Resolutions for representations of reductive ehBadic groups via their buildings

Abstract

Schneider–Stuhler and Vignéras have used cosheaves on the affine Bruhat–Tits building to construct natural projective resolutions of finite type for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building and to establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.