On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

TL;DR

This paper proves that the reducibility of certain induced representations for classical p-adic groups depends only on specific support conditions, and it establishes an equivalence of categories with modules over affine Hecke algebras, extending Jantzen's work.

Contribution

It demonstrates that reducibility is independent of certain supercuspidal representations and extends categorical equivalences to broader settings involving affine Hecke algebras.

Findings

Reducibility depends only on the supercuspidal support being 'separated'.

Categories of representations are equivalent to modules over affine Hecke algebras.

Extends Jantzen's bijection to more general categories.

Abstract

Let $\pi $ be an irreducible smooth complex representation of a general linear $p$-adic group and let $\sigma $ be an irreducible complex supercuspidal representation of a classical $p$-adic group of a given type, so that $\pi\otimes\sigma $ is a representation of a standard Levi subgroup of a $p$-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate $p$-adic classical group obtained by (normalized) parabolic induction from $\pi\otimes\sigma $ does not depend on $\sigma $, if $\sigma $ is "separated" from the supercuspidal support of $\pi $. (Here, "separated" means that, for each factor $\rho $ of a representation in the supercuspidal support of $\pi $, the representation parabolically induced from $\rho\otimes\sigma $ is irreducible.) This was conjectured by E. Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set $I$ of inertial orbits of supercuspidal representations of $p$-adic general linear groups, the category $\CC _{I,\sigma}$ of smooth complex finitely generated representations of classical $p$-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by $\sigma $ and $I$, show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type $A$, $B$ and $D$ and establish functoriality properties, relating categories with disjoint $I$'s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato's exotic geometry.