad \mathrm{rad}_G^i(M)|_G = \mathrm{rad}_G^i(M)|_G. $$

*Proof.* The statement on the socle filtration for $\mathcal{G}^{\mathrm{der}}$ and $\mathcal{G}^{\mathrm{der}}$ follows from the proof of [Pil97, Lemma 3.1]. While loc. cit. assumes that $\mathcal{G}_0^{\mathrm{der}}$ is split over $\mathbb{F}_p$, the proof applies setting $n = 1$ and using that $L(p\lambda_1)|_{\mathcal{G}^{\mathrm{der}}}$ is isomorphic to $L(\pi\lambda_1)|_{\mathcal{G}^{\mathrm{der}}}$ rather than $L(\lambda_1)|_{\mathcal{G}^{\mathrm{der}}}$. By duality, noting that M is $3(h-1)$-deep if and only if its linear dual $M^*$ is $3(h-1)$-deep and that $\mathrm{soc}^{\bullet}_{\ell\ell(M)-i}(M^*) = (M/\mathrm{rad}^i_\bullet(M))^*$ for $\bullet \in \{\mathcal{G}^{\mathrm{der}}, \mathcal{G}^{\mathrm{der}}\}$, we obtain the analogous statement for the radical filtration (recall that $\ell\ell(M)$ is the Loewy length of M). The general case follows from Claim 1 in the proof of Theorem 4.2.1. $\square$

COROLLARY 4.2.3. Let $\mu \in X_1(\mathcal{T})$ such that $Q_1(\mu)$ is $3(h-1)$-deep. Then

$$ \mathrm{soc}_i^G(Q_1(\mu))|_G = \mathrm{soc}_i^G(\mathrm{Inj}(F(\mu))), \quad \mathrm{rad}_G^i(Q_1(\mu))|_G = \mathrm{rad}_G^i(\mathrm{Inj}(F(\mu))). $$

*Proof.* This follows from Theorem 4.2.1 and Lemma 4.2.2. $\square$

If $\nu, \kappa \in X^*(\mathcal{T})$, we let $m_\kappa(\nu) \stackrel{\mathrm{def}}{=} \dim_{\mathbb{F}}(L(\kappa))_\nu$. Moreover, we write $\nu \in L(\kappa)$ as a shorthand for $(L(\kappa))_\nu \neq 0$.

LEMMA 4.2.4 (Translation principle). Assume that $p \ge 2(h-1)$. Let $\lambda, \xi \in X_1(\mathcal{T})$. Assume that for all weights $\nu \in L(\xi)$, the weights $\lambda + \nu$ belong to the same alcove as $\lambda$. Then we have the following isomorphism of $\mathcal{G}$-modules:

$$ (1) \quad L(\lambda) \otimes_{\mathbb{F}} L(\xi) = \bigoplus_{\nu \in L(\xi)} L(\lambda + \nu)^{\oplus m_{\xi}(\nu)}, $$

$$ (2) \quad Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi) = \bigoplus_{\nu \in L(\xi)} Q_1(\lambda + \nu)^{\oplus m_{\xi}(\nu)}. $$

*Proof.* We first prove item (1). The isomorphism holds upon restriction to $\mathcal{G}_1^{\mathrm{der}}$ by [Pil93, Lemma 5.1]. As the LHS and RHS of (1) have the same central characters, the $\mathcal{G}_1^{\mathrm{der}}$-isomorphisms extend to $\mathcal{G}$-isomorphisms by (4.3).

We now switch to item (2). By the same argument as above, we deduce from [Pil93, Lemma 5.1] a $\mathcal{G}_1\mathcal{T}$-equivariant isomorphism

$$ \hat{Q}_1(\lambda) \otimes_{\mathbb{F}} \hat{L}_1(\xi) \cong \bigoplus_{\nu \in \hat{L}_1(\xi)} \hat{Q}_1(\lambda + \nu)^{\oplus m_{\xi}(\nu)}. \quad (4.6) $$
By item (1) and the isomorphism $\mathrm{soc}_{\mathcal{G}}(Q_1(\lambda)) \cong L(\lambda)$ from Theorem 4.2.1, we have a $\mathcal{G}$-equivariant injection

$$ \begin{aligned} \mathrm{soc}_{\mathcal{G}} \left( \bigoplus_{\nu \in L(\xi)} Q_1(\lambda + \nu)^{\oplus m_{\xi}(\nu)} \right) &= \bigoplus_{\nu \in L(\xi)} L(\lambda + \nu)^{\oplus m_{\xi}(\nu)} \\ &= L(\lambda) \otimes_{\mathbb{F}} L(\xi) \hookrightarrow Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi). \end{aligned} \quad (4.7) $$

We claim that in the full subcategory of $p$-bounded $\mathcal{G}$-modules, the functor $\mathrm{Hom}_{\mathcal{G}}(\bullet, Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi))$ is exact. Granting the claim we deduce from (4.7) a $\mathcal{G}$-equivariant morphism

$$ \bigoplus_{\nu \in L(\xi)} Q_1(\lambda + \nu)^{\oplus m_{\xi}(\nu)} \hookrightarrow Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi) $$

which is injective since it is injective on socles. The morphism is hence an isomorphism by (4.6) (note that $\hat{L}_1(\xi)_v = L(\xi)_v$ for all $v \in X_1(T)$ since $\xi \in X_1(T)$) and item (2) follows.

We prove the claim. It will be enough to prove that for any irreducible, $p$-bounded $\mathcal{G}$-module $L(\kappa)$, one has

$$ \mathrm{Ext}^1_{\mathcal{G}}(L(\kappa), Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi)) = 0. \quad (4.8) $$

As $L(\kappa)$ is $p$-bounded, we can write $\kappa = \kappa^{(0)} + p\omega_\kappa$ where $\kappa^{(0)} \in X_1(T)$ and $\omega_\kappa \in X_+^*(T)$ satisfies $\langle \omega_\kappa, \alpha^\vee \rangle < 2(h-1)$ for all coroots $\alpha^\vee \in \mathbb{R}^\vee$. Recall that $Q_1(\lambda)|_{\mathcal{G}\mathcal{I}} \cong \hat{Q}_1(\lambda)$. As $\hat{Q}_1(\lambda)$ is injective as a $\mathcal{G}_1$-module, the Lyndon–Hochschild–Serre spectral sequence [Jan03, I.6.6(3), I.6.5(2)], together with (4.6), provides us with an isomorphism

$$ \begin{aligned} \mathrm{Ext}^1_{\mathcal{G}}(L(\kappa), Q_1(\lambda) \otimes_{\mathbb{F}} L(\xi)) & \\ & \cong \mathrm{Ext}^1_{\mathcal{G}/\mathcal{G}_1}(L(p\omega_\kappa), \mathrm{Hom}_{\mathcal{G}_1}(L_1(\kappa^{(0)}), \hat{Q}_1(\lambda) \otimes_{\mathbb{F}} L_1(\xi))) \\ & \cong \mathrm{Ext}^1_{\mathcal{G}/\mathcal{G}_1}\left(L(p\omega_\kappa), \mathrm{Hom}_{\mathcal{G}_1}\left(L_1(\kappa^{(0)}), \bigoplus_{\nu \in \hat{L}_1(\xi)} \hat{Q}_1(\lambda + \nu)\right)^{\oplus m_{\xi}(\nu)}\right) \\ & \cong \left\{ \begin{array}{ll} \mathrm{Ext}^1_{\mathcal{G}}\left(L(\omega_\kappa), L\left(\frac{\lambda + \nu - \kappa^{(0)}}{p}\right)\right)^{\oplus m_{\xi}(\nu)} & \text{if } \lambda + \nu - \kappa^{(0)} \in pX^0(T) \\ 0 & \text{else.} \end{array} \right. \end{aligned} $$

As $\langle \omega_\kappa, \alpha^\vee \rangle < 2(h-1) \le p-2$ for all coroots $\alpha^\vee \in \mathbb{R}^\vee$, it follows that $\omega_\kappa$ lies in the lower $p$-restricted alcove; in particular, there are no algebraic extensions between $L(\omega_\kappa)$ and $L(\omega)$ for any $\omega \in X^0(T)$. This establishes (4.8). $\square$
4.2.2. The case of GL<sub>3</sub>. We now describe the modules Q<sub>1</sub>(μ)|<sub>G</sub> in more detail in the case G<sub>0</sub> is Π<sub>˜v∈S</sub> Res<sub>k<sub>˜v</sub>/F<sub>p</sub></sub> GL<sub>3</sub> as in Section 1.4 and μ ∈ X<sub>1</sub>(T). We recall the alcove labeling for SL<sub>3</sub> in [Hum06, Section 13.9] and write

$$
\mathcal{A} \stackrel{\text{def}}{=} \{A, B, C, D, E, F, G, H, I, J\}.
$$

If $X \in \mathcal{A}$, let $\tilde{w}_X \in W_a$ be the unique element such that $\tilde{w}_X \cdot C_0 = X$. For $i \in \mathcal{J}$, we define $\tilde{w}_{X,i} \in W_a$ in the evident way. For $\underline{X} \in \mathcal{A}^{\mathcal{J}}$, we also define $\tilde{w}_{\underline{X}} \in W_a$ in the evident way. In what follows, we let $f \stackrel{\text{def}}{=} \#\mathcal{J}$.

Assume from now on that $p \ge 5$ and $\mu$ is 2-deep. Let $\mu^{\text{op}}$ be $\tilde{w}_B \cdot \mu$ and write $\mu^{\text{op}}$ as the sum $\sum_i \mu_i^{\text{op}}$. Let $Q_1(\mu_i)$ be the $\text{GL}_{3/\mathbb{F}}$-module defined in Theorem 4.2.1. It is rigid with Loewy length $\ell\ell(Q_1(\mu)) = 6+1$ and is endowed with a Weyl filtration (see [Jan03, II.11.13, II.11.5(5), II.4.19] and also [Hum06, Section 13.9]) with submodule $V_{\mu_i} \stackrel{\text{def}}{=} V(\mu_i^{\text{op}} + p\eta'_i)$ and the $\text{GL}_{3/\mathbb{F}}$-module obtained by extension of scalars from the Weyl module for $\text{GL}_{3/\mathbb{F}}$ with the highest weight $\mu_i^{\text{op}} + p\eta'_i$. Moreover, the socle filtration of $Q_1(\mu_i)$ (see [Hum06, Section 13.9] for a concise reference, [AM01, Proposition 8.4] and its proof and [Jan03, Section II.D.4]) and $V_{\mu_i}$ (cf. [BDM15, Section 4], and Table 6) is known. (The condition that $\mu$ is 2-deep is to guarantee that all $V_{\mu_i}$ has maximal length; their Loewy length in this case is 3 + 1.) In particular, one sees that $V_{\mu_i}$ is a multiplicity-free submodule of $Q_1(\mu_i)$. Then the $\mathcal{G}$-modules $Q_1(\mu)$ and $V_{\mu}$ are defined to be the tensor products $\otimes_i Q_1(\mu_i)$ and $\otimes_i V_{\mu_i}$, respectively.

The module $Q_1(\mu)$ is rigid with Loewy length $\ell\ell(Q_1(\mu)) = 6f+1$. The socle filtration on $Q_1(\mu)$ is the tensor product of the socle filtrations Fil on $Q_1(\mu_i)$ for $i \in \mathcal{J}$, and the graph $\Gamma(V_\mu)$ is the product $\prod_i \Gamma(V_{\mu_i})$. In particular, $V_\mu$ is rigid. Let Fil be the unique increasing Loewy series for $Q_1(\mu)$; its restriction to $V_\mu$ is the unique Loewy series for $V_\mu$.

Recall that an irreducible $\mathbb{F}[\mathcal{G}]$-module $F$ is said to be *n-deep* if we can write $F \cong L(\mu)|_{\mathbb{F}(\mathbb{F}_p)}$ for some $\mu \in X_1(T)$ which is *n-deep*. A $\mathbb{F}[\mathcal{G}]$-module is defined to be *n-deep* if all its Jordan–Hölder constituents are *n-deep*.

LEMMA 4.2.5. Let $\mu \in X^*(T)$, $n \in \mathbb{N}$, and $\tilde{w} \in \tilde{W}$. If $\mu$ is $n$-deep in alcove $a$, then $\tilde{w} \cdot \mu$ is $n$-deep in alcove $\tilde{w} \cdot a$. In particular, if $\mu \in X_1(T)$ is $n$-deep, then $Q_1(\mu)$ and $V_\mu$ are $n$-deep. If $\mu \in X_1(T)$ is $n+2$-deep, then $Q_1(\mu)|_G$ and $V_\mu|_G$ are all $n$-deep.

*Proof.* The first two claims are easy. To prove the final claim, first note that the Jordan–Hölder factors of $Q_1(\mu)|_G$ and $V_\mu|_G$ are the same, and so it suffices to prove the claim for $L|_G$ where $L \in \text{JH}(V_\mu)$. Suppose that $L$ is isomorphic to
$L(\lambda)$ and $\lambda = \lambda^0 + p\omega_\lambda$. Then

$$L|_{\mathbb{G}} \cong L(\lambda^0 + \pi \omega_{\lambda}) \cong \bigoplus_{\varepsilon \in L(\pi \omega_{\lambda})} L(\lambda + \varepsilon)^{m_{\pi \omega_{\lambda}}(\varepsilon)}$$

by Lemma 4.2.4(1). The result now follows from the fact that $|\langle\varepsilon, \alpha^{\vee}\rangle| \le 2$ for any $\varepsilon \in L(\pi\omega_\lambda)$ and any positive root $\alpha \in \mathbb{R}^+$ of a simple factor of $\mathfrak{G}$. $\square$

Assume that $\mu$ is 6-deep. By Corollary 4.2.3, the socle filtration on $\mathrm{Inj}(F(\mu))$ is given by $(\mathrm{Fil}_n Q_1(\mu))|_{\mathbb{G}}$. Since $\mathrm{Inj}(F(\mu))$ is rigid (it is isomorphic to $\mathrm{Proj}(F(\mu))$), this is the unique increasing Loewy series. One can use Lemma 4.2.4 to compute $\mathrm{gr}_n \mathrm{Inj}(F(\mu))$. We do this in the case $n = 1$ to compute $\mathbb{G}$-extensions, justifying the name ‘extension graph’ introduced in Section 2.

**LEMMA 4.2.6.** Assume that $\mu_i$ is 6-deep for all $i$ and let $\sigma \stackrel{\mathrm{def}}{=} F(\mu)$. Then $\mathrm{d}_{\mathrm{gph}}(\sigma, \kappa) = 1$ if and only if $\mathrm{Ext}^1_{\mathbb{G}}(\kappa, \sigma) \ne 0$, in which case the dimension of the $\mathrm{Ext}^1$ group is 1.

*Proof.* Since $\mu$ is 6-deep, $Q_1(\mu)$ is 6-deep by Lemma 4.2.5 so that $Q_1(\mu)|_{\mathbb{G}} \cong \mathrm{Inj}(F(\mu))$ by Corollary 4.2.3. It suffices to show that $[\mathrm{gr}_1 Q_1(\mu)|_{\mathbb{G}} : \kappa] \le 1$ and that $\mathrm{d}_{\mathrm{gph}}(\sigma, \kappa) = 1$ if and only if $[\mathrm{gr}_1 Q_1(\mu)|_{\mathbb{G}} : \kappa] = 1$. Note that $\mathrm{gr}_1 Q_1(\mu) = \bigoplus_i (\mathrm{gr}_1 Q_1(\mu_i)) \otimes \bigotimes_{j \ne i} L(\mu_j)$. Let $\tilde{w}_\mu$ be the element of $\mathcal{W}_a$ so that $\lambda \stackrel{\mathrm{def}}{=} \tilde{w}_\mu^{-1} \cdot \mu$ is in $\mathcal{A}$. The length of $\mathrm{gr}_1 Q_1(\mu)$ is 3f with Jordan–Hölder factors of the form $L(\tilde{w} \cdot \mu)$ for 3f choices of $\tilde{w}$. Writing $\tilde{w}$ as $t_{\omega_-} \tilde{w}_+$ with $\tilde{w}_+ \in \tilde{W}_1^+$, the 3f choices of $\tilde{w}$ correspond to $\omega_- = 0$, $\varepsilon'_{1,i}$, or $\varepsilon'_{2,i}$ for some $i \in \mathcal{J}$, with $\tilde{w}_+$ the unique element in $\tilde{W}_1^+$ so that $t_{\omega_-} \tilde{w}_+$ is in $\mathcal{W}_a$ and $\tilde{w}_+ \tilde{w}_\mu \cdot \mathcal{A} = \tilde{w}_{B,i} \tilde{w}_\mu \cdot \mathcal{A}$.

It suffices to show that

$$L(\tilde{w} \cdot \mu)|_{\mathbb{G}}$$

is multiplicity free and contains exactly the weights of the form $F(\mathfrak{Tr}_{\lambda+\eta}(\omega, \pi(\tilde{w}_{B,i}\tilde{w}_\mu \cdot \mathcal{A})))$ with $\omega$ a permutation of $\pi\omega_-$. (Here, $\omega$ as the first argument of $\mathfrak{Tr}_{\lambda+\eta}$ is understood to be the image of $\omega$ in $\mathcal{A}_W$.)

We have isomorphisms

$$\begin{align}
L(\tilde{w} \cdot \mu)|_{\mathbb{G}} &\cong L(\tilde{w}_+ \cdot \mu) \otimes L(\pi \omega_{-,i})|_{\mathbb{G}} \nonumber \\
&\cong \bigoplus_{\omega \in L(\pi \omega_{-,i})} F(\tilde{w}_+ \cdot \mu + \omega)^{\oplus m_{\pi \omega_{-,i}}(\omega)}, \tag{4.9}
\end{align}$$

where the second isomorphism follows from Lemma 4.2.4. On the other hand, the pair $(\omega, \pi(\tilde{w}_{+,i}\tilde{w}_\mu \cdot \mathcal{A}))$ is $\beta(\omega, \tilde{w}_{+,i}\tilde{w}_\mu)$ with $\beta$ as in Lemma 2.1.1 since we have that $t_\omega \pi(\tilde{w}_{+,i}\tilde{w}_\mu) \in \mathcal{W}_a$. Then by definition, we have

$$F(\mathfrak{Tr}_{\lambda+\eta}(\omega, \pi(\tilde{w}_{+} \tilde{w}_{\mu} \cdot \mathcal{A}))) \cong F(\tilde{w}_{+} \tilde{w}_{\mu} \cdot (\lambda+\omega)) \cong F(\tilde{w}_{+} \cdot \mu + w_{+} w_{\mu} \omega), \quad (4.10)$$
where $w_+w_\mu$ is the image of $\tilde{w}_+\tilde{w}_\mu$ in $W$. We conclude by combining (4.9) and (4.10). $\square$

**4.2.3.** *Study of the Weyl module $V_\mu|_G$.* We now assume that $\mu \in X_1(T)$ is such that $\mu \in \bar{B}$ and push the analysis in Lemma 4.2.6 further to describe $V_\mu|_G$ in this case. Recall that $\underline{G}_0 \stackrel{\text{def}}{=} \text{Res}_{\bar{k}_{\bar{\nu}}/\mathbb{F}_p} \text{GL}_3$ and that $\underline{G} \stackrel{\text{def}}{=} \underline{G}_0 \times_{\mathbb{F}_p} \mathbb{F}$. Recall that $\underline{T} \subset \underline{G}$ is the diagonal torus, and $\underline{\Delta}_W$ is the weight lattice of $\underline{G}^{\text{der}}$. Let $\underline{\Delta}_{\eta'} \subseteq \underline{\Delta}_W$ be the convex hull of the $\underline{W}$-orbit of $\underline{\eta'}$. Explicitly, we have

$$ \underline{\Delta}_{\eta'} = \{ (v_i) \in \underline{\Delta}_W, v_i \in \{0, \pm\varepsilon_{1,i}, \pm\varepsilon_{2,i}, \pm\eta'_i, \pm(\varepsilon_{1,i} - \varepsilon_{2,i}), \\ \qquad \pm(2\varepsilon_{1,i} - \varepsilon_{2,i}), \pm(\varepsilon_{1,i} - 2\varepsilon_{2,i})\} \}. $$

We define the subgraph $\underline{\Delta}_{\le(\eta',0)} \subseteq \underline{\Delta}_{\eta'} \times \mathcal{A}$ as follows:

$$ \underline{\Delta}_{\le(\eta',0)} \stackrel{\text{def}}{=} \{ (\omega, a) \in \underline{\Delta}_{\eta'} \times \mathcal{A} : a_i = 0 \text{ if } \omega_i = w\eta'_i \text{ for some } w \in W \}. $$

The main result concerning $V_\mu$ is the following.

**THEOREM 4.2.7.** Let $\mu \in X_1(T)$ be a $p$-restricted weight such that $\mu$ is 2-deep in alcove $\bar{B}$.

(1) The translation map $\mathfrak{Tr}_{\mu^{\text{op}}+\eta} : \underline{\Delta}_W^{(\mu^{\text{op}}+\eta)} \times \mathcal{A} \to X_1(\underline{T})/\langle(p-\pi)X^0(\underline{T})\rangle$ induces a bijection:

$$ \begin{gathered} \mathfrak{Tr}_{V_\mu} : \underline{\Delta}_{\le(\eta',0)} \to \text{JH}(V_\mu|_G) \\ (\omega, a) \mapsto \sigma_{(\omega,a)}. \end{gathered} $$

(2) We have

$$ [\text{gr}_d(V_\mu)|_G : \sigma_{(\omega,a)}] = \begin{cases} 1 & \text{if } d = \text{d}_{\text{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)}) \\ 0 & \text{if } d < \text{d}_{\text{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)}) \end{cases} $$

for all $(\omega, a) \in \underline{\Delta}_{\le(\eta',0)}$.

(3) Assume that $\mu$ is 6-deep. Then there exists a $G$-submodule $U \subset V_\mu|_G$ such that:

(a) $U$ is multiplicity free;

(b) $\text{JH}(U) = \text{JH}(V_\mu|_G)$;

(c) if we denote the restriction $\text{Fil}|_U$ by $\text{Fil}$, for any $\sigma \in \text{JH}(U)$, we have $[\text{gr}_d(U) : \sigma] = 1$ if and only if $d = \text{d}_{\text{gph}}(\sigma_{(0,1)}, \sigma)$; and
(d) if $(\omega, a)$, $(\omega', a') \in \underline{\Lambda}_{\leq(\eta',0)}$, then $\sigma_{(\omega,a)}$ points to $\sigma_{(\omega',a')}$ (with respect to $U$) if and only if $\mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)}) \geq \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega',a')})$, and $\mathrm{d}_{\mathrm{gph}}(\sigma_{(\omega,a)}, \sigma_{(\omega',a')}) = 1$. In particular, $\Gamma(U)^{\mathrm{op}}$ (that is, the graph obtained from $\Gamma(U)$ by reversing the direction of the edges) is predicted by the extension graph with respect to $\sigma_{(0,1)}$.

Proof of 4.2.7(1) and (2). We first show that the image of $\mathfrak{Tr}_{V_\mu}$ contains $\mathrm{JH}(V_\mu|_G)$. Let $\tilde{w} \in \underline{W}_a$ such that $[V_\mu : L(\tilde{w} \cdot \mu)] \neq 0$. It suffices to show that $\mathfrak{Tr}_{V_\mu}$ contains

$$ \mathrm{JH}(L(\tilde{w} \cdot \mu)|_G). \qquad (4.11) $$

The proof is similar to that of Lemma 4.2.6. There is a decomposition $\tilde{w} = t_{\omega_-} \tilde{w}_+$ where $\tilde{w}_+ \in \tilde{W}_1^+$. Again, $L(\tilde{w} \cdot \mu)|_G$ is isomorphic to $L(\tilde{w}_+ \cdot \mu) \otimes L(\pi\omega_-)|_G$, which is isomorphic to

$$ \bigoplus_{\epsilon \in L(\pi \omega_-)} F(\tilde{w}_+ \cdot \mu + \epsilon)^{\oplus m_{\pi \omega_-}(\epsilon)} $$

by Lemma 4.2.4. As in the proof of Lemma 4.2.6, the summand $F(\tilde{w}_+ \cdot \mu + \epsilon)$ is $\sigma_{(w_0\epsilon, \pi(\tilde{w}_+ \cdot \tilde{w}_h \cdot A))}$. Then (4.11) is contained in the image of $\mathfrak{Tr}_{V_\mu}$ by an analysis of the weights of $L(\pi\omega_-)$. Indeed, $\tilde{w} \cdot \mu$ is in one of the alcoves in the set $\{A, B, C, D, E, F, G\}^{\mathcal{J}}$ so that $\omega_{-,i}$ can be taken to be one of 0, $\epsilon_{1,i}$, $\epsilon_{2,i}$, and $\eta'_i$ for all $i$, and if $\omega_{-,i}$ is $\eta'_i$, then $\tilde{w}_{+,i} = \tilde{w}_{B,i}$.

