Op\'erateurs d'entrelacement et alg\`ebres de Hecke avec param\`etres d'un groupe r\'eductif $p$-adique - le cas des groupes classiques

TL;DR

This paper computes endomorphism algebras of certain projective generators in the smooth representation category of classical $p$-adic groups, showing they are isomorphic to semi-direct products of Hecke algebras with finite group algebras, with broader applicability.

Contribution

It provides explicit descriptions of endomorphism algebras for classical $p$-adic groups, extending the understanding of their representation theory and algebraic structures.

Findings

Endomorphism algebras are isomorphic to semi-direct products of Hecke algebras with finite groups.

Results apply to symplectic, orthogonal, and inner forms of general linear $p$-adic groups.

Methodology has potential applications to general reductive $p$-adic groups.

Abstract

For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators \`a la Bernstein of the category of smooth representations of $G$ and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected $p$-adic group.