The monodromy operator $N_{\mathfrak{m},\varrho}^{\text{coho}}$ acting on $H^0(\text{Sh}_{K,\bar{\eta}_v}, \Psi_{K,\varrho})_\mathfrak{m}$ is such that

$$N_{\mathfrak{m},\varrho}^{\text{coho}} \otimes_{\mathbb{Z}_l} \overline{\mathbb{Q}}_l \cong \text{Id} \otimes N_{\mathfrak{m},\varrho, \overline{\mathbb{Q}}_l},$$

i.e. it acts trivially on the first factor $\sigma_{\mathfrak{m},K,\varrho}$. We then deduce that $N_{\mathfrak{m},\varrho}^{\text{coho}}$ induces a nilpotent operator $N_{\mathfrak{m},\varrho}$ (resp. $\overline{N}_{\mathfrak{m},\varrho}$) on $\rho_{\mathfrak{m},K,\varrho}$ (resp. $\overline{\rho}_{\mathfrak{m},K,\varrho} := \rho_{\mathfrak{m},K,\varrho} \otimes_{\mathbb{Z}_l} \overline{\mathbb{F}}_l$).

In the following, we will work with the following levels at $v$.

**Notation 4.3.4.** — For $\alpha \in \mathbb{N} \setminus \{0\}$, we will denote by

$$K_v(\alpha) := \ker(\text{GL}_d(\mathcal{O}_v) \to \text{GL}_d(\mathcal{O}_v/\varpi_v^\alpha)),$$

the open compact subgroup of $\text{GL}_d(\mathcal{O}_v)$. For a fixed level $K^v$ outside $v$, we denote by $R_{K,\varrho}(\alpha)$ the image of $\mathbb{T}(K^v)_\mathfrak{m}$ acting on the direct factor $H^0(\text{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \Psi_\varrho)_\mathfrak{m}$ of $H^0(\text{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \Psi_v)_\mathfrak{m}$.

We now focus on the modulo $l$ reduction of the previous objects

$$H^0(\text{Sh}_{K^v K_v(\alpha), \bar{\eta}_v}, \Psi_\varrho)_\mathfrak{m} \otimes_{\mathbb{Z}_l} \overline{\mathbb{F}}_l \cong \overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{R}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha),$$

where $\overline{\sigma}_{K,\varrho}(\alpha) := \sigma_{\mathfrak{m},K^v K_v(\alpha),\varrho} \otimes_{\mathbb{Z}_l} \overline{\mathbb{F}}_l$, similarly $\overline{\rho}_{K,\varrho}(\alpha) := \rho_{\mathfrak{m},K^v K_v(\alpha),\varrho} \otimes_{\mathbb{Z}_l} \overline{\mathbb{F}}_l$ and $\overline{R}_{K,\varrho}(\alpha) := R_{K,\varrho}(\alpha) \otimes_{\mathbb{Z}_l} \overline{\mathbb{F}}_l$.

Note that $\overline{R}_{K,\varrho}(\alpha)$ is an artinian local commutative ring with maximal ideal $\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ and with finite length $r_{K,\varrho}(\alpha)$.

As $\overline{\rho}_{K,\varrho}(\alpha)$ is $\overline{R}_{K,\varrho}(\alpha)^d$ provided with a Galois action, it is flat as a $\overline{R}_{K,\varrho}(\alpha)$-module, we then have

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

**Notation 4.3.5.** — Let $E_{K,\varrho}(\alpha)$ be the injective hull of $R_{K,\varrho}(\alpha)/\mathfrak{m}_{K,\varrho}(\alpha)$ and

$$D: M \to D(M) = \text{hom}_{R_{K,\varrho}(\alpha)}(M, E_{K,\varrho}(\alpha))$$

be the Maltis duality functor for the local artinian ring $\overline{R}_{K,\varrho}(\alpha)$ where $M$ is any finitely generated $\overline{R}_{K,\varrho}(\alpha)$-module. We have the following properties:

(a) $D$ is exact and $D \circ D(M)$ is canonically isomorphic to $M$;

(b) $\text{length}_{\overline{R}_{K,\varrho}(\alpha)} D(M) = \text{length}_{\overline{R}_{K,\varrho}(\alpha)} M$;

