Steinberg representations and harmonic cochains for split adjoint quasi-simple groups

TL;DR

This paper establishes an isomorphism between the dual of the Steinberg representation of a split adjoint quasi-simple group over a non-archimedean local field and a space of harmonic cochains on its Bruhat-Tits building, with coefficients in any ring.

Contribution

It proves a new duality result linking Steinberg representations to harmonic cochains for split adjoint quasi-simple groups over local fields.

Findings

Dual of Steinberg representation is isomorphic to harmonic cochains

Result holds with coefficients in any commutative ring

Provides new insights into representation theory of p-adic groups

Abstract

Let $G$ be an adjoint quasi-simple group defined and split over a non-archimedean local field $K$. We prove that the dual of the Steinberg representation of $G$ is isomorphic to a certain space of harmonic cochains on the Bruhat-Tits building of $G$. The Steinberg representation is considered with coefficients in any commutative ring.