Item (1) follows from (2) and the above paragraph. We now prove item (2). With $\tilde{w}$ as above, we define $n_i(\tilde{w}_i) \in \mathbb{N}$ by

$$ [\mathrm{gr}_{n_i(\tilde{w}_i)}(V_{\mu_i}) : L(\tilde{w}_i \cdot \mu_i)] \neq 0. $$

Let $0 \leq d \leq 3f$. Since $V_\mu$ is multiplicity free, it suffices to show that

$$ \bigoplus_{\sum_i n_i(\tilde{w}_i)=d} L(\tilde{w} \cdot \mu)|_G \qquad (4.12) $$

contains weights of the form $\sigma_{(\omega,a)}$ with multiplicity one if $(\omega, a) \in \underline{\Lambda}_{\leq(\eta',0)}$ and the distance between $(\omega_i, a_i)$ and $(0_i, 1_i)$ is $n_{\pi^{-1}i}(\tilde{w}_{\pi^{-1}i})$ for all $i$. If $\nu \in L(\pi\omega_-)$ is a permutation of $\pi\omega_-$, then it appears in $L(\pi\omega_-)$ with multiplicity one. Thus, $\sigma_{(\epsilon, \pi(\tilde{w}_+ \cdot \tilde{w}_h) \cdot A)}$ appears in (4.12) with multiplicity one. A casewise analysis, using the fifth column of Table 5 and the description of the socle layers of $V_{\mu_i}$ in [BDM15], shows that the distance from $(\nu_i, \pi(\tilde{w}_{+,\pi^{-1}i} \cdot \tilde{w}_{B,\pi^{-1}i}) \cdot A)$ to $(0_i, 1_i)$ is $n_{\pi^{-1}i}(\tilde{w}_{\pi^{-1}i})$. For example, if $n_{\pi^{-1}i}(\tilde{w}_{\pi^{-1}i}) = 2$, then $\omega_{-,\pi^{-1}i}$ is $\epsilon'_{1,\pi^{-1}i}$ or $\epsilon'_{2,\pi^{-1}i}$ and $\tilde{w}_{+,\pi^{-1}i}$ is trivial. Then $\nu_i$ is a permutation of $\epsilon_{1,i}$ or $\epsilon_{2,i}$. $\square$

We now move to the proof of Theorem 4.2.7(3). We start with the following preliminary lemma.
<table><caption>Table 5. Comparison between alcoves $C_i$, affine Weyl group elements $\tilde{w}_i$ and the graph.</caption><thead><tr><th>$\tilde{w}_i$</th><th>$\tilde{w}_{+,i}$</th><th>$\omega_{-,i}$</th><th>$\tilde{w}_i \cdot A$</th><th>$(\omega, a)_{\pi i}$</th></tr></thead><tbody><tr><td>id</td><td>id</td><td>$t_0$</td><td>A</td><td>(0, 0)</td></tr><tr><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>$t_0$</td><td>B</td><td>(0, 1)</td></tr><tr><td>(123)$t_{-\alpha_2}$</td><td>(123)$t_{-\varepsilon'_1}$</td><td>$t_{\varepsilon'_2}$</td><td>C</td><td>$(s(\varepsilon'_2), 0)$; $s \in S_3$</td></tr><tr><td>(12)$t_{\alpha_2}$</td><td>(12)$t_{-(\varepsilon'_2-\varepsilon'_2-1)}$</td><td>$t_{\varepsilon'_2}$</td><td>E</td><td>$(s(\varepsilon'_2), 1)$; $s \in S_3$</td></tr><tr><td>(132)$t_{-\alpha_1}$</td><td>(132)$t_{-\varepsilon'_1}$</td><td>$t_{\varepsilon'_1}$</td><td>D</td><td>$(s(\varepsilon'_1), 0)$; $s \in S_3$</td></tr><tr><td>(23)$t_{\alpha_1}$</td><td>(23)$t_{-(\varepsilon'_2-\varepsilon'_2+1)}$</td><td>$t_{\varepsilon'_1}$</td><td>F</td><td>$(s(\varepsilon'_1), 1)$; $s \in S_3$</td></tr><tr><td>(13)$t_{-2(\varepsilon'_1+\varepsilon'_2)}$</td><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>$t_{(\varepsilon'_1+\varepsilon'_2)}$</td><td>J</td><td>$(s(\varepsilon'_1+\varepsilon'_2), 1)$; $s \in S_3$ and (0, 1)</td></tr></tbody></table><table><thead><tr><th>$\tilde{w}_i$</th><th>$\tilde{w}_{i,+}$</th><th>$\tilde{w}_{i,-}$</th><th>$\tilde{w}_i \cdot B$</th><th>$(\omega, a)_{\pi i}$</th></tr></thead><tbody><tr><td>id</td><td>id</td><td>$t_0$</td><td>B</td><td>(0, 1)</td></tr><tr><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>$t_0$</td><td>A</td><td>(0, 0)</td></tr><tr><td>(132)$t_{-(2\alpha_1+\alpha_2)}$</td><td>(132)$t_{-(2\varepsilon'_1-1)}$</td><td>$t_{\varepsilon'_2}$</td><td>E</td><td>$(s(\varepsilon'_2), 1)$; $s \in S_3$</td></tr><tr><td>(23)$t_{-\alpha_2}$</td><td>(23)$t_{-\varepsilon'_1}$</td><td>$t_{\varepsilon'_2}$</td><td>C</td><td>$(s(\varepsilon'_2), 0)$; $s \in S_3$</td></tr><tr><td>(123)$t_{-(\alpha_1+2\alpha_2)}$</td><td>(123)$t_{-(2\varepsilon'_2+1)}$</td><td>$t_{\varepsilon'_1}$</td><td>F</td><td>$(s(\varepsilon'_1), 1)$; $s \in S_3$</td></tr><tr><td>(12)$t_{-\alpha_1}$</td><td>(12)$t_{-\varepsilon'_1}$</td><td>$t_{\varepsilon'_1}$</td><td>D</td><td>$(s(\varepsilon'_1), 0)$; $s \in S_3$</td></tr><tr><td>(13)$t_{-2(\varepsilon'_1+\varepsilon'_2)}$</td><td>(13)$t_{-(\varepsilon'_1+\varepsilon'_2)}$</td><td>$t_{(\varepsilon'_1+\varepsilon'_2)}$</td><td>G</td><td>$(s(\varepsilon'_1+\varepsilon'_2), 0)$; $s \in S_3$ and (0, 0)</td></tr></tbody></table>

For each element $\tilde{w}^*$ in the first column, we consider the decomposition $\tilde{w}_i = \omega_{i,-} \cdot \tilde{w}_{i,+}$ with $(\tilde{w}_{i,+})_i \in \tilde{W}^+$. In the fourth column, we write the alcove containing $\tilde{w}_i$. In the fifth column, we write the $\pi$th coordinate of the points in the graph (with origin $\mu + \eta'$) corresponding to an irreducible constituent of $(\otimes_i L(\tilde{w}_i \cdot \mu_i))|_G$. Similar comments apply to the second half of the table. Note that in this case, we consider the decomposition $\tilde{w}_i = \omega_{i,-} \cdot \tilde{w}_{+,i}$ with $\tilde{w}_{+,i} \in \tilde{W}_1^+ \tilde{w}_B$ and the graph has origin in $\mu^{op} + \eta$. Finally, we have set $\alpha_1$ and $\alpha_2 \in X^*(T)$ to be $(1, -1, 0)$ and $(0, 1, -1)$, respectively.

**LEMMA 4.2.8.** Let $\mu \in X_1(T)$ be a $p$-restricted weight which is 2-deep in alcove $\underline{B}$. There exists a $G$-submodule $U \subset V_\mu|_G$ such that $\mathrm{JH}(U) = \mathrm{JH}(V_\mu|_G)$ and

$$ [\mathrm{gr}_d(U)|_G : \sigma_{(\omega,a)}] = \begin{cases} 1 & \text{if } d = \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)}) \\ 0 & \text{if } d \neq \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)}), \end{cases} \quad (4.13) $$

where gr is with respect to Fil $\stackrel{\mathrm{def}}{=} \mathrm{Fil}|_U$. In particular, $U$ is multiplicity free.

*Proof.* The notation in this proof is complicated by necessity. To illustrate the simple underlying idea, we first present the proof in the case that $f = \#\mathcal{J} = 1$. We have the following:

$$ \mathrm{gr}_3(V_\mu) \cong L(\tilde{w}_G \tilde{w}_B \cdot \mu) $$

$$ \mathrm{gr}_2(V_\mu) \cong L(\tilde{w}_E \tilde{w}_B \cdot \mu) \oplus L(\tilde{w}_F \tilde{w}_B \cdot \mu) $$
<figure><img src="image_10.png" /><figcaption>Table 6. Graph of $V(\mu^{\text{op}} + p\eta')$.</figcaption></figure>

The graphs of $V(\mu^{\text{op}} + p\eta')$ when $f = 1$ (and $\mu$ is 2-deep). In the first (respectively second) diagram, $\mu \in X_1(\mathcal{T})$ is upper alcove (respectively lower alcove). For each alcove, we write below the unique weight $L$ such that $L(x, y, z) \uparrow L$. We remark that $\text{gr}_1(V(\mu^{\text{op}} + p\eta') = \text{gr}_1(Q_1(\mu))$ if $\mu = (x, y, z)$ is upper alcove, while $\text{gr}_1(V(\mu^{\text{op}} + p\eta') \oplus L(\mu^{\text{op}}) = \text{gr}_1(Q_1(\mu))$ if $\mu = (x, y, z)$ is lower alcove.

$$ \text{gr}_1(V_\mu) \cong L(\tilde{w}_C \tilde{w}_B \cdot \mu) \oplus L(\tilde{w}_B \cdot \mu) \oplus L(\tilde{w}_D \tilde{w}_B \cdot \mu) $$

$$ \text{gr}_0(V_\mu) \cong L(\mu). $$

For the alcove $a$, there is a decomposition $\tilde{w}_a$ as the product $t_{\omega_{a,-}} \tilde{w}_{a,+}$ where $\tilde{w}_{a,+} \in \tilde{W}_1^+$ and $\omega_{a,-} \in X^*(T)$. Let $\lambda$ be $\tilde{w}_B \cdot \mu$. Then

$$ \begin{align*} L(\tilde{w}_a \tilde{w}_B \cdot \mu)|_G &\cong L(\tilde{w}_a \cdot \lambda) \cong L(\tilde{w}_{a,+} \cdot \lambda + p\omega_{a,-})|_G \\ &\cong (L(\tilde{w}_{a,+} \cdot \lambda) \otimes L(p\omega_{a,-}))|_G \\ &\cong (L(\tilde{w}_{a,+} \cdot \lambda) \otimes L(\omega_{a,-}))|_G \\ &\cong \bigoplus_{\omega \in L(\omega_{a,-})} L(\tilde{w}_{a,+} \cdot \lambda + \omega)|_G^{\oplus m_{\omega_{a,-}}(\omega)} \\ &\cong \bigoplus_{\omega \in L(\omega_{a,-})} F(\tilde{\mathfrak{r}}_{\lambda+\eta}(\omega, \bar{a}))^{\oplus m_{\omega_{a,-}}(\omega)}, \end{align*} $$

where $\bar{a}$ is the unique $pX^*(T)$-translate of $a$ lying in $\{A, B\}$. The third isomorphism above follows from the Steinberg tensor product theorem, and the fifth isomorphism follows from Lemma 4.2.4(1). We see then that $\text{gr}_1(V_\mu)|_G$ is isomorphic to the multiplicity-free direct sum of $F(\tilde{\mathfrak{r}}_{\lambda+\eta}(\omega, 0))$ where $\omega$ is a permutation of 0, $\varepsilon_1$, or $\varepsilon_2$. Similarly, $\text{gr}_2(V_\mu)|_G$ is isomorphic to the multiplicity-free direct sum of $F(\tilde{\mathfrak{r}}_{\lambda+\eta}(\omega, 1))$ where $\omega$ is a permutation of $\varepsilon_1$ or $\varepsilon_2$. Finally,
$\mathrm{gr}_3(V_\mu)|_G$ is isomorphic to $\bigoplus_{\omega \in L(\eta')} F(\mathfrak{tr}_{\lambda+\eta}(\omega, 0))^{m_{\eta'}(\omega)}$. Then, we can take $U$ to be the preimage of $\bigoplus_{\omega \in L(\eta')} F(\mathfrak{tr}_{\lambda+\eta}(\omega, 0))$ in $V_\mu|_G$. It is then easy to check that $U$ satisfies the required properties.

We now proceed to the general case. Let $i \in \mathcal{J}$. We have an exact sequence of $\mathcal{G}$-modules

$$0 \to \mathrm{rad}(V_{\mu_i}) \to V_{\mu_i} \xrightarrow{p_i} L(\mu_i^{\mathrm{op}} + p\eta_i') \to 0,$$

which gives the exact sequence

$$0 \to \mathrm{rad}(V_{\mu_i}) \otimes \bigotimes_{j \neq i} V_{\mu_j} \to V_{\mu} \to L(\mu_i^{\mathrm{op}} + p\eta_i') \otimes \bigotimes_{j \neq i} V_{\mu_j} \to 0.$$

Then $L(\mu_i^{\mathrm{op}} + p\eta_i') \otimes \bigotimes_{j \neq i} V_{\mu_j}|_G$ is isomorphic to

$$L(\mu_i^{\mathrm{op}}) \otimes (V_{\mu_{\pi i}} \otimes L(\eta_{\pi i}')) \otimes \bigotimes_{j \neq i, \pi i} V_{\mu_j}|_G.$$

Then we claim that

$$V_{\mu_{\pi i}} \otimes L(\eta'_{\pi i}) \cong \bigoplus_{\omega \in L(\eta'_{\pi i})} V_{\mu_{\pi i} + \omega}^{\oplus m_{\eta'_{\pi i}}(\omega)}$$

as $\mathcal{G}$-representations. We have an embedding

$$V_{\mu_{\pi i}} \otimes L(\eta'_{\pi i}) \hookrightarrow Q_1(\mu_{\pi i}) \otimes L(\eta'_{\pi i}) \cong \bigoplus_{\omega \in L(\eta'_{\pi i})} Q_1(\mu_{\pi i} + \omega)^{\oplus m_{\eta'_{\pi i}}(\omega)}, \quad (4.14)$$

where the last isomorphism follows from Lemma 4.2.4. Since

(1) $V_{\mu_{\pi i}} \otimes L(\eta'_{\pi i})$ contains all the Jordan–Hölder factors of the right-hand side of (4.14) with highest weights in alcove $G$ (using Lemma 4.2.4) and

(2) $\bigoplus_{\omega \in L(\eta'_{\pi i})} V_{\mu_{\pi i} + \omega}^{\oplus m_{\eta'_{\pi i}}(\omega)}$ is the minimal submodule of the right-hand side of (4.14) containing all the Jordan–Hölder factors (counted with multiplicities) with highest weights in alcove $G$ (using the cosocle filtration of $V_{\mu_{\pi i} + \omega}$),

there is an injective map $\bigoplus_{\omega \in L(\eta'_{\pi i})} V_{\mu_{\pi i} + \omega}^{\oplus m_{\eta'_{\pi i}}(\omega)} \hookrightarrow V_{\mu_{\pi i}} \otimes L(\eta'_{\pi i})$ which must be an isomorphism since the domain and codomain have the same length.

Let $U_i$ be the preimage of

$$\bigoplus_{\omega \in L(\eta_i'), \omega \neq 0} L(\mu_i^{\mathrm{op}}) \otimes V_{\mu_{\pi i} + \omega} \otimes \bigotimes_{j \neq i, \pi i} V_{\mu_j}|_G$$
in $V_\mu|_G$. Let $U$ be the intersection $\cap_i U_i$. We claim that $U$ has the desired properties. If $d < \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)})$, then (4.13) holds by Theorem 4.2.7(2). If $d = \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)})$, then $[\mathrm{gr}_d(V_\mu|_G) : \sigma_{(\omega,a)}] = 1$ and $[\mathrm{gr}_d(U_i) : \sigma_{(\omega,a)}] = 1$ for all $i$ by the proof of Theorem 4.2.7(2). We conclude that $[\mathrm{gr}_d(U) : \sigma_{(\omega,a)}] = 1$.

We now suppose that $d > \mathrm{d}_{\mathrm{gph}}(\sigma_{(0,1)}, \sigma_{(\omega,a)})$ and use the notation of the proof of Theorem 4.2.7(2). Suppose that $\sigma_{(\omega,a)}$ is a Jordan–Hölder factor of $L(\tilde{w} \cdot \mu)|_G \cap \mathrm{gr}_d(U)$. Then there is some $i$ such that $n_{\pi^{-1}i}(\tilde{w}_{\pi^{-1}i}) > d_i$ where $d_i$ is the distance from $(\omega_i, a_i)$ to $(0_i, 1_i)$. More precisely, $n_{\pi^{-1}i}(\tilde{w}_{\pi^{-1}i})$ is 3 and $(\omega_i, a_i)$ is $(0_i, 0_i)$. However, one can check as in the proof of Theorem 4.2.7(2) that $L(\tilde{w} \cdot \mu)|_G \cap \mathrm{gr}_d(U_{\pi^{-1}i})$ does not contain any weight of the form $\sigma_{(\omega,a)}$ with $(\omega_i, a_i) = (0_i, 0_i)$. $\square$

**PROPOSITION 4.2.9.** Let $\lambda, \theta$ be 6-deep in an alcove in $\{A, B, C, D, E, F, G\}^{\mathcal{J}}$. Then $\mathrm{Ext}_G^1(L(\theta), L(\lambda))$ is at most one-dimensional, and it is one-dimensional if and only if $\lambda$ and $\theta$ are linked and lie in adjacent alcoves (that is, there exists $i_0 \in \mathcal{J}$ such that $\lambda_i = \theta_i$ for all $i \neq i_0$ and $\lambda_{i_0}$ and $\theta_{i_0}$ lie in different alcoves sharing a face).

*Proof.* We immediately reduce to the case of $\mathrm{GL}_3$. Using (4.3), it suffices to consider the case of $\mathrm{SL}_3$, where the result follows from [And87, Section 4.1]. $\square$

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

Let $\mu \in X_1(\mathcal{T})$ be a $p$-restricted weight 6-deep in alcove $\mathcal{B}$. Let $\tilde{w}, \tilde{y} \in \mathcal{W}_a$ be elements such that the alcoves containing $\lambda \stackrel{\mathrm{def}}{=} \tilde{w} \cdot \mu, \theta \stackrel{\mathrm{def}}{=} \tilde{y} \cdot \mu$ are in $\{A, B, C, D, E, F, G\}^{\mathcal{J}}$ and $\mathrm{Ext}_G^1(L(\theta), L(\lambda)) \neq 0$. Note that $\lambda, \theta$ are both 6-deep in their alcove by Lemma 4.2.5.

Let $i_0 \in \mathcal{J}$ be as in Proposition 4.2.9. Let $M$ be a nonsplit extension of $L(\theta)$ by $L(\lambda)$, which is unique up to isomorphism by Proposition 4.2.9.

**PROPOSITION 4.2.10.** If $F_0$ and $F_1$ are in the socle and cosocle of $M|_G$, respectively, with $[M|_G : F_0] = 1 = [M|_G : F_1]$ and $\mathrm{d}_{\mathrm{gph}}(F_0, F_1) = 1$, then there is a subquotient of $M|_G$ which is a nonsplit extension, unique up to isomorphism by Lemma 4.2.6, of $F_1$ by $F_0$.

**Proposition 4.2.10** will be proven in several steps. Let $\lambda$ be $\lambda^0 + p\omega_\lambda$ and $\theta = \theta^0 + p\omega_\theta$ so that $\lambda^0$ and $\theta^0$ are in $X_1(\mathcal{T})$ and $\omega_\lambda$ and $\omega_\theta$ belong to $\{0, \varepsilon'_{1,i}, \varepsilon'_{2,i}, \eta'_i\}^{\mathcal{J}}$. We begin with an algebraization lemma.

**LEMMA 4.2.11.** With $M$ and $i_0$ defined as above, assume, moreover, that $\lambda_i$ is $p$-restricted for all $i \neq i_0$. Then there is an injection $M \hookrightarrow Q_1(\lambda^0) \otimes L(p\omega_\lambda) \cong (\otimes_{i \neq i_0} Q_1(\lambda_i)) \otimes Q_1(\lambda_{i_0}^0) \otimes L(p\omega_\lambda)$ whose restriction to $G$ is an injective hull.
**Proof.** The proof follows closely the argument of [Pil97, Lemma 3.1] (which is, in turn, based on [And87, Lemma 2.2]). There is an injection

$$ \mathrm{soc} M \cong L(\lambda^0) \otimes L(p\omega_\lambda) \hookrightarrow Q_1(\lambda^0) \otimes L(p\omega_\lambda). $$

This extends to an injection $M \hookrightarrow Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ since

$$ \mathrm{Ext}^1_G(L(\theta), Q_1(\lambda^0) \otimes L(p\omega_\lambda)) = 0 $$

by the proof of (4.8). Both $Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ and $M$ are 6-deep by Lemma 4.2.5. By Lemma 4.2.2, the restriction to $G$ of the map $M \hookrightarrow Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ is an isomorphism on socles and is, thus, essential. The restriction $Q_1(\lambda^0) \otimes L(p\omega_\lambda)|_G$ is an injective object by Lemma 4.2.4 and Theorem 4.2.1. $\square$

We begin with the following special case of Proposition 4.2.10, from which the general case will follow by the translation principle.

**PROPOSITION 4.2.12.** If $\lambda_i$ is $p$-restricted for all $i \neq i_0$, the conclusion of Proposition 4.2.10 holds.

**Proof.** We assume that $\mathrm{d}_{\mathrm{gph}}(F_0, F_1) = 1$. Fix nonzero maps $\mathrm{Proj}(F_1) \to M|_G$ and $M|_G \to \mathrm{Inj}(F_0)$, unique up to scalar. If suffices to show that the composition

$$ \mathrm{Proj}(F_1) \to M|_G \to \mathrm{Inj}(F_0) \qquad (4.15) $$

is nonzero. The first map factors through $\mathrm{Proj}(F_1)/\mathrm{rad}^2(\mathrm{Proj}(F_1))$ since the Loewy length of $M|_G$ is 2. The second map factors as $M|_G \hookrightarrow \mathrm{Inj}(M|_G) \to \mathrm{Inj}(F_0)$ where the second map is a projection to a direct summand. Applying $\mathrm{gr}_1$ to the composition

$$ \mathrm{Proj}(F_1)/\mathrm{rad}^2(\mathrm{Proj}(F_1)) \to M|_G \hookrightarrow \mathrm{Inj}(M|_G) \to \mathrm{Inj}(F_0), $$

we obtain

$$ F_1 \hookrightarrow \mathrm{gr}_1(M|_G) \hookrightarrow \mathrm{gr}_1 \mathrm{Inj}(M|_G) \to \mathrm{gr}_1 \mathrm{Inj}(F_0), \qquad (4.16) $$

where the grading is with respect to the socle filtration. It suffices to show that the composition (4.16) is nonzero.

By Lemma 4.2.4, $F_0$ (respectively $F_1$) is isomorphic to $F(\lambda')$ where $\lambda' = \lambda^0 + \varepsilon_\lambda$ (respectively $F(\theta^0 + \varepsilon_\theta)$) for some $\varepsilon_\lambda \in L(\pi\omega_\lambda)$ (respectively $\varepsilon_\theta \in L(\pi\omega_\theta)$). Then using Lemma 4.2.11, (4.16) can be rewritten as

$$ L(\theta^0 + \varepsilon_\theta)|_G \hookrightarrow L(\theta)|_G \hookrightarrow (\mathrm{gr}_1(Q_1(\lambda^0) \otimes L(p\omega_\lambda)))|_G \to (\mathrm{gr}_1 Q_1(\lambda^0 + \varepsilon_\lambda))|_G, $$

where the second map is the restriction of a map of $G$-modules. By Lemma 4.2.2, the socle filtrations of $Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ and $Q_1(\lambda^0) \otimes L(\pi\omega_\lambda)$ both induce the
socle filtration on $Q_1(\lambda^0) \otimes L(p\omega_\lambda)|_G$. Applying Lemma 4.2.4 to $Q_1(\lambda^0) \otimes L(\pi\omega_\lambda)$ and using the description of $Q_1(\lambda^0 + \nu)$ in [Hum06, Section 13.9], we see that the graded pieces of the socle filtration of $Q_1(\lambda^0) \otimes L(\pi\omega_\lambda)$ are $\text{gr}_k(Q_1(\lambda^0)) \otimes L(\pi\omega_\lambda)$. We claim that the graded pieces of the socle filtration of $Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ are $\text{gr}_k(Q_1(\lambda^0)) \otimes L(p\omega_\lambda)$. Taking the tensor product of the socle filtration on $Q_1(\lambda^0)$ with $L(p\omega_\lambda)$ gives a semisimple filtration $\mathcal{F}$ (as follows from Steinberg's tensor product theorem). Moreover, since the restrictions $Q_1(\lambda^0) \otimes L(\pi\omega_\lambda)|_G$ and $Q_1(\lambda^0) \otimes L(p\omega_\lambda)|_G$ are isomorphic, Lemma 4.2.2 implies that the dimensions of the graded pieces of the socle filtrations of $Q_1(\lambda^0) \otimes L(\pi\omega_\lambda)$ and $Q_1(\lambda^0) \otimes L(p\omega_\lambda)$ agree. A dimension consideration implies that $\mathcal{F}$ is the socle filtration.

In particular, we have

$$ \text{gr}_1(Q_1(\lambda^0) \otimes L(p\omega_\lambda)) \cong \bigoplus_i \left( L(p\omega_\lambda) \otimes (\text{gr}_1 Q_1(\lambda_i^0)) \otimes \bigotimes_{j \neq i} L(\lambda_j^0) \right). $$

The algebraic map $L(\theta) \hookrightarrow \text{gr}_1(Q_1(\lambda^0) \otimes L(p\omega_\lambda))$ factors through the direct summand

$$ L(\lambda^{i_0}) \otimes (\text{gr}_1 Q_1(\lambda_{i_0}^0)) \otimes L(p\omega_\lambda) $$

by alcove considerations, where $\lambda^{i_0} \stackrel{\text{def}}{=} \sum_{i \neq i_0} \lambda_i$. (Note that $\lambda_i = \lambda_i^0$ and $\theta_i = \lambda_i$ for all $i \neq i_0$.) Additionally, the map $F_1 \to \text{gr}_1 \text{Inj}(F_0) \cong (\text{gr}_1 Q_1(\lambda'))|_G$ factors through the direct summand

$$ L(\lambda'^{i_0}) \otimes (\text{gr}_1 Q_1(\lambda_{i_0}'))|_G, $$

where $\lambda'^{i_0} = \sum_{i \neq i_0} \lambda'_i$ since if $j \neq i_0$, then the highest weights of the Jordan–Hölder factors of

$$ \left( \left( \bigotimes_{i \neq j} L(\lambda_i') \right) \otimes (\text{gr}_1 Q_1(\lambda_j')) \right)|_G $$

lie in the same alcove as the highest weight of $L(\lambda^0)$ except exactly at embedding $j$. Thus, it suffices to show that the composition

$$ \begin{aligned} L(\theta^0 + \epsilon_\theta)|_G &\hookrightarrow L(\theta)|_G \hookrightarrow L(\lambda^{i_0}) \otimes (\text{gr}_1 Q_1(\lambda_{i_0}^0)) \otimes L(p\omega_\lambda)|_G \\ &\twoheadrightarrow L(\lambda'^{i_0}) \otimes (\text{gr}_1 Q_1(\lambda_{i_0}'))|_G \end{aligned} \quad (4.17) $$

is nonzero.

