Local Langlands correspondence for classical groups and affine Hecke algebras

Abstract

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 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.

A functoriality principle for blocks of 𝑝-adic linear groups

“…For i = 1, 2, the unipotent factor of Rep Q ℓ (G i φ ) consists of representations that are trivial on U(1), therefore (G). We hope that the recent results of Heiermann [9] will enable us to confirm the above "expectations" for groups of classical type. In this case, the disconnected centralizers are explained by even orthogonal factors (in the last example, the groups G φ are "pure" inner forms of O 2 ).…”

“…In this subsection we speculate on how it should work in an "ideal world", in which Langland's parametrization is known and satisfies some natural properties. In a forthcoming work, we will treat groups of classical type, meaning groups which are products of restriction of scalars of quasi-split classical groups, where all we need is available, and the desired equivalence of categories will be extracted from the work of Heiermann [9].…”

Spectral transfer morphisms for unipotent affine Hecke algebras

“…The existence of the desired cuspidal unipotent STMs is now an easy task; one considers the list of all cuspidal unipotent representations of all inner forms of G. This means that we need to make a list of all maximal F u -stable parahoric subgroups of the inner forms G u , consider their reductive quotients over k, and for those quotients which admit a cuspidal unipotent character, compute the normalization of the associated Hecke algebra H u,s,e := (L, τ s,e ) according to (25). Of course the main part of this formula is the degree of the unipotent cuspidal characters of the simple finite groups of Lie type, which is due to Lusztig [32][33][34] and conveniently tabulated in [9].…”

“…It is quite clear that the definition of the notion of STM could be generalized to Bernstein components [2,[23][24][25] in greater generality than only for the unipotent Bernstein components. It would be interesting to investigate the above mentioned induction principle in general.…”

Parameters of Hecke algebras for Bernstein components of p-adic groups

“…The category of right modules of End G (Π s ) is naturally equivalent with Rep(G) s . Heiermann [Hei2,Hei3] showed that for symplectic groups, special orthogonal groups, unitary groups and inner forms of GL n (F ), End G (Π s ) is always Morita equivalent with an (extended) affine Hecke algebra.…”