Abstract
In the computation of the intersection cohomology of Shimura varieties, or of the L 2 cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.
Highlights
The goal of this paper is to give more natural and geometric proofs of some combinatorial identities that appear when one calculates the commuting actions of the Hecke algebra and the absolute Galois group of Q on the cohomology of a Siegel modular variety
We provide some details about the manner our combinatorial identities appear in [8]
We present a geometric proof of this result which involves the geometry of the Coxeter complex of the symmetric group Sn
Summary
The goal of this paper is to give more natural and geometric proofs of some combinatorial identities that appear when one calculates the commuting actions of the Hecke algebra and the absolute Galois group of Q on the cohomology of a Siegel modular variety. The cohomology that is used is the intersection cohomology of the minimal compactification These identities, which appear in the calculation of weighted orbital integrals at the real place, are the technical heart of the paper [8], but were relegated to an appendix and proved by brute force. The second expression, which comes from Arthur and Kottwitz’s expression for the spectral side of the stable trace formula, involves averaged characters of discrete series representations of G(R); see [8, Section 3.1]. Both expressions can Manuscript received 24th June 2018, revised and accepted 18th February 2019. Permutahedron, intersection cohomology, ordered set partitions, shellability
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