Stable Trace Formulas and Discrete Series Representations

TL;DR

This paper proposes a conjectural formula for the traces of Hecke operators on spaces of automorphic forms for reductive groups, linking stable distributions, orbital integrals, and endoscopic transfer, with evidence from specific cases.

Contribution

It introduces a conjectural stable trace formula for Hecke operators on automorphic forms, connecting it to endoscopic transfer and orbital integrals, extending Arthur's and Kottwitz's work.

Findings

The conjectural formula aligns with known unipotent contributions in specific cases.

Evidence is provided through comparison with Wakatsuki's computations for GSp_4.

The approach reduces the problem to elliptic p-adic orbital integrals and endoscopy theory.

Abstract

Let G be a reductive algebraic group over Q, and suppose that Gamma is an arithmetic subgroup of G(R) defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in L^2(Gamma \ G(R)), and in general to determine the traces of Hecke operators on these spaces. In this paper we give a conjectural formula for the traces of Hecke operators, in terms of stable distributions. It is based on a stable version of Arthur's formula for L^2-Lefschetz numbers, which is due to Kottwitz. We reduce this formula to the computation of elliptic p-adic orbital integrals and the theory of endoscopic transfer. As evidence for this conjecture, we demonstrate the agreement of the central terms of this formula with the unipotent contributions to the multiplicity coming from Selberg's trace formula computed by Wakatsuki, in the case G=GSp_4 and Gamma=GSp_4(Z).