The G-module $\text{gr}_1 Q_1(\lambda_{i_0}^0)$ is the direct sum of three irreducible modules in alcoves A, C, and D (respectively B, E, and F) if $\lambda_{i_0}^0$ is in alcove B (respectively A). Let $\tilde{w}_1$, $\tilde{w}_2$, and $\tilde{w}_3$ be the elements of $W_a$ such that $\tilde{w}_1 \cdot A$, $\tilde{w}_2 \cdot A$, and $\tilde{w}_3 \cdot A$ are the alcoves A, C, and D (respectively B, E, and F), respectively. Let $\tilde{w}_B \in W_a$ be
the element such that $B = \tilde{w}_B \cdot A$. The natural identification $X^*(T)\backslash\tilde{W}/\Omega \cong \{A, B\}$ where $\Omega = \text{Stab}_{\tilde{W}}(A)$ and the fact that $\theta_{i_0}^0$ is in alcove A (respectively B) if $\lambda_{i_0}^0$ is in alcove B (respectively A) shows that $X^*(T)\tilde{y}_{i_0}\tilde{w}_B\Omega = X^*(T)\tilde{w}_j\Omega$ for any $j \in \{1, 2, 3\}$. Moreover, it is not difficult to check that

$$X^*(T)\tilde{y}_{i_0}\tilde{w}_B\Omega = \coprod_{j=1}^3 X^*(T)\tilde{w}_j.$$

This implies that there exist unique $j \in \{1, 2, 3\}$ and $\omega \in X^*(T)$ such that

$$t_{-\omega_{\theta, i_0}} \tilde{y}_{i_0} \tilde{w}_B (\tilde{w}_{\lambda} t_{-\omega_{\lambda, i_0}} \tilde{w}_{i_0} \tilde{w}_B)^{-1} = t_{-\omega} \tilde{w}_j, \quad (4.18)$$

where $\tilde{w}_{\lambda} \in W_a$ is the unique element such that $\tilde{w}_{\lambda} \cdot \lambda_{i_0}^0 \in A$ or, equivalently, that $\tilde{w}_{\lambda} t_{-\omega_{\lambda, i_0}} \tilde{w}_{i_0} \tilde{w}_B \in \Omega$. Then we have that

$$\theta_{i_0}^0 + p\omega = t_{\omega} t_{-\omega_{\theta, i_0}} \tilde{y}_{i_0} \cdot \mu_{i_0} = \tilde{w}_j \tilde{w}_{\lambda} t_{-\omega_{\lambda, i_0}} \tilde{w}_{i_0} \cdot \mu_{i_0} = \tilde{w}_j \tilde{w}_{\lambda} \cdot \lambda_{i_0}^0$$

so that, by construction, $\omega \in X^*(T)$ is the unique weight such that $L(\theta_{i_0}^0 + p\omega)$ is a Jordan–Hölder factor of $\text{gr}_1 Q_1(\lambda_{i_0}^0)$. We now consider $\omega$ as an element of $X^*(\mathcal{T})$ in the $i_0$-embedding.

We now proceed casewise. If $\omega_{\lambda}$ (respectively $\omega_{\theta}$) is 0, then $\text{gr}_0(M|_G)$ (respectively $\text{gr}_1(M|_G)$) is irreducible, and so the map $M|_G \to \text{Inj}(F_0)$ is injective (respectively the map $\text{Proj}(F_1) \to M|_G$ is surjective) and the composition (4.15) is nonzero.

Now assume that $\omega_{\lambda}$ and $\omega_{\theta}$ are both nonzero. At most one of $\omega_{\lambda}$ and $\omega_{\theta}$ can be $\eta'_{i_0}$. By duality, we can and will assume that $\omega_{\lambda} \neq \eta'_{i_0}$. Then $\omega_{\lambda}$ is either $\varepsilon'_{1, i_0}$ or $\varepsilon'_{2, i_0}$. We assume that $\omega_{\lambda} = \varepsilon'_{2, i_0}$, the other case being symmetric. So if $\lambda_{i_0}$ is in alcove D (respectively F), then $\theta_{i_0}$ is in alcove E or F (respectively C, D, or G) by Proposition 4.2.9. We claim that if $\omega_{\theta}$ is $\varepsilon'_{1, i_0}$ (respectively $\varepsilon'_{2, i_0}$ or $\eta'_{i_0}$), then the $\omega$ defined above is $\varepsilon'_{2, i_0} + (1, 1, 1)_{i_0}$ (respectively 0 or $\varepsilon'_{1, i_0}$). Indeed, since $t_{-\omega}\tilde{w}_j \cdot A \in \{A, B\}$ by (4.18) and $\tilde{w}_j \cdot A \in \{A, B, C, D, E, F\}$ by definition, $\omega_{i_0}$ is in $\{0, \varepsilon'_1, \varepsilon'_2\} + X^0(T)$. From (4.18), we see that $\omega \equiv \omega_{\theta} - \omega_{\lambda}$ (mod $\underline{\Delta}_R$). These two facts determine $\omega$.

A morphism

$$L(\theta_{i_0}) \cong L(\theta_{i_0}^0 + p\omega_{\theta}) \to (\text{gr}_1 Q_1(\lambda_{i_0}^0)) \otimes L(p\omega_{\lambda})$$

must factor through $L(\theta_{i_0}^0 + p\omega') \otimes L(p\omega_{\lambda})$ for some $\omega'$. By construction, $\omega'$ must be $\omega$. We conclude that the map

$$L(\theta) \hookrightarrow L(\lambda^{i_0}) \otimes (\text{gr}_1 Q_1(\lambda^{i_0})) \otimes L(p\omega_{\lambda})$$
factors through $L(\theta^0 + p\omega) \otimes L(p\omega_\lambda)$. It suffices to show that the composition

$$L(\theta^0 + \varepsilon_\theta)|_G \hookrightarrow L(\theta)|_G \hookrightarrow L(\theta^0) \otimes L(p\omega) \otimes L(p\omega_\lambda)|_G \\ \rightarrow L(\lambda^{\prime,i_0}) \otimes (\mathrm{gr}_1 Q_1(\lambda_{i_0}^{\prime}))|_G$$

is nonzero.

Let $\varepsilon'_\lambda \in L(\pi\omega_\lambda)$ be such that $\lambda^0 + \varepsilon_\lambda$ and $\theta^0 + \varepsilon'_\lambda$ are in the same $\tilde{W}$-orbit under the $p$-dot action. Similar to an earlier argument of the three simple Jordan–Hölder factors of $L(\lambda^{\prime,i_0}) \otimes (\mathrm{gr}_1 Q_1(\lambda_{i_0}^{\prime}))$, only the restriction to $G$ of one of the form $L(\theta^0 + \varepsilon'_\lambda) \otimes L(p\omega')$ for some $\omega'$ contains $F_1 \cong F(\theta^0 + \varepsilon_\theta)$ as a Jordan–Hölder factor. By construction, $\omega'$ must be $\omega$. It now suffices to show that the composition

$$L(\theta^0 + \varepsilon_\theta)|_G \hookrightarrow L(\theta^0 + \pi\omega_\theta)|_G \cong L(\theta)|_G \hookrightarrow L(\theta^0) \otimes L(p\omega) \otimes L(p\omega_\lambda)|_G \\ \cong L(\theta^0) \otimes L(\pi\omega_\lambda) \otimes L(p\omega)|_G \rightarrow L(\theta^0 + \varepsilon'_\lambda) \otimes L(p\omega)|_G \qquad (4.19)$$

is nonzero. Moreover, the multiplicity $[L(\theta^0 + \varepsilon'_\lambda) \otimes L(p\omega)|_G : F_1]$ is one.

Assume now that $\omega_\theta$ is not $\varepsilon'_{1,i_0}$. Then using the assumption that $[M|_G : F_1] = 1$ so that $\varepsilon_\theta \neq 0$, one can check that $F_1$ appears in $L(\theta^0) \otimes L(p\omega) \otimes L(p\omega_\lambda)|_G$ with multiplicity one, and we see that (4.19) is nonzero.

Finally, we assume that $\omega_\theta$ is $\varepsilon'_{1,i_0}$. Let $\varepsilon$ be $\varepsilon_\theta - \varepsilon'_\lambda$ which is in $L(\pi\omega)$ by the composition (4.19). Note that $\varepsilon \neq \varepsilon'_\lambda$ (mod $X^0(\mathcal{T})$). The weight $F_1$ appears with multiplicity two in $\bigoplus_{\nu_\lambda \in L(\pi\omega_\lambda)} L(\theta^0 + \nu_\lambda) \otimes L(p\omega)|_G$ and with multiplicity one in each of $L(\theta^0 + \varepsilon'_\lambda) \otimes L(p\omega)|_G$ and $L(\theta^0 + \varepsilon - \pi(1, 1, 1)_{i_0}) \otimes L(p\omega)|_G$. Assume for the sake of contradiction that the composition (4.19) is zero. Then the image of the composition of the first four maps of (4.19) is contained in $L(\theta^0 + \varepsilon - \pi(1, 1, 1)_{i_0}) \otimes L(p\omega)|_G$. In particular, since $\varepsilon \neq \varepsilon'_\lambda$ (mod $X^0(\mathcal{T})$), the image of the composition of the first three maps of (4.19) is not stable under the involution action on $L(\theta^0) \otimes L(p\omega) \otimes L(p\omega_\lambda)|_G \cong L(\theta^0 + p(1, 1, 1)_{i_0}) \otimes L(p\omega_\lambda) \otimes L(p\omega_\lambda)|_G$ which permutes the last two tensor factors. Since $F_1$ appears with multiplicity one in $L(\theta)|_G$, this implies that the image of

$$L(\theta^0 + p(1, 1, 1)_{i_0}) \otimes L(p\omega_\theta - p(1, 1, 1)_{i_0}) \\ \hookrightarrow L(\theta^0 + p(1, 1, 1)_{i_0}) \otimes L(p\omega_\lambda) \otimes L(p\omega_\lambda),$$

which induces the second map of (4.19), is not stable under this involution action (note the key role played by Lemma 4.2.11). However, the unique submodule of $L(p\omega_\lambda) \otimes L(p\omega_\lambda)$ isomorphic to $L(p\omega_\theta - p(1, 1, 1)_{i_0})$ is the submodule where this involution acts by $-1$. This is a contradiction. $\square$
**Proof of Proposition 4.2.10.** We write $\lambda$ and $\theta$ as $\lambda^0 + p\omega_\lambda$ and $\theta^0 + p\omega_\theta$, respectively, where $\lambda^0$ and $\theta^0$ are $p$-restricted weights. Write $\omega_\lambda$ as $\omega_{\lambda,i_0} + \omega_\lambda^{i_0}$ where $\omega_{\lambda,i_0}$ (respectively $\omega_\lambda^{i_0}$) is 0 away from (respectively at) embedding $i_0$. Similarly, write $\omega_\theta$ as $\omega_{\theta,i_0} + \omega_\theta^{i_0}$. Then $\omega_\lambda^{i_0}$ equals $\omega_\theta^{i_0}$ by assumption, and so we set $\omega_\lambda^{i_0} \stackrel{\text{def}}{=} \omega_\theta^{i_0} = \omega_\theta^{i_0}$. Let $M_{i_0}$ be the unique up to isomorphism nontrivial extension of $L(\theta_{i_0})$ by $L(\lambda_{i_0})$ and let $M^{i_0}$ be $\bigotimes_{i \neq i_0} L(\lambda_i)$ so that $M$ is isomorphic to $M_{i_0} \otimes M^{i_0}$. Then $M|_G$ is isomorphic to

$$ \begin{aligned} M_{i_0} \otimes M^{i_0}|_G &\cong M_{i_0} \otimes \left( \bigotimes_{i \neq i_0} L(\lambda_i^0) \right) \otimes L(p\omega^{i_0})|_G \\ &\cong M_{i_0} \otimes \left( \bigotimes_{i \neq i_0} L(\lambda_i^0) \right) \otimes L(\pi\omega^{i_0})|_G. \end{aligned} $$

Let $M'$ be $M_{i_0} \otimes (\bigotimes_{i \neq i_0} L(\lambda_i^0)) \otimes L(\pi\omega^{i_0})$. By Lemma 4.2.2, we have that $\text{soc}(M')|_G \cong \text{soc}(M'|_G) \cong \text{soc}(M|_G) \cong \text{soc}(M)|_G$ so that

$$ \text{soc}(M') \cong L(\lambda_{i_0}) \otimes \left( \bigotimes_{i \neq i_0} L(\lambda_i^0) \right) \otimes L(\pi\omega^{i_0}). $$

Similarly, we have that

$$ \text{cosoc}(M') \cong L(\theta_{i_0}) \otimes \left( \bigotimes_{i \neq i_0} L(\lambda_i^0) \right) \otimes L(\pi\omega^{i_0}). $$

The socle and cosocle of $M'$ can be decomposed using Lemma 4.2.4. Fix a direct sum decomposition $\text{cosoc}(M') = \bigoplus_j M'_{1,j}$ into simple modules. Using the known dimensions of $\text{Ext}_G^1$ groups between simple modules and the fact that $M'$ is rigid of Loewy length 2, one sees that the minimal submodule $M'_j$ of $M'$ whose projection to $\text{cosoc}(M')$ contains $M'_{1,j}$ has length exactly 2. Thus, the natural surjection $\bigoplus M'_j \to \sum M'_j = M'$ is an isomorphism by length considerations. More explicitly, if $M'_{1,j}$ is isomorphic to $L(\lambda^0 + p\omega_{\lambda,i_0} + \pi\varepsilon^{i_0})$ for some $\varepsilon^{i_0} \in L(\omega^{i_0})$, then $M'_{0,j}$ is isomorphic to $L(\theta^0 + p\omega_{\theta,i_0} + w\pi\varepsilon^{i_0})$ for the unique element $w \in \mathbb{W}$ such that these weights are linked.

The alcoves of $F_1$ and $F_0$ differ only in embedding $i_0$. Then by Lemma 4.2.6, the corresponding elements of $\Delta_W^\nu$ under $\mathfrak{T}\tau_\nu$ for any appropriate $\nu$ must be the same for all embeddings except for $\pi(i_0)$. By the above explicit description, we conclude that $F_1$ is in $\text{JH}(M'_{1,j}|_G)$ if and only if $F_0$ is in $\text{JH}(M'_{0,j}|_G)$. So it suffices to prove the proposition for $M'_j$ in place of $M$. This now follows from Proposition 4.2.12. $\square$
*Proof of Theorem 4.2.7(3).* Let $\Gamma$ be the directed graph with vertices $JH(U) = JH(V_\mu|_G)$ so that there is a directed edge $\sigma_1 \to \sigma_2$ for $\sigma_1$ and $\sigma_2 \in JH(V_\mu|_G)$ if and only if $d_{gph}(\sigma_{(0,1)}, \sigma_1) > d_{gph}(\sigma_{(0,1)}, \sigma_2)$ and $d_{gph}(\sigma_1, \sigma_2) = 1$. Note that the first condition ensures that $\Gamma$ is acyclic.

We claim that $\Gamma$ is a subgraph of $\Gamma(U)$. Let $\sigma_1$ and $\sigma_2$ be in $JH(U)$ with $d_{gph}(\sigma_1, \sigma_2) = 1$. Let $d_j = d_{gph}(\sigma_{(0,1)}, \sigma_j)$ for $j = 1$ and $2$ and suppose that $d_1 > d_2$ so that $d_1 = d_2 + 1$. Then by Lemma 4.2.8, $d_U^{\text{soc}}(\sigma_j)$ is $d_j$ for $j = 1$ and $2$. Moreover, by Theorem 4.2.7(2) and parity reasons, $[\text{gr}_{d_j}(V_\mu)|_G : \sigma_k] = \delta_{jk}$ for $j$ and $k$ in $\{1, 2\}$. Using that every nontrivial extension which can occur in the layers of $V_\mu$ does occur (see Table 6), it is easy to check that there is a unique length 2 subquotient $M$ of $\text{Fil}_{d_1}(V_\mu)/\text{Fil}_{d_2-1}(V_\mu)$ such that $\sigma_1$ and $\sigma_2$ appear in $JH(M|_G)$ with multiplicity one. By Proposition 4.2.10, there is a subquotient of $M|_G$ which is a nonsplit extension of $\sigma_1$ by $\sigma_2$. For multiplicity reasons, this must also be a subquotient of $U$. Hence, there is a directed edge from $\sigma_1$ to $\sigma_2$ in $\Gamma(U)$.

We now claim that if $\Gamma$ is a subgraph of a directed graph $\Gamma'$ such that

(1) $\Gamma$ and $\Gamma'$ have the same vertices,

(2) $\Gamma'$ is acyclic, and

(3) $\sigma_1 \to \sigma_2$ is a subgraph of $\Gamma'$ only if $d_{gph}(\sigma_1, \sigma_2) = 1$,

then $\Gamma' = \Gamma$. Since $\Gamma(U)$ satisfies these conditions (3) follows from Lemmas 4.2.5 and 4.2.6), this would complete the proof.

Assume that $\sigma_1 \to \sigma_2$ is a subgraph of $\Gamma'$. By (3) and the geometry of the extension graph, we have that $d_{gph}(\sigma_{(0,1)}, \sigma_1) = d_{gph}(\sigma_{(0,1)}, \sigma_2) \pm 1$, and, hence, it is enough to prove that $d_{gph}(\sigma_{(0,1)}, \sigma_1) = d_{gph}(\sigma_{(0,1)}, \sigma_2) + 1$. Suppose otherwise. Then $\sigma_2 \to \sigma_1$ is a subgraph of $\Gamma$ by definition and thus a subgraph of $\Gamma'$. But this contradicts (2). $\square$

### 4.2.4. The embedding construction.
We start with the following observation.

**LEMMA 4.2.13.** Let $R$ be an $n$-generic Deligne–Lusztig representation. Then $R$ is $n-2$-deep.

*Proof.* If we write $R = R_s(\mu + \eta)$ with $\mu$ being $n$-deep, it is hence enough to prove that $\sigma_{r(\omega,a)}$ is $n-2$-deep for any obvious weight $\sigma_{(\omega,a)}$ in $JH(\bar{R})$. This follows from Proposition 2.3.4. $\square$

From now onward, we assume that $\mu$ is $6$-deep in alcove $\mathcal{B}$. (By Lemma 4.2.13, the Deligne–Lusztig representation $R \stackrel{\text{def}}{=} R_s(\mu^{\text{op}} + \eta')$ is therefore $4$-deep.) We write $\sigma^{\text{op}} \stackrel{\text{def}}{=} F(\mu) = F(\mathfrak{t}_{\mu^{\text{op}}+\eta}(0, 1))$. Note from Proposition 2.3.5 that the
unique element $\sigma \in \text{JH}(\bar{R})$ satisfying $\text{d}_{\text{gph}}(\sigma, \sigma^{\text{op}}) = 3f$ (that is, having maximal graph distance from $\sigma^{\text{op}}$) is

$$ \sigma \stackrel{\text{def}}{=} F(\mu^{\text{op}} + s(\eta')) = F(\bar{\tau}_{\mu^{\text{op}}+\eta}(s(\eta'), 0)). $$

To ease notation, we write $\lambda \stackrel{\text{def}}{=} \mu^{\text{op}} + s(\eta')$ (hence, $\sigma = F(\lambda)$).

**LEMMA 4.2.14.** Let $\mu \in X_1(\bar{T})$ be a weight such that $\mu \in \bar{B}$ is 6-deep and $R \stackrel{\text{def}}{=} R_s(\mu^{\text{op}} + \eta')$. Set $\sigma^{\text{op}} \stackrel{\text{def}}{=} F(\mu)$ and let $\sigma \stackrel{\text{def}}{=} F(\lambda) \in \text{JH}(\bar{R})$ be the unique constituent at maximal graph distance. We have

$$ [\text{Inj}(\sigma^{\text{op}}) : \sigma] = [V_{\mu}|_{\bar{G}} : \sigma] = [U : \sigma] = [\text{gr}_{3f}(V_{\mu}|_{\bar{G}}) : \sigma] = 1. $$

*Proof.* The equations $[V_{\mu}|_{\bar{G}} : \sigma] = [U : \sigma] = [\text{gr}_{3f}(V_{\mu}|_{\bar{G}}) : \sigma] = 1$ follow from Theorem 4.2.7(2). The equation $[\text{Inj}(\sigma^{\text{op}}) : \sigma] = 1$ is proved similarly using Theorem 4.2.1. $\square$

**PROPOSITION 4.2.15.** Let $\mu \in X_1(\bar{T})$ be a weight such that $\mu \in \bar{B}$ is 6-deep and $R \stackrel{\text{def}}{=} R_s(\mu^{\text{op}} + \eta')$. Set $\sigma^{\text{op}} \stackrel{\text{def}}{=} F(\mu)$ and let $\sigma \stackrel{\text{def}}{=} F(\lambda) \in \text{JH}(\bar{R})$ be the unique constituent at maximal graph distance. Let $R^{\sigma}$ be an $\mathcal{O}$-lattice in $R$ with irreducible cosocle isomorphic to $\sigma$. There is an injection $\iota : \bar{R}^{\sigma} \to U$.

Note that the map $\iota$ is unique up to scalar as there is a unique up to scalar nonzero morphism $\text{Proj}(\sigma) \to U$ by Lemma 4.2.14.

*Proof.* Since $\sigma^{\text{op}} \in \text{JH}(\bar{R}^{\sigma})$, there is a unique up to scalar nonzero map

$$ \bar{R}^{\sigma} \to \text{Inj}(\sigma^{\text{op}}). \qquad (4.20) $$

We first prove that the map (4.20) factors through the embedding

$$ U \stackrel{\alpha}{\hookrightarrow} \text{Inj}(\sigma^{\text{op}}) \qquad (4.21) $$

or, equivalently, that the composite of (4.20) with the natural projection $\text{Inj}(\sigma^{\text{op}}) \to \text{coker}(\alpha)$ is zero. As formation of cosocle is right exact and $\bar{R}^{\sigma}$ has irreducible cosocle isomorphic to $\sigma$, the image of the composite map $\bar{R}^{\sigma} \to \text{coker}(\alpha)$ is either zero or has irreducible cosocle $\sigma$. By Lemma 4.2.14, we know that $\sigma$ is not a Jordan–Hölder constituent of $\text{coker}(\alpha)$, and, therefore, $\text{Hom}_{\bar{G}}(\bar{R}^{\sigma}, \text{coker}(\alpha)) = 0$. Now the image of this nonzero map contains $\sigma$ as a Jordan–Hölder factor. Since the minimal submodule of $U$ containing $\sigma$ as a Jordan–Hölder factor has length $9f$, which is the length of $\bar{R}^{\sigma}$, by Theorem 4.2.7(3) and Proposition 4.1.1, this map must be an injection. $\square$
**THEOREM 4.2.16.** With the hypotheses of Theorem 4.1.9, assume, moreover, that $\sigma$ is a lowest alcove weight with $\mathrm{Def}_R(\sigma) = 0$. Then Theorem 4.1.9(1)–(4) hold.

*Proof.* By Remark 4.1.10, it suffices to prove Theorem 4.1.9(3). This follows from Theorem 4.2.7(3) and Propositions 4.2.15 and 4.1.2. □

**4.3. Proof of the structure theorem in the general case.** The aim of this section is to deduce Theorem 4.1.9 from the particular case in Theorem 4.2.16. For this, we introduce a third notion of distance called the saturation distance (see Definition 4.3.4). The first step is to show an inequality between the graph and saturation distances (Corollary 4.3.8) and then proceed further in Section 4.3.2 to prove that under appropriate conditions on $\sigma \in \mathrm{JH}(\bar{R})$ (that is, when $\sigma$ is maximally saturated), the notions of graph, saturation, and cosocle distances ($d_{\mathrm{rad}}^{\sigma}(\kappa)$) actually coincide (Proposition 4.3.16). The agreement of the three distances is equivalent to Theorem 4.1.9, taking into account Lemma 4.3.12. In Section 4.3.3, we show that all weights are maximally saturated, thus concluding the proof of Theorem 4.1.9.

**4.3.1. Notions of distances.** In what follows, $R$ is a 3-generic Deligne–Lusztig representation. Recall that if $\sigma \in \mathrm{JH}(\bar{R})$, we write $d_{\mathrm{rad}}^{\sigma}$ as a shorthand for $d_{\mathrm{rad}}^{\bar{R}^{\sigma}}$. We establish some properties of $d_{\mathrm{rad}}^{\sigma}$. We deduce the following from Lemma 4.2.6.

**COROLLARY 4.3.1.** Assume that $\bar{R}$ is 6-deep and $\sigma \in \mathrm{JH}(\bar{R})$. Then $d_{\mathrm{gph}}(\sigma, \sigma') \le d_{\mathrm{rad}}^{\sigma}(\sigma')$ for all $\sigma' \in \mathrm{JH}(\bar{R})$.

