Alg\`ebres de Hecke avec param\`etres et repr\'esentations d'un groupe p-adique classique: pr\'eservation du spectre temp\'er\'e

TL;DR

This paper proves that the equivalence between Bernstein components of certain p-adic groups and modules over Hecke algebras with parameters preserves the tempered spectrum and discrete series representations, extending understanding of their representation theory.

Contribution

It demonstrates that the established equivalence maintains the tempered spectrum and discrete series, providing new insights into the structure of representations of p-adic groups.

Findings

Preservation of the tempered spectrum under the equivalence.

Preservation of discrete series representations.

Extension of previous results to non-split and inner form groups.

Abstract

Let G be an orthogonal or symplectic p-adic group (not necessarily split) or an inner form of a general linear p-adic group. In a previous paper, it was shown that the Bernstein components of the category of smooth representations of G are equivalent to the category of right modules over some Hecke algebra with parameters, or more general over the semi-direct product of such an algebra with a finite group algebra. The aim of the present paper is to show that this equivalence preserves the tem- pered spectrum and the discrete series representations.