On the exactness of ordinary parts over a local field of characteristic $p$

TL;DR

This paper proves the exactness of the ordinary part functor for admissible smooth representations of a reductive group over a local field of characteristic p, under specific ring conditions, and explores related extension properties.

Contribution

It establishes the exactness of the ordinary part functor in a new setting where the residue characteristic matches the characteristic of the field, with implications for extension groups.

Findings

Exactness of the ordinary part functor under specified conditions

Results on Yoneda extensions between admissible smooth representations

Conditions when p is nilpotent in the coefficient ring

Abstract

Let $G$ be a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$, $P$ be a parabolic subgroup of $G$, and $R$ be a commutative ring. When $R$ is artinian, $p$ is nilpotent in $R$, and $\mathrm{char}(F)=p$, we prove that the ordinary part functor $\mathrm{Ord}_P$ is exact on the category of admissible smooth $R$-representations of $G$. We derive some results on Yoneda extensions between admissible smooth $R$-representations of $G$.

Equations

$${\mathrm{Ext}}_M^n(\sigma ,{\mathrm{Ord}}_P(\pi ))\stackrel{\sim }{\to }{\mathrm{Ext}}_G^n({\mathrm{Ind}}_{\bar{P}}^G(\sigma ),\pi ).$$

$$H^{\bullet }(N_0,\pi )\stackrel{m}{\to }{H}^{\bullet }(m{N}_0{m}^{-1},\pi )\stackrel{\mathrm{cor}}{\to }{H}^{\bullet }(N_0,\pi )$$

$$\begin{gathered} (m\cdot \varphi )(n_0,\dots ,n_k) \\ =\sum _{\bar{n}\in \overline{N_0/m{N}_0{m}^{-1}}}\bar{n}m\cdot \varphi (m^{-1}{\bar{n}}^{-1}{n}_0\overline{n_0^{-1}\bar{n}}m,\dots ,m^{-1}{\bar{n}}^{-1}{n}_k\overline{n_k^{-1}\bar{n}}m) \end{gathered}$$

$$S:=\sum _{\bar{n}\in \overline{N_0/m{N}_0{m}^{-1}}}\bar{n}m\in {\mathbb{F}}_p[M^{+}⋉N_0]$$

$$0\to N_1^{\prime }\to N_0^{\prime }\to N_0^{\prime }/N_1^{\prime }\to 0$$

$$S^{\prime }:=\sum _{{\bar{n}}^{\prime }\in \overline{N_0^{\prime }/m{N}_0^{\prime }{m}^{-1}}}{\bar{n}}^{\prime }m\in {\mathbb{F}}_p[M^{+}⋉N_0^{\prime }].$$

$$n_0^{\prime }{S}^{\prime }=\sum _{{\bar{n}}^{\prime }\in \overline{N_0^{\prime }/m{N}_0^{\prime }{m}^{-1}}}({\bar{n}}_0^{\prime }{\bar{n}}^{\prime })m(m^{-1}({\bar{n}}_0^{\prime -1}{n}_0^{\prime })m)=S^{\prime }(m^{-1}({\bar{n}}_0^{\prime -1}{n}_0^{\prime })m)$$

$$S:=\sum _{{\bar{n}}^{\prime \prime }\in \overline{N_0^{\prime \prime }/m{N}_0^{\prime \prime }{m}^{-1}}}[{\bar{n}}^{\prime \prime }]S^{\prime }\in {\mathbb{F}}_p[M^{+}⋉N_0]$$

$$\overline{N_0/m{N}_0{m}^{-1}}:=\{[{\bar{n}}^{\prime \prime }]{\bar{n}}^{\prime }:{\bar{n}}^{\prime \prime }\in \overline{N_0^{\prime \prime }/m{N}_0^{\prime \prime }{m}^{-1}},{\bar{n}}^{\prime }\in \overline{N_0^{\prime }/m{N}_0^{\prime }{m}^{-1}}\}\subseteq N_0$$

$$S=\sum _{\bar{n}\in \overline{N_0/m{N}_0{m}^{-1}}}\bar{n}m.$$

$$(n^{\prime \prime }\cdot \varphi )(n_0^{\prime },\dots ,n_j^{\prime })=[n^{\prime \prime }]\cdot \varphi (n_0^{\prime },\dots ,n_j^{\prime })$$

$$(m\cdot \varphi )(n_0^{\prime },\dots ,n_j^{\prime })=S^{\prime }\cdot \varphi (m^{-1}{n}_0^{\prime }{\bar{n}}_0^{\prime -1}m,\dots ,m^{-1}{n}_j^{\prime }{\bar{n}}_j^{\prime -1}m)$$

$$(m\cdot \varphi )(n_0^{\prime },\dots ,n_j^{\prime })=S\cdot \varphi (m^{-1}{n}_0^{\prime }{\bar{n}}_0^{\prime -1}m,\dots ,m^{-1}{n}_j^{\prime }{\bar{n}}_j^{\prime -1}m)$$

$$H^i(N_0^{\prime \prime },H^j(N_0^{\prime },\pi ))\Rightarrow H^{i+j}(N_0,\pi )$$

$$0\to p\pi \to \pi \to \pi /p\pi \to 0.$$

$$0\to (p\pi {)}^{N_0}\to {\pi }^{N_0}\to (\pi /p\pi {)}^{N_0}\to H^1(N_0,p\pi )\to \cdots .$$

$${\mathrm{Ord}}_P(\pi ):=({\mathrm{Ind}}_{M^{+}}^M({\pi }^{N_0}){)}^{Z_M-l.fin}$$

$${\mathrm{Ord}}_P(\pi )\stackrel{\sim }{\to }R[z^{\pm 1}]{\otimes }_{R[z]}{\pi }^{N_0}$$

$$C_{\mathrm{c}}^{\infty }(N,-):{\mathrm{Mod}}_M^{\infty }(R{)}^{Z_M-l.fin}\to {\mathrm{Mod}}_P^{\infty }(R)$$

$${\mathrm{Ord}}_P:{\mathrm{Mod}}_P^{\infty }(R)\to {\mathrm{Mod}}_M^{\infty }(R{)}^{Z_M-l.fin}$$