*Proof.* The assumption that $\bar{R}$ is 6-deep implies that all the elements in $\mathrm{JH}(\bar{R})$ satisfy the hypotheses of Lemma 4.2.6. If $\kappa \in \mathrm{JH}(\bar{R})$ and $k+1 = d_{\mathrm{rad}}^{\sigma}(\kappa)$, then $\mathrm{Ext}_G^1(\mathrm{gr}^k(\bar{R}^{\sigma}), \kappa) \ne 0$. Hence, $\mathrm{Ext}_G^1(\kappa', \kappa) \ne 0$ for some $\kappa' \in \mathrm{JH}(\bar{R})$ with $d_{\mathrm{rad}}^{\sigma}(\kappa') = k$. Equivalently, $d_{\mathrm{gph}}(\kappa, \kappa') = 1$ by Lemma 4.2.6. If $d = d_{\mathrm{rad}}^{\sigma}(\sigma')$, we conclude that there is a sequence of weights $\sigma_i$ for $0 \le i \le d-1$ such that $\sigma_0 = \sigma, \sigma_d = \sigma'$ and $d_{\mathrm{gph}}(\sigma_i, \sigma_{i+1}) = 1$ for all $i$. Hence, $d_{\mathrm{gph}}(\sigma, \sigma') \le d$. □

**LEMMA 4.3.2.** Let $\sigma$, $\sigma_1 \in \mathrm{JH}(\bar{R})$. There is a unique minimal subrepresentation $Q_{\sigma}(\sigma_1) \subseteq \bar{R}^{\sigma}$ containing $\sigma_1$ as a Jordan–Hölder factor. Moreover, we have that

$$ (1) \cosoc(Q_{\sigma}(\sigma_1)) \cong \sigma_1; $$

$$ (2) \ Q_{\sigma}(\sigma_1) \text{ is the image of any nonzero map } \bar{R}^{\sigma_1} \to \bar{R}^{\sigma}. $$
*Proof.* Let $Q_{\sigma}(\sigma_1)$ be the image of any nonzero map $\overline{R}^{\sigma_1} \to \overline{R}^{\sigma}$. It is clear that $\cosoc(Q_{\sigma}(\sigma_1)) \cong \sigma_1$. Suppose that $M \subset \overline{R}^{\sigma}$ is a subrepresentation containing $\sigma_1$ as a Jordan–Hölder factor. As $\overline{R}^{\sigma}$ is multiplicity free, $\overline{R}^{\sigma}/M$ does not contain $\sigma_1$ as a Jordan–Hölder factor, and, hence, the composition $Q_{\sigma}(\sigma_1) \hookrightarrow \overline{R}^{\sigma} \twoheadrightarrow \overline{R}^{\sigma}/M$ is 0 or, equivalently, $Q_{\sigma}(\sigma_1) \subset M$. $\square$

**LEMMA 4.3.3.** Let $\sigma$ and $\sigma_1 \in \text{JH}(\overline{R})$ and let $\sigma_2 \in \text{JH}(Q_{\sigma}(\sigma_1))$. If $n = d_{\text{rad}}^{\sigma_1}(\sigma_2)$, then we have $\text{Hom}_{\text{G}}(\sigma_2, \text{gr}_{\text{rad}}^{n}(Q_{\sigma}(\sigma_1))) \neq 0$. Moreover, $d_{\text{rad}}^{\sigma}(\sigma_2) \geq d_{\text{rad}}^{\sigma}(\sigma_1) + d_{\text{rad}}^{\sigma_1}(\sigma_2)$.

Note that, a priori, the inequality may be strict so that what we call the cosocle distance $d_{\text{rad}}^{\sigma}(\kappa)$ from $\sigma$ to $\kappa$ is not a priori necessarily a metric.

*Proof.* Since formation of cosocle is right exact, we see that the map $\overline{R}^{\sigma_1} \to Q_{\sigma}(\sigma_1)$ is strictly compatible with the radical filtrations. This proves the first claim.

Let $\mathcal{F} = \{\text{Fil}^k(\overline{R}^\sigma)\}_k$ be the decreasing filtration defined by $\text{Fil}^k(\overline{R}^\sigma) = \text{rad}^{k+d_{\text{rad}}^\sigma(\sigma_1)}(\overline{R}^\sigma)$ (that is, $\mathcal{F}$ is the radical filtration of $\overline{R}^\sigma$ shifted by $d_{\text{rad}}^\sigma(\sigma_1)$). Under the inclusion $Q_\sigma(\sigma_1) \hookrightarrow \overline{R}^\sigma$, $\mathcal{F}$ induces a semisimple filtration on $Q_\sigma(\sigma_1)$, beginning at 0. So, with the radical filtration on the domain and $\mathcal{F}$ on the codomain, this inclusion is compatible with the filtrations. This immediately gives $d_{\text{rad}}^\sigma(\sigma_2) - d_{\text{rad}}^\sigma(\sigma_1) \geq d_{\text{rad}}^{\sigma_1}(\sigma_2)$ for any $\sigma_2 \in \text{JH}(Q_\sigma(\sigma_1))$. $\square$

We now introduce a third notion of distance on the set $\text{JH}(\overline{R})$ and give a comparison result with the graph distance (Corollary 4.3.8). To do this, we make crucial use of a *global* input (Proposition 4.3.7). We also use these results to deduce some basic results about $\Gamma_{\text{rad}}(\overline{R}^\sigma)$ (see, for example, Lemma 4.3.12).

**DEFINITION 4.3.4.** Let $\sigma_1, \sigma_2 \in \text{JH}(\overline{R})$. Let $R^{\sigma_1}$ be an *O*-lattice with irreducible cosocle $\sigma_1$ and fix a saturated inclusion of lattices $R^{\sigma_2} \subseteq R^{\sigma_1}$. The saturation distance (with respect to $R$) between $\sigma_1$ and $\sigma_2$, noted by $d_{\text{sat}}(\sigma_1, \sigma_2)$, is defined to be the unique integer $d$ such that $p^d R^{\sigma_1} \subseteq R^{\sigma_2}$ is a saturated inclusion.

**REMARK 4.3.5.** We indicate a justification for the existence of the integer $d$ in Definition 4.3.4. Given two *O*-lattices $\Lambda_1$ and $\Lambda_2$ in an *E*-vector space $V$, there is a unique $d \in \mathbb{Z}$ such that $\omega^d \Lambda_2 \subset \Lambda_1$ is a saturated inclusion. If $V$, $\Lambda_1$, and $\Lambda_2$ are obtained from base change from an unramified subfield of $E$, then, in fact, $\omega^d$ is a power of $p$ up to units. Since $R, R^{\sigma_1}$, and $R^{\sigma_2}$ are defined over an unramified extension of $\mathbb{Q}_p$, the integer $d$ in Definition 4.3.4 exists.

The following lemma is clear.
**LEMMA 4.3.6.** The saturation distance is a metric on JH($\bar{R}$).

The following proposition is the main global input.

**PROPOSITION 4.3.7.** Let $R$ be a 13-generic Deligne–Lusztig representation. Let $\sigma_1, \sigma_2 \in \text{JH}(\bar{R})$ be such that $\text{d}_{\text{gph}}(\sigma_1, \sigma_2) = 1$. Then $\text{d}_{\text{sat}}(\sigma_1, \sigma_2) = 1$.

*Proof.* Let $\tau_S$ be the collection of tame inertial type such that $\sigma(\tau_S) = R$. We can and do fix a collection $\bar{\rho}_S$ of semisimple 10-generic Galois representation such that if $\tilde{w} \stackrel{\text{def}}{=} \tilde{w}(\bar{\rho}_S, \tau_S)$, then $\ell(\tilde{w}_i^*) \ge 2$ for all $i \in \mathcal{J}$ and $\sigma_1$ and $\sigma_2 \in W^\ell(\bar{\rho}_S, \tau_S)$. (Note that the condition that $R$, hence $\tau_S$, is 13-generic guarantees that $\bar{\rho}_S$ is 10-generic by the argument of Proposition 3.4.5.) Fix a weak minimal patching functor $M_\infty$ for $\bar{\rho}_S$, which exists by Corollary 3.5.16. Recall $R_{\bar{\rho}_v, \bar{\rho}_v}^{\bar{\tau}_v, \bar{\beta}_v, \square}$ from [LLHLM18, Definition 5.10].

For $i \in \{1, 2\}$, define the modules

$$M'_{\infty}(R^{\sigma_i}) \stackrel{\text{def}}{=} M_{\infty}(R^{\sigma_i}) \widehat{\otimes}_{(\hat{\otimes}_{\bar{v} \in S} R_{\bar{\rho}_v}^{\square})} (\widehat{\otimes}_{\bar{v} \in S} R_{\bar{\rho}_v, \bar{\rho}_v}^{\bar{\tau}_v, \bar{\beta}_v, \square})$$

$$M'_{\infty}(\sigma_i) \stackrel{\text{def}}{=} M_{\infty}(\sigma_i) \widehat{\otimes}_{(\hat{\otimes}_{\bar{v} \in S} R_{\bar{\rho}_v}^{\square})} (\widehat{\otimes}_{\bar{v} \in S} R_{\bar{\rho}_v, \bar{\rho}_v}^{\bar{\tau}_v, \bar{\beta}_v, \square}).$$

Similarly, define $R'_\infty(\tau_S)$ and $\bar{R}'_\infty(\tau_S)$. Then $M'_\infty(R^{\sigma_i})$ is $p$-torsion-free and maximally Cohen–Macaulay over $R'_\infty(\tau_S)$, and generically of rank one, and similarly $M'_\infty(\sigma_i)$ is maximally Cohen–Macaulay over $\bar{R}'_\infty(\tau_S)$.

We recall the setup of Lemma 3.6.10. The ring $R'_\infty(\tau_S)$ is formally smooth over $\widehat{\otimes}_{\bar{v} \in S} R_{\bar{\rho}_v, \bar{\rho}_v}^{\text{expl}, \nabla}$, where we write $\bar{\mathfrak{M}}_{\bar{v}} \in Y^{\eta, \tau_v}(\mathbb{F})$ for the unique Kisin module corresponding to $\bar{\rho}_{\bar{v}}$ as in Theorem 3.3.12. By letting $N \stackrel{\text{def}}{=} \sum_{i \in \mathcal{J}} (4 - \ell(\tilde{w}_i^*))$, the latter ring is formally smooth over $R_N \stackrel{\text{def}}{=} \widehat{\otimes}_{j=1}^N \mathcal{O}(\llbracket x_j, y_j \rrbracket / (x_j y_j - p))$ by [LLHLM18, §5.3.2] using that $\ell(\tilde{w}_i^*) > 1$ for all $i \in \mathcal{J}$. Fix an isomorphism $R_N[\llbracket t_1, \dots, t_k \rrbracket] \to R'_\infty(\tau_S)$. Let $S \subset R_N[\llbracket t_1, \dots, t_k \rrbracket]$ be the subring $\mathbb{Z}_p[\llbracket (x_j, y_j)^{N}, (t_j)^k \rrbracket] / (x_j y_j - p)^N$ for $M'$ is a maximal Cohen–Macaulay $R'_\infty(\tau_S)$-module, it is a maximal Cohen–Macaulay $S$-module as well by [Gro65, Corollaire 5.7.10] and using that the maximal ideal of $R'_\infty(\tau_S)$ is the only prime above the maximal ideal of $S$. The maximality follows from the fact that $R'_\infty(\tau_S)$ is finite over $S$. It is convenient to work over $S$ below since Spec $S/pS$ is reduced.

For $\sigma \in \text{JH}(\bar{R})$, $\text{Ann}_{R'_\infty(\tau_S)}(M'_\infty(\sigma))$ is $p(\sigma)R'_\infty(\tau_S)$ by Proposition 3.6.1(2). For $\sigma \in W^\ell(\bar{\rho}_S, \tau_S)$, let $\mathfrak{p}(\sigma)$ be $(z_j(\sigma))^N_{j=1} + (\sigma)$ where $z_j(\sigma) \in \{x_j, y_j\}$ for each $1 \le j \le N$. Let $\mathfrak{p}_S(\sigma) \subset S$ be the preimage of $\mathfrak{p}(\sigma)R'_\infty(\tau_S)$ for all $\sigma \in \text{JH}(\bar{R})$ so that $\mathfrak{p}_S(\sigma) = (z_j(\sigma))^N_{j=1} + (\sigma)$ and $\text{Supp}_S M'_\infty(\sigma) = \text{Spec}(S/\mathfrak{p}_S(\sigma))$.
For all $1 \le j \le N$, assume, without loss of generality, that $z_j(\sigma_1) = x_j$ and let $z_j \stackrel{\text{def}}{=} z_j(\sigma_2)$. Then by Lemma 3.6.10, $\#\{(x_j)_j \Delta\{z_j\}_j\} = 2$. We assume, without loss of generality, that $z_1 = y_1$ and $z_j = x_j$ for $j \ne 1$. To simplify notation, let $R_i = R^{\sigma_i}$ for $i = 1, 2$. We fix a chain of saturated inclusion of lattices $p^k R_1 \subseteq R_2 \subseteq R_1$ with $k \ge 1$. Since $R$ is residually multiplicity free, $C \stackrel{\text{def}}{=} \text{coker}(R_2 + pR_1 \hookrightarrow R_1)$ does not contain $\sigma_2$ as a Jordan–Hölder factor (as can be seen from descent to an unramified coefficient ring). Thus,

$$ \text{Supp}_S M'_{\infty}(C) \subset \bigcup_{\sigma \in \text{JH}(\bar{R}), \sigma \neq \sigma_2} \text{Spec } S/\mathfrak{p}_S(\sigma). $$

The scheme-theoretic support of $M'_{\infty}(C)$ in $\text{Spec } S$ is contained in $\text{Spec } S/\mathfrak{p}S$ and is thus generically reduced so that by the proof of Lemma 3.6.2, the scheme-theoretic support of $M'_{\infty}(C)$ in $\text{Spec } S$ is a closed subscheme of

$$ \text{Spec} \left( S / \bigcap_{\sigma \in \text{JH}(\bar{R}), \sigma \neq \sigma_2} \mathfrak{p}_S(\sigma) \right). $$

Since $x_1y_2 \cdots y_N \in \mathfrak{p}_S(\sigma)$ for all $\sigma \in \text{JH}(\bar{R})$ with $\sigma \neq \sigma_2$, $x_1y_2 \cdots y_N$ annihilates $M'_{\infty}(C)$ or, equivalently,

$$ x_1y_2 \cdots y_N M'_{\infty}(R_1) \subset M'_{\infty}(R_2) + pM'_{\infty}(R_1). $$

Symmetrically, we have

$$ y_1y_2 \cdots y_N M'_{\infty}(R_2) \subset M'_{\infty}(p^k R_1) + pM'_{\infty}(R_2). $$

Combining these, we have

$$ \begin{aligned} & x_1y_2 \cdots y_N y_1y_2 \cdots y_N M'_{\infty}(R_1) \\ & \subset y_1y_2 \cdots y_N M'_{\infty}(R_2) + y_1y_2 \cdots y_N pM'_{\infty}(R_1) \\ & \subset M'_{\infty}(p^k R_1) + pM'_{\infty}(R_2) + y_1y_2 \cdots y_N pM'_{\infty}(R_1). \end{aligned} $$

Simplifying and canceling $p$, we have

$$ y_2^2 \cdots y_N^2 M'_{\infty}(R_1) \subset p^{k-1} M'_{\infty}(R_1) + M'_{\infty}(R_2) + y_1y_2 \cdots y_N M'_{\infty}(R_1). $$

Assume that $k > 1$. Then projecting via $M'_{\infty}(R_1) \to M'_{\infty}(\bar{R}_1) \to M'_{\infty}(\sigma_1)$, we have the inclusion

$$ y_2^2 \cdots y_N^2 M'_{\infty}(\sigma_1) \subset y_1y_2 \cdots y_N M'_{\infty}(\sigma_1). $$

This is a contradiction since $M'_{\infty}(\sigma_1)$ is free over (a power series ring over) $\mathbb{F}[y_1, \dots, y_N]$, being maximal Cohen–Macaulay over it. $\square$
We deduce the following inequality.

**COROLLARY 4.3.8.** Assume that $R$ is 13-generic and let $\sigma_1, \sigma_2 \in \text{JH}(\overline{R})$. Then $d_{\text{gph}}(\sigma_1, \sigma_2) \ge d_{\text{sat}}(\sigma_1, \sigma_2)$.

*Proof.* The proof proceeds by induction on $d_{\text{gph}}(\sigma_1, \sigma_2)$, the case $d_{\text{gph}}(\sigma_1, \sigma_2) = 1$ being covered by Proposition 4.3.7. Pick a weight $\kappa \in \text{JH}(\overline{R})$ distinct from $\sigma_1$ and $\sigma_2$ such that $d_{\text{gph}}(\sigma_1, \kappa) + d_{\text{gph}}(\kappa, \sigma_2) = d_{\text{gph}}(\sigma_1, \sigma_2)$. Then we have

$$ \begin{aligned} d_{\text{gph}}(\sigma_1, \sigma_2) &= d_{\text{gph}}(\sigma_1, \kappa) + d_{\text{gph}}(\kappa, \sigma_2) \\ &\ge d_{\text{sat}}(\sigma_1, \kappa) + d_{\text{sat}}(\kappa, \sigma_2) \ge d_{\text{sat}}(\sigma_1, \sigma_2). \end{aligned} \quad \square $$

We conclude this section showing that, under appropriate genericity conditions on the Deligne–Lusztig representation $R$, the graph $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$ is predicted by the extension graph if its Loewy strata are predicted by the extension graph (which is a weaker assumption, a priori).

**LEMMA 4.3.9.** Assume that $R$ is 13-generic. Let $\sigma_1, \sigma_2 \in \text{JH}(\overline{R})$ be such that $d_{\text{gph}}(\sigma_1, \sigma_2) = 1$. Let us fix $\sigma \in \text{JH}(\overline{R})$ as well as two saturated inclusions of lattices $R^{\sigma_2} \subseteq R^{\sigma}$ and $R^{\sigma_1} \subseteq R^{\sigma}$. Then either $R^{\sigma_2} \subseteq R^{\sigma_1}$ or $R^{\sigma_1} \subseteq R^{\sigma_2}$.

*Proof.* We have $d_{\text{gph}}(\sigma_1, \sigma_2) = d_{\text{sat}}(\sigma_1, \sigma_2) = 1$ by Proposition 4.3.7. Hence, there exists an integer $k \in \mathbb{Z}$ such that

$$ p^{k+1} R^{\sigma_1} \subseteq R^{\sigma_2} \subseteq p^k R^{\sigma_1} $$

is a chain of saturated inclusions of lattices (see Remark 4.3.5). The first inclusion implies that $p^{k+1} R^{\sigma_1} \subseteq R^{\sigma}$ so that $k+1 \ge 0$. The second inclusion implies that $p^{-k} R^{\sigma_2} \subseteq R^{\sigma}$ so that $-k \ge 0$. Hence, $k=0$ or $-1$ and the result follows. $\square$

**LEMMA 4.3.10.** Assume that $R$ is 13-generic. Let $\sigma, \sigma_1, \sigma_2 \in \text{JH}(\overline{R})$ and fix saturated inclusions $R^{\sigma_i} \subseteq R^{\sigma}$ for $i=1, 2$. If $d_{\text{rad}}^{\sigma}(\sigma_2) \ge d_{\text{rad}}^{\sigma}(\sigma_1)$ and $d_{\text{gph}}(\sigma_1, \sigma_2) = 1$, then $R^{\sigma_2} \subseteq R^{\sigma_1}$.

*Proof.* By Lemma 4.3.9, either $R^{\sigma_1} \subseteq R^{\sigma_2}$ or $R^{\sigma_2} \subseteq R^{\sigma_1}$. Suppose that the former holds. Then $Q_{\sigma}(\sigma_1) \subseteq Q_{\sigma}(\sigma_2)$, and thus $d_{\text{rad}}^{\sigma}(\sigma_1) > d_{\text{rad}}^{\sigma}(\sigma_2)$ by Lemma 4.3.3. This is a contradiction. $\square$

**DEFINITION 4.3.11.** Assume that $R$ is 2-generic and let $\sigma \in \text{JH}(\overline{R})$. We say that the radical strata of $\overline{R}^{\sigma}$ are predicted by the extension graph if $d_{\text{gph}}(\sigma, \sigma') = d_{\text{rad}}^{\sigma}(\sigma')$ for all $\sigma' \in \text{JH}(\overline{R})$.
LEMMA 4.3.12. Assume that $R$ is 13-generic and let $\sigma \in \text{JH}(\bar{R})$. If the radical strata of $\bar{R}^\sigma$ are predicted by the extension graph, then $\Gamma_{\text{rad}}(\bar{R}^\sigma)$ is predicted by the extension graph with respect to $\sigma$.

*Proof.* Let $\sigma_i$ and $\sigma_{i+1}$ be elements of $\text{JH}(\bar{R})$ such that $d_{\text{gph}}(\sigma, \sigma_i) = i$ and $d_{\text{gph}}(\sigma, \sigma_{i+1}) = i + 1$. As the radical strata of $\bar{R}^\sigma$ are predicted by the extension graph, we have $d_{\text{gph}}(\sigma, \sigma_i) = d_{\text{rad}}^\sigma(\sigma_i)$ and $d_{\text{gph}}(\sigma, \sigma_{i+1}) = d_{\text{rad}}^\sigma(\sigma_{i+1})$.

If $d_{\text{gph}}(\sigma_i, \sigma_{i+1}) = 1$, then $\sigma_{i+1} \in \text{JH}(Q_\sigma(\sigma_i))$ by Lemma 4.3.10. By Lemma 4.3.3, we have

$$i + 1 = d_{\text{rad}}^{\sigma}(\sigma_{i+1}) \ge d_{\text{rad}}^{\sigma}(\sigma_i) + d_{\text{rad}}^{\sigma_i}(\sigma_{i+1}) = i + d_{\text{rad}}^{\sigma_i}(\sigma_{i+1}).$$

This implies that $\sigma_{i+1}$ appears in the second layer of the radical filtration of $Q_\sigma(\sigma_i)$, whose cosocle is isomorphic to $\sigma_i$ by Lemma 4.3.2(1). Then there is an edge from $\sigma_i$ to $\sigma_{i+1}$ in $\Gamma_{\text{rad}}(\bar{R}^\sigma)$.

Conversely, if there is an edge from $\sigma_i$ to $\sigma_{i+1}$ in $\Gamma_{\text{rad}}(\bar{R}^\sigma)$, then $d_{\text{gph}}(\sigma_i, \sigma_{i+1})$ is 1 by Lemma 4.2.6. $\square$

### 4.3.2. Distance equalities.
In this subsection, we define when a weight $\sigma \in \text{JH}(\bar{R})$ is maximally saturated. Crucially using Lemma 4.2.6, we show that if $\sigma$ is maximally saturated, then the graph, saturation, and cosocle distances from $\sigma$ are equal, and, therefore, the structure Theorem 4.1.9 holds for $\bar{R}^\sigma$. From now on, we assume that the Deligne–Lusztig representation $R$ is 13-generic.

From Proposition 2.3.5 and the definition of defect (Definition 2.8), it follows that

$$\max \{d_{\text{gph}}(\sigma, \kappa), \kappa \in \text{JH}(\bar{R})\} = 3f - \text{Def}_R(\sigma). \quad (4.22)$$

Furthermore, if $\text{Def}_R(\sigma) = 0$, then there is a unique $\sigma^{\text{op}} \in \text{JH}(\bar{R})$ which has maximal graph distance from $\sigma$.

### DEFINITION 4.3.13. Let $\sigma \in \text{JH}(\bar{R})$.
We say that the weight $\sigma$ is maximally saturated in $R$ if the following property holds:

If $\kappa \in \text{JH}(\bar{R})$ verifies $d_{\text{gph}}(\sigma, \kappa) = 3f - \text{Def}_R(\sigma)$, then $d_{\text{sat}}(\sigma, \kappa) = 3f - \text{Def}_R(\sigma)$. (4.23)

The following proposition motivates Definition 4.3.13.

### PROPOSITION 4.3.14. A weight $\sigma \in \text{JH}(\bar{R})$ is maximally saturated if and only if $d_{\text{gph}}(\sigma, \kappa) = d_{\text{sat}}(\sigma, \kappa)$ for all $\kappa \in \text{JH}(\bar{R})$.
Proof. The ‘if’ part is clear. Let $\kappa \in \mathrm{JH}(\bar{R})$ be any weight and write $d \stackrel{\mathrm{def}}{=} d_{\mathrm{gph}}(\sigma, \kappa)$ and $D \stackrel{\mathrm{def}}{=} 3f - \mathrm{Def}_R(\sigma)$. There are weights $\sigma = \sigma_0, \sigma_1, \dots, \sigma_D$ so that $\kappa \in \{\sigma_0, \sigma_1, \dots, \sigma_D\} \subset \mathrm{JH}(\bar{R})$, $d_{\mathrm{gph}}(\sigma_i, \sigma_{i+1}) = 1$ for all $0 \le i \le D-1$ and $d_{\mathrm{gph}}(\sigma, \sigma_D) = D$. It is then easy to see that $d_{\mathrm{gph}}(\sigma_i, \sigma_j) = j-i$ if $0 \le i \le j \le D$. We have the following chain of a priori inequalities

$$ 
\begin{aligned}
d_{\mathrm{sat}}(\sigma, \sigma_D) &\le d_{\mathrm{sat}}(\sigma, \kappa) + d_{\mathrm{sat}}(\kappa, \sigma_D) \\
&\le d_{\mathrm{gph}}(\sigma, \kappa) + d_{\mathrm{gph}}(\kappa, \sigma_D) \\
&= d_{\mathrm{gph}}(\sigma, \sigma_D) \\
&= d_{\mathrm{sat}}(\sigma, \sigma_D),
\end{aligned}
$$

using Corollary 4.3.8 where the last equality holds by assumption. We conclude that the above inequalities are in fact equalities and that $d_{\mathrm{sat}}(\sigma, \kappa) = d_{\mathrm{gph}}(\sigma, \kappa)$. $\square$

The following lemma will be the key in relating the notions of saturation and cosocle distance.

