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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