$$C_{\mathrm{c}}^{\infty }(N,\sigma )\to \pi .$$

$$C_{\mathrm{c}}^{\infty }(N,\sigma {)}^{N_0}\to {\pi }^{N_0}.$$

$${\mathrm{Ord}}_P(C_{\mathrm{c}}^{\infty }(N,\sigma ))\to {\mathrm{Ord}}_P(\pi )$$

$$0\to \kappa \to C_{\mathrm{c}}^{\infty }(N,\sigma )\to \iota \to 0$$

$$0\to \iota \to \pi \to \pi /\iota \to 0$$

$$C_{\mathrm{c}}^{\infty }(N,\sigma )$$

$$C_{\mathrm{c}}^{\infty }(N,\sigma {)}^{N_0}$$

$$0\to {\kappa }^{N_0}\to C_{\mathrm{c}}^{\infty }(N,\sigma {)}^{N_0}\to {\iota }^{N_0}\to H^1(N_0,\kappa )\to 0$$

$$H^k(N_0,\iota )\stackrel{\sim }{\to }{H}^{k+1}(N_0,\kappa )$$

$$0\to {\iota }^{N_0}\to {\pi }^{N_0}\to (\pi /\iota {)}^{N_0}\to H^1(N_0,\iota )\to \cdots .$$

$$0\to {\pi }_1\to {\pi }_2\to {\pi }_3\to 0$$

$$0\to {\pi }_1^{N_0}\to {\pi }_2^{N_0}\to {\pi }_3^{N_0}\to H^1(N_0,{\pi }_1).$$

Full text

This research was partly supported by EPSRC grant EP/L025302/1.

1. Results

Let $F$ be a non-archimedean local field of residue characteristic $p$. Let $\mathbf{G}$ be a connected reductive algebraic $F$-group and $G$ denote the topological group $\mathbf{G}(F)$. We let $\mathbf{P}=\mathbf{M}\mathbf{N}$ be a parabolic subgroup of $\mathbf{G}$. We write $\bar{\mathbf{P}}=\mathbf{M}\bar{\mathbf{N}}$ for the opposite parabolic subgroup.

Let $R$ be a commutative ring. We write $\operatorname{Mod}_{G}^{\infty}(R)$ for the category of smooth $R$-representations of $G$ (i.e. $R[G]$-modules $\pi$ such that for all $v\in\pi$ the stabiliser of $v$ is open in $G$) and $R[G]$-linear maps. It is an $R$-linear abelian category. When $R$ is noetherian, we write $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ for the full subcategory of $\operatorname{Mod}_{G}^{\infty}(R)$ consisting of admissible representations (i.e. those representations $\pi$ such that $\pi^{H}$ is finitely generated over $R$ for any open subgroup $H$ of $G$). It is closed under passing to subrepresentations and extensions, thus it is an $R$-linear exact subcategory, but quotients of admissible representations may not be admissible when $\operatorname{char}(F)=p$ (see [AHV17, Example 4.4]).

Recall the smooth parabolic induction functor $\operatorname{Ind}_{\bar{P}}^{G}:\operatorname{Mod}_{M}^{\infty}(R)\to\operatorname{Mod}_{G}^{\infty}(R)$, defined on any smooth $R$-representation $\sigma$ of $M$ as the $R$-module $\operatorname{Ind}_{\bar{P}}^{G}(\sigma)$ of locally constant functions $f:G\to\sigma$ satisfying $f(m\bar{n}g)=m\cdot f(g)$ for all $m\in M$, $\bar{n}\in\bar{N}$, and $g\in G$, endowed with the smooth action of $G$ by right translation. It is $R$-linear, exact, and commutes with small direct sums. In the other direction, there is the ordinary part functor $\operatorname{Ord}_{P}:\operatorname{Mod}_{G}^{\infty}(R)\to\operatorname{Mod}_{M}^{\infty}(R)$ ([Eme10a, Vig16]). It is $R$-linear and left exact. When $R$ is noetherian, $\operatorname{Ord}_{P}$ also commutes with small inductive limits, both functors respect admissibility, and the restriction of $\operatorname{Ord}_{P}$ to $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ is right adjoint to the restriction of $\operatorname{Ind}_{\bar{P}}^{G}$ to $\operatorname{Mod}_{M}^{\mathrm{adm}}(R)$.

Theorem 1**.**

If $R$ is artinian, $p$ is nilpotent in $R$, and $\operatorname{char}(F)=p$, then $\operatorname{Ord}_{P}$ is exact on $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$.

Thus the situation is very different from the case $\operatorname{char}(F)=0$ (see [Eme10b]). On the other hand if $R$ is artinian and $p$ is invertible in $R$, then $\operatorname{Ord}_{P}$ is isomorphic on $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ to the Jacquet functor with respect to $P$ (i.e. the $N$-coinvariants) twisted by the inverse of the modulus character $\delta_{P}$ of $P$ ([AHV17, Corollary 4.19]), so that it is exact on $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ without any assumption on $\operatorname{char}(F)$.

Remark*.*

Without any assumption on $R$, $\operatorname{Ind}_{P}^{G}:\operatorname{Mod}_{M}^{\infty}(R)\to\operatorname{Mod}_{G}^{\infty}(R)$ admits a left adjoint $\operatorname{L}_{P}^{G}:\operatorname{Mod}_{G}^{\infty}(R)\to\operatorname{Mod}_{M}^{\infty}(R)$ (the Jacquet functor with respect to $P$) and a right adjoint $\operatorname{R}_{P}^{G}:\operatorname{Mod}_{G}^{\infty}(R)\to\operatorname{Mod}_{M}^{\infty}(R)$ ([Vig16, Proposition 4.2]). If $R$ is noetherian and $p$ is nilpotent in $R$, then $\operatorname{R}_{P}^{G}$ is isomorphic to $\operatorname{Ord}_{\bar{P}}$ on $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ ([AHV17, Corollary 4.13]). Thus under the assumptions of Theorem 1, $\operatorname{R}_{P}^{G}$ is exact on $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$. On the other hand if $R$ is noetherian and $p$ is invertible in $R$, then $\operatorname{R}_{P}^{G}$ is expected to be isomorphic to $\delta_{P}\operatorname{L}_{\bar{P}}^{G}$ (‘second adjointness’), and this is proved in the following cases: when $R$ is the field of complex numbers ([Ber87]) or an algebraically closed field of characteristic $\ell\neq p$ ([Vig96, II.3.8 2)]); when $\mathbf{G}$ is a Levi subgroup of a general linear group or a classical group with $p\neq 2$ ([Dat09, Théorème 1.5]); when $\mathbf{P}$ is a minimal parabolic subgroup of $\mathbf{G}$ (see also [Dat09]). In particular, $\operatorname{L}_{P}^{G}$ and $\operatorname{R}_{P}^{G}$ are exact in all these cases.