LEMMA 4.3.15. Let $d < \ell\ell(\bar{R}^\sigma)$ and let

$$ \sigma_d \leftarrow \sigma_{d-1} \leftarrow \dots \leftarrow \sigma_1 \leftarrow \sigma_0 \stackrel{\mathrm{def}}{=} \sigma $$

be an extension path in $\Gamma_{\mathrm{rad}}(\bar{R}^\sigma)$ (note that $d_{\mathrm{rad}}^\sigma(\sigma_i) = i$ for all $i \in \{0, \dots, d\}$). For each $i \in \{0, \dots, d\}$, let us fix a saturated inclusion $R^{\sigma_i} \subseteq R^\sigma$. Then we have a chain of (saturated) inclusions

$$ R^{\sigma_d} \subseteq R^{\sigma_{d-1}} \subseteq \dots \subseteq R^{\sigma_1} \subseteq R^{\sigma_0}. $$

Proof. Since $d_{\mathrm{rad}}^\sigma(\sigma_{i+1}) = i+1 \ge i = d_{\mathrm{rad}}^\sigma(\sigma_i)$ for $0 \le i \le d-1$, the result follows from Lemma 4.3.10. $\square$

We can use Lemma 4.3.15 to show that all three notions of distance from $\sigma$ agree in some particular situations.

PROPOSITION 4.3.16. If $\sigma \in \mathrm{JH}(\bar{R})$ is maximally saturated, then $\Gamma_{\mathrm{rad}}(\bar{R}^\sigma)$ is predicted by the extension graph with respect to $\sigma$.

Proof. Assume that $\sigma$ is maximally saturated. By Lemma 4.3.12, it suffices to show that $d_{\mathrm{gph}}(\sigma, \sigma') = d_{\mathrm{rad}}^\sigma(\sigma')$ for all $\sigma' \in \mathrm{JH}(\bar{R})$. Let $\sigma' \in \mathrm{JH}(\bar{R})$ and $d = d_{\mathrm{gph}}(\sigma, \sigma')$. We have that $\mathrm{Hom}_G(\sigma', \mathrm{gr}^k(\bar{R}^\sigma)) \ne 0$ for some $k \ge d$ by Corollary
4.3.1. Assume for the sake of contradiction that $k > d$. By the definition of the
radical filtration, we may, and do, fix an extension path of length $k+1$ in $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$:

$$
\sigma' = \sigma_k \leftarrow \sigma_{k-1} \leftarrow \dots \leftarrow \sigma_1 \leftarrow \sigma_0 = \sigma.
$$

Since $d_{\text{gph}}(\sigma_0, \sigma_k) = d < k$ and $d_{\text{gph}}(\sigma_0, \sigma_i) \le d_{\text{rad}}^{\sigma_0}(\sigma_i) = i$ for all $i$ by Corollary 4.3.1, there exists an index $i \in \{0, \dots, k-1\}$ such that

$$
(1) \quad d_{\text{gph}}(\sigma_0, \sigma_i) = i \text{ and}
$$

$$
(2) \quad d_{\text{gph}}(\sigma_0, \sigma_{i+1}) < i + 1.
$$

Moreover, since the extension graph is bipartite, we have that $d_{\text{gph}}(\sigma_0, \sigma_{i+1}) \le i - 1$. By Lemma 4.3.15, there is a chain of saturated inclusions:

$$
R^{\sigma_{i+1}} \subseteq R^{\sigma_i} \subseteq \dots \subseteq R^{\sigma_1} \subseteq R^{\sigma},
$$

where, in particular, $R^{\sigma_i} \subseteq R^{\sigma}$ is saturated as well. As $d_{\text{gph}}(\sigma_0, \sigma_{i+1}) = d_{\text{sat}}(\sigma_0, \sigma_{i+1})$ by Proposition 4.3.14, we further have $p^{i-1}R^{\sigma} \subseteq R^{\sigma_{i+1}}$. We conclude that $p^{i-1}R^{\sigma} \subseteq R^{\sigma_i} \subseteq R^{\sigma}$ and, hence, that $d_{\text{sat}}(\sigma_0, \sigma_i) \le i - 1$. This contradicts (1) since $d_{\text{gph}}(\sigma_0, \sigma_i) = d_{\text{sat}}(\sigma_0, \sigma_i)$ by Proposition 4.3.14. $\square$

We now use Theorem 4.2.16 to prove that lower alcove weights of defect zero
are maximally saturated.

PROPOSITION 4.3.17. Let $R$ be a 13-generic Deligne–Lusztig representation and let $\sigma \in \text{JH}(\overline{R})$ be a constituent with $\text{Def}_R(\sigma) = 0$. Assume that $\sigma \cong F(\lambda)$ where $\lambda \in X_1(\overline{T})$ is in alcove $\overline{A}$. Then $\sigma$ is maximally saturated in $\overline{R}$.

*Proof.* By Theorem 4.2.16, we can and do fix an extension path in $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$ with starting point $\sigma_0 \stackrel{\text{def}}{=} \sigma$: $\sigma^{\text{op}} = \sigma_{3f} \leftarrow \sigma_{3f-1} \leftarrow \dots \leftarrow \sigma_1 \leftarrow \sigma_0$. By Lemma 4.3.15, we have a sequence of saturated inclusions:

$$
R^{\sigma} \supseteq R^{\sigma_1} \supseteq \dots \supseteq R^{\sigma_{3f-1}} \supseteq R^{\sigma^{\text{op}}},
$$

where $R^{\sigma^{\text{op}}} \subseteq R^{\sigma}$ is itself saturated. For each $0 \le i \le 3f-1$, let $n_i$ be $d_{\text{sat}}(\sigma_i, \sigma^{\text{op}})$. It suffices to show that $n_0 = d_{\text{gph}}(\sigma_0, \sigma^{\text{op}}) = 3f$.

By Theorem 4.2.16, the reduction of the lattice in the dual Deligne–Lusztig representation $R^*$ with cosocle $\sigma^\vee$ is rigid and $\Gamma((\overline{R^*})^{\sigma^\vee})$ is predicted by the extension graph with respect to $\sigma^\vee$. Noting that the reduction of the dual of a lattice is the dual of the reduction of a lattice and using Proposition 4.1.7, we see that $\Gamma_{\text{rad}}(\overline{R}^{\sigma^{\text{op}}})$ is predicted by the extension graph with respect to $\sigma^{\text{op}}$.
In particular, $\sigma_0 \leftarrow \sigma_1 \leftarrow \dots \leftarrow \sigma_{3f-1} \leftarrow \sigma_{3f} = \sigma^{\text{op}}$ is an extension path in $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$, and, hence, from Lemma 4.3.15, we deduce a chain of (saturated) inclusions

$$R^{\sigma^{\text{op}}} \supseteq p^{n_{3f-1}} R^{\sigma_{3f-1}} \supseteq \dots \supseteq p^{n_1} R^{\sigma_1} \supseteq p^{n_0} R^{\sigma}$$

where we necessarily have $n_{i-1} > n_i$ for all $i$ (as $R^{\sigma_{i+1}} \subseteq R^{\sigma_i}$ is saturated). In particular, $n_0 \ge 3f$. On the other hand, Corollary 4.3.8 implies that $n_0 \le 3f$ so that $n_0 = 3f$. $\square$

### 4.3.3. Induction on defect.
In this subsection, we show inductively that all weights are maximally saturated, starting from lower alcove weights as in Proposition 4.3.17. We conclude the section with the proof of Theorem 4.1.9. We first start with the defect zero case.

**LEMMA 4.3.18.** Let $R$ be a 13-generic Deligne–Lusztig representation. If $\sigma \in \text{JH}(\overline{R})$ and $\text{Def}_R(\sigma) = 0$, then $\sigma$ is maximally saturated.

*Proof.* We claim that if $\sigma \stackrel{\text{def}}{=} \sigma_0$, $\sigma_1 \in \text{JH}(\overline{R})$ such that $\text{Def}_R(\sigma_0) = 0$, $\text{Def}_R(\sigma_1) = 0$, $\text{d}_{\text{gph}}(\sigma_0, \sigma_1) = 1$, and $\sigma_0$ is maximally saturated, then $\sigma_1$ is maximally saturated. The result then follows from Proposition 4.3.17 and an easy induction argument.

By Proposition 4.3.16, the graph $\Gamma_{\text{rad}}(\overline{R}^{\sigma})$ is predicted by the extension graph. By duality (cf. the proof of Proposition 4.3.17), the graph $\Gamma_{\text{rad}}(\overline{R}^{\sigma^{\text{op}}})$ is also predicted by the extension graph. Hence, we may and do choose two extension paths, of starting point $\sigma$, in the graph $\Gamma_{\text{rad}}(\overline{R}^{\sigma_0})$ having the form: $\sigma_0^{\text{op}} \leftarrow \sigma'_{3f-1} \leftarrow \dots \leftarrow \sigma_2' \leftarrow \sigma_1 \leftarrow \sigma_0$ and $\sigma_0^{\text{op}} \leftarrow \sigma_1^{\text{op}} \leftarrow \sigma'_{3f-2} \leftarrow \dots \leftarrow \sigma_1'' \leftarrow \sigma_0$ where $\sigma_i' \in \text{JH}(\overline{R}^{\sigma})$. As the graph $\Gamma_{\text{rad}}(\overline{R}^{\sigma^{\text{op}}})$ is predicted by the extension graph, the extension paths above induce extensions paths in $\overline{R}^{\sigma^{\text{op}}}$ by ‘reversing the arrows and the endpoints’. Let us fix saturated inclusions of lattices $R^{\sigma_i}$, $R^{\sigma_i''} \subseteq R^{\sigma}$, $R^{\sigma_1}$, $R^{\sigma_1''} \subseteq R^{\sigma^{\text{op}}}$, $R^{\sigma_1^{\text{op}}} \subseteq R^{\sigma^{\text{op}}}$, $p^{3f-i} R^{\sigma_i'}$, $p^{3f-i} R^{\sigma_i''} \subseteq R^{\sigma^{\text{op}}}$, $p^{3f-1} R^{\sigma_1}$, $p R^{\sigma_1^{\text{op}}} \subseteq R^{\sigma^{\text{op}}}$ are saturated inclusions as well.

By Lemma 4.3.15, we deduce the following chain of saturated inclusions:

$$ \begin{align} p^{3f} R^{\sigma_1} &\subseteq \dots \subseteq p R^{\sigma'_{3f-1}} & R^{\sigma'_{3f-1}} &\subseteq \dots \subseteq R^{\sigma_1} & R^{\sigma} \nonumber \\ p^{3f} R^{\sigma_1} &\subseteq p^{3f} R^{\sigma} & R^{\sigma^{\text{op}}} &\subseteq R^{\sigma_1^{\text{op}}} & R^{\sigma_1} & R^{\sigma_1''} \subseteq R^{\sigma^{\text{op}}}, \nonumber \\ p^{3f} R^{\sigma_1''} &\subseteq \dots \subseteq p R^{\sigma_1^{\text{op}}} & R^{\sigma_1^{\text{op}}} &\subseteq \dots \subseteq R^{\sigma_1''} & \subseteq & \nonumber \\ & & & & & \end{align} \tag{4.24} $$
We claim that the inclusions $p^{3f}R^{\sigma_1} \subseteq pR^{\sigma_1^{\text{op}}}$ and $pR^{\sigma_1^{\text{op}}} \subseteq R^{\sigma_1}$, obtained by composing the saturated inclusions above, are saturated.

We first show that $p^{3f}R^{\sigma_1} \subseteq pR^{\sigma_1^{\text{op}}}$ is saturated. If not, then we obtain a chain of saturated inclusion $p^{3f-1}R^{\sigma_1} \subseteq pR^{\sigma_1^{\text{op}}} \subseteq R^{\sigma^{\text{op}}}$. We deduce that $\sigma_1$ is a constituent in image $Q_{\sigma^{\text{op}}}(\sigma_1^{\text{op}})$ and hence $d_{\text{rad}}^{\sigma^{\text{op}}}(\sigma_1) \ge d_{\text{rad}}^{\sigma^{\text{op}}}(\sigma_1^{\text{op}}) + d_{\text{rad}}^{\sigma_1^{\text{op}}}(\sigma_1) \ge d_{\text{gph}}(\sigma^{\text{op}}, \sigma_1^{\text{op}}) + d_{\text{gph}}(\sigma_1^{\text{op}}, \sigma_1) \ge 3f + 1$ by Corollary 4.3.1 and Lemma 4.3.3. On the other hand, as the graph $\Gamma_{\text{rad}}(\bar{R}^{\sigma_1^{\text{op}}})$ is predicted by the extension graph, we have that $d_{\text{rad}}^{\sigma^{\text{op}}}(\sigma_1) = d_{\text{gph}}(\sigma^{\text{op}}, \sigma_1) = 3f - 1$ contradiction.

The evident analogue of the previous argument shows that $pR^{\sigma_1^{\text{op}}} \subseteq R^{\sigma_1}$ is saturated as well. Hence, $d_{\text{sat}}(\sigma_1, \sigma_1^{\text{op}}) = 3f$. $\square$

We now give the induction argument.

**PROPOSITION 4.3.19.** Let $R$ be a 13-generic Deligne–Lusztig representation. Then any constituent $\sigma \in \text{JH}(\bar{R})$ is maximally saturated in $\bar{R}$.

*Proof.* We induct on the defect $\delta \stackrel{\text{def}}{=} \text{Def}_R(\sigma)$ for $\sigma \in \text{JH}(\bar{R})$. The case $\delta = 0$ holds by Lemma 4.3.18. Suppose that $\delta > 0$. To ease notation, let $d \stackrel{\text{def}}{=} 3f - \delta$ and pick a weight $\sigma_d \in \text{JH}(\bar{R})$ such that $d_{\text{gph}}(\sigma, \sigma_d) = d$ (in other words, $\sigma_d \in \text{JH}(\bar{R})$ is at maximal graph distance from $\sigma$); we will show that $d = d_{\text{sat}}(\sigma, \sigma_d)$. Note that $\text{Def}_R(\sigma_d) \le \text{Def}_R(\sigma)$. If $\text{Def}_R(\sigma_d) < \text{Def}_R(\sigma)$, then $\sigma_d$ is maximally saturated (by induction on $\text{Def}_R(\sigma_d)$), and, hence, $d_{\text{gph}}(\sigma, \sigma_d) = d_{\text{sat}}(\sigma, \sigma_d)$ by Proposition 4.3.14.

We now consider the case $\text{Def}_R(\sigma_d) = \text{Def}_R(\sigma)$. By a direct check on the extension graph, we see that there exists $\sigma_{d-1} \in \text{JH}(\bar{R})$ with $d_{\text{gph}}(\sigma_d, \sigma_{d-1}) = 1$ and $\text{Def}_R(\sigma_{d-1}) = \text{Def}_R(\sigma) - 1$. Note that $d_{\text{gph}}(\sigma_{d-1}, \sigma) = d - 1$. By induction, the weight $\sigma_{d-1}$ is maximally saturated; hence, $d_{\text{sat}}(\sigma_{d-1}, \kappa) = d_{\text{gph}}(\sigma_{d-1}, \kappa) = d_{\text{rad}}^{\sigma_{d-1}}(\kappa)$ for all $\kappa \in \text{JH}(\bar{R})$ by Propositions 4.3.14 and 4.3.16.

Let $R^{\sigma_{d-1}} \subseteq R^{\sigma}$, $R^{\sigma_d} \subseteq R^{\sigma}$ be saturated inclusions of lattices. By Lemma 4.3.9, we are in one of the following situations:

$$ (1) \quad R^{\sigma_d} \subseteq R^{\sigma_{d-1}} \subseteq R^{\sigma}; $$

$$ (2) \quad R^{\sigma_{d-1}} \subseteq R^{\sigma_d} \subseteq R^{\sigma}, $$

where the inclusions are all saturated.

*Case (1).* Taking $\kappa = \sigma$ in the above, we have that $d_{\text{sat}}(\sigma, \sigma_{d-1}) = d - 1$ and, hence, obtain chains of saturated inclusions:

$$ \begin{array}{c} p^{d-1} R^{\sigma} \subseteq R^{\sigma_{d-1}} \subseteq R^{\sigma} \\ \cup \\ R^{\sigma_d} \end{array} $$
$$ (2) \quad \mathrm{d}_{\mathrm{gph}}(\kappa_0, \kappa_{i+1}) = \mathrm{d}_{\mathrm{sat}}(\kappa_0, \kappa_{i+1}) < i+1, \text{ and actually } \mathrm{d}_{\mathrm{gph}}(\kappa_0, \kappa_{i+1}) \le i-1. $$

Fix a chain of saturated inclusions $R^{\kappa_0} \subseteq R^{\kappa_1} \subseteq \dots \subseteq R^{\kappa_k}$. Item (2) implies that $R^{\kappa_0} \subseteq R^{\kappa_{i+1}} \subseteq p^{-i+1}R^{\kappa_0}$. The induced inclusion $R^{\kappa_i} \subseteq R^{\kappa_{i+1}} \subseteq p^{-i+1}R^{\kappa_0}$ contradicts item (1). □

*Proof of Theorem 4.1.9.* (5) follows from Propositions 4.3.14 and 4.3.19. Items (1) and (2) follow from Propositions 4.3.16 and 4.3.19. Note that $\Gamma_{\mathrm{rad}}(\overline{R}^{\sigma})$ is a subgraph of $\Gamma(\overline{R}^{\sigma})$. Using Lemma 4.2.6 and the fact that $\Gamma(\overline{R}^{\sigma})$ is acyclic, $\Gamma(\overline{R}^{\sigma})$ must in fact be $\Gamma_{\mathrm{rad}}(\overline{R}^{\sigma})$. This implies (3). □

# 5. Breuil's Conjectures

In this section, we deduce generalizations of Breuil's conjectures on mod $p$ multiplicity one and lattices from the results in Sections 3-4. In Sections 5.1-5.2, we prove a version of Breuil's conjectures for abstract patching functors. Finally, we deduce the main results in Section 5.3.

## 5.1. Cyclicity for patching functors.
In this subsection, we show that certain patched modules for tame types are locally free of rank one over the corresponding local deformation space using several inductive steps. The base case is Lemma 5.1.3. Each inductive step uses one of the two arguments in [EGS15, Section 10]. It is here that the results of Section 3.6.3 enter.

Recall from Section 1.4 that $\mathcal{S}$ is a finite set so that for each $\tilde{v} \in \mathcal{S}$, $F_{\tilde{v}}$ is a finite unramified extension of $\mathbb{Q}_p$ of degree $f_{\tilde{v}}$. For each $\tilde{v}$, let $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \mathrm{GL}_3(\mathbb{R})$ be a 10-generic semisimple continuous Galois representation and let $\bar{\rho}_{\mathcal{S}}$ be $(\bar{\rho}_{\tilde{v}})_{\tilde{v} \in \mathcal{S}}$. Recall the weak minimal patching functor setting of Section 3.5. Suppose that $\bar{\rho}_{\mathcal{S}}|_{I_{\mathcal{S}}} \cong \bar{\tau}_{\mathcal{S}}(s, \mu)$ for $s \in \mathcal{W}$ and $\mu \in X^*(\mathcal{T})$.

Let $K$ be $\prod_{\tilde{v} \in \mathcal{S}} K_{\tilde{v}}$. Suppose that $M_{\infty}$ is a weak minimal patching functor for $\bar{\rho}_{\mathcal{S}}$ (cf. Definition 3.5.1). For each $\tilde{v} \in \mathcal{S}$, let $\tau_{\tilde{v}}$ be a 13-generic tame inertial type for $F_{\tilde{v}}$. Recall $R_{\mathfrak{M}_{\tilde{v}}, \bar{\rho}_{\tilde{v}}}^{\tau_{\tilde{v}}, \bar{\rho}_{\tilde{v}}, \square}$ from [LLHLM18, Definition 5.10]. We let $\tau_{\mathcal{S}}$ be $(\tau_{\tilde{v}})_{\tilde{v}}$ so that $\sigma(\tau_{\mathcal{S}})$ is the $K$-module $\otimes_{\tilde{v} \in \mathcal{S}} \sigma(\tau_{\tilde{v}})$. Let

$$ R'_{\infty}(\tau_{\mathcal{S}}) \stackrel{\mathrm{def}}{=} R_{\infty}(\tau_{\mathcal{S}}) \hat{\otimes}_{(\otimes_{\tilde{v} \in \mathcal{S}} R_{\tilde{v}}^{\tau_{\tilde{v}}})} \left( \otimes_{\tilde{v} \in \mathcal{S}} R_{\mathfrak{M}_{\tilde{v}}, \bar{\rho}_{\tilde{v}}}^{\tau_{\tilde{v}}, \bar{\rho}_{\tilde{v}}, \square} \right) $$

and

$$ M'_{\infty}(V) \stackrel{\mathrm{def}}{=} M_{\infty}(V) \hat{\otimes}_{(\otimes_{\tilde{v} \in \mathcal{S}} R_{\tilde{v}}^{\tau_{\tilde{v}}})} \left( \otimes_{\tilde{v} \in \mathcal{S}} R_{\mathfrak{M}_{\tilde{v}}, \bar{\rho}_{\tilde{v}}}^{\tau_{\tilde{v}}, \bar{\rho}_{\tilde{v}}, \square} \right) $$

for any subquotient $V$ of a lattice in $\sigma(\tau_{\mathcal{S}})$.
We assume that $R_{\infty}(\tau_S)$ is nonzero. Then there exists $\tilde{w} = (\tilde{w}_{\tilde{v}})_{\tilde{v} \in S} = (\tilde{w}_i)_{i \in \mathcal{J}} \in \mathrm{Adm}^{\vee}(\eta)$ such that $\tau_S \cong \tau_S(s(w^*)^{-1}, \mu + s(\tilde{w}^*)^{-1}(0))$ with the lowest alcove presentation $(s(w^*)^{-1}, \mu + s(\tilde{w}^*)^{-1}(0) - \eta)$ by Theorem 3.3.12 and Proposition 3.4.1. Note that $\tilde{w}_{\tilde{v}} = \tilde{w}(\bar{\rho}_{\tilde{v}}, \tau_{\tilde{v}})$. Then, as explained in Section 3.6.1, $\mathrm{Spf} R'_{\infty}(\tau_S)$ (respectively $\mathrm{Spf} R'_{\infty}(\tau_S)$) is formally smooth over $\mathrm{Spf}(\bigotimes_{i \in \mathcal{J}} \bar{R}_{\overline{\mathfrak{m}_{\tilde{v}}}, \tilde{w}_i}^{\mathrm{exp}, \vee})$ (respectively $\mathrm{Spf}(\bigotimes_{i \in \mathcal{J}} \bar{R}_{\overline{\mathfrak{m}_{\tilde{v}}}, \tilde{w}_i}^{\mathrm{exp}, \vee})$ if $\ell(\tilde{w}_i) > 1$ for all $i \in \mathcal{J}$), where $i = (\tilde{v}, i_{\tilde{v}})$. We consider $\bar{R}'_{\infty}(\tau_S)$ (respectively $R'_{\infty}(\tau_S)$) both as a $\bigotimes_{i \in \mathcal{J}} \bar{R}_{\overline{\mathfrak{m}_{\tilde{v}}}, \tilde{w}_i}^{\mathrm{exp}, \vee}$-algebra and a $\bigotimes_{\tilde{v} \in S} \bar{R}_{\bar{\rho}_{\tilde{v}}}^{\Box}$-algebra (respectively a $\bigotimes_{i \in \mathcal{J}} \bar{R}_{\overline{\mathfrak{m}_{\tilde{v}}}, \tilde{w}_i}^{\mathrm{exp}, \vee}$-algebra and a $\bigotimes_{\tilde{v} \in S} \bar{R}_{\bar{\rho}_{\tilde{v}}}^{\Box}$-algebra). It is easy to see that $M'_{\infty}(V)$ is a cyclic $R'_{\infty}(\tau_S)$-module if and only if $M_{\infty}(V)$ is a cyclic $R_{\infty}(\tau_S)$-module. Moreover, $M'_{\infty}(V)$ is always maximal Cohen–Macaulay over its support. It will often be more convenient to prove that $M'_{\infty}(V)$ is a cyclic $R'_{\infty}(\tau_S)$-module, and we will switch to $M'_{\infty}(V)$ without comment. If $\sigma \in \mathrm{JH}(\sigma(\tau_S))$, then recall from Section 4.1.2 that $\sigma(\tau_S)^\sigma$ is the unique $\mathcal{O}$-lattice up to homothety in $\sigma(\tau_S)$ with cosocle $\sigma$.

**THEOREM 5.1.1.** Let $\tau_S$ be a 13-generic tame type and $\sigma \stackrel{\mathrm{def}}{=} F(\lambda) \in W^\vee(\bar{\rho}_S, \tau_S)$ such that for all $i \in \mathcal{J}$,

$$ \lambda_{\pi^{-1}(i)} \in X_1(T) \text{ is in alcove } C_1 \text{ if } \ell(\tilde{w}_i^*) \le 1. \quad (5.1) $$

Then $M_{\infty}(\sigma(\tau_S)^\sigma)$ is free of rank one as a $R_{\infty}(\tau_S)$-module.

The proof of Theorem 5.1.1 proceeds by proving cases of increasing complexity. We distinguish five steps in the argument.

**LEMMA 5.1.2.** If $\sigma \in W^\vee(\bar{\rho}_S)$, then $M_{\infty}(\sigma)$ is a cyclic $R_{\infty}(\tau_S)$-module.

*Proof.* The proof follows the methods of Diamond and Fujiwara [Dia97]. Note that the support of $M_{\infty}(\sigma)$ is formally smooth as can be checked from Theorem 3.5.2. Since $M_{\infty}(\sigma)$ is maximal Cohen–Macaulay over its formally smooth support by Definition 3.5.1(3), it is free over its support by the Auslander–Buchsbaum–Serre theorem and the Auslander–Buchsbaum formula. Since $e(M_{\infty}(\sigma)) = 1$ by Theorem 3.5.2, the free rank is one. $\square$

If $V$ is a $\sigma$-torsion $K$-module satisfying the hypotheses of Lemma 3.6.2, we write $R_{\infty}(V)$ to denote the quotient $R_{\infty}/I(V)$. For the rest of this subsection, we assume that the weight $\sigma$ satisfies the conditions of Theorem 5.1.1.

