Bernstein components 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 Rep$(G)^s$ in the category of smooth $G$-representations. We address the question: what does Rep$(G)^s$ look like?

Usually this is investigated with Bushnell--Kutzko types, but these are not always available. Instead, we approach it via the endomorphism algebra of a progenerator of Rep$(G)^s$. We will show that 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 Rep$(G)^s$ in terms of the complex torus and the finite groups that are canonically associated to this Bernstein component.

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.