Question*.*

Are $\operatorname{L}_{P}^{G}$ and $\operatorname{R}_{P}^{G}$ exact when $R$ is noetherian, $p$ is nilpotent in $R$, and $\operatorname{char}(F)=p$?

We derive from Theorem 1 some results on Yoneda extensions between admissible $R$-representations of $G$. We compute the $R$-modules $\operatorname{Ext}_{G}^{\bullet}$ in $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$.

Corollary 2**.**

Assume $R$ artinian, $p$ nilpotent in $R$, and $\operatorname{char}(F)=p$. Let $\sigma$ and $\pi$ be admissible $R$-representations of $M$ and $G$ respectively. For all $n\geq 0$, there is a natural $R$-linear isomorphism

[TABLE]

This is in contrast with the case $\operatorname{char}(F)=0$ (see [Hau16b]). A direct consequence of Corollary 2 is that under the same assumptions, $\operatorname{Ind}_{\bar{P}}^{G}$ induces an isomorphism between the $\operatorname{Ext}^{n}$ for all $n\geq 0$ (Corollary 5). When $R=C$ is an algebraically closed field of characteristic $p$ and $\operatorname{char}(F)=p$, we determine the extensions between certain irreducible admissible $C$-representations of $G$ using the classification of [AHHV17] (Proposition 6). In particular, we prove that there exists no non-split extension of an irreducible admissible $C$-representation $\pi$ of $G$ by a supersingular $C$-representation of $G$ when $\pi$ is not the extension to $G$ of a supersingular representation of a Levi subgroup of $G$ (Corollary 7). When $\mathbf{G}=\mathrm{GL}_{2}$, this was first proved by Hu ([Hu17, Theorem A.2]).

Acknowledgements

The author would like to thank F. Herzig and M.-F. Vignéras for several comments on the first version of this paper.

2. Proofs

2.1. Hecke action

In this subsection, $\mathbf{M}$ denotes a linear algebraic $F$-group and $\mathbf{N}$ denotes a split unipotent algebraic $F$-group (see [CGP15, Appendix B]) endowed with an action of $\mathbf{M}$ that we identify with the conjugation in $\mathbf{M}\ltimes\mathbf{N}$. We fix an open submonoid $M^{+}$ of $M$ and a compact open subgroup $N_{0}$ of $N$ stable under conjugation by $M^{+}$.

If $\pi$ is a smooth $R$-representation of $M^{+}\ltimes N_{0}$, then the $R$-modules $\operatorname{H}^{\bullet}(N_{0},\pi)$, computed using the homogeneous cochain complex $\operatorname{C}^{\bullet}(N_{0},\pi)$ (see [NSW08, § I.2]), are naturally endowed with the Hecke action of $M^{+}$, defined as the composite

[TABLE]

for all $m\in M^{+}$. At the level of cochains, this action is explicitly given as follows (see [NSW08, § I.5]). We fix a set of representatives $\overline{N_{0}/mN_{0}m^{-1}}\subseteq N_{0}$ of the left cosets $N_{0}/mN_{0}m^{-1}$ and we write $n\mapsto\bar{n}$ for the projection $N_{0}\twoheadrightarrow\overline{N_{0}/mN_{0}m^{-1}}$. For $\phi\in\operatorname{C}^{k}(N_{0},\pi)$, we have

[TABLE]

for all $(n_{0},\dots,n_{k})\in N_{0}^{k+1}$.

Lemma 3**.**

Assume $p$ nilpotent in $R$ and $\operatorname{char}(F)=p$. Let $\pi$ be a smooth $R$-representation of $M^{+}\ltimes N_{0}$ and $m\in M^{+}$. If the Hecke action $h_{N_{0},m}$ of $m$ on $\pi^{N_{0}}$ is locally nilpotent (i.e. for all $v\in\pi^{N_{0}}$ there exists $r\geq 0$ such that $h_{N_{0},m}^{r}(v)=0$), then the Hecke action of $m$ on $\operatorname{H}^{k}(N_{0},\pi)$ is locally nilpotent for all $k\geq 0$.

Proof.

First, we prove the lemma when $pR=0$, i.e. $R$ is a commutative $\mathbb{F}_{p}$-algebra. We assume that the Hecke action of $m$ on $\pi^{N_{0}}$ is locally nilpotent and we prove the result together with the following fact: there exists a set of representatives $\overline{N_{0}/mN_{0}m^{-1}}\subseteq N_{0}$ of the left cosets $N_{0}/mN_{0}m^{-1}$ such that the action of

[TABLE]

on $\pi$ is locally nilpotent.