**LEMMA 5.1.3.** Assume that $\ell(\tilde{w}_i^*) > 1$ for all $i \in \mathcal{J}$. Let $\sigma \in W^\vee(\bar{\rho}_S, \tau_S)$ and let $V$ be a nonzero quotient of $\bar{\sigma}(\tau_S)^\sigma$. Then $M_{\infty}(V)$ is a cyclic $R_{\infty}(\tau_S)$-module.
Proof. If $R'_{\infty}(\tau_S)$ is formally smooth over $\mathcal{O}$, then the result follows from Lemma 5.1.2. Suppose otherwise. We use the notation of the proof of Lemma 4.3.7 with $R = \sigma(\tau_S)$. Recall that for $\sigma \in \mathrm{JH}(\sigma(\tau_S))$, $p_S(\sigma) = (z_j(\sigma))_{j=1}^N + (p)$. By Lemma 3.6.10, for $\sigma_1$ and $\sigma_2 \in W^?(\bar{\rho}_S, \tau_S)$, we have

$$ \#\{(z_j(\sigma_1))_j \Delta\{z_j(\sigma_2)\}_j\} = 2\mathrm{d}_{\mathrm{gph}}(\sigma_1, \sigma_2). \quad (5.2) $$

Let $\sigma_1 \in W^?(\bar{\rho}_S, \tau_S)$ be such that $\mathrm{d}_{\mathrm{gph}}(\sigma, \sigma_1) = 1$. Furthermore, fix a saturated inclusion $\sigma(\tau_S)^{\sigma_1} \hookrightarrow \sigma(\tau_S)^{\sigma}$. Letting $M \stackrel{\mathrm{def}}{=} M'_{\infty}(\sigma(\tau_S)^{\sigma})$ and $M_1 \stackrel{\mathrm{def}}{=} M'_{\infty}(\sigma(\tau_S)^{\sigma_1})$, we have a map $M_1 \to M$. By Proposition 4.3.7, $\sigma(\tau_S)^{\sigma} / \sigma(\tau_S)^{\sigma_1}$ is $p$-torsion, with Jordan–Hölder factors determined by Theorem 4.1.9. By the proofs of Lemmas 3.6.2 and 4.3.7, the scheme-theoretic support of $M/M_1$ in Spec $S$ is

$$ \mathrm{Spec} \left( S / \bigcap_{\sigma' \in \mathrm{JH}(\sigma(\tau_S)^{\sigma}) / \sigma(\tau_S)^{\sigma_1}} \mathfrak{p}_S(\sigma') \right). \quad (5.3) $$

The sets $\{z_j(\sigma_1)\}_j$ and $\{z_j(\sigma)\}_j$ differ at exactly one component, say $j_1$. The equation (5.2) determines $p_S(\sigma')$ for all $\sigma' \in \mathrm{JH}(\sigma(\tau_S))$ from which one checks that $z_{j_1}(\sigma)$ is in $p_S(\sigma')$ for any $\sigma' \in \mathrm{JH}(\sigma(\tau_S)^{\sigma} / \sigma(\tau_S)^{\sigma_1})$. By (5.3), we see that coker($M_1 \to M$) is killed by $z_{j_1}(\sigma)$. Similarly, we have that coker($pM \to M_1$) is annihilated by $z_{j_1}(\sigma_1)$. Hence, we have inclusions $z_{j_1}(\sigma)M \subset M_1$ and $z_{j_1}(\sigma_1)M_1 \subset pM$. Combining these, we have that $pM = z_{j_1}(\sigma)z_{j_1}(\sigma_1)M \subset z_{j_1}(\sigma_1)M_1 \subset pM$. We conclude that the above inclusions are equalities.

Applying the above argument for all $\sigma_1 \in W^?(\bar{\rho}_S, \tau_S)$ at distance one from $\sigma$, we have that $M'_{\infty}(\sigma) \cong M'_{\infty}(\sigma(\tau_S)^{\sigma}) / (\omega, \{z_i(\sigma)\}_i)$. The left-hand side, and hence the right-hand side, is cyclic by Lemma 5.1.2. We conclude that $M'_{\infty}(\bar{\sigma}(\tau_S)^{\sigma})$ is cyclic by Nakayama's lemma. Since $M'_{\infty}(V)$ is a nonzero quotient of $M'_{\infty}(\bar{\sigma}(\tau_S)^{\sigma})$, it too is cyclic. $\square$

As in Section 2.3, let $\sigma_{(\omega,a)} \stackrel{\mathrm{def}}{=} F(\mathfrak{F}\tau_{\mu}(s\omega, a))$ (where $(s\omega, a) \in \underline{\Delta}_W^{\mu} \times A$). Then $W^?(\bar{\rho}_S) = \{\sigma_{(\omega,a)} : (\omega, a) \in r(\Sigma)\}$. Similarly, for $\tilde{w} \in \mathrm{Adm}^{\vee}(\eta)$, we have a bijection

$$ \begin{gathered} (\tilde{w}^*)^{-1}(\Sigma) \xrightarrow{\sim} \mathrm{JH}(\bar{\sigma}(\tau_S)) \\ (\omega, a) \mapsto \sigma_{(\omega,a)}, \end{gathered} $$

where $\tau_S \stackrel{\mathrm{def}}{=} \tau(s(w^*)^{-1}, \mu - s(\tilde{w}^*)^{-1}(0))$. In what follows, $\sigma \in W^?(\bar{\rho}_S, \tau_S)$ is a Serre weight satisfying (5.1) and $V$ is a nonzero quotient of $\bar{\sigma}(\tau_S)^{\sigma}$. We assume that for each $i \in \mathcal{J}$, there exists a subset $\Sigma_{V,i} \subseteq (\tilde{w}_i^*)^{-1}(\Sigma_0)$ such that

$$ \prod_{i \in \mathcal{J}} \Sigma_{V,i} \xrightarrow{\sim} \mathrm{JH}(V) \quad (5.4) $$
$$ (\omega, a) \mapsto \sigma_{(\omega,a)} = F(\mathfrak{F}\mathfrak{V}_{\mu}(s\omega, a)) $$

is a bijection. All representations we consider below satisfy this assumption.

In the following lemmas, we use a gluing procedure to show that $M_\infty(V)$ is cyclic in cases where the set $W^\dagger(\bar{\rho}_S, \tau_S)$ is of increasing complexity. In Figures 3-7, we give a pictorial realization of the gluing procedure employed in the proofs of Lemma 5.1.4-5.1.7, respectively.

**LEMMA 5.1.4.** Suppose that for each $i \in \mathcal{J}$, either $\ell(\tilde{w}_i^*) > 1$ or $\Sigma_{V,i} \subset \{(\omega, 1), (0, 0), (\varepsilon_1, 0), (\varepsilon_2, 0)\}$ for some $\omega \in \{0, \varepsilon_1, \varepsilon_2\}$. Then $M_\infty(V)$ is a cyclic $R_\infty(\tau_S)$-module.

*Proof.* Note that by condition (5.1), one has $(\omega, 1) \in \Sigma_{V,i}$ whenever $\ell(\tilde{w}_i^*) \le 1$. In this case, we assume, without loss of generality, that $(\omega, 0) \in \Sigma_{V,i}$ (because the quotient of a cyclic module is cyclic). Let $\bar{\Sigma}_{V,i}$ be the image of $\Sigma_{V,i}$ in $\Lambda_W$ under the natural projection. We proceed by induction on

$$ n = \#\{i \in \mathcal{J} : \ell(\tilde{w}_i^*) \le 1 \text{ and } \#\bar{\Sigma}_{V,i} = 3\}. $$

If $n=0$, then a casewise check shows that there exists $\tilde{w}' \in \mathrm{Adm}(\eta)$ such that for all $i \in \mathcal{J}$, $\ell(\tilde{w}_i') > 1$ and $\Sigma_{V,i} \subset \Sigma_{(\tilde{w}_i')^{-1}}$. For example, if $\Sigma_{V,i} \subset \{(0,1), (0, 0), (\varepsilon_1, 0)\}$, then one takes $\tilde{w}_i' = \beta\alpha t_1$. Let $\tau_S'$ be $\tau_S(s(w')^{-1}, \mu + s(\tilde{w}')^{-1}(0))$ where $w' \in W$ is the image of $\tilde{w}'$. Then $M_\infty(V)$ is a cyclic $R_\infty(\tau_S')$-module by Lemma 5.1.3. Hence, $M_\infty(V)$ is a cyclic $R_\infty$-module, and, thus, it is a cyclic $R_\infty(\tau_S)$-module as well.

Suppose now that $n > 0$ so that there is an $i' \in \mathcal{J}$ such that $\#\bar{\Sigma}_{V_{i'}} = 3$ and $\ell(\tilde{w}_{i'}^*) \le 1$. By Theorem 4.1.9 and Propositions 4.1.1 and 4.1.2, there are quotients $V^1$ and $V^2$ of $V$ satisfying (5.4) with the following additional properties:

$$ (1) \quad \Sigma_{V,i'} = \Sigma_{V^1,i'} \cup \Sigma_{V^2,i'} $$

$$ (2) \quad (\omega, 0) \in \Sigma_{V^1,i'} \cap \Sigma_{V^2,i'} $$

$$ (3) \quad \#\bar{\Sigma}_{V^1,i'}, \#\bar{\Sigma}_{V^2,i'} = 2, \text{ and} $$

$$ (4) \quad \Sigma_{V,j,i} = \Sigma_{V,i} \text{ for all } i \neq i' \text{ and } j = 1, 2. $$

For example, if $\Sigma_{V,i'} = \{(0,1), (0,0), (\varepsilon_1, 0), (\varepsilon_2, 0)\}$, then one takes $\Sigma_{V^1,i'} = \{(0,1), (0,0), (\varepsilon_1, 0)\}$ and $\Sigma_{V^2,i'} = \{(0,1), (0,0), (\varepsilon_2, 0)\}$ (cf. Figure 3).

Let $V^3$ be the quotient of $V^1$ and $V^2$ satisfying (5.4) such that $\Sigma_{V^3,i} = \Sigma_{V^1,i} \cap \Sigma_{V^2,i}$ for all $i$. By the inductive hypothesis, $M_\infty'(V^1)$, $M_\infty'(V^2)$, and $M_\infty'(V^3)$ are cyclic. Let $I_j$ be $\mathrm{Ann}_{R_\infty'(\tau_S)} M_\infty'(V^j)$ for $j = 1, 2, 3$. By (1), we have an injection
$V \hookrightarrow V^1 \oplus V^2$. In fact, this injection lands in $V^1 \times_{V^3} V^2$. Comparing lengths, we see that the injection $V \hookrightarrow V^1 \times_{V^3} V^2$ is in fact an isomorphism, which, using the exactness of $M_\infty$ (see Definition 3.5.1), gives the first isomorphism below:

$$ 
\begin{aligned}
M'_\infty(V) &\cong M'_\infty(V^1) \times_{M'_\infty(V^3)} M'_\infty(V^2) \\
&\cong R'_\infty(\tau_S)/I_1 \times_{R'_\infty(\tau_S)/I_3} R'_\infty(\tau_S)/I_2 \\
&\cong R'_\infty(\tau_S)/I_1 \times_{R'_\infty(\tau_S)/(I_1+I_2)} R'_\infty(\tau_S)/I_2 \\
&\cong R'_\infty(\tau_S)/(I_1 \cap I_2).
\end{aligned}
$$

The second isomorphism follows from Lemma 3.6.2. The third isomorphism follows from the fact that $I_1 + I_2 = I_3$ by Theorem 3.6.4, Table 3, and Lemmas 3.6.12 and 3.6.16(3.19). We deduce in particular that $M'_\infty(V)$ is cyclic. $\square$

We remind the reader that our parametrization of Serre weights $\sigma_{(\omega,a)}$ is ‘centered at $\bar{\rho}_S$’. Thus, $\sigma_{(\omega,a)} \in W^\dagger(\bar{\rho}_S)$ exactly when $(\omega, a) \in r(\Sigma)$, and $\sigma_{(\omega,a)} \in \mathbb{JH}(\bar{\sigma}(\tau_S))$ exactly when $(\omega, a) \in (\tilde{w}^*)^{-1}(\Sigma)$.

LEMMA 5.1.5. Suppose that for all $i \in \mathcal{J}$, either $\ell(\tilde{w}_i^*) > 1$ or

$$ \Sigma_{V,i} \subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}), $$

where $(v_1, v_2, v_3)$ is $(\varepsilon_1 - \varepsilon_2, \varepsilon_1, \varepsilon_1 + \varepsilon_2)$, $(\varepsilon_2 - \varepsilon_1, \varepsilon_2, \varepsilon_1 + \varepsilon_2)$, or $(\varepsilon_1 - \varepsilon_2, 0, \varepsilon_2 - \varepsilon_1)$. Then $M_\infty(V)$ is a cyclic $R_\infty(\tau_S)$-module.

Proof. We note that $\#(\Sigma_{V,i} \cap r(\Sigma_0)) \le 5$ for all $i \in \mathcal{J}$. We proceed by induction on $n = \#\{i \in \mathcal{J} : \#(\Sigma_{V,i} \cap r(\Sigma_0)) = 5\}$. There is a unique $\tilde{w}_i^* \in \mathrm{Adm}(2, 1, 0)$ such that

$$ (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) = (\tilde{w}_i^*)^{-1}(\Sigma_0) \cap (\tilde{w}_i')^{-1}(\Sigma_0). \quad (5.5) $$

One can check that if $\#((\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) \cap r(\Sigma_0)) \le 4$, then one of $\ell(\tilde{w}_i^*)$ and $\ell(\tilde{w}_i')$ is strictly greater than one. If we change the type $\tau_S$ so that $\tilde{w}_i^*$ is replaced by $\tilde{w}_i'$, but $\tilde{w}_i'$ are unchanged for $i' \ne i$, there is still a surjection $\bar{\sigma}(\tau_S)^\sigma \to V$ by (5.5) and Theorem 4.1.9. We can therefore assume, without loss of generality, that for each $i$, either $\ell(\tilde{w}_i^*) > 1$ or $\#((\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) \cap r(\Sigma_0)) = 5$.

Suppose that $n = 0$. If $\ell(\tilde{w}_i^*) \le 1$ for some $i \in \mathcal{J}$, then $\#((\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) \cap r(\Sigma_0)) = 5$ by assumption. On the other hand, $\Sigma_{V,i} \cap r(\Sigma_0) < 5$, which implies that $\Sigma_{V,i} \cap r(\Sigma_0) \subset \{(\omega, 1), (0, 0), (\varepsilon_1, 0), (\varepsilon_2, 0)\}$ where $\omega = 0, \varepsilon_1$, or $\varepsilon_2$ by considerations of submodule structure. Hence, there is a quotient $V'$ of $V$ such that $V'$ is of the form in Lemma 5.1.4 and the induced
map $M_\infty(V) \to M_\infty(V')$ is an isomorphism by exactness of $M_\infty$ and Theorem 3.5.2. We are then done by Lemma 5.1.4.

Suppose now that $n > 0$. Suppose that $\#(\Sigma_{V,i'} \cap r(\Sigma_0)) = 5$ for some $i' \in \mathcal{J}$. This, in particular, implies that $\#(\Sigma_{V,i'}) = 6$, by consideration of submodule structure given in Theorem 4.1.9. Then $V$ has a unique quotient $V^1$ such that $\Sigma_{V^1,i} = \Sigma_{V,i}$ if $i \neq i'$ and $\Sigma_{V,i'}^1 = \Sigma_{V,i'} \setminus \{\langle \tilde{w}_i^* \rangle^{-1}(v_4, 1)\}$ where $v_4 = 0, \varepsilon_1$, or $\varepsilon_2$ and $v_4 \neq v_2$ (cf. Figure 4 for an example in the particular case where $\tilde{w}_{i'} = t_1$). Then $M_\infty(V^1)$ is cyclic by the induction hypothesis. There is also a submodule $V^2 \subset V$ such that $\Sigma_{V^2,i} = \Sigma_{V,i}$ if $i \neq i'$ and $\#\Sigma_{V^2,i'} = \#(\Sigma_{V^2,i'} \cap r(\Sigma_0)) = 2$ (cf. Figure 4; note also that $(\tilde{w}_i^*)^{-1}(v_4, 1) \in \Sigma_{V^2,i'}$ since $\text{soc } V^2 \subseteq \text{soc } V$). One can check that $M_\infty(V^2)$ is cyclic by the inductive hypothesis. Then letting $M'' = \ker(M_\infty(V) \to M_\infty(V^1))$ and $M' = M_\infty(V^2)$, [EGS15, Lemma 10.1.13] implies that $M_\infty(V)$ is cyclic. $\square$

**LEMMA 5.1.6.** Suppose that for all $i \in \mathcal{J}$, either $\ell(\tilde{w}_i) > 1$ or $\Sigma_{V,i} \subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v, 0)\})$ where $v$ is $\varepsilon_1 - \varepsilon_2$, $\varepsilon_2 - \varepsilon_1$, or $\varepsilon_1 + \varepsilon_2$. Then $M_\infty(V)$ is a cyclic $R_\infty(\tau_S)$-module.

In the setting of Lemma 5.1.6, note that when $\ell(\tilde{w}_i) \le 1$, the condition $\Sigma_{V,i} \subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v, 0)\})$, $v \in \{\varepsilon_1 - \varepsilon_2, \varepsilon_2 - \varepsilon_1, \varepsilon_1 + \varepsilon_2\}$ is equivalent to $\#\Sigma_{V,i} < 9$.

*Proof.* We proceed by induction on

$n = \#\{i \in \mathcal{J} : \ell(\tilde{w}_i^*) \le 1 \text{ and } \Sigma_{V,i} \not\subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) \text{ for} \\ (v_1, v_2, v_3) = (\varepsilon_1 - \varepsilon_2, \varepsilon_1, \varepsilon_1 + \varepsilon_2), (\varepsilon_2 - \varepsilon_1, \varepsilon_2, \varepsilon_1 + \varepsilon_2), \\ \text{and } (\varepsilon_1 - \varepsilon_2, 0, \varepsilon_2 - \varepsilon_1)\}.$

The case $n = 0$ is covered by Lemma 5.1.5. Suppose that $n > 0$ and that $i' \in \mathcal{J}$ with $\ell(\tilde{w}_{i'}^*) \le 1$ and

$$ \Sigma_{V,i'} \not\subset (\tilde{w}_{i'}^*)^{-1}(\Sigma_0 \setminus \{(v_1, 0), (v_2, 1), (v_3, 0)\}) $$

for $(v_1, v_2, v_3) \in \{(\varepsilon_1 - \varepsilon_2, \varepsilon_1, \varepsilon_1 + \varepsilon_2), (\varepsilon_2 - \varepsilon_1, \varepsilon_2, \varepsilon_1 + \varepsilon_2), (\varepsilon_1 - \varepsilon_2, 0, \varepsilon_2 - \varepsilon_1)\}$. Assume, without loss of generality, that $\Sigma_{V,i'} = (\tilde{w}_{i'}^*)^{-1}(\Sigma_0 \setminus \{(v, 0)\})$, as the quotient of a cyclic module is again cyclic. We can further assume, without loss of generality, that $v = \varepsilon_1 + \varepsilon_2$ (which implies that if $\sigma = \sigma_{(\omega,a)}$, then $(\omega_{i'}, a_{i'}) = (\tilde{w}_{i'}^*)^{-1}(0, 1)$). Then there are quotients $V^1$ and $V^2$ of $V$ such that

(1) $\Sigma_{V,i'} = \Sigma_{V^1,i'} \cup \Sigma_{V^2,i'}$,

(2) $\Sigma_{V^j,i} = \Sigma_{V,i}$ for $i \neq i'$ and $j = 1, 2$,

(3) $\Sigma_{V^1,i'} \subset (\tilde{w}_{i'}^*)^{-1}(\Sigma_0 \setminus \{(\varepsilon_1 - \varepsilon_2, 0), (\varepsilon_1, 1), (\varepsilon_1 + \varepsilon_2, 0)\})$, and
$$ (4) \quad \Sigma_{V^2,i'} \subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(\varepsilon_2 - \varepsilon_1, 0), (\varepsilon_2, 1), (\varepsilon_1 + \varepsilon_2, 0)\}) $$

(cf. Figure 5 for an example when $\tilde{w}_{i'} = \alpha$). We now argue by induction as in the proof of Lemma 5.1.4 using Theorem 3.6.4 and Lemmas 3.6.14, 3.6.16(3.17), and 3.6.16(3.18). Note that the analogue of Lemma 3.6.16 for the shapes $\beta$ and $\gamma$ hold symmetrically (see Remark 3.6.5). $\square$

**LEMMA 5.1.7.** With $V$ as described before Lemma 5.1.4, $M_\infty(V)$ is a cyclic $R_\infty(\tau_S)$-module.

*Proof.* We proceed by induction on

$$ n = \#\{i \in \mathcal{J} : \ell(\tilde{w}_i^*) \le 1 \text{ and } \#\Sigma_{V,i} = 9\}. $$

(In the definition of $n$, note that the condition $\#\Sigma_{V,i} = 9$ can be replaced by 'for all $\nu \in \{\varepsilon_1 - \varepsilon_2, \varepsilon_2 - \varepsilon_1, \varepsilon_1 + \varepsilon_2\}$, one has $\Sigma_{V,i} \subset (\tilde{w}_i^*)^{-1}(\Sigma_0 \setminus \{(\nu, 0)\})$.) The case $n = 0$ is covered by Lemma 5.1.6. Suppose that $n > 0$. Let $i' \in \mathcal{J}$ be such that $\ell(\tilde{w}_{i'}^*) \le 1$ and $\#\Sigma_{V,i'} = 9$ (equivalently, $\ell(\tilde{w}_i^*) \le 1$ and $\Sigma_{V,i} = (\tilde{w}_i^*)^{-1}(\Sigma_0)$). Then $V$ has a quotient $V^1$ such that $\Sigma_{V^1,i} = \Sigma_{V,i}$ if $i \ne i'$ and $\Sigma_{V^1,i'} = \Sigma_{V,i'} \setminus \{(\tilde{w}_{i'}^*)^{-1}(\nu, 0)\}$ where $\nu = \varepsilon_1 - \varepsilon_2, \varepsilon_2 - \varepsilon_1$, or $\varepsilon_1 + \varepsilon_2$ (cf. Figure 7 for an example when $\tilde{w}_i^* = \gamma^+ t_1$). If the map $M_\infty(V) \to M_\infty(V^1)$ is an isomorphism, we are done. Otherwise, there is a submodule $V^2$ of $V$ such that $\Sigma_{V^2,i} = \Sigma_{V,i}$ if $i \ne i'$, $\#\Sigma_{V^2,i'} = \#(\Sigma_{V^2,i'} \cap r(\Sigma_0)) = 2$, and $(\tilde{w}_i^*)^{-1}(\nu, 0) \in \Sigma_{V^2,i'} \subset r(\Sigma_0)$. Then one argues as in the proof of Lemma 5.1.5. $\square$

**Proof of Theorem 5.1.1.** Lemma 5.1.7 implies that $M_\infty(\bar{\sigma}(\tau_S)^\sigma)$ is a cyclic $\bar{R}_\infty(\tau)$-module. Nakayama's lemma implies that $M_\infty(\sigma(\tau_S)^\sigma)$ is a cyclic $R_\infty(\tau_S)$-module. By Theorem 3.5.2, $M_\infty(\sigma)$, and thus $M_\infty(\sigma(\tau_S)^\sigma)$, is nonzero. By Theorem 3.5.3, $R_\infty(\tau_S)$ is irreducible and reduced so that Definition 3.5.1(3) implies that $M_\infty(\sigma(\tau_S)^\sigma)$ is a faithful $R_\infty(\tau_S)$-module. Since a faithful cyclic module is free of rank one, we are done. $\square$

Finally, we show that the hypothesis of Theorem 5.1.1 is actually necessary.

**PROPOSITION 5.1.8.** Let $\tau_S$ be a 13-generic tame type and $\sigma \stackrel{\text{def}}{=} F(\lambda) \in W'(\bar{\rho}_S, \tau_S)$. Assume that for some $i \in \mathcal{J}$,

$$ \lambda_{\pi^{-1}(i)} \in X_1(T) \text{ is in alcove } C_0 \text{ and } \ell(\tilde{w}_i^*) \le 1. \quad (5.6) $$

Then $M_\infty(\sigma(\tau_S)^\sigma)$ is not free as an $R_\infty(\tau_S)$-module.

*Proof.* As we will not make use of this, we only give a sketch of proof. Unlike the rest of this section, we will parametrize the weights by centering around
$\tau_S$ so that $\tau_S = \tau(s, \mu)$ and $\sigma_{(\omega,a)}$ means $\sigma_{(\omega,a)} \stackrel{\text{def}}{=} F(\mathfrak{F}\tau_\mu(s\omega, a))$. In particular, $W^?(\bar{\rho}_S, \tau_S)$ consists of $\sigma_{(\omega,a)}$ such that $(\omega, a) \in \Sigma_{\tilde{w}^*}$.

We write $\sigma = \sigma_{(\omega,a)}$, so $(\omega, a) \in \Sigma$. For each $i$ satisfying (5.6), we have $a_i = 0$. If moreover $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$, we set $(\omega'_i, a'_i) = (\omega_i, 1)$, otherwise we set $(\omega'_i, a'_i)$ to be one of the two elements in $\Sigma_0$ that are adjacent to $(\omega_i, a_i)$. For $i$ not satisfying (5.6), we set $(\omega'_i, a'_i) = (\omega_i, a_i)$. Thus, we obtain a weight $\sigma' = \sigma_{(\omega',a')}$ which satisfies the hypothesis of Theorem 5.1.1.

We now fix a saturated inclusion $\iota : \sigma(\tau_S)^\sigma \hookrightarrow \sigma(\tau_S)^{\sigma'}$, and let $C$ denote the cokernel. Since $M_\infty(\sigma(\tau_S)^{\sigma'}) \cong R_\infty(\tau_S)$, $M_\infty(\iota)$ identifies $M_\infty(\sigma(\tau_S)^\sigma)$ with an ideal $I(C)$ of $R_\infty(\tau_S)$. Thus, we need to show that $I(C)$ is not a principal ideal, and to do so, it suffices to show that the image of $I(C)$ in $\bar{R}_\infty(\tau_S)$ is not principal.

