Resolutions for representations of reductive p-adic groups via their buildings

TL;DR

This paper develops new methods using cosheaves and sheaves on affine buildings to construct resolutions of representations of reductive p-adic groups, leading to insights into their homological properties and trace formulas.

Contribution

It introduces a novel approach using idempotent endomorphisms to construct acyclic sheaves and cosheaves, extending previous work and deriving new trace formulas.

Findings

Constructed acyclic sheaves and cosheaves on buildings.

Computed homology and cohomology of these sheaves.

Derived trace formulas for admissible representations.

Abstract

Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions 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. We 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.