(c) for every ideal $I$ of $\overline{R}_{K,\varrho}(\alpha)$, we have $D(M[I]) \cong D(M)/\text{ID}(M)$ and $D(M/IM) \cong D(M)[I]$;

(d) $D(\overline{R}_{K,\varrho}(\alpha)) = E_{K,\varrho}(\alpha)$ and $D(\overline{R}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)) \cong \overline{R}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$;

(e) $M \otimes_{\overline{R}_{K,\varrho}(\alpha)} D(N) \cong D(\text{hom}_{\overline{R}_{K,\varrho}(\alpha)}(M, N))$.
**Lemma 4.3.6.** — With the previous notations, we have

$$ \begin{aligned} \left( \bar{\sigma}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha) \right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] &\cong \bar{\sigma}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\bar{\mathbb{F}}_l} \bar{p}_{K,\varrho}(\alpha) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \\ &\cong \bar{\sigma}_{K,\varrho}(\alpha) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \otimes_{\bar{\mathbb{F}}_l} \bar{p}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha). \end{aligned} $$

*Proof.* — From the previous properties $(\bar{\sigma}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha))[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]$ is isomorphic to

$$ \begin{aligned} &\cong \left( D^2(\bar{\sigma}_{K,\varrho}(\alpha)) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha) \right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha)))\right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha))) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{R}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha)\right) \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha)) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha))\right) \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}((\bar{p}_{K,\varrho}(\alpha)), D(\bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)])\right) \\ &\cong \bar{p}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} D^2(\bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]) \\ &\cong \bar{p}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong \bar{p}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)/\bar{\mathfrak{m}}_{K,\varrho}} \bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \end{aligned} $$

The second isomorphism is obtained by the symmetry of the proof. □

With the notations of §4.2, for a fixed $\varrho$, theorem 4.2.2 gives us the existence of $\tilde{\mathfrak{m}} \subseteq \mathfrak{m}$ such that $\Pi_{\tilde{\mathfrak{m}},v} \cong \text{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v) \times ?$ where $\pi_v$ is a lift of $\varrho$ and $v$ is a representation of $\text{GL}_{d-l_{\varrho}(\mathfrak{m})g}(F_v)$ such that the supercuspidal support of its modulo $l$ reduction does not contain $\varrho$. We consider any $\alpha_0$ big enough such that $\Pi_{\tilde{\mathfrak{m}},v}$ has non trivial vectors invariant under $K_v(\alpha_0)$.

For $\alpha \ge \alpha_0$, let $k$ be maximal such that $\bar{\mathfrak{m}}_{K,\varrho}(\alpha)^k \bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1}$ is non zero and so has its image contained in $\bar{p}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]$. Consider any $a \in \bar{\mathfrak{m}}_{K,\varrho}(\alpha)^k$ such that $a \bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1} \ne 0$. (16) It then induces a map

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

<footnote>(16) think that $\bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1}$ is given by a matrix in $\text{GL}_d(\bar{R}_{\mathfrak{m},\varrho}(\alpha))$</footnote>
**Lemma 4.3.6.** — With the previous notations, we have

$$ \begin{aligned} \left( \bar{\sigma}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha) \right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] &\cong \bar{\sigma}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\bar{\mathbb{F}}_l} \bar{p}_{K,\varrho}(\alpha) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \\ &\cong \bar{\sigma}_{K,\varrho}(\alpha) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \otimes_{\bar{\mathbb{F}}_l} \bar{p}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha). \end{aligned} $$

*Proof.* — From the previous properties $(\bar{\sigma}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha))[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]$ is isomorphic to

$$ \begin{aligned} &\cong \left( D^2(\bar{\sigma}_{K,\varrho}(\alpha)) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{p}_{K,\varrho}(\alpha) \right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha)))\right) [\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha))) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{R}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha)\right) \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}(\bar{p}_{K,\varrho}(\alpha), D(\bar{\sigma}_{K,\varrho}(\alpha)) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha))\right) \\ &\cong D\left(\operatorname{hom}_{\bar{R}_{K,\varrho}(\alpha)}((\bar{p}_{K,\varrho}(\alpha)), D(\bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)])\right) \\ &\cong \bar{p}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} D^2(\bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]) \\ &\cong \bar{p}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)} \bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong \bar{p}_{K,\varrho}(\alpha) / \bar{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\bar{R}_{K,\varrho}(\alpha)/\bar{\mathfrak{m}}_{K,\varrho}} \bar{\sigma}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)] \end{aligned} $$

