Automorphic representations and harmonic cochains for $GL_{n+1}$

TL;DR

This paper establishes an explicit isomorphism linking automorphic forms and harmonic cochains on the Bruhat-Tits building for $GL_{n+1}$ over a global function field, enhancing understanding of their structural relationship.

Contribution

It provides a novel explicit isomorphism connecting automorphic forms with harmonic cochains on the Bruhat-Tits building for $GL_{n+1}$ over positive characteristic fields.

Findings

Explicit isomorphism between automorphic forms and harmonic cochains.

Characterization of cusp forms via harmonic cochains with finite support.

Enhanced understanding of automorphic representations in positive characteristic.

Abstract

Let $K$ be a global field of positive characteristic. Let $\infty$ be a fixed place of $K$. This paper gives an explicit isomorphism between the space of automorphic forms (resp. cusp forms) for $GL_{n+1}(K)$ that transform like the special representations and certain spaces of harmonic cochains (resp. those with finite support) defined on the Bruhat-Tits building of $GL_{n+1}(K_\infty)$.