Let G be a reductive algebraic
group over ℚ, and suppose that Γ ⊂ G(ℝ) is an arithmetic subgroup defined by
congruence conditions. A basic problem in arithmetic is to determine the
multiplicities of discrete series representations in L2(Γ∖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 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(ℤ).
Keywords
discrete series, Hecke operators, orbital integrals,
Shimura varieties, endoscopy, fundamental lemma, stable
trace formula