The second isomorphism is obtained by the symmetry of the proof. $\square$

With the notations of §4.2, for a fixed $\varrho$, theorem 4.2.2 gives us the existence of $\tilde{\mathfrak{m}} \subseteq \mathfrak{m}$ such that $\Pi_{\tilde{\mathfrak{m}},v} \cong \text{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v) \times ?$ where $\pi_v$ is a lift of $\varrho$ and $v$ is a representation of $\text{GL}_{d-l_{\varrho}(\mathfrak{m})g}(F_v)$ such that the supercuspidal support of its modulo $l$ reduction does not contain $\varrho$. We consider any $\alpha_0$ big enough such that $\Pi_{\tilde{\mathfrak{m}},v}$ has non trivial vectors invariant under $K_v(\alpha_0)$.

For $\alpha \ge \alpha_0$, let $k$ be maximal such that $\bar{\mathfrak{m}}_{K,\varrho}(\alpha)^k \bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1}$ is non zero and so has its image contained in $\bar{p}_{K,\varrho}(\alpha)[\bar{\mathfrak{m}}_{K,\varrho}(\alpha)]$. Consider any $a \in \bar{\mathfrak{m}}_{K,\varrho}(\alpha)^k$ such that $a \bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1} \ne 0$. $^{(16)}$ It then induces a map

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

<footnote>$^{(16)}$ think that $\bar{N}_{\mathfrak{m},\varrho}^{l_{\varrho}(\mathfrak{m})-1}$ is given by a matrix in $\text{GL}_d(\bar{R}_{\mathfrak{m},\varrho}(\alpha))$</footnote>
We now want to understand $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)\otimes_{\overline{\mathbb{F}}_l}\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module. Consider

$$x \in \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \text{ with } 0 \neq z := a \overline{N}_{\overline{\mathfrak{m}},\varrho}^{l_{\varrho}(\mathfrak{m})-1}(x) \in \overline{\rho}_{K,\varrho}(\alpha)[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)].$$

The image of

$$g : \overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \to \\ \overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{R}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha))[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)]$$

contains $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l z$ which is then also contained in the image of $(\overline{N}_{\overline{\mathfrak{m}},\varrho}^{coho})^{l_{\varrho}(\mathfrak{m})-1}$ i.e., up to multiplicities,

$$(21) \qquad H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=l_{\varrho}(\mathfrak{m})g} H T_{\overline{\mathbb{F}}_l}(\varrho, \mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\varrho)))_m.$$

Indeed, recall that, cf. the fact after 4.2.1,

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=tg} H T_{\overline{\mathbb{F}}_l}(\pi_v, \Pi_t))_m = (0),$$

for every $t > l_{\varrho}(\mathfrak{m})$ so that, as these cohomology groups are torsion free,

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \Psi_{\varrho, \overline{\mathbb{F}}_l})_m \cong H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{Fil}_l^{l_{\varrho}(\mathfrak{m})}(\Psi_{\varrho}))_m \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l$$

and

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{gr}_l^{l_{\varrho}(\mathfrak{m})}(\Psi_{\varrho, \overline{\mathbb{F}}_l}))_m \cong \\ \bigoplus_{\pi_v \in \mathrm{Cusp}(\varrho)} H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=l_{\varrho}(\mathfrak{m})g} H T(\pi_v, \mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v))(\frac{1-l_{\varrho}(\mathfrak{m})}{2}))_m \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l.$$

Moreover $(\overline{N}_{\overline{\mathfrak{m}},\varrho}^{coho})^{l_{\varrho}(\mathfrak{m})-1}$ is zero on $H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{Fil}_l^{l_{\varrho}(\mathfrak{m})-1}(\Psi_{\varrho}))_m \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l$.

