Local Langlands correspondence for classical groups and affine Hecke algebras

Abstract

Using the results of Colette Moeglin on the representations of p-adic classical groups (based on methods of James Arthur) and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a quasi-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. A statement of this kind had been conjectured 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 sheds some light on what is known as the stable Bernstein center.