Hecke action on tamely ramified Eisenstein series over $\mathbb{P}^1$

TL;DR

This paper investigates the structure of Eisenstein series in the context of automorphic functions over a rational function field, focusing on their interaction with the affine Hecke algebra and providing explicit results for specific groups.

Contribution

It introduces a conjecture describing the generators and relations of the Eisenstein series module and proves it for PGL(2) and SL(3).

Findings

Conjecture on generators and relations of the Eisenstein series module.

Proof of the conjecture for G=PGL(2) and SL(3).

Analysis of the Hecke action on tamely ramified Eisenstein series.

Abstract

We study the space of automorphic functions for the rational function field $\mathbb{F}_q(t)$ tamely ramified at three places. Eisenstein series are functions induced from the maximal torus. The space of Eisenstein series generates a trimodule for the affine Hecke algebra. We conjecture a generators and relations description of this module and prove the conjecture when $G=\mathrm{PGL}(2)$ and $\mathrm{SL}(3)$.