Concerning the action of $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$, recall that the image of (21), in the Grothendieck group of $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-modules, is equal to

$$(22) \qquad \sum_{\tilde{\mathfrak{m}} \subseteq \mathfrak{m}} n_{\tilde{\mathfrak{m}}} r_l \left( \Pi_{\tilde{\mathfrak{m}}^{\infty}} \right)^{K^v K_v(\alpha)},$$

where

- $n_{\tilde{\mathfrak{m}}}$ is a positive integer we do not need to precise,

- and the sum goes on the set of $\tilde{\mathfrak{m}} \subseteq \mathfrak{m}$ such that $\Pi_{\tilde{\mathfrak{m}},v}$ is of the form $\mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v) \times$? where $\pi_v$ is a lift of $\varrho$ and ? is a representation of $\mathrm{GL}_{d-l_{\varrho}(\mathfrak{m})g}(F_v)$ whose modulo $l$ reduction does not contain $\varrho$ in its supercuspidal support.

We then also deduce that

$$\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l x \subseteq \overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$$

as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module, is, up to multiplicities, equal to (22). As the action of $\mathbb{T}_v(K_v(\alpha))$ commutes with those of $\mathrm{Gal}(\overline{\mathbb{F}}/F)$, the same is true for the
We now want to understand $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)\otimes_{\overline{\mathbb{F}}_l}\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module. Consider

$$x \in \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \text{ with } 0 \neq z := a \overline{N}_{\overline{\mathfrak{m}},\varrho}^{l_{\varrho}(\mathfrak{m})-1}(x) \in \overline{\rho}_{K,\varrho}(\alpha)[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)].$$

The image of

$$g : \overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \to \\ \overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)] \cong (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{R}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha))[\overline{\mathfrak{m}}_{K,\varrho}(\alpha)]$$

contains $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l z$ which is then also contained in the image of $(\overline{N}_{\overline{\mathfrak{m}},\varrho}^{coho})^{l_{\varrho}(\mathfrak{m})-1}$ i.e., up to multiplicities,

$$(21) \qquad H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=l_{\varrho}(\mathfrak{m})g} H T_{\overline{\mathbb{F}}_l}(\varrho, \mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\varrho)))_\mathfrak{m}.$$

Indeed, recall that, cf. the fact after 4.2.1,

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=tg} H T_{\overline{\mathbb{F}}_l}(\pi_v, \Pi_t))_\mathfrak{m} = (0),$$

for every $t > l_{\varrho}(\mathfrak{m})$ so that, as these cohomology groups are torsion free,

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \Psi_{\varrho, \overline{\mathbb{F}}_l})_\mathfrak{m} \cong H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{Fil}_!^{l_{\varrho}(\mathfrak{m})}(\Psi_{\varrho}))_\mathfrak{m} \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l$$

and

$$H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{gr}_!^{l_{\varrho}(\mathfrak{m})}(\Psi_{\varrho, \overline{\mathbb{F}}_l}))_\mathfrak{m} \cong \\ \bigoplus_{\pi_v \in \mathrm{Cusp}(\varrho)} H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, {}^p j_{!*}^{=l_{\varrho}(\mathfrak{m})g} H T(\pi_v, \mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v))(\frac{1-l_{\varrho}(\mathfrak{m})}{2}))_\mathfrak{m} \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l.$$

Moreover $(\overline{N}_{\overline{\mathfrak{m}},\varrho}^{coho})^{l_{\varrho}(\mathfrak{m})-1}$ is zero on $H^0(\mathrm{Sh}_{K^v K_v(\alpha), \bar{s}_v}, \mathrm{Fil}_!^{l_{\varrho}(\mathfrak{m})-1}(\Psi_{\varrho}))_\mathfrak{m} \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l$.

