Anne-Marie Aubert

The Hecke algebra of a reductive p-adic group: a geometric conjecture

Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n) and SO(5). The conjecture, if true, leads to a parametrization of the smooth dual of G by the points in a complex affine locally algebraic variety.

Geometric structure in the principal series of the p-adic group G_2

In the representation theory of reductive -adic groups , the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in (2007), that there exists a simple geometric structure underlying this intricate theory. We will illustrate here the conjecture with some detailed computations in the principal series of . A feature of this article is the role played by cocharacters attached to two-sided cells in certain extended affine Weyl groups. The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space is a model of the smooth dual . In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by the cocharacters . The cocharacters themselves appear to be closely related to Langlands parameters.

Hecke algebras for inner forms of p-adic special linear groups

Let F be a non-archimedean local field and let G^# be the group of F-rational points of an inner form of SL_n. We study Hecke algebras for all Bernstein components of G^#, via restriction from an inner form G of GL_n (F). For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth G^#-representations. This algebra comes from an idempotent in the full Hecke algebra of G^#, and the idempotent is derived from a type for G. We show that the Hecke algebras for Bernstein components of G^# are similar to affine Hecke algebras of type A, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

Depth and the local Langlands correspondence

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

GENERALIZATIONS OF THE SPRINGER CORRESPONDENCE AND CUSPIDAL LANGLANDS PARAMETERS

Let

On L-packets and depth for SL2(K) and its inner form

{\mathcal H}

Geometric structure in smooth dual and local Langlands conjecture

H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for