Bernstein Components and Hecke Algebras for $p$-adic Groups

Abstract: Suppose that one has a supercuspidal representation of a Levi subgroup of some reductive $p$-adic group $G$. Bernstein associated to this a block $\mathrm{Rep}(G)^s$ in the category of smooth $G$-representations. We address the question: what does $\mathrm{Rep}(G)^s$ look like? Usually this is investigated with Bushnell--Kutzko types, but these are not always available. Instead, we approach it via a progenerator of $\mathrm{Rep}(G)^s.$ We will discuss the structure of the $G$ -endomorphism algebra of such a progenerator in detail. We will show that $\mathrm{Rep}(G)^s$ is "almost" equivalent with the module category of an affine Hecke algebra -- a statement that will be made precise in several ways. In the end, this leads to a classification of the irreducible representations in $\mathrm{Rep}(G)^s$ in terms of the complex torus and the finite group that are canonically associated to this Bernstein component.

number theoryrepresentation theory

Audience: researchers in the topic

Geometry, Number Theory and Representation Theory Seminar