An Euler-Poincare formula for a depth zero Bernstein projector

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. We establish an Euler-Poincaré formula for natural sums of depth zero Bernstein projectors (which is often the projector of a single Bernstein component) in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups GF associated to a block of the reductive quotient GF /G+F