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

Ciubotaru, D
Heiermann, V

7 June 2019

## Journal:

Israel Journal of Mathematics

## Last Updated:

2020-07-25T08:23:28.897+01:00

1

231

## DOI:

10.1007/s11856-019-1857-7

379–417-

## abstract:

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⊗ σ 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 π ⊗ σ does not depend on σ, if σ is ”separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⊗ σ 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 CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ 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.

907423

Submitted

Journal Article