Euler-PoincarÉ Formulae for Positive Depth Bernstein Projectors

Abstract

ABSTRACT Work of Bezrukavnikov–Kazhdan–Varshavsky uses an equivariant system of trivial idempotents of Moy–Prasad groups to obtain an Euler–Poincaré formula for the r-depth Bernstein projector. Barbasch–Ciubotaru–Moy use depth-zero cuspidal representations of parahoric subgroups to decompose the Euler–Poincaré presentation of the depth-zero projector. For positive depth r, we establish a decomposition of the Euler–Poincaré presentation of the r-depth Bernstein projector based on a notion of associate classes of cuspidal pairs for Moy–Prasad quotients. We apply these new Euler–Poincaré presentations to obtain decompositions of the resolutions of Schneider–Stuhler and Bestvina–Savin.