Concerning the action of $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$, recall that the image of (21), in the Grothendieck group of $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-modules, is equal to

$$(22) \qquad \sum_{\tilde{\mathfrak{m}} \subseteq \mathfrak{m}} n_{\tilde{\mathfrak{m}}} r_l \left( \Pi_{\tilde{\mathfrak{m}}^{\infty}} \right)^{K^v K_v(\alpha)},$$

where

- $n_{\tilde{\mathfrak{m}}}$ is a positive integer we do not need to precise,

- and the sum goes on the set of $\tilde{\mathfrak{m}} \subseteq \mathfrak{m}$ such that $\Pi_{\tilde{\mathfrak{m}},v}$ is of the form $\mathrm{st}_{l_{\varrho}(\mathfrak{m})}(\pi_v) \times$? where $\pi_v$ is a lift of $\varrho$ and ? is a representation of $\mathrm{GL}_{d-l_{\varrho}(\mathfrak{m})g}(F_v)$ whose modulo $l$ reduction does not contain $\varrho$ in its supercuspidal support.

We then also deduce that

$$\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l x \subseteq \overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$$

as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module, is, up to multiplicities, equal to (22). As the action of $\mathbb{T}_v(K_v(\alpha))$ commutes with those of $\mathrm{Gal}(\overline{\mathbb{F}}/F)$, the same is true for the
$\overline{\mathbb{F}}_l[\mathrm{Gal}(\overline{F}/F)]$-module generated by $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l x$. Recall that by hypothesis, $\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ is irreducible, so that the $\overline{\mathbb{F}}_l[\mathrm{Gal}(\overline{F}/F)]$-module generated by $x$ is the all of $\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ and

$$ (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{R}}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha)) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) \cong \overline{\sigma}_{K,\varrho}(\alpha) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) $$

as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module, is, up to multiplicities, equal to (22). We then conclude that as a $\overline{\mathbb{F}}_l$-representation of $\mathrm{GL}_d(F_v)$, every irreducible constituent of

$$ H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m / \mathfrak{m} = \lim_{\alpha \to \alpha} (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{R}}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha)) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) $$

is of the form $\mathrm{st}_{l_\varrho(m)}(\varrho) \times? $ where $\varrho$ does not belong to the supercuspidal support of $?$. Varying $\varrho$ we conclude that every irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-constituent of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m / \overline{\mathfrak{m}}_{K,\varrho}$ is isomorphic to $\times_\varrho \mathrm{st}_{l_\varrho(m)}(\varrho)$, i.e. is generic.

Consider finally an irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-quotient $V$ of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m$: then $V \cong V/\mathfrak{m}V$ is a quotient of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m/\mathfrak{m}$ and so is necessary generic as a $\overline{\mathbb{F}}_l$-representation of $\mathrm{GL}_d(F_v)$.

Using proposition 3.7 (3) of [NT16], through Verdier duality, we obtain the gener-
icity of any irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-submodules of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m$, which finishes
the proof.

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[Boy23] __________, *Galois irreducibility implies cohomology freeness for KHT shimura varieties*, Journal de l'Ecole Polytechnique **10** (2023), 199–232.

[Boyne] __________, *La cohomologie des espaces de Lubin-Tate est libre*, Duke Math. Journal (2023 June), 1531–1622.
$\overline{\mathbb{F}}_l[\mathrm{Gal}(\overline{\mathbb{F}}/F)]$-module generated by $\overline{\sigma}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\mathbb{F}}_l x$. Recall that by hypothesis, $\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ is irreducible, so that the $\overline{\mathbb{F}}_l[\mathrm{Gal}(\overline{\mathbb{F}}/F)]$-module generated by $x$ is the all of $\overline{\rho}_{K,\varrho}(\alpha)/\overline{\mathfrak{m}}_{K,\varrho}(\alpha)$ and

$$ (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{R}}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha)) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) \cong \overline{\sigma}_{K,\varrho}(\alpha) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{F}}_l} \overline{\rho}_{K,\varrho}(\alpha) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) $$

as a $\overline{\mathbb{F}}_l[\mathbb{T}_v(K_v(\alpha))]$-module, is, up to multiplicities, equal to (22). We then conclude that as a $\overline{\mathbb{F}}_l$-representation of $\mathrm{GL}_d(F_v)$, every irreducible constituent of

$$ H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m / \mathfrak{m} = \lim_{\alpha \to \infty} (\overline{\sigma}_{K,\varrho}(\alpha) \otimes_{\overline{\mathbb{R}}_{K,\varrho}(\alpha)} \overline{\rho}_{K,\varrho}(\alpha)) / \overline{\mathfrak{m}}_{K,\varrho}(\alpha) $$

is of the form $\mathrm{st}_{l_\varrho(m)}(\varrho) \times? $ where $\varrho$ does not belong to the supercuspidal support of $?$. Varying $\varrho$ we conclude that every irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-constituent of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m / \overline{\mathfrak{m}}_{K,\varrho}$ is isomorphic to $\times_\varrho \mathrm{st}_{l_\varrho(m)}(\varrho)$, i.e. is generic.

