An Euler-Poincar\'e formula for depth zero Bernstein projector

TL;DR

This paper derives an Euler-Poincaré formula for the depth zero Bernstein projector using Peter-Weyl idempotents, extending previous work on Moy-Prasad groups to a more specific setting.

Contribution

It introduces a new Euler-Poincaré formula for the depth zero Bernstein projector based on Peter-Weyl idempotents of parahoric subgroups, refining prior approaches.

Findings

Established an Euler-Poincaré formula for the depth zero Bernstein projector.

Connected the formula to Peter-Weyl idempotents of parahoric subgroups.

Extended the framework of Bezrukavnikov--Kazhdan--Varshavsky to depth zero components.

Abstract

Work of Bezrukavnikov--Kazhdan--Varshavsky uses an equivariant system of trivial idempotents of Moy--Prasad groups to obtain an Euler--Poincar\'e formula for the r--depth Bernstein projector. We establish an Euler--Poincar\'e formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter--Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.