We proceed by induction on the dimension of $\mathbf{N}$ (recall that $\mathbf{N}$ is split so that it is smooth and connected). If $\mathbf{N}=1$, then the (Hecke) action of $m$ on $\pi^{N_{0}}=\pi$ is locally nilpotent by assumption, so that the result and the fact are trivially true. Assume $\mathbf{N}\neq 1$ and that the result and the fact are true for groups of smaller dimension. Since $\mathbf{N}$ is split, it admits a non-trivial central subgroup isomorphic to the additive group. We let $\mathbf{N}^{\prime}$ be the subgroup of $\mathbf{N}$ generated by all such subgroups. It is a non-trivial vector group (i.e. isomorphic to a direct product of copies of the additive group) which is central (hence normal) in $\mathbf{N}$ and stable under conjugation by $\mathbf{M}$ (since it is a characteristic subgroup of $\mathbf{N}$). We set $\mathbf{N}^{\prime\prime}\coloneqq\mathbf{N}/\mathbf{N}^{\prime}$. It is a split unipotent algebraic $F$-group endowed with the induced action of $\mathbf{M}$ and $\dim(\mathbf{N}^{\prime\prime})<\dim(\mathbf{N})$. Since $\mathbf{N}^{\prime}$ is split, we have $N^{\prime\prime}=N/N^{\prime}$. We write $N^{\prime}_{0}$ and $N^{\prime\prime}_{0}$ for the compact open subgroups $N^{\prime}\cap N_{0}$ and $N_{0}/N^{\prime}_{0}$ of $N^{\prime}$ and $N^{\prime\prime}$ respectively. They are stable under conjugation by $M^{+}$. We fix a set-theoretic section $[-]:N^{\prime\prime}_{0}\hookrightarrow N_{0}$.

Since $\mathbf{N}^{\prime}$ is commutative and $p$-torsion, $N^{\prime}_{0}$ is a compact $\mathbb{F}_{p}$-vector space. Thus for any open subgroup $N^{\prime}_{1}$ of $N^{\prime}_{0}$, the short exact sequence of compact $\mathbb{F}_{p}$-vector spaces

[TABLE]

splits. Indeed, it admits an $\mathbb{F}_{p}$-linear splitting (since $\mathbb{F}_{p}$ is a field) which is automatically continuous (since $N^{\prime}_{0}/N^{\prime}_{1}$ is discrete). In particular with $N^{\prime}_{1}=mN^{\prime}_{0}m^{-1}$, we may and do fix a section $N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}\hookrightarrow N^{\prime}_{0}$. We write $\overline{N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}}$ for its image, so that $N^{\prime}_{0}=\overline{N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}}\times mN^{\prime}_{0}m^{-1}$, and $n^{\prime}\mapsto\bar{n}^{\prime}$ for the projection $N^{\prime}_{0}\twoheadrightarrow\overline{N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}}$. We set

[TABLE]

For all $n^{\prime}_{0}\in N^{\prime}_{0}$, we have $n^{\prime}_{0}=\bar{n}^{\prime}_{0}(\bar{n}_{0}^{\prime-1}n^{\prime}_{0})$ with $\bar{n}_{0}^{\prime-1}n^{\prime}_{0}\in mN^{\prime}_{0}m^{-1}$, thus

[TABLE]

with $m^{-1}(\bar{n}_{0}^{\prime-1}n^{\prime}_{0})m\in N^{\prime}_{0}$ (in the first equality we use the fact that $N^{\prime}_{0}$ is commutative and in the second one we use the fact that $\overline{N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}}$ is a group). Therefore, there is an inclusion $\mathbb{F}_{p}[N^{\prime}_{0}]S^{\prime}\subseteq S^{\prime}\mathbb{F}_{p}[N^{\prime}_{0}]$.

The $R$-module $\pi^{N^{\prime}_{0}}$, endowed with the induced action of $N^{\prime\prime}_{0}$ and the Hecke action of $M^{+}$ with respect to $N^{\prime}_{0}$, is a smooth $R$-representation of $M^{+}\ltimes N^{\prime\prime}_{0}$ (see the proof of [Hau16a, Lemme 3.2.1] in degree [math]). On $\pi^{N^{\prime}_{0}}$, the Hecke action of $m$ with respect to $N^{\prime}_{0}$ coincides with the action of $S^{\prime}$ by definition. On $(\pi^{N^{\prime}_{0}})^{N^{\prime\prime}_{0}}=\pi^{N_{0}}$, the Hecke action of $m$ with respect to $N^{\prime\prime}_{0}$ coincides with the Hecke action of $m$ with respect to $N_{0}$ (see the proof of [Hau16a, Lemme 3.2.2]) which is locally nilpotent by assumption. Thus by the induction hypothesis, there exists a set of representatives $\overline{N^{\prime\prime}_{0}/mN^{\prime\prime}_{0}m^{-1}}\subseteq N^{\prime\prime}_{0}$ of the left cosets $N^{\prime\prime}_{0}/mN^{\prime\prime}_{0}m^{-1}$ such that the action of

[TABLE]

on $\pi^{N^{\prime}_{0}}$ is locally nilpotent. Moreover, there is an inclusion $\mathbb{F}_{p}[N^{\prime}_{0}]S\subseteq S\mathbb{F}_{p}[N^{\prime}_{0}]$ (because $N^{\prime}_{0}$ is central in $N_{0}$ and $\mathbb{F}_{p}[N^{\prime}_{0}]S^{\prime}\subseteq S^{\prime}\mathbb{F}_{p}[N^{\prime}_{0}]$).

We prove the fact. By [Hau16c, Lemme 2.1],

[TABLE]

is a set of representatives of the left cosets $N_{0}/mN_{0}m^{-1}$, and by definition

[TABLE]

We prove that the action of $S$ on $\pi$ is locally nilpotent. We proceed as in the proof of [Hu12, Théorème 5.1 (i)]. Let $v\in\pi$ and set $\pi_{r}\coloneqq\mathbb{F}_{p}[N^{\prime}_{0}]\cdot(S^{r}\cdot v)$ for all $r\geq 0$. Since $\mathbb{F}_{p}[N^{\prime}_{0}]S\subseteq S\mathbb{F}_{p}[N^{\prime}_{0}]$, we have $\pi_{r+1}\subseteq S\cdot\pi_{r}$ for all $r\geq 0$. Since $N^{\prime}_{0}$ is compact, we have $\dim_{\mathbb{F}_{p}}(\pi_{r})<\infty$ for all $r\geq 0$. If $S^{r}\cdot v\neq 0$, i.e. $\pi_{r}\neq 0$, for some $r\geq 0$, then $\pi_{r}^{N^{\prime}_{0}}\neq 0$ (because $N^{\prime}_{0}$ is a pro-$p$ group and $\pi_{r}$ is a non-zero $\mathbb{F}_{p}$-vector space) so that $\dim_{\mathbb{F}_{p}}(S\cdot\pi_{r})<\dim_{\mathbb{F}_{p}}\pi_{r}$ (because the action of $S$ on $\pi^{N^{\prime}_{0}}$ is locally nilpotent). Therefore $\pi_{r}=0$, i.e. $S^{r}\cdot v=0$, for all $r\geq\dim_{\mathbb{F}_{p}}(\pi_{0})$.

