A categorical approach to the stable center conjecture

TL;DR

This paper introduces a categorical framework to address the stable center conjecture in p-adic representation theory, demonstrating the stability of the Bernstein projector to the depth zero spectrum.

Contribution

It presents a novel categorical approach to the depth zero part of the stable center conjecture, providing new insights and proofs of stability for Bernstein projectors.

Findings

Bernstein projector to depth zero spectrum is stable

Categorical approach offers new perspective on the conjecture

Advances understanding of the stable center in p-adic groups

Abstract

The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.