On the depth r Bernstein projector

TL;DR

This paper provides an explicit formula for the Bernstein projector for representations of a given depth, revealing its support, stability, and relation to the Steinberg character, with local proofs for integral depths.

Contribution

It introduces a new explicit formula for the Bernstein projector of depth r and demonstrates its support, stability, and connection to the Steinberg representation.

Findings

Depth zero projector supported on topologically unipotent elements

Depth zero projector equals Steinberg character restriction

Depth r projector is stable

Abstract

In this paper we prove an explicit formula for the Bernstein projector to representations of depth at most r. As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth r Bernstein projector is stable. Moreover, for integral depths our proof is purely local.