We prove the result. The $R$-modules $\operatorname{H}^{\bullet}(N^{\prime}_{0},\pi)$, endowed with the induced action of $N^{\prime\prime}_{0}$ and the Hecke action of $M^{+}$, are smooth $R$-representations of $M^{+}\ltimes N^{\prime\prime}_{0}$ (see the proof of [Hau16a, Lemme 3.2.1]111We do not know whether [Eme10b, Proposition 2.1.11] holds true when $\operatorname{char}(F)=p$, but [Hau16a, Lemme 3.1.1] does and any injective object of $\operatorname{Mod}_{M^{+}\ltimes N_{0}}^{\infty}(R)$ is still $N_{0}$-acyclic.). At the level of cochains, the actions of $n^{\prime\prime}\in N^{\prime\prime}_{0}$ and $m$ are explicitly given as follows. For $\phi\in\operatorname{C}^{j}(N^{\prime}_{0},\pi)$, we have

[TABLE]

for all $(n^{\prime}_{0},\dots,n^{\prime}_{j})\in N_{0}^{\prime j+1}$ (for (2) we use the fact that $N^{\prime}_{0}$ is central in $N_{0}$, for (3) we use (1) and the fact that $n^{\prime}\mapsto\bar{n}^{\prime}$ is a group homomorphism $N^{\prime}_{0}\to\overline{N^{\prime}_{0}/mN^{\prime}_{0}m^{-1}}$). Using (2) and (3), we can give explicitly the Hecke action of $m$ on $\operatorname{H}^{\bullet}(N^{\prime}_{0},\pi)^{N^{\prime\prime}_{0}}$ at the level of cochains as follows. For $\phi\in\operatorname{C}^{j}(N^{\prime}_{0},\pi)$, we have

[TABLE]

for all $(n^{\prime}_{0},\dots,n^{\prime}_{j})\in N_{0}^{\prime j+1}$. Since the action of $S$ on $\pi$ is locally nilpotent and the image of a locally constant cochain is finite by compactness of $N^{\prime}_{0}$, we deduce that the Hecke action of $m$ on $\operatorname{H}^{j}(N^{\prime}_{0},\pi)^{N^{\prime\prime}_{0}}$ is locally nilpotent for all $j\geq 0$. Thus the Hecke action of $m$ on $\operatorname{H}^{i}(N^{\prime\prime}_{0},\operatorname{H}^{j}(N^{\prime}_{0},\pi))$ is locally nilpotent for all $i,j\geq 0$ by the induction hypothesis. We conclude using the spectral sequence of smooth $R$-representations of $M^{+}$

[TABLE]

(see the proof of [Hau16a, Proposition 3.2.3] and footnote 1).

Now, we prove the lemma without assuming $pR=0$. We proceed by induction on the degree of nilpotency $r$ of $p$ in $R$. If $r\leq 1$, then the lemma is already proved. We assume $r>1$ and that we know the lemma for rings in which the degree of nilpotency of $p$ is $r-1$. There is a short exact sequence of smooth $R$-representations of $M^{+}\ltimes N_{0}$

[TABLE]

Taking the $N_{0}$-cohomology yields a long exact sequence of smooth $R$-representations of $M^{+}$

[TABLE]

If the Hecke action of $m$ on $\pi^{N_{0}}$ is locally nilpotent, then the Hecke action of $m$ on $(p\pi)^{N_{0}}$ is also locally nilpotent so that the Hecke action of $m$ on $\operatorname{H}^{k}(N_{0},p\pi)$ is locally nilpotent for all $k\geq 0$ by the induction hypothesis (since $p\pi$ is an $R/p^{r-1}R$-module). Using (4), we deduce that the Hecke action of $m$ on $(\pi/p\pi)^{N_{0}}$ is also locally nilpotent so that the Hecke action of $m$ on $\operatorname{H}^{k}(N_{0},\pi/p\pi)$ is locally nilpotent for all $k\geq 0$ (since $\pi/p\pi$ is an $\mathbb{F}_{p}$-vector space). Using again (4), we conclude that the Hecke action of $m$ on $\operatorname{H}^{k}(N_{0},\pi)$ is locally nilpotent for all $k\geq 0$. ∎

2.2. Proof of the main result

We fix a compact open subgroup $N_{0}$ of $N$ and we let $M^{+}$ be the open submonoid of $M$ consisting of those elements $m$ contracting $N_{0}$ (i.e. $mN_{0}m^{-1}\subseteq N_{0}$). We let $\mathbf{Z}_{\mathbf{M}}$ denote the centre of $\mathbf{M}$ and we set $Z_{M}^{+}\coloneqq Z_{M}\cap M^{+}$. We fix an element $z\in Z_{M}^{+}$ strictly contracting $N_{0}$ (i.e. $\cap_{r\geq 0}z^{r}N_{0}z^{-r}=1$).

Recall that the ordinary part of a smooth $R$-representation $\pi$ of $P$ is the smooth $R$-representation of $M$

[TABLE]

where $\operatorname{Ind}_{M^{+}}^{M}(\pi^{N_{0}})$ is defined as the $R$-module of functions $f:M\to\pi^{N_{0}}$ such that $f(mm^{\prime})=m\cdot f(m^{\prime})$ for all $m\in M^{+}$ and $m^{\prime}\in M$, endowed with the action of $M$ by right translation, and the superscript ${}^{Z_{M}-\mathrm{l.fin}}$ denotes the subrepresentation consisting of locally $Z_{M}$-finite elements (i.e. those elements $f$ such that $R[Z_{M}]\cdot f$ is contained in a finitely generated $R$-submodule). The action of $M$ on the latter is smooth by [Vig16, Remark 7.6]. If $R$ is artinian and $\pi^{N_{0}}$ is locally $Z_{M}^{+}$-finite (i.e. it may be written as the union of finitely generated $Z_{M}^{+}$-invariant $R$-submodules), then there is a natural $R$-linear isomorphism

[TABLE]

(cf. [Eme10b, Lemma 3.2.1 (1)], whose proof also works when $\operatorname{char}(F)=p$ and over any artinian ring).

