Vol. 256, No. 2, 2012

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Stable trace formulas and discrete series multiplicities

Steven Spallone

Vol. 256 (2012), No. 2, 435–488
Abstract

Let G be a reductive algebraic group over , and suppose that Γ G() is an arithmetic subgroup defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in L2G()), 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 L2-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 of Wakatsuki, in the case G = GSp4 and Γ = GSp4().

Keywords
discrete series, Hecke operators, orbital integrals, Shimura varieties, endoscopy, fundamental lemma, stable trace formula
Mathematical Subject Classification 2010
Primary: 11F46, 11F72, 22E55, 32N10
Milestones
Received: 26 April 2011
Revised: 5 December 2011
Accepted: 5 December 2011
Published: 30 May 2012
Authors
Steven Spallone
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Rd
Colaba
Mumbai 400005
India