Maarten Solleveld

Resolutions for representations of reductive p-adic groups via their buildings

Abstract Schneider–Stuhler and Vignéras have used cosheaves on the affine Bruhat–Tits building to construct natural projective resolutions of finite type for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building and to establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernsteins result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.

On the classification of irreducible representations of affine Hecke algebras with unequal parameters

Introduction 31 Preliminaries 101.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Affine Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Graded Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Analytic localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 The relation with reductive p-adic groups . . . . . . . . . . . . . . . 192 Classification of irreducible representations 242.1 Two reduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The Langlands classification . . . . . . . . . . . . . . . . . . . . . . . 282.3 An affine Springer correspondence . . . . . . . . . . . . . . . . . . . 353 Parabolically induced representations 373.1 Unitary representations and intertwining operators . . . . . . . . . . 383.2 The Schwartz algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Parametrization of representations with induction data . . . . . . . . 483.4 The geometry of the dual space . . . . . . . . . . . . . . . . . . . . . 524 Parameter deformations 584.1 Scaling Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Preserving unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Scaling intertwining operators . . . . . . . . . . . . . . . . . . . . . . 684.4 Scaling Schwartz algebras . . . . . . . . . . . . . . . . . . . . . . . . 725 Noncommutative geometry 775.1 Topological K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Periodic cyclic homology . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Weakly spectrum preserving morphisms . . . . . . . . . . . . . . . . 855.4 The Aubert–Baum–Plymen conjecture . . . . . . . . . . . . . . . . . 885.5 Example: type C

Periodic cyclic homology of reductive p-adic groups

Let G be a reductive p-adic group, H(G) its Hecke algebra and S(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This provides an alternative proof of the Baum{Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly, a general comparison theorem for the periodic cyclic homology of nite type algebras and certain Fr echet completions thereof. Sec

Characters and growth of admissible representations of reductive p -adic groups

We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

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

Topological K-theory of affine Hecke algebras

{\mathcal H}