A Combinatorial Formula for Test Functions with Pro-p Iwahori Level Structure

TL;DR

This paper generalizes a formula for test functions with pro-p Iwahori level structure, enabling explicit computation for a broader class of split groups using combinatorics on Coxeter groups.

Contribution

It extends the Haines-Rapoport formula from the Drinfeld case to a wider class of split groups, providing a new combinatorial method for explicit test function calculation.

Findings

Derived a new combinatorial formula for test functions

Provided explicit examples for low-rank groups

Enabled broader applications in the Langlands-Kottwitz method

Abstract

The Test Function Conjecture due to Haines and Kottwitz predicts that the geometric Bernstein center is a source of test functions required by the Langlands-Kottwitz method for expressing the local semisimple Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions. Haines and Rapoport found an explicit formula for such test functions in the Drinfeld case with pro-p Iwahori level structure. This article generalizes the Haines-Rapoport formula for the Drinfeld case to a broader class of split groups. The main theorem presents a new formula for test functions with pro-p Iwahori level structure, which can be computed through some combinatorics on Coxeter groups. Explicit descriptions of the test function in certain low-rank general linear and symplectic group examples are included.