Regular Bernstein blocks

TL;DR

This paper studies regular Bernstein blocks in the representation theory of reductive groups over non-archimedean fields, establishing isomorphisms of Bernstein centers and proving the ABPS Conjecture in new cases.

Contribution

It demonstrates that the Bernstein center of a regular Bernstein block is isomorphic to that of a depth-zero block in a Levi subgroup, and proves the ABPS Conjecture in certain cases.

Findings

Bernstein centers of regular blocks are isomorphic under mild conditions.

In some cases, the blocks themselves are equivalent.

The ABPS Conjecture is proved in new cases.

Abstract

For a connected reductive group $G$ defined over a non-archimedean local field $F$, we consider the Bernstein blocks in the category of smooth representations of $G(F)$. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called $\textit{regular}$ Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of $F$ is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of $G(F)$ is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of $G^{0}(F)$, where $G^{0}$ is a certain twisted Levi subgroup of $G$. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.