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On the Euler characteristic of the discrete spectrum. (English) Zbl 1080.11041

Let \(G\) be a simply connected, semi-simple algebraic group defined over \(\mathbb Q\). Let \(\mathbb A\) be the ring of adeles of \(\mathbb Q\). Let \(dg\) be a fixed Haar measure on \(G(\mathbb A)\). The group \(G(\mathbb A)\) acts unitarily, by right translation, on the Hilbert space \[ L^2=L^2(G(\mathbb Q)G(\mathbb A),dg). \] Let \[ L=L^2_{\text{disc}}\subset L^2 \] be the sum of all irreducible \(G(\mathbb A)\)-subspaces of \(L^2\). \(L\) is called the discrete spectrum of \(G(\mathbb A)\). If \(S\) is a finite set of places of \(\mathbb Q\) which contains the real place and all finite primes \(p\) where \(G\) is ramified, there exists an integral model \(\underline G\) for \(G\) over the ring \({\mathbb Z}_S\) of \(S\)-integers, with \({\underline G}\) having good reduction at all primes \(p\) outside of \(S\). Fix such a finite set \(S\). The product \[ G_S(\mathbb A)=\prod_{v\in S}G({\mathbb Q}_v)\times\prod_{p\not\in S}{\underline G}({\mathbb Z}_p) \] is locally compact, open in \(G(\mathbb A)\). Let \(V\) be an irreducible, finite-dimensional representation of the real Lie group \(G(\mathbb R)\), such that \(V\) has trivial central character. The tensor product \(L\otimes V\) is a continuous, complex representation of the group \(G_S(\mathbb A)\). The continuous cohomology groups \[ H^i(G_S(\mathbb A), L\otimes V) \] are finite dimensional complex vector spaces, zero for \(i\gg 0\). Let \[ \chi=\chi(G, S, V)=\sum_{i\geq 0}\dim H^i(G_S(\mathbb A), L\otimes V) \] be the Euler characteristic of \(L\otimes V\). The goal of the authors is to give an explicit formula for \(\chi\) under the hypotheses that \(S\) contains a finite prime and \(G(\mathbb R)\) contains a maximal compact torus. This Euler characteristic is first expressed as a trace of a certain test function on the space of automorphic forms, and then, by the stable trace formula, is converted into a sum of orbital integrals, which are calculated in terms of the values of certain Artin \(L\)-functions at negative integers.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E41 Continuous cohomology of Lie groups
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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