Let $V$ denote the cokernel of $\iota \mod \omega$. Using Theorem 4.1.9, we see that $V$ is multiplicity free, and $JH(V) \cap W^?(\bar{\rho}_S)$ consists of $\sigma_{(v,c)}$ in $(v, c) \in \Sigma_{\tilde{w}^*} \setminus \prod_i \Sigma_{V^c, i}$, where

* if $i$ does not satisfy (5.6), $\Sigma_{V^c, i} = \Sigma_{\tilde{w}_i^*}$;

* if $i$ satisfies (5.6) and $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$, $\Sigma_{V^c, i} = \{(\omega_i, 0), (\mu_1, 1), (\mu_2, 1)\}$ where $\{\omega_i, \mu_1, \mu_2\} = \{0, \varepsilon_1, \varepsilon_2\}$; and

* if $i$ satisfies (5.6) and $(\omega_i, a_i) \notin \Sigma_0^{\text{inn}}$, $\Sigma_{V^c, i} = \{(\omega_i, 0), (\omega_i'', 1)\}$ where $\omega_i''$ is such that $\{\omega_i, \omega_i', \omega_i''\} = \{\varepsilon_2 - \varepsilon_1, 0, \varepsilon_2\}, \{\varepsilon_1 - \varepsilon_2, 0, \varepsilon_1\}, \text{ or } \{\varepsilon_1 + \varepsilon_2, \varepsilon_1, \varepsilon_2\}$.

**Lemma 3.6.2** shows that $M_\infty(V) \cong \bar{R}_\infty(\tau_S)/I(V)$, where $I(V)$ is the intersection, over $\kappa \in \text{JH}(V) \cap W^?(\bar{\rho}_S)$, of the ideals $\mathfrak{p}(\kappa)$ defined in Proposition 3.6.1. Note that the image of $I(C)$ in $\bar{R}_\infty(\tau_S)$ is $I(V)\bar{R}_\infty(\tau_S)$. Thus, we need to show $I(V)\bar{R}_\infty(\tau_S)$ or, equivalently, $I(V)\bar{R}_\infty'(\tau_S)$ is not principal. Recall that there is a formally smooth map $\widehat{\otimes}_{i \in \mathcal{J}} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_i}}^{\text{expl}, \nabla} \to \bar{R}_\infty'(\tau_S)$. We claim that the ideal $I(V)$ comes from $I \stackrel{\text{def}}{=} \widehat{\otimes}_i \left( \bigcap_{(v_i, c_i) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V^c, i}} \mathfrak{c}_{(v_i, c_i)} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla} \right)$. To check the claim, let $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}}}^{\text{expl}, \nabla} \stackrel{\text{def}}{=} \widehat{\otimes}_{i \in \mathcal{J}} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_i}}^{\text{expl}, \nabla}$. We first check that $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}}}^{\text{expl}, \nabla}/I$ has the expected cycle. To do this, we check using Table 3 that $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla} / \left( \bigcap_{(v_i, c_i) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V^c, i}} \mathfrak{c}_{(v_i, c_i)} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla} \right)$ is filtered by $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla} / \mathfrak{c}_{(v_j, c_j)}$ for $(v_j, c_j) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V^c, i}$ and that $\bigcap_{(v_i, c_i) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V^c, i}} \mathfrak{c}_{(v_i, c_i)} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla}$ is filtered by $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_{f_{\bar{v}}-1-i}}}^{\text{expl}, \nabla} / \mathfrak{c}_{(v_j, c_j)}$ for $(v_j, c_j) \in \Sigma_{V^c, i}$. Thus, we can filter the quotient $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}}}^{\text{expl}, \nabla}/I$ with factors of the form $\widehat{\otimes}_{i \in \mathcal{J}} \bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_i}}^{\text{expl}, \nabla}/\mathfrak{p}_i$, where the $\mathfrak{p}_i$ are minimal primes of $\bar{R}_{\widehat{\mathfrak{M}_{\bar{v}}, \tilde{w}_i}}^{\text{expl}, \nabla}$ (note that taking completion is an exact operation). As $\Gamma(\text{im}(\iota$
(mod $\omega$))) decomposes as a product over $\mathcal{J}$, an inductive argument using Theorem 3.6.4 and the above filtration on $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}}^{\text{expl}, \nabla} / I$ shows that the cycle of $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}}^{\text{expl}, \nabla} / I$ is given by the components of $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}}^{\text{expl}, \nabla}$ labeled by the Serre weights corresponding to $\Gamma(\text{im}(t \pmod \omega))$. In order to prove the claim, we are left to show that $I$ is a radical ideal. To see this, we make the following observation: If $R, S$ are two reduced local Noetherian rings over a perfect field $k$ and $J, L$ are radical ideals of $R, S$, then $J \otimes_k L$ is a radical ideal of $R \otimes_k S$. This is because $(R \otimes S)/(J \otimes L)$ embeds into $((R/J) \otimes S) \times (R \otimes (S/L))$ (as can be seen by choosing bases of $J, L$ as $k$-vector spaces and extending them to bases of $R, S$), and the latter ring is reduced. Finally, we note that in the situation above, the property of being a radical ideal is preserved by completion so that $J \widehat{\otimes}_k L$ is also radical in $R \widehat{\otimes}_k S$. The claim is proven.

By Nakayama's lemma, the size of a minimal set of generators of $I$ is given by the dimension over $\mathbb{F}$ of

$$I \otimes_{\widehat{\otimes}_i \in \mathcal{J}} \overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{m}}, \tilde{w}_i} \mathbb{F} = \bigotimes_i \left( \bigcap_{(v_i, c_i) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V_c, i}} \mathbf{c}_{(v_i, c_i)} \otimes \overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{m}}, \tilde{w}, f_{\tilde{v}-1-i}} \mathbb{F} \right)$$

Hence, it suffices to show that for $i$ satisfying (5.6),

$$\bigcap_{(v_i, c_i) \in \Sigma_{\tilde{w}_i^*} \setminus \Sigma_{V_c, i}} \mathbf{c}_{(v_i, c_i)} \otimes \overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{m}}, \tilde{w}, f_{\tilde{v}-1-i}} \mathbb{F}$$

has dimension greater than one, which can be checked from Table 3. $\square$

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

**5.2. Gauges for patching functors.** In this subsection, we compute the image of maps between patched modules for lattices in Deligne–Lusztig representations. This can be viewed as a calculation of lattice gauges in families. The main ingredient for this section is Theorem 5.1.1, after which algebraic geometric arguments using the projection formula and the Cohen–Macaulay property prove Theorem 5.2.3. We learned these arguments from M. Emerton.

We continue to use the weak minimal patching functor setting of Section 3.5. Let $S, \bar{\rho}_S, R_\infty, X_\infty$, and $M_\infty$ be as in Section 5.1. Let $\tau_S$ be $(\tau_{\tilde{v}})_{\tilde{v} \in S}$, where $\tau_{\tilde{v}}$ is a 13-generic tame type. Let $X_\infty(\tau_S) \stackrel{\text{def}}{=} \text{Spec } R_\infty(\tau_S)$. It is a normal and Cohen–Macaulay scheme by (the proof of) Theorem 3.5.3. Let $Z \subset X_\infty(\tau_S)$ be the locus of points lying on two irreducible components of the special fiber of $X_\infty(\tau_S)$. Note that the codimension of $Z \subset X_\infty(\tau_S)$ is two. Let

$$j : U \stackrel{\text{def}}{=} X_\infty(\tau_S) \setminus Z \hookrightarrow X_\infty(\tau_S)$$
be the natural open immersion. Note that $j$ and $U$ come from pulling back the open immersion $j_0: U_0 \hookrightarrow \text{Spf}(\bigotimes_{\tilde{v} \in S} R_{\tilde{v}}^{\tau_{\tilde{v}}})$ via the formally smooth map $X_\infty(\tau_S) \to \text{Spf}(\bigotimes_{\tilde{v} \in S} R_{\tilde{v}}^{\tau_{\tilde{v}}})$, where $U_0$ is defined in an analogous way as $U$.

**LEMMA 5.2.1.** *The scheme $U$ is regular.*

*Proof.* The irreducible components of the special fiber of $X_\infty(\tau_S)$ are formally smooth over $\mathbb{F}$; hence, the special fiber $\bar{U}$ of $U$ is regular by the last part of Theorem 3.5.3. The dimension of the tangent space of $U$ at a characteristic $p$ point is at most one more than the dimension of the tangent space of $\bar{U}$ at that point. By $p$-flatness, the Krull dimension of $U$ is one more than the Krull dimension of $\bar{U}$, and so $U$ is regular at characteristic $p$ points. Since the generic fiber of $U$, which is isomorphic to the generic fiber of $X_\infty(\tau_S)$, is regular, $U$ is regular. $\square$

We now use the notation $j^*$ and $j_*$ which take quasicoherent sheaves on $X_\infty(\tau_S)$ to those on $U$ and vice versa, respectively.

**LEMMA 5.2.2.** Let $\sigma, \kappa \in \text{JH}(\overline{\sigma(\tau_S)})$ and let $\iota: \sigma(\tau_S)^\kappa \hookrightarrow \sigma(\tau_S)^\sigma$ be a saturated injection. For any $\theta = \bigotimes_{\tilde{v} \in S} \theta_{\tilde{v}} \in W^!(\bar{\rho}_S)$, let $m(\theta)$ be the multiplicity with which $\theta$ appears in the cokernel of $\iota$. Then the induced injection $j^*M_\infty(\iota): j^*M_\infty(\sigma(\tau_S)^\kappa) \hookrightarrow j^*M_\infty(\sigma(\tau_S)^\sigma)$ has image

$$j^*\left(\prod_{\theta \in W^!(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} M_\infty(\sigma(\tau_S)^\sigma)\right),$$

$$\text{where } \mathfrak{p}(\theta) = \sum_{\tilde{v} \in S} \mathfrak{p}(\theta_{\tilde{v}}).$$

*Proof.* Note that $j^*(M_\infty(\sigma(\tau_S)^\kappa))$ and $j^*(M_\infty(\sigma(\tau_S)^\sigma))$ are locally free (of rank one) since they are Cohen–Macaulay of full support over the regular scheme $U$ (see Definition 3.5.1). Then the image of $j^*M_\infty(\iota)$ is $J \otimes_{\mathcal{O}_U} j^*(M_\infty(\sigma(\tau_S)^\sigma))$, where $J$ is the ideal sheaf of the Cartier divisor corresponding to the cokernel of $j^*M_\infty(\iota)$. It is easily seen that $J = j^*\left(\prod_{\theta \in W^!(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S)\right)$. Finally, we have that

$$\begin{align*}
& j^*\left(\prod_{\theta \in W^!(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S)\right) \otimes_{\mathcal{O}_U} j^*\left(M_\infty(\sigma(\tau_S)^\sigma)\right) \\
& \cong j^*\left(\prod_{\theta \in W^!(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S)\right) j^*\left(M_\infty(\sigma(\tau_S)^\sigma)\right)
\end{align*}$$
$$= j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} M_\infty(\sigma(\tau_S)^\sigma) \right)$$

where the isomorphism follows from the fact that $j^*(M_\infty(\sigma(\tau_S)^\sigma))$ is locally free. □

Let $\sigma$ be as in Theorem 5.1.1 so that $M_\infty(\sigma(\tau_S)^\sigma)$ is cyclic.

**THEOREM 5.2.3.** With the notation of Lemma 5.2.2, suppose further that $\sigma$ is as in Theorem 5.1.1. Then the induced injection $M_\infty(\iota) : M_\infty(\sigma(\tau_S)^\kappa) \hookrightarrow M_\infty(\sigma(\tau_S)^\sigma)$ has image

$$j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S) \right) M_\infty(\sigma(\tau_S)^\sigma).$$

*Proof.* If $\mathcal{M}$ is a Cohen–Macaulay sheaf on $X_\infty(\tau_S)$, then $j_*j^*\mathcal{M} = \mathcal{M}$ since the codimension of $Z$ is two (cf. [HK04, Proposition 3.5]). Hence, the image of $M_\infty(\iota)$ is

$$j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} M_\infty(\sigma(\tau_S)^\sigma) \right)$$

by Lemma 5.2.2. Since $M_\infty(\sigma(\tau_S)^\sigma)$ is free over $R_\infty(\tau_S)$, we have that

$$\begin{aligned} & j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} M_\infty(\sigma(\tau_S)^\sigma) \right) \\ & = j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S) \right) M_\infty(\sigma(\tau_S)^\sigma). \end{aligned}$$

□

**REMARK 5.2.4.** As $j$ comes from pulling back $j_0$ via $X_\infty(\tau_S) \to \text{Spf}(\widehat{\otimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\widehat{\otimes}})$ and $\mathfrak{p}(\theta)$ are ideals of $\text{Spf}(\widehat{\otimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\widehat{\otimes}})$, we see that the ideal

$$j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S) \right) = j_{0*} j_0^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} \widehat{\otimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\widehat{\otimes}} \right) R_\infty(\tau_S)$$

comes from an ideal in $\widehat{\otimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\widehat{\otimes}}$.

**5.3. Global applications.** In this subsection, we deduce generalizations of conjectures of Dembélé and Breuil on mod $p$ multiplicity one and lattices, respectively (see [Bre14, Conjectures B.1 and 1.2]). Theorem 5.3.4 follows immediately from Theorem 5.1.1. While Theorem 5.3.5 also follows from
**Theorem 5.2.3**, it is crucial that the image of the map between two patched modules given in Theorem 5.2.3 is described by an ideal. This is far from formal and relies crucially on Theorem 5.1.1.

We use the setup in [LLHLM18, Section 7.1]. Let $F/\mathbb{Q}$ be a CM field with maximal totally real subfield $F^+ \neq \mathbb{Q}$ and let $\Sigma_p^+$ (respectively $\Sigma_p$) be the set of places of $F^+$ (respectively of $F$) lying above $p$. Let $G_{/F^+}$ be a reductive group which is an outer form for $\text{GL}_3$, which is quasisplit at all finite places of $F^+$, and which splits over $F$. Suppose that $G_{/F^+}$ is definite, that is, that $G(F_v^+) \cong U_3(\mathbb{R})$ for all $v|\infty$. Recall from [EGH13, Section 7.1] that $G$ admits a reductive model $\mathcal{G}$ defined over $\mathcal{O}_{F^+}[1/N]$, for some $N \in \mathbb{N}$ which is prime to $p$, together with an isomorphism

$$ \iota: \mathcal{G}_{/\mathcal{O}_F[1/N]} \xrightarrow{\iota} \text{GL}_3/\mathcal{O}_F[1/N] \qquad (5.7) $$

which specializes to $\iota_w: \mathcal{G}(\mathcal{O}_{F_v^+}) \xrightarrow{\sim} \mathcal{G}(\mathcal{O}_{F_w}) \xrightarrow{\iota} \text{GL}_3(\mathcal{O}_{F_w})$ for all places $w \in \Sigma_p$ with $w|_{F^+} = v$.

Let $U = U^p U_p \leq G(\mathbb{A}_{F^+}^{(\infty,p)}) \times \mathcal{G}(\mathcal{O}_{F^+,p})$ be a compact open subgroup. If $W$ is a finite $\mathcal{O}$-module endowed with a continuous action of $U$, we write $S(U, W)$ to denote the space of algebraic automorphic forms with coefficients in $W$:

$$ S(U, W) \stackrel{\text{def}}{=} \left\{ \begin{array}{l} f: G(F^+) \backslash G(\mathbb{A}_{F^+}^{\infty}) \to W \mid f(gu) \\ = u^{-1}f(g) \quad \forall g \in G(\mathbb{A}_{F^+}^{\infty}), u \in U \end{array} \right\}. $$

We define

$$ S(U^p, W) \stackrel{\text{def}}{=} \lim_{\underset{U_p}{\to}} S(U^p U_p, W) \quad \text{and} \quad \tilde{S}(U^p, W) \stackrel{\text{def}}{=} \lim_{\underset{s}{\leftarrow}} S(U^p, W/\sigma^s) $$

where in the first limit, the subgroups $U_p \leq \mathcal{G}(\mathcal{O}_{F^+,p})$ run over the compact open neighborhoods of $1 \in \mathcal{G}(\mathcal{O}_{F^+,p})$.

For $U$ as above, let $\mathcal{P}_U$ be the set of finite places $w$ in $F$ whose restriction $v \stackrel{\text{def}}{=} w|_{F^+}$ is a place that splits in $F$ and at which $U$ is unramified. Let $\mathcal{P} \subset \mathcal{P}_U$ be a subset of finite complement. Then the universal Hecke algebra $\mathbb{T}_{\mathcal{P}} = \mathcal{O}[T_w^{(i)}$, $w \in \mathcal{P}$, $0 \leq i \leq n]$ on $\mathcal{P}$ acts naturally on $S(U, W)$. Let $\bar{r}: G_F \to \text{GL}_3(\mathbb{F})$ be a continuous Galois representation. We let $m \subseteq \mathbb{T}_{\mathcal{P}}$ be the maximal ideal which is the kernel of the system of Hecke eigenvalues $\bar{\alpha}: \mathbb{T}_{\mathcal{P}} \to \mathbb{F}$ associated with $\bar{r}$, that is, $\bar{\alpha}$ satisfies

$$ \det(1 - \bar{r}^{\vee}(\text{Frob}_w)X) = \sum_{j=0}^{3} (-1)^j (\mathbf{N}_{F/\mathbb{Q}}(w))^{\binom{j}{2}} \bar{\alpha}(T_w^{(j)})X^j $$

for all $w$ as above. Then we say that $\bar{r}$ is automorphic if $S(U, W)_m$ is nonzero for some $U$ and $W$.
For technical reasons, we choose a place $v_1$ of $F^+$ as in [**CEG+16**, Section 2.3]. We now fix $U^p \le G(\mathbb{A}_{F^+}^{(\infty, p)})$ to be the subgroup with:

(1) $(U^p)_v = \mathcal{G}(\mathcal{O}_{F_v^+})$ for all finite places $v$ of $F^+$ which split in $F$ and do not belong to $\Sigma_p^+ \cup \{v_1\}$;

(2) $(U^p)_{v_1}$ is the preimage of the upper triangular matrices under the map

$$ \mathcal{G}(\mathcal{O}_{F_{v_1}^+}) \to \mathcal{G}(k_{v_1}) \xrightarrow{\iota_{\tilde{v}_1}} \mathrm{GL}_3(k_{\tilde{v}_1}); $$

(3) $(U^p)_v$ is hyperspecial maximal compact in $G(F_v^+)$ if $v$ is inert in $F$.

Let $\bar{r}$ be an automorphic Galois representation. Let $\Sigma_0^+$ denote the set of finite places of $F^+$ which are the restriction of the finite places of $F$ away from $p$ where $\bar{r}$ ramifies. For each $v \in \Sigma_0^+$, we let $\tau_{\tilde{v}}$ be the minimally ramified type in the sense of [**CHT08**, Definition 2.4.14] corresponding to $\bar{r}|_{G_{F_{\tilde{v}}}}$ and $\sigma(\tau_{\tilde{v}})$ be the $\mathrm{GL}_3(\mathcal{O}_{F_{\tilde{v}}})$-representation over $E$ associated with it (cf. the beginning of Section 5.3.1). We write $\sigma(\tau_v) \stackrel{\mathrm{def}}{=} \sigma(\tau_{\tilde{v}}) \circ \iota_{\tilde{v}}$, which is a $\mathcal{G}(\mathcal{O}_{F_v^+})$-representation independent of the choice of $\tilde{v}|v$. For each $v \in \Sigma_0^+$, fix a $\mathcal{O}$-lattice $\sigma(\tau_v)^\circ$ in $\sigma(\tau_v)$ and let $W_{\Sigma_0^+}$ be $\otimes_{v \in \Sigma_0^+} \sigma(\tau_v)^\circ$.

We let $\mathbb{T}^{\mathrm{univ}}$ denote the abstract Hecke algebra over $\mathcal{O}$ generated by the formal variables $T_w^{(j)}$, where $w$ runs over the finite places of $F$ such that $w|_{F^+}$ is split in $F$ and $w|_{F^+} \notin \Sigma_0^+ \cup \Sigma_p^+ \cup \{v_1\}$, and by $T_{v_1}^{(j)}$ for $j = 1, 2, 3$. For a $G(\mathbb{A}_{F^+}^{(\infty, p)})$-module $V$ over $\mathcal{O}$, $\mathbb{T}^{\mathrm{univ}}$ acts naturally on $S(U^p U_p, W)$, $S(U^p, W)$, and $\tilde{S}(U^p, W)$ where $W = W_{\Sigma_0^+} \otimes V$ (cf. [**CEG+16**, Section 2.3]), and we let $\mathfrak{m} \subseteq \mathbb{T}^{\mathrm{univ}}$ be the maximal ideal as before. We will now assume that $S(U, W)_\mathfrak{m}$ is nonzero for some choice of $V$ above. In fact, one can show, by the proof of Proposition 6.0.2, that this is a consequence of the hypothesis that $\bar{r}$ is automorphic.

In the remainder of this section, we let $\bar{r} : G_F \to \mathrm{GL}_3(\mathbb{F})$ be an automorphic Galois representation that satisfies the Taylor–Wiles conditions in the sense of [**LLHLM18**, Definition 7.3]. We assume furthermore that

(i) the extension $F/F^+$ is unramified at all finite places;

(ii) (split ramification) if $\bar{r} : G_F \to \mathrm{GL}_3(\mathbb{F})$ is ramified at a place $w$ of $F$, then $v = w|_{F^+}$ splits as $ww^c$;

(iii) $p$ is unramified in $F^+$ and all places in $F^+$ above $p$ split in $F$; and

(iv) $\bar{r}|_{G_{F_w}}$ is semisimple for all $w \in \Sigma_p$.
For each $v \in \Sigma_p^+$, we choose a place $\tilde{v}|v$ of $F$ and let $S$ be the set $\{\tilde{v}|v \in \Sigma_p^+\}$. Let $\bar{\rho}_{\tilde{v}}$ be $\bar{r}|_{G_{F\tilde{v}}} \text{ and } \bar{\rho}_S$ be $(\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$. We set $K \stackrel{\text{def}}{=} \prod_{\tilde{v} \in S} \text{GL}_3(\mathcal{O}_{F_{\tilde{v}}})$. Let $\tilde{M}_{\infty}$ be the weak minimal patching functor for $\bar{r}$ in the sense of [LLHLM18, Definition 7.11] constructed in [LLHLM18, Proposition 7.15]. As in Section 3.5, we let $M_{\infty}$ be $\tilde{M}_{\infty} \circ \prod_{\tilde{v} \in S} l_{\tilde{v}}$, which is a weak minimal patching functor for $\bar{\rho}_S$ by Proposition 3.5.14.

Along with the construction of $\tilde{M}_{\infty}$ (cf. [CEG+16, Section 2.8], [Le18, Section 4.2], and [LLHLM18, Section 7.3]), one has a ring homomorphism $R_{\infty} = \bigotimes_{\tilde{v} \in S} R_{\tilde{v}}^{\square}[[x_1, \dots, x_h]] \to \text{T}_{U^p G(O_{F^{+,p}})}^{\text{univ}}(W_{\Sigma_0^+})_m$, where $\text{T}_{U^p G(O_{F^{+,p}})}^{\text{univ}}(W_{\Sigma_0^+})$ is the image of $\text{T}^{\text{univ}}$ in

$$ \text{End}_{\mathcal{O}}(S(U^p G(O_{F^{+,p}}), W_{\Sigma_0^+})). $$

We also write $\mathfrak{m}$ for the pullback in $R_{\infty}$ of the maximal ideal of $\text{T}_{U^p G(O_{F^{+,p}})}^{\text{univ}}(W_{\Sigma_0^+})_m$. Then if $\bar{W}_{\Sigma_0^+}$ is a smooth, finite-dimensional $\mathcal{G}(O_{F^{+,p}})$-representation over $\mathbb{F}$, one has

$$ (\tilde{M}_{\infty}(\bar{W}_{\Sigma_0^+})/\mathfrak{m})^{\vee} = S(U^p\mathcal{G}(O_{F^{+,p}}), \bar{W}_{\Sigma_0^+} \otimes \bar{W}_{\Sigma_0^+}^{\vee})[\mathfrak{m}] \quad (5.8) $$

where $\vee$ denotes Pontrjagin duals (cf. [Le18, Theorem 4.1.5]).

**5.3.1. Automorphy lifting.** Recall from Proposition 2.2.6 that if $\tilde{v} \in S$ and $\tau_{\tilde{v}}$ is a 1-generic tame inertial type for $I_{F_{\tilde{v}}}$, we defined a $\text{GL}_3(\mathcal{O}_{F_{\tilde{v}}})$-representation $\sigma(\tau_{\tilde{v}})$ over $E$ corresponding to $\tau_{\tilde{v}}$ by results toward inertial local Langlands. We again let $\sigma(\tau_S)$ be $\otimes_{\tilde{v} \in S} \sigma(\tau_{\tilde{v}})$.

**THEOREM 5.3.1.** Let $r : G_F \to \text{GL}_3(E)$ be an absolutely irreducible Galois representation such that

(1) for all places $w \in \Sigma_p$, $\bar{r}|_{I_{Fw}}$ is semisimple and 10-generic;

(2) $r$ is unramified almost everywhere and satisfies $r^c \cong r^\vee \epsilon^{-2}$;

(3) for all places $\tilde{v} \in S$, the representation $r|_{G_{F\tilde{v}}}$ is potentially crystalline, with parallel Hodge–Tate weights (2, 1, 0) and with tame inertial type $\tau_{\tilde{v}}$ (see [LLHLM18, Definition 2.1]);

(4) $\bar{r}$ satisfies the Taylor–Wiles conditions as above and $\bar{r}$ has split ramification; and

(5) $\bar{r} \cong \bar{r}_i(\pi)$ for a regular algebraic conjugate self dual cuspidal (RACSDC) representation $\pi$ of $\text{GL}_3(\mathbb{A}_F)$ with trivial infinitesimal character such that $\otimes_{\tilde{v} \in S} \sigma(\tau_{\tilde{v}})$ is a K-type for $\otimes_{\tilde{v} \in S} \pi_{\tilde{v}}$.
Then $r$ is automorphic in the sense of [LLHLM18, Section 7.2].