If $\sigma$ is a smooth $R$-representation of $M$, then the $R$-module $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,\sigma)$ of locally constant functions $f:N\to\sigma$ with compact support, endowed with the action of $N$ by right translation and the action of $M$ given by $(m\cdot f):n\mapsto m\cdot f(m^{-1}nm)$ for all $m\in M$, is a smooth $R$-representation of $P$. Thus we obtain a functor $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,-):\operatorname{Mod}_{M}^{\infty}(R)\to\operatorname{Mod}_{P}^{\infty}(R)$. It is $R$-linear, exact, and commutes with small direct sums. The results of [Eme10a, § 4.2] hold true when $\operatorname{char}(F)=p$ and over any ring, thus the functors

[TABLE]

are adjoint and the unit of the adjunction is an isomorphism.

Lemma 4**.**

Assume $R$ artinian, $p$ nilpotent in $R$, and $\operatorname{char}(F)=p$. Let $\pi$ be a smooth $R$-representation of $P$. If $\pi^{N_{0}}$ is locally $Z_{M}^{+}$-finite, then the Hecke action of $z$ on $\operatorname{H}^{k}(N_{0},\pi)$ is locally nilpotent for all $k\geq 1$.

Proof.

We set $\sigma\coloneqq\operatorname{Ord}_{P}(\pi)$. The counit of the adjunction between $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,-)$ and $\operatorname{Ord}_{P}$ induces a natural morphism of smooth $R$-representations of $P$

[TABLE]

Taking the $N_{0}$-invariants yields a morphism of smooth $R$-representations of $M^{+}$

[TABLE]

By definition, $\sigma$ is locally $Z_{M}$-finite so it may be written as the union of finitely generated $Z_{M}$-invariant $R$-submodules $(\sigma_{i})_{i\in I}$. Thus $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,\sigma)^{N_{0}}$ is the union of the finitely generated $Z_{M}^{+}$-invariant $R$-submodules $(\operatorname{\mathcal{C}}^{\infty}(z^{-r}N_{0}z^{r},\sigma_{i})^{N_{0}})_{r\geq 0,i\in I}$, so it is locally $Z_{M}^{+}$-finite. By assumption, $\pi^{N_{0}}$ is also locally $Z_{M}^{+}$-finite. Therefore, using (5) and its analogue with $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,\sigma)$ instead of $\pi$, the localisation with respect to $z$ of (7) is the natural morphism of smooth $R$-representations of $M$

[TABLE]

induced by applying the functor $\operatorname{Ord}_{P}$ to (6), and it is an isomorphism since the unit of the adjunction between $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,-)$ and $\operatorname{Ord}_{P}$ is an isomorphism.

Let $\kappa$ (resp. $\iota$) be the kernel (resp. image) of (6), hence two short exact sequences of smooth $R$-representations of $P$

[TABLE]

such that the third arrow of (8) and the second arrow of (9) fit into a commutative diagram of smooth $R$-representations of $P$

[TABLE]

whose upper arrow is (6). Taking the $N_{0}$-invariants yields a commutative diagram of smooth $R$-representations of $M^{+}$

[TABLE]

whose upper arrow is (7). Since the localisation with respect to $z$ of the latter is an isomorphism, the localisation with respect to $z$ of the injection $\iota^{N_{0}}\hookrightarrow\pi^{N_{0}}$ is surjective, thus it is an isomorphism (as it is also injective by exactness of localisation). Therefore the localisation with respect to $z$ of the morphism $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,\sigma)^{N_{0}}\to\iota^{N_{0}}$ is an isomorphism.

Since $\operatorname{\mathcal{C}}_{\mathrm{c}}^{\infty}(N,\sigma)\cong\bigoplus_{n\in N/N_{0}}\operatorname{\mathcal{C}}^{\infty}(nN_{0},\sigma)$ as a smooth $R$-representation of $N_{0}$, it is $N_{0}$-acyclic (see [NSW08, § I.3]). Thus the long exact sequence of $N_{0}$-cohomology induced by (8) yields an exact sequence of smooth $R$-representations of $M^{+}$

[TABLE]

and an isomorphism of smooth $R$-representations of $M^{+}$

[TABLE]

for all $k\geq 1$. Since the localisation with respect to $z$ of the third arrow of (10) is an isomorphism, the Hecke action of $z$ on $\kappa^{N_{0}}$ is locally nilpotent. Thus the Hecke action of $z$ on $\operatorname{H}^{k}(N_{0},\kappa)$ is locally nilpotent for all $k\geq 0$ by Lemma 3. Using (11), we deduce that the Hecke action of $z$ on $\operatorname{H}^{k}(N_{0},\iota)$ is locally nilpotent for all $k\geq 1$.

Taking the $N_{0}$-cohomology of (9) yields a long exact sequence of smooth $R$-representations of $M^{+}$

[TABLE]

Since the localisation with respect to $z$ of the second arrow is an isomorphism and the Hecke action of $z$ on $\operatorname{H}^{1}(N_{0},\iota)$ is locally nilpotent, the Hecke action of $z$ on $(\pi/\iota)^{N_{0}}$ is locally nilpotent. Thus the Hecke action of $z$ on $\operatorname{H}^{k}(N_{0},\pi/\iota)$ is locally nilpotent for all $k\geq 0$ by Lemma 3. We conclude using (12) and the fact that the Hecke action of $z$ on $\operatorname{H}^{k}(N_{0},\iota)$ is locally nilpotent for all $k\geq 1$. ∎

Proof of Theorem 1.