Consider finally an irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-quotient $V$ of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m$: then $V \cong V/\mathfrak{m}V$ is a quotient of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m/\mathfrak{m}$ and so is necessary generic as a $\overline{\mathbb{F}}_l$-representation of $\mathrm{GL}_d(F_v)$.

Using proposition 3.7 (3) of [NT16], through Verdier duality, we obtain the gener-
icity of any irreducible $\overline{\mathbb{F}}_l[\mathrm{GL}_d(F_v)]$-submodules of $H^0(\mathrm{Sh}_{K^v(\infty), \bar{\eta}_v}, \overline{\mathbb{F}}_l)_m$, which finishes
the proof.

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<footnote>PASCAL BOYER • E-mail : boyer@math.univ-paris13.fr, Université Paris 13, Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse (France), Coloss: ANR-19-PRC</footnote>
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[CEG$^{+}$16] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, and S. W. Shin, *Patching and the p-adic local Langlands correspondence*, Camb. J. Math. **4** (2016), no. 2, 197-287. MR 3529394

[CHT08] L. Clozel, M. Harris, and R. Taylor, *Automorphy for some l-adic lifts of automorphic mod l Galois representations*, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 1-181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR 2470687

[DT94] F. Diamond and R. Taylor, *Non-optimal levels of mod l modular representations*, Invent. Math. **115**, no. 3 (1994), 435-462.

[Eme] M. Emerton, *Local-global compatibility in the p-adic Langlands program for GL$_2$/Q*, http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf, preprint (2011).

[Gee11] T. Gee, *Automorphic lifts of prescribed types*, Math. Ann. **350** (2011), no. 1, 107-144. MR 2785764 (2012c:11118)

[HT01] M. Harris and R. Taylor, *The geometry and cohomology of some simple Shimura varieties*, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich. MR 1876802 (2002m:11050)

[Ill94] L. Illusie, *Autour du théorème de monodromie locale*, Périodes p-adiques, Astérisque, no. 223, 1994.

[Mos21] G. Moss, *The universal unramified module for GL(n) and the Ihara conjecture*, Algebra and number theory **15** (2021), 1181-1212.

[MS21] J. Manning and J. Shotton, *Ihara's lemma for Shimura curves over totally real fields via patching*, Math. Ann. **379** (2021), 187-234.

[NF19] J. Nekovar and K. Fayad, *Semisimplicity of certain Galois representations occurring in étale cohomology of unitary Shimura varieties*, Amer. J. of Math. **141** (2019), 503-530.

[NT16] J. Newton and J. A. Thorne, *Torsion galois representations over CM fields and hecke algebras in the derived category*, Forum of Mathematics, Sigma **4** (2016), e21.

[Sch18] P. Scholze, *On the p-adic cohomology of the Lubin-Tate tower*, Ann. Sci. Ec. Norm. Supér. (**4**) **51** (2018), no. 4, 811-863.

[Tay08] R. Taylor, *Automorphy for some l-adic lifts of automorphic mod l representations*. ii, Pub. Math. IHES **108** (2008), 183-239.

[Vig96] M.-F. Vignéras, *Représentations l-modulaires d'un groupe réductif p-adique avec $l 
eq p$*, Progress in Mathematics, vol. 137, Birkhäuser Boston Inc., Boston, MA, 1996. MR MR1395151 (97g:22007)

[Zel80] A. V. Zelevinsky, *Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n)*, Ann. Sci. École Norm. Sup. (**4**) **13** (1980), no. 2, 165-210. MR MR584084 (83g:22012)