*Proof.* Given Theorems 5.3.3 and 3.5.3, the proof of [LLHLM18, Theorem 7.4] goes through unchanged. □

**REMARK 5.3.2.** Compared to [LLHLM18, Theorem 7.4], we relaxed the hypothesis that $p$ splits completely to $p$ being unramified. However, we also assumed that $\bar{r}$ is semisimple at all places above $p$. The reason is that we only established the connectedness of the generic fiber of $R_{\bar{\rho}}^r$ when $\bar{\rho}$ is semisimple (though we do know it for non-semisimple $\bar{\rho}$ in the case that all shapes have length $\ge 2$). In work in progress, we will establish a counterpart of [LLHLM18, Theorem 7.4] for non-semisimple representations. This will allow us to remove the semisimplicity hypothesis in a manner similar to [LLHLM18, Theorem 7.4].

**5.3.2. The Serre weight conjecture.** Let $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ be as in the beginning of Section 5.3. Recall that for each $v \in \Sigma_p^+$, we chose a place $\tilde{v}|v$ of $F$ and set $S$ to be the set $\{\tilde{v}|v \in \Sigma_p^+\}$. Furthermore, we let $\bar{\rho}_{\tilde{v}}$ be $\bar{r}|_{G_{F_{\tilde{v}}}}$ and $\bar{\rho}_S$ be $(\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$. Recall that $K$ is the product $\prod_{\tilde{v} \in S} \text{GL}_3(\mathcal{O}_{F_{\tilde{v}}})$. Let $W(\bar{r})$ be the set of irreducible $K$-representations $\sigma$ over $\mathbb{F}$ such that

$$S\left( U^p \mathcal{G}(\mathcal{O}_{F^+, p}), \bar{W}_{\Sigma_0^+} \otimes \left( \sigma^\vee \circ \prod_{\tilde{v} \in S} \iota_{\tilde{v}} \right) \right)_m \neq 0.$$

We have the following version of the weight part of Serre’s conjecture.

**THEOREM 5.3.3.** Let $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ be a continuous Galois representation, satisfying the Taylor–Wiles conditions. Assume that $\bar{\rho}_{\tilde{v}} \stackrel{\text{def}}{=} \bar{r}|_{G_{F_{\tilde{v}}}}$ is semisimple and 10-generic for all $\tilde{v} \in S$, that $\bar{r}$ is automorphic, and that $\bar{r}$ has split ramification. Then $W(\bar{r}) \circ \prod_{\tilde{v} \in S} \iota_{\tilde{v}} = W^2(\bar{\rho}_S)$.

*Proof.* We have that $W^{\text{BM}}(\bar{\rho}_S) = W(\bar{r}) \circ \prod_{\tilde{v} \in S} \iota_{\tilde{v}}$ by (5.8). The result now follows from Theorem 3.5.2. □

**5.3.3. Mod $p$ multiplicity one.** We continue using the setup from the beginning of Section 5.3. We have the following mod $p$ multiplicity one result.

**THEOREM 5.3.4.** Let $\tau_S$ and $\sigma \in W^2(\bar{\rho}_S, \tau_S)$ be as in the statement of Theorem 5.1.1. Then

$$S\left( U^p \mathcal{G}(\mathcal{O}_{F^+, p}), \left( \bar{\sigma}(\tau_S)^\sigma \circ \prod_{v \in \Sigma_p^+} \iota_{\tilde{v}} \right)^\vee \otimes_{\mathcal{O}_E} W_{\Sigma_0^+} \right)[m]$$
is one-dimensional over $\mathbb{F}$.

*Proof.* By (5.8),

$$ M_{\infty}(\bar{\sigma}(\tau_S)^{\sigma})/m \cong \tilde{M}_{\infty}\left(\bar{\sigma}(\tau_S)^{\sigma} \circ \prod_{v \in \Sigma_p^+} \iota_{\tilde{v}}\right)/m \\ \cong \left( S\left(U^p G(O_{F^+, p}), \left(\bar{\sigma}(\tau_S)^{\sigma} \circ \prod_{v \in \Sigma_p^+} \iota_{\tilde{v}}\right)^{\vee} \otimes_O W_{\Sigma_0^+}\right)[m] \right)^{\vee}. $$

By Theorem 5.1.1, the dimension of $M_{\infty}(\bar{\sigma}(\tau_S)^{\sigma})/m$ is one. $\square$

<figure><img src="image_13.png" /></figure>

5.3.4. Lattices in cohomology. Let $r : G_F \to \text{GL}_3(E)$ be an automorphic Galois representation as in Theorem 5.3.1. We say that $r$ is minimally ramified if $r|_{G_{F_{\tilde{v}}}}$ is minimally ramified in the sense of [CHT08, Definition 2.4.14] for all $v \in \Sigma_0^+$. Following the notation of [LLHLM18, Section 7.1], let $\lambda$ be the kernel of the system of Hecke eigenvalues $\alpha : \mathbb{T}^{\text{univ}} \to \mathcal{O}$ associated with $r$, that is, $\alpha$ satisfies

$$ \det (1 - r^{\vee}(\text{Frob}_w)X) = \sum_{j=0}^{3} (-1)^j (\mathbf{N}_{F/Q}(w))^{\binom{j}{2}} \alpha(T_w^{(j)})X^j $$

for all $w$ as above. We now set $W \stackrel{\text{def}}{=} W_{\Sigma_0^+}$ as in Section 5.3.3. By Theorem 5.3.1, $\tilde{S}(U^p, W)[\lambda]$ is nonzero. Since $r$ is minimally ramified, $r$ corresponds to a prime ideal of $R_{\infty}$ as in [HLM17, Theorem 5.2.1]. By an abuse of notation, we call this ideal $\lambda$. Note that we have that $M_{\infty}/\lambda \cong \tilde{S}(U^p, W)^{\text{d}}/\lambda$ by (the proof of) [CEG+16, Corollary 2.11].

THEOREM 5.3.5. Let $r : G_F \to \text{GL}_3(E)$ be as in Theorem 5.3.1. Assume furthermore that $r$ is minimally ramified. Let $\{\tau_{\tilde{v}}\}_{\tilde{v} \in S}$ be a 13-generic tame type. The lattice $\sigma(\tau)^0 \stackrel{\text{def}}{=} \sigma(\tau) \cap \tilde{S}(U^p, W)[\lambda] \subset \sigma(\tau) \cap \tilde{S}(U^p, W)[\lambda] \otimes_O E \cong \sigma(\tau)$ depends only on $\{r|_{G_{F_{\tilde{v}}}}\}_{\tilde{v} \in S}$.

*Proof.* Let $\bar{\rho}_S$ be $(\bar{r}|_{G_{F_{\tilde{v}}}})_{\tilde{v} \in S}$. Fix $\sigma \in \text{JH}(\overline{\sigma(\tau_S)})$ as in Theorem 5.1.1, a saturated inclusion $\sigma(\tau_S)^{\sigma} \subset \sigma(\tau_S)^0$, and saturated inclusions $\sigma(\tau_S)^{\kappa} \subset \sigma(\tau_S)^{\sigma}$ for all $\kappa \in \text{JH}(\overline{\sigma(\tau_S)})$. Let $\gamma(\kappa) \in \mathcal{O}$ so that $\gamma(\kappa)^{-1}\sigma(\tau_S)^{\kappa} \subset \sigma(\tau_S)^0$ is saturated. Then $\sigma(\tau_S)^0 = \sum_{\kappa \in \text{JH}(\overline{\sigma(\tau_S)})} \gamma(\kappa)^{-1}\sigma(\tau_S)^{\kappa}$ for some $\gamma(\kappa) \in \mathcal{O}$ by [EGS15, Lemma 4.1.2]. It suffices to show that for each $\kappa \in \text{JH}(\overline{\sigma(\tau_S)})$, the ideal $(\gamma(\kappa)) \subset \mathcal{O}$ depends only on $\bar{\rho}_S$.
Observe that the image of the inclusion

$$ \mathcal{O} \cong \operatorname{Hom}_{U_p}(\sigma(\tau_S)^\sigma, \tilde{S}(U^p, W)[\lambda]) \to \operatorname{Hom}_{U_p}(\sigma(\tau_S)^\kappa, \tilde{S}(U^p, W)[\lambda]) \cong \mathcal{O}, $$

is given by the ideal $(\gamma(\kappa))$. By Schikhov duality, the natural inclusion

$$ \begin{aligned} \mathcal{O} &\cong \operatorname{Hom}_{U_p}((\tilde{S}(U^p, W)[\lambda])^d, (\sigma(\tau_S)^\sigma)^d) \\ &\to \operatorname{Hom}_{U_p}((\tilde{S}(U^p, W)[\lambda])^d, (\sigma(\tau_S)^\kappa)^d) \cong \mathcal{O} \end{aligned} $$

is also given by the ideal $(\gamma(\kappa))$. By another application of Schikhov duality, the natural inclusion

$$ \begin{aligned} \mathcal{O} &\cong \operatorname{Hom}_{U_p}((\tilde{S}(U^p, W)[\lambda])^d, (\sigma(\tau_S)^\kappa)^d)^d \\ &\to \operatorname{Hom}_{U_p}((\tilde{S}(U^p, W)[\lambda])^d, (\sigma(\tau_S)^\sigma)^d)^d \cong \mathcal{O} \end{aligned} $$

is also given by the ideal $(\gamma(\kappa))$. Since $\tilde{S}(U^p, W)[\lambda]^d$ is canonically isomorphic to the torsion-free part of $\tilde{S}(U^p, W)^d/\lambda$, the ideal $(\gamma(\kappa))$ gives the inclusion

$$ \operatorname{Hom}_{U_p}(M_\infty/\lambda, (\sigma(\tau_S)^\kappa)^d)^d \to \operatorname{Hom}_{U_p}(M_\infty/\lambda, (\sigma(\tau_S)^\sigma)^d)^d. $$

Again, note that $\operatorname{Hom}_{U_p}(M_\infty/\lambda, (\sigma(\tau_S)^\sigma)^d)^d$ is isomorphic to (the $p$-torsion-free part of)

$$ \operatorname{Hom}_{U_p}(M_\infty, (\sigma(\tau_S)^\sigma)^d)^d/\lambda = M_\infty(\sigma(\tau_S)^\sigma)/\lambda, $$

and similarly for $\kappa$. So the image of

$$ \mathcal{O} \cong M_\infty(\sigma(\tau_S)^\kappa)/\lambda \to M_\infty(\sigma(\tau_S)^\sigma)/\lambda \cong \mathcal{O} \quad (5.9) $$

is given by the ideal $(\gamma(\kappa))$.

On the other hand, the image of (5.9) is given by

$$ \left( j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S) \right) \right) \left( M_\infty(\sigma(\tau_S)^\sigma)/\lambda \right) $$

by Theorem 5.2.3. Then $(\gamma(\kappa))$ is generated by elements in the ideal

$$ j_* j^* \left( \prod_{\theta \in W^2(\bar{\rho}_S)} \mathfrak{p}(\theta)^{m(\theta)} R_\infty(\tau_S) \right) $$

modulo the ideal $\lambda$. By Remark 5.2.4, the above ideal comes from $\hat{\otimes}_{\tilde{v} \in S} R_{\tilde{v}}^2$; hence, the ideal generated by its generators modulo $\lambda$ depends only on $\{r|_{G_{\bar{F}_{\tilde{v}}}}\}_{\tilde{v} \in S}$. $\square$
REMARK 5.3.6. In the hypotheses of Theorem 5.3.5, assume further that $\tilde{w}(\overline{\tau_v}, \tau_v)_i$ has length at least 2 for all $\tilde{v} \in S$ and $i \in \mathbb{Z}/f_v$. Then the lattice $\sigma(\tau)^0$ can be described explicitly as in [Le18, Theorem 14].

In the following figures, we give a pictorial realization of the gluing procedure
appearing in the proofs of Lemmas 5.1.4–5.1.7. Recall that $\tilde{w}_{i'}^* = \tilde{w}(\overline{\rho_S}, \tau_S)_i^*$.

<figure><img src="image_14.png" /><figcaption>Figure 3. Comparison of $\Sigma_{V_{j,i'}}$ in Lemma 5.1.4.</figcaption></figure>

From left to right with arrows pointing down, we have $\Sigma_{V^1,i'}$, $\Sigma_{V^3,i'}$, and $\Sigma_{V^2,i'}$.
The edges correspond to adjacent pairs.

<figure><img src="image_15.png" /><figcaption>Figure 4. Comparison of $\Sigma_{V,j,i'}$ and $\Sigma_{V,j,i'}$ in Lemma 5.1.5 when $\tilde{w}_{i'}^* t_{-1} = \text{id}$.</figcaption></figure>

Assume $(\nu_1, \nu_2, \nu_3) = (\varepsilon_1 - \varepsilon_2, \varepsilon_1, \varepsilon_1 + \varepsilon_2)$ and $\tilde{w}_i^* t_{-1} = \text{id}$. Then $\nu_4 = \varepsilon_2$. From left to right with arrows pointing down, we have $\Sigma_{V_{j,i'}}$, $\Sigma_{V^1,i'}$, and $\Sigma_{V^2,i'}$ (a Weyl segment). In red are the elements in $r(\Sigma_0)$.

<figure><img src="image_16.png" /><figcaption>Figure 5. Comparison of $\Sigma_{V_{j,i'}}$ in Lemma 5.1.6 when $\tilde{w}_i^* t_{-1} = \alpha$.</figcaption></figure>

In the notation of Lemma 5.1.6 consider the case where $\nu = \varepsilon_1 + \varepsilon_2$ and $\tilde{w}_i^* t_{-1} = \alpha$. On the left, we have $\Sigma_{V^2,i'}$, where we write the elements in $r(\Sigma_0)$ in red. Similarly on the right for $\Sigma_{V^1,i'}$.
<figure><img src="image_17.png" /><figcaption>Figure 6. Comparison of $\Sigma_{V_{j,i'}}$ in Lemma 5.1.6 when $\tilde{w}_i^* t_{-1} = \gamma^+$.</figcaption></figure>

In the notation of Lemma 5.1.6, consider the case where $\nu = \varepsilon_1 + \varepsilon_2$ and $\tilde{w}_i^* t_{-1} = \gamma^+$. On the left, we have $\Sigma_{V_{2,i'}}$, where we write the elements in $r(\Sigma_0)$ in red. Similarly on the right for $\Sigma_{V^{1,i'}}$.

<figure><img src="image_18.png" /><figcaption>Figure 7. Comparison of $\Sigma_{V_{j,i'}}$ in Lemma 5.1.7 when $\tilde{w}_i^* t_{-1} = \gamma^+$.</figcaption></figure>

In the notation of Lemma 5.1.7, consider the case where $\nu = \varepsilon_1 + \varepsilon_2$ and $\tilde{w}_i^* t_{-1} = \gamma^+$. On the left, we have $\Sigma_{V_{i'}}$ (respectively $\Sigma_{V^{1,i'}}$ removing the dotted lower part), where we write the elements in $r(\Sigma_0)$ in red. On the right, we have one of the possible choices for $\Sigma_{V^{2,i'}}$.

# 6. Addendum to [LLHLM18]

(1) In Theorem 1.1, the statement that ‘its special fiber is as predicted by the geometric Breuil–Mézard conjecture’ means the following: under the assumptions of Theorem 1.1, the special fiber of $R_{\bar{\rho}}^{(2,1,0),\tau}$ is reduced and the number of irreducible components is less or equal to $\#(W^?(\bar{\rho}^{ss}) \cap JH(\bar{\sigma}(\tau)))$, with equality if $\bar{\rho}$ is semisimple. This can be checked directly using Theorem 6.14, Tables 3, 7, 8, Proposition 8.5, 8.6, and 8.12, and the results of Section 8.2.2.

(2) In Proposition 3.4, the codomain of $T_{\text{tan}}$ should be replaced by

$$ \{ (\rho, \gamma_0) \mid \rho \in \text{Rep}_{\mathbb{F}[\epsilon]/(\epsilon^2)}(G_{K_\infty}), \gamma_0 : \rho \quad \text{mod} \quad \varepsilon \to T_{\text{dd}}^*(\overline{\mathfrak{M}}) \}. $$

(3) After Equation (3.7), we remark that if $(\mathfrak{M}_A, \rho_A, \delta_A) \in D_{\overline{\mathfrak{M}}, \bar{\rho}}^{\tau, \square}$, then we have a canonical isomorphism $\mathfrak{M}_A \otimes_A \mathbb{F} \to \overline{\mathfrak{M}}$.
(4) After Definition 4.15, the symbol $Y_{\overline{\mathfrak{M}}}^{[0,2],\tau}(R)$ denotes the category of pairs $(\mathfrak{M}_R, J_R)$ where $\mathfrak{M}_R \in Y^{[0,h],\tau}(R)$ and $J_R : \mathfrak{M}_R \otimes_R \mathbb{F} \xrightarrow{\sim} \overline{\mathfrak{M}}$ is an isomorphism in $Y^{[0,h],\tau}(\mathbb{F})$. A similar comment applies to $Y_{\overline{\mathfrak{M}}}^{\mu,\tau}(R)$.

(5) Proof of Theorem 4.17: the ring $R$ is $p$-flat and reduced by [Cal18, Lemma 2.6].

(6) The formula in Lemma 5.2 still converges in $\frac{1}{p}$-Mat$(R[1/p]\otimes\mathbb{U})$ for $R$ a complete local Noetherian flat $\mathcal{O}$-algebra $R$. While it is possible to show that it lies in $\mathcal{O}_R^{\text{rig}}$, for the computations and the arguments in this paper, we only need that its formation is compatible with base change.

(7) In Corollary 5.13, $T_1, \dots, T_8$ should be replaced by $T_1, \dots, T_9$. Similar comment applies to the displayed equation before Theorem 6.14.

(8) In Section 5.3.2, 'c$_{11}$ ≡ 0 modulo $\overline{\sigma}$' should read 'c$_{11}$ and c$_{13}$ ≡ 0 modulo the maximal ideal'.

(9) In Section 5.3.3, line -6 and -4, the $c_{13}^*$ in the displayed equations must be replaced by $c_{13}$.

(10) Definition 7.1 should also define automorphic of weight $V$, level $U$, and coefficients $W$ as follows.

DEFINITION 6.0.1. Let $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ be a continuous Galois representation. Let $V$ be a Serre weight for $\mathcal{G}$, $U$ be a compact open subgroup of $\mathcal{G}(\mathbb{A}_F^{\infty,p}) \times \mathcal{G}(\mathcal{O}_{F^+,p})$ which is unramified at places $v|p$, and $W$ be an $\mathcal{O}$-module with a $U$-action for which the factor $\mathcal{G}(\mathcal{O}_{F^+,p})$ acts trivially. We say that $\bar{r}$ is *automorphic of weight $V$, level $U$, and coefficients $W$* if there exists a cofinite subset $\mathcal{P} \subset \mathcal{P}_U$ such that

$$S(U, V \otimes W)_{\overline{m}} \neq 0,$$

where $\overline{m}$ is the kernel of the system of Hecke eigenvalues $\overline{\alpha} : \text{\textit{T}}_\mathcal{P} \to \mathbb{F}$ associated with $\bar{r}$, and $\overline{\alpha}$ satisfies the equality

$$\det(1 - \bar{r}^{\vee}(\text{Frob}_w)X) = \sum_{j=0}^{3} (-1)^j (\mathbf{N}_{F/\mathbb{Q}}(w))^{\binom{j}{2}} \overline{\alpha}(T_w^{(j)}) X^j$$

for all $w \in \mathcal{P}$. We say that $\bar{r}$ is *automorphic of weight $V$* (or that $V$ is a Serre weight of $\bar{r}$) if $\bar{r}$ is *automorphic of weight $V$, level $U$, and coefficients $W$* for some subgroup $U$ and coefficients $W$ as above. We write $W(\bar{r})$ for the set of all Serre weights of $\bar{r}$. We say that $\bar{r}$ is *automorphic if $W(\bar{r}) \neq \emptyset$*.
(11) In Definition 7.11(2), 'automorphic of weight V' should be replaced with 'automorphic of weight V, level U, and coefficients W' where U is a fixed compact open subgroup of $G(\mathbb{A}_F^{\infty, p}) \times \mathcal{G}(\mathcal{O}_{F^{+}, p})$ which is unramified at places $v|p$ and $W$ is a fixed $\mathcal{O}$-module with a $U$-action on which the factor $\mathcal{G}(\mathcal{O}_{F^{+}, p})$ acts trivially. This definition of patching functor depends on the implicit choices of $U$ and $W$.

(12) The definition of $e(M)$, given before Proposition 7.14, is incorrect. The correct definition of $e(M)$ is the following: given a finitely generated $R_\infty$-module $M$ with scheme-theoretic support Spec $A$ of dimension at most $d$, define $e(M)$ to be $d!$ times the coefficient of the degree $d$-term of the Hilbert polynomial of $M$ (considered as an $A$-module).

(13) In the paragraph following the proof of Proposition 7.14, the definition of $\Sigma_0$ should exclude primes dividing $p$.

(14) In the proof of Proposition 7.15, 'automorphic of weight $V$' should be replaced with 'automorphic of weight $V$, level $U$, and coefficients $W$', where $U = \prod_{v \nmid \infty} U_v$ and $U_v$ is

* $\mathcal{G}(\mathcal{O}_v)$ for $v$ which split in $F$ except for $v_1$,

* the preimage of the upper triangular matrices under the map

$$ \mathcal{G}(\mathcal{O}_v) \to \mathcal{G}(k_v) \xrightarrow{\sim} \mathrm{GL}_3(k_{\tilde{v}}) $$

if $v = v_1$, or

* a maximal hyperspecial maximal compact open subgroup of $G(F_v)$ if $v$ is inert in $F$,

and $W$ is an $\mathcal{O}$-lattice in $\otimes_{v \in \Sigma_0^+} \sigma(\tau_{\tilde{v}}) \circ l_{\tilde{v}}$.

(15) In the proof of Proposition 7.16, the tame inertial type $\tau'$ should read: $\tau' \stackrel{\text{def}}{=} \omega_2^{-(b+1)+p(b+1)} \oplus \omega_2^{-(c-1)+pa} \oplus \omega_2^{-(a+p(c-1))}$.

(16) The proof of Theorem 7.8 also requires the following proposition, which is a level-lowering result based on techniques in [Tay08]. The proof of Theorem 7.4, which was omitted, uses the same techniques.

**PROPOSITION 6.0.2.** Let $\bar{r} : G_F \to \mathrm{GL}_3(\mathbb{F})$ be a continuous Galois representation with split ramification outside $p$, which is automorphic and satisfies the Taylor-Wiles conditions. If $\bar{r}$ is automorphic of a reachable weight, then it is automorphic of a reachable weight and level $U$ and coefficients $W$, where $U = \prod_{v \nmid \infty} U_v$ and $U_v$ is
• $\mathcal{G}(\mathcal{O}_v)$ for $v$ which split in $F$ except for $v_1$,

• the preimage of the upper triangular matrices under the map

$$ \mathcal{G}(\mathcal{O}_v) \to \mathcal{G}(k_v) \xrightarrow{\sim} \mathrm{GL}_3(k_{\tilde{v}}) $$

if $v = v_1$, or

• a maximal hyperspecial maximal compact open subgroup of $\mathcal{G}(F_v)$ if $v$ is inert in $F$,

and $W$ is an $\mathcal{O}$-lattice in $\bigotimes_{v \in \Sigma_0^+} \sigma(\tau_{\tilde{v}}) \circ \iota_{\tilde{v}}$.

*Proof.* If $\bar{r}$ is automorphic of a reachable weight $V$, then $S(U, V \otimes W)_m$ is nonzero for some level $U$ and coefficients $W$. Let $V$ be $\bigotimes_v V_v$ where $V_v = V_{\tilde{v}} \circ \iota_{\tilde{v}}$. Choose tame types $\tau_v$ such that

• $\bar{\rho}_v$ is admissible with respect to $\tau_v$;

• $\ell(\mathbf{w}(\bar{\rho}_v, \tau_v)) \ge 3$; and

• $V_{\tilde{v}} \in \mathrm{JH}(\bar{\sigma}(\tau_v))$.

Letting $\sigma$ be an $\mathcal{O}$-lattice in $\bigotimes_v \sigma(\tau_v) \circ \iota_{\tilde{v}}$, we have that $S(U, \sigma \otimes W)_m$ is nonzero. By [EGH13, Lemma 7.1.6 and Theorem 7.2.1], $\bar{r} \otimes_F \bar{F}_p$ is isomorphic to the reduction of $r_\pi : G_F \to \mathrm{GL}_n(\bar{Q}_p)$ for some $\pi$ as in [EGH13, Theorem 7.2.1]. Let $\pi_F$ be the regular, algebraic, (conjugate) self dual, cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ obtained from $\pi$ through base change. Choose a totally real extension $L^+$ of $F^+$ such that

• $4|[L^+ : \mathbb{Q}]$;

• $L^+/F^+$ is Galois and solvable;

• $L \stackrel{\mathrm{def}}{=} L^+F$ is linearly disjoint from $\bar{F}^{\ker \bar{r}}(\zeta_p)$ over $F$;

• $L/L^+$ is everywhere unramified;

• $p$ is unramified in $L^+$;

• $v_1$ splits completely in $L$;

• if $\pi_L$ is the base change of $\pi_F$ to $L$ and $w$ is a place of $L$ lying above $\tilde{v}$ for $v \in \Sigma_0^+$, then $\pi_{L,w}$ has Iwahori fixed vectors and $\tau_v|_{L,w}$ is trivial.

One can define analogues of $R_\tau^{\mathrm{univ}}$ and $R_S^{\mathrm{univ}}$ from [Gee11, Section 5] as follows. Let $S$ be the union of $\{v_1\}$ and $\Sigma^+$. Let $\tilde{S}$ be the union of $\{\tilde{v}_1\}$ and $\{\tilde{v} : v \in \Sigma^+\}$. Let $S_L$ be the set of places in $L^+$ lying above places in $S$, and let $\tilde{S}_L$ be the set of places in $L$ lying above places in $\tilde{S}$. Let $\tilde{\Sigma}_{L,p}$ be the set