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

2017/08/27 by Marc Horn, Horn, Marc · 1 citation
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Polynomial and algebraic computation #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.1708.08111

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.

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