Local Langlands Correspondence for Classical Groups and Affine Hecke Algebras

TL;DR

This paper establishes a natural decomposition of smooth representations of p-adic classical groups into subcategories linked to unipotent representations, connecting the local Langlands correspondence with affine Hecke algebras.

Contribution

It proves a conjecture by Lusztig on the categorical decomposition of representations of classical groups and provides new parameterizations of affine Hecke algebra representations.

Findings

Decomposition of representation categories into tensor products of unipotent categories

Parameterizations of affine Hecke algebra representations

Insights into the stable Bernstein center

Abstract

Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). A statement of this kind had been conjecture by G. Lusztig. All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context. We get also parameterizations of representations of affine Hecke algebras, which seem not all to be in the literature yet. All this should also shed some light on what is known as the stable Bernstein center.