Simple characters and coefficient systems on the building

TL;DR

This paper constructs a canonical subset and coefficient system on the Bruhat-Tits building associated with certain representations of GL(N,F), providing new tools for explicit character computations and a conjecture on projective resolutions.

Contribution

It introduces a canonical subset and coefficient system on the building for representations in Bernstein blocks, and formulates a conjecture relating these to projective resolutions, with proofs for discrete series.

Findings

Constructed a G-equivariant coefficient system C[] on the building.

Proved a technical lemma for irreducible discrete series representations.

Derived an explicit Lefschetz-type formula for character values.

Abstract

Let F be a non-archimedean local field and G be the group GL(N,F). Let \pi be a smooth complex representation of G lying in the Bernstein block B(\pi) of some simple type in the sense of Bushnell and Kutzko. Refining the approach of the second author and U. Stuhler, we canonically attach to \pi a subset X_\pi of the Bruhat-Tits building X of G, as well as a G-equivariant coefficient system C[\pi ] on X_\pi. Roughly speaking the coefficient system is obtained by taking isotypic components of \pi according to some representations constructed from the Bushnell and Kutzko type of \pi . We conjecture that when \pi has central character, the augmented chain complex associate to C[\pi ] is a projective resolution of \pi in the category B(\pi). Moreover we reduce this conjecture to a technical lemma of representation theoretic nature. We prove this lemma when \pi is an irreducible discrete series of G. We then attach to any irreducible discrete series \pi of G an explicit pseudo-coefficient f_\pi and obtain a Lefschetz type formula for the value of the Harish-Chandra character of \pi at a regular elliptic element. In contrast to that obtained by U. Stuhler and the second author, this formula allows explicit character value computations.