Assume $R$ artinian, $p$ nilpotent in $R$, and $\operatorname{char}(F)=p$. Let

[TABLE]

be a short exact sequence of admissible $R$-representations of $G$. Taking the $N_{0}$-invariants yields an exact sequence of smooth $R$-representations of $M^{+}$

[TABLE]

The terms $\pi_{1}^{N_{0}},\pi_{2}^{N_{0}},\pi_{3}^{N_{0}}$ are locally $Z_{M}^{+}$-finite (cf. [Eme10b, Theorem 3.4.7 (1)], whose proof in degree [math] also works when $\operatorname{char}(F)=p$ and over any noetherian ring) and the Hecke action of $z$ on $\operatorname{H}^{1}(N_{0},\pi_{1})$ is locally nilpotent by Lemma 4. Therefore, using (5), the localisation with respect to $z$ of (14) is the short sequence of admissible $R$-representations of $M$

[TABLE]

induced by applying the functor $\operatorname{Ord}_{P}$ to (13), and it is exact by exactness of localisation. ∎

2.3. Results on extensions

We assume $R$ noetherian. The $R$-linear category $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ is not abelian in general, but merely exact in the sense of Quillen ([Qui73]). An exact sequence of admissible $R$-representations of $G$ is an exact sequence of smooth $R$-representations of $G$

[TABLE]

such that the kernel and the cokernel of every arrow are admissible. In particular, each term of the sequence is also admissible.

For $n\geq 0$ and $\pi,\pi^{\prime}$ two admissible $R$-representations of $G$, we let $\operatorname{Ext}_{G}^{n}(\pi^{\prime},\pi)$ denote the $R$-module of $n$-fold Yoneda extensions ([Yon60]) of $\pi^{\prime}$ by $\pi$ in $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$, defined as equivalence classes of exact sequences

[TABLE]

We let $D(G)$ denote the derived category of $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ ([Nee90, Kel96, Büh10]). The results of [Ver96, § III.3.2] on the Yoneda construction carry over to this setting (see e.g. [Pos11, Proposition A.13]), hence a natural $R$-linear isomorphism

[TABLE]

Proof of Corollary 2.

Since $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$ are exact adjoint functors between $\operatorname{Mod}_{M}^{\mathrm{adm}}(R)$ and $\operatorname{Mod}_{G}^{\mathrm{adm}}(R)$ by Theorem 1, they induce adjoint functors between $D(M)$ and $D(G)$, hence natural $R$-linear isomorphisms

[TABLE]

for all $n\geq 0$. ∎

Remark*.*

We give a more explicit proof of Corollary 2. The exact functor $\operatorname{Ind}_{\bar{P}}^{G}$ and the counit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$ induce an $R$-linear morphism

[TABLE]

In the other direction, the exact (by Theorem 1) functor $\operatorname{Ord}_{P}$ and the unit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$ induce an $R$-linear morphism

[TABLE]

We prove that (16) is the inverse of (15). For $n=0$ this is the unit-counit equations. Assume $n\geq 1$ and let

[TABLE]

be an exact sequence of admissible $R$-representations of $M$. By [Yon60, § 3], the image of the class of (17) under (15) is the class of any exact sequence of admissible $R$-representations of $G$

[TABLE]

such that there exists a commutative diagram of admissible $R$-representations of $G$

[TABLE]

in which the upper row is obtained from (17) by applying the exact functor $\operatorname{Ind}_{\bar{P}}^{G}$, the lower row is (18), and the leftmost vertical arrow is the natural morphism induced by the counit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$. Applying the exact functor $\operatorname{Ord}_{P}$ to the diagram and using the unit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$ yields a commutative diagram of admissible $R$-representations of $M$

[TABLE]

in which the lower row is obtained from (18) by applying the exact functor $\operatorname{Ord}_{P}$, the upper row is (17), and the rightmost vertical arrow is the natural morphism induced by the unit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$. The leftmost vertical morphism is the identity by the unit-counit equations. Thus the image of the class of (18) under (16) is the class of (17) by [Yon60, § 3]. We have proved that (16) is a left inverse of (15). The proof that it is a right inverse is dual.

Corollary 5**.**

Assume $R$ artinian, $p$ nilpotent in $R$, and $\operatorname{char}(F)=p$. Let $\sigma$ and $\sigma^{\prime}$ be two admissible $R$-representations of $M$. The functor $\operatorname{Ind}_{\bar{P}}^{G}$ induces an $R$-linear isomorphism

[TABLE]

for all $n\geq 0$.

Proof.

The isomorphism in the statement is the composite

[TABLE]

where the first isomorphism is induced by the unit of the adjunction between $\operatorname{Ind}_{\bar{P}}^{G}$ and $\operatorname{Ord}_{P}$, which is an isomorphism, and the second one is the isomorphism of Corollary 2 with $\sigma^{\prime}$ and $\operatorname{Ind}_{\bar{P}}^{G}(\sigma)$ instead of $\sigma$ and $\pi$ respectively. ∎

We fix a minimal parabolic subgroup $\mathbf{B}\subseteq\mathbf{G}$, a maximal split torus $\mathbf{S}\subseteq\mathbf{B}$, and we write $\Delta$ for the set of simple roots of $\mathbf{S}$ in $\mathbf{B}$. We say that a parabolic subgroup $\mathbf{P}=\mathbf{M}\mathbf{N}$ of $\mathbf{G}$ is standard if $\mathbf{B}\subseteq\mathbf{P}$ and $\mathbf{S}\subseteq\mathbf{M}$. In this case, we write $\Delta_{\mathbf{P}}$ for the corresponding subset of $\Delta$, and given $\alpha\in\Delta_{\mathbf{P}}$ (resp. $\alpha\in\Delta\backslash\Delta_{\mathbf{P}}$) we write $\mathbf{P}^{\alpha}=\mathbf{M}^{\alpha}\mathbf{N}^{\alpha}$ (resp. $\mathbf{P}_{\alpha}=\mathbf{M}_{\alpha}\mathbf{N}_{\alpha}$) for the standard parabolic subgroup corresponding to $\Delta_{\mathbf{P}}\backslash\{\alpha\}$ (resp. $\Delta_{\mathbf{P}}\sqcup\{\alpha\}$).

