[Paper Review] On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups

Noriyuki Abe, Guy Henniart+1 more|arXiv (Cornell University)|Mar 30, 2017

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes a precise correspondence between pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups and modules over pro-$p$-Iwahori Hecke algebras, extending previous work on parabolic induction. It proves that for an algebraically closed field $R$ of characteristic $p$, any irreducible admissible $R$-representation of $G$ with non-zero smooth dual must be finite-dimensional, resolving a key case in the classification of such representations.

ABSTRACT

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf{G}(F)$. Fix a commutative ring $R$. If $π$ is a smooth $R[G]$-representation, the space of invariants $π^{\mathcal{U}}$ is a right module over the Hecke algebra $\mathcal{H}$ of $\mathcal{U}$ in $G$. Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P = MN$ adapted to $\mathcal{U}$. We complement previous investigation of Ollivier-Vignéras on the relation between taking $\mathcal{U}$-invariants and various functor like $\mathrm{Ind}_P^G$ and right and left adjoints. More precisely the authors' previous work with Herzig introduce representations $I_G(P,σ,Q)$ where $σ$ is a smooth representation of $M$ extending, trivially on $N$, to a larger parabolic subgroup $P(σ)$, and $Q$ is a parabolic subgroup between $P$ and $P(σ)$. Here we relate $I_G(P,σ,Q)^{\mathcal{U}}$ to an analogously defined $\mathcal{H}$-module $I_\mathcal{H}(P,σ^{\mathcal{U}_M},Q)$, where $\mathcal{U}_M = \mathcal{U}\cap M$ and $σ^{\mathcal{U}_M}$ is seen as a module over the Hecke algebra $\mathcal{H}_M$ of $\mathcal{U}_M$ in $M$. In the reverse direction, if $\mathcal{V}$ is a right $\mathcal{H}_M$-module, we relate $I_\mathcal{H}(P,\mathcal{V},Q)\otimes extrm{c-Ind}_\mathcal{U}^G\mathbf{1}$ to $I_G(P,\mathcal{V}\otimes_{\mathcal{H}_M} extrm{c-Ind}_{\mathcal{U}_M}^M\mathbb{1},Q)$. As an application we prove that if $R$ is an algebraically closed field of characteristic $p$, and $π$ is an irreducible admissible representation of $G$, then the contragredient of $π$ is $0$ unless $π$ has finite dimension.

Motivation & Objective

• To relate the pro-$p$-Iwahori invariants of $I_G(P,\sigma,Q)$ to Hecke modules over the pro-$p$-Iwahori Hecke algebra $\mathcal{H}$.
• To extend the framework of Ollivier-Vignéras and Abe-Henniart-Vignéras to include invariants under pro-$p$-Iwahori subgroups.
• To prove that irreducible admissible $R$-representations of $G$ with non-zero smooth dual are finite-dimensional when $R$ is an algebraically closed field of characteristic $p$.
• To establish a duality result in the context of mod-$p$ representation theory, generalizing results from characteristic 0.

Proposed method

• Constructs a correspondence between $I_G(P,\sigma,Q)^\mathcal{U}$ and an $\mathcal{H}$-module $I_\mathcal{H}(P,\sigma^{\mathcal{U}_M},Q)$, where $\sigma^{\mathcal{U}_M}$ is the $\mathcal{U}_M$-invariant space of $\sigma$.
• Uses the Hecke algebra $\mathcal{H}$ of the pro-$p$-Iwahori subgroup $\mathcal{U}$ and its Levi subalgebra $\mathcal{H}_M$ for $\mathcal{U}_M = \mathcal{U} \cap M$, and studies extension properties of $\mathcal{H}_M$-modules.
• Applies the right adjoint of $(-)^{\mathcal{U}}$ via the functor $\mathcal{V} \mapsto \mathcal{V} \otimes_{\mathcal{H}_M} \operatorname{c-Ind}_{\mathcal{U}_M}^M \mathbf{1}$ to relate $\mathcal{H}$-modules to $G$-representations.
• Employs the classification of supersingular $\mathcal{H}_R$-modules and the structure of the universal representation $I_\mathcal{H}(P,\mathcal{V},Q) \otimes_\mathcal{H} R[\mathcal{U} \backslash G]$ to analyze invariants.
• Applies the theory of parabolic induction and coinduction in the Hecke algebra setting, particularly using the module $\mathrm{St}_Q^{P(\sigma)}$ as a building block.
• Uses the character $\chi$ associated to a supersingular representation to derive constraints on coefficients $a_w$, leading to a contradiction unless the Weyl group is trivial.

Experimental results

Research questions

• RQ1 How do the pro-$p$-Iwahori invariants of $I_G(P,\sigma,Q)$ relate to Hecke modules over the pro-$p$-Iwahori Hecke algebra $\mathcal{H}$?
• RQ2 Can the $\mathcal{H}$-module structure of $I_G(P,\sigma,Q)^\mathcal{U}$ be explicitly described in terms of $\sigma^{\mathcal{U}_M}$ and the Hecke algebra $\mathcal{H}_M$?
• RQ3 Under what conditions does the smooth dual of an irreducible admissible $R$-representation of $G$ vanish, and when is it non-zero?
• RQ4 What is the role of supersingular $\mathcal{H}_M$-modules in determining the finiteness of $I_G(P,\sigma,Q)$?
• RQ5 Can the finite-dimensionality of irreducible admissible representations with non-zero smooth dual be established in positive characteristic using Hecke algebra methods?

Key findings

• The space of pro-$p$-Iwahori invariants $I_G(P,\sigma,Q)^\mathcal{U}$ is isomorphic to the $\mathcal{H}$-module $I_\mathcal{H}(P,\sigma^{\mathcal{U}_M},Q)$, establishing a precise link between representation-theoretic and Hecke-algebraic invariants.
• For any right $\mathcal{H}_M$-module $\mathcal{V}$, the tensor product $I_\mathcal{H}(P,\mathcal{V},Q) \otimes_{\mathcal{H}} \operatorname{c-Ind}_{\mathcal{U}}^G \mathbf{1}$ is isomorphic to $I_G(P, \mathcal{V} \otimes_{\mathcal{H}_M} \operatorname{c-Ind}_{\mathcal{U}_M}^M \mathbf{1}, Q)$, showing compatibility between parabolic induction and Hecke algebra operations.
• If $R$ is an algebraically closed field of characteristic $p$, then any irreducible admissible $R$-representation $\pi$ of $G$ with non-zero smooth dual must be finite-dimensional.
• The proof relies on the existence of a supersingular $\mathcal{H}_M$-submodule in $\sigma^{\mathcal{U}_M}$ when $\sigma$ is supercuspidal, which forces the Levi subgroup $M$ to be central, hence finite.
• The argument uses the non-vanishing of the smooth dual of $\sigma$ to deduce that $M = Z$, the center, implying $\sigma$ is finite-dimensional and hence $\pi = I_G(P,\sigma,G)$ is finite-dimensional.
• The result holds without restriction on the residue characteristic of $F$, and does not depend on results from characteristic 0, making it valid in the full mod-$p$ setting.