Let $C$ be an algebraically closed field of characteristic $p$. Given a standard parabolic subgroup $P=MN$ and a smooth $C$-representation $\sigma$ of $M$, there exists a largest standard parabolic subgroup $P(\sigma)=M(\sigma)N(\sigma)$ such that the inflation of $\sigma$ to $P$ extends to a smooth $C$-representation $\prescript{\mathrm{e}}{}{\sigma}$ of $P(\sigma)$, and this extension is unique ([AHHV17, II.7 Corollary 1]). We say that a smooth $C$-representation of $G$ is supercuspidal if it is irreducible, admissible, and does not appear as a subquotient of $\operatorname{Ind}_{P}^{G}(\sigma)$ for any proper parabolic subgroup $P=MN$ of $G$ and any irreducible admissible $C$-representation $\sigma$ of $M$. A supercuspidal standard $C[G]$-triple is a triple $(P,\sigma,Q)$ where $P=MN$ is a standard parabolic subgroup, $\sigma$ is a supercuspidal $C$-representation of $M$, and $Q$ is a parabolic subgroup of $G$ such that $P\subseteq Q\subseteq P(\sigma)$. To such a triple is attached in [AHHV17] a smooth $C$-representation of $G$

[TABLE]

where $\mathrm{St}_{Q}^{P(\sigma)}\coloneqq\operatorname{Ind}_{Q}^{P(\sigma)}(1)/\sum_{Q\subsetneq Q^{\prime}\subseteq P(\sigma)}\operatorname{Ind}_{Q^{\prime}}^{P(\sigma)}(1)$ (here $1$ denotes the trivial $C$-representation) is the inflation to $P(\sigma)$ of the generalised Steinberg representation of $M(\sigma)$ with respect to $M(\sigma)\cap Q$ ([GK14, Ly15]). It is irreducible and admissible ([AHHV17, I.3 Theorem 1]).

Proposition 6**.**

Assume $\operatorname{char}(F)=p$. Let $(P,\sigma,Q)$ and $(P^{\prime},\sigma^{\prime},Q^{\prime})$ be two supercuspidal standard $C[G]$-triples. If $Q\not\subseteq Q^{\prime}$, then the $C$-vector space

[TABLE]

is non-zero if and only if $P^{\prime}=P$, $\sigma^{\prime}\cong\sigma$, and $Q^{\prime}=Q^{\alpha}$ for some $\alpha\in\Delta_{Q}$, in which case it is one-dimensional and the unique (up to isomorphism) non-split extension of $\operatorname{I}_{G}(P^{\prime},\sigma^{\prime},Q^{\prime})$ by $\operatorname{I}_{G}(P,\sigma,Q)$ is the admissible $C$-representation of $G$

[TABLE]

Proof.

There is a natural short exact sequence of admissible $C$-representations of $G$

[TABLE]

Note that we can restrict the sum to those $Q^{\prime\prime}$ that are minimal, i.e. of the form $Q^{\prime}_{\alpha}$ for some $\alpha\in\Delta_{P(\sigma^{\prime})}\backslash\Delta_{Q^{\prime}}$. Moreover, we deduce from [AHV17, Theorem 3.2] that its cosocle is isomorphic to $\bigoplus_{\alpha\in\Delta_{P(\sigma^{\prime})}\backslash\Delta_{Q^{\prime}}}\operatorname{I}_{G}(P^{\prime},\sigma^{\prime},Q^{\prime}_{\alpha})$. Now if $Q\not\subseteq Q^{\prime}$, then $\operatorname{Ord}_{\bar{Q}^{\prime}}(\operatorname{I}_{G}(P,\sigma,Q))=0$ by [AHV17, Theorem 1.1 (ii) and Corollary 4.13] so that using Corollary 2, we see that the long exact sequence of Yoneda extensions obtained by applying the functor $\operatorname{Hom}_{G}(-,\operatorname{I}_{G}(P,\sigma,Q))$ to (19) yields a natural $C$-linear isomorphism

[TABLE]

for all $n\geq 1$. In particular, with $n=1$ and using the identification of the cosocle of the sum and [AHHV17, I.3 Theorem 2], we deduce that the $C$-vector space in the statement is non-zero if and only if $P^{\prime}=P$, $\sigma^{\prime}\cong\sigma$, and $Q=Q^{\prime}_{\alpha}$ for some $\alpha\in\Delta_{P(\sigma^{\prime})}\backslash\Delta_{Q^{\prime}}$ (or equivalently $Q^{\prime}=Q^{\alpha}$ for some $\alpha\in\Delta_{Q}$), in which case it is one-dimensional. Finally, using again [AHV17, Theorem 3.2], we see that for all $\alpha\in\Delta_{Q}$ the admissible $C$-representation of $G$ in the statement is a non-split extension of $\operatorname{I}_{G}(P,\sigma,Q^{\alpha})$ by $\operatorname{I}_{G}(P,\sigma,Q)$. ∎

Corollary 7**.**

Assume $\operatorname{char}(F)=p$. Let $\pi$ and $\pi^{\prime}$ be two irreducible admissible $C$-representations of $G$. If $\pi$ is supercuspidal and $\pi^{\prime}$ is not the extension to $G$ of a supercuspidal representation of a Levi subgroup of $G$, then $\operatorname{Ext}_{G}^{1}(\pi^{\prime},\pi)=0$.

Proof.

By [AHHV17, I.3 Theorem 3], there exist two supercuspidal standard $C[G]$-triples $(P,\sigma,Q)$ and $(P^{\prime},\sigma^{\prime},Q^{\prime})$ such that $\pi\cong\operatorname{I}_{G}(P,\sigma,Q)$ and $\pi^{\prime}\cong\operatorname{I}_{G}(P^{\prime},\sigma^{\prime},Q^{\prime})$. The assumptions on $\pi$ and $\pi^{\prime}$ are equivalent to $P=G$ and $Q^{\prime}\neq G$. In particular, $Q\not\subseteq Q^{\prime}$ and $P\neq P^{\prime}$ so that $\operatorname{Ext}_{G}^{1}(\pi^{\prime},\pi)=0$ by Proposition 6. ∎