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On the cohomology of Kottwitz’s arithmetic varieties. (English) Zbl 0974.11019

This paper is about the cohomology of certain co-compact arithmetic subgroups \(\Gamma\) of unitary groups \(\text{U}(p,q)\). Many authors have previously obtained vanishing results for arbitrary \(\Gamma\), using vanishing results for relative Lie algebra cohomology. One purpose of the present article is to prove stronger, in fact stably sharp, vanishing results for specially chosen \(\Gamma\), by exploiting their arithmetic properties. This was inspired by an example found by Rapoport and Zink of an arithmetic \(\Gamma\subseteq \text{U}(2,1)\) and a degree \(i\) in which there is nontrivial \(({\mathfrak g},K)\)-cohomology, for which all finite index subgroups of \(\Gamma\) have zero \(i\)th Betti number. Rapoport and Zink went on to study generalizations to higher dimensional unitary groups, leading to a conjecture by Rapoport that for certain \(\mathbb{Q}\)-forms \(\mathbf G\) of \(\text{U}(p,q)\), having a twist of the Steinberg representation at a finite place of an automorphic representation of \(\mathbf G(\mathbf A)\) implies that the infinite component has vanishing \(({\mathfrak g},K)\)-cohomology in all but one dimension. This conjecture is proved in the article under review.
To give the flavor of the other main results we mention a simpler case. Suppose \(p+q\) is an odd prime. With the appropriate \(\mathbb{Q}\)-structure on \(\text{U}(p,q)\), an arithmetic subgroup \(\Gamma\) then has cohomology \(H^\bullet(\Gamma,\mathbb{C})=H^\bullet_{\text{cst}}(\Gamma)\oplus H^{pq}_{\text{var}}(\Gamma)\), where \(H^\bullet_{\text{cst}}(\Gamma)\) comes from \(\text{U}(p,q)\)-invariant differential forms on its symmetric space (i.e., automorphic forms with trivial infinite component), and \(H^\bullet_{\text{var}}(\Gamma)\) comes from automorphic forms with nontrivial infinite component. The point is that this last is concentrated in the middle dimension \(pq\) and is therefore determined by the Euler characteristic of \(\Gamma\). It is nonzero for sufficiently small \(\Gamma\), by the Hirzebruch proportionality principle. There is an analogous but more complicated result for arbitrary \(p,q\), and nontrivial coefficient systems are also considered.
The proofs use recent results of Kottwitz relating the eigenvalues of Frobenius on the etale cohomology of Shimura varieties to eigenvalues of Hecke operators on \(L^2(\Gamma\backslash\text{U}(p,q))\), extensions of the author’s earlier work [Publ. Math., Inst. Hautes Étud. Sci. 73, 97-145 (1991; Zbl 0739.11020)], and the determination of the residual spectrum for \(\text{GL}_n\) by Moeglin-Waldspurger. A key step is the determination of possible Hecke eigenvalues for automorphic forms on certain noncompact arithmetic quotients of \(\text{GL}_n(\mathbb{C})^d\).

MSC:

11F75 Cohomology of arithmetic groups
11F55 Other groups and their modular and automorphic forms (several variables)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E40 Discrete subgroups of Lie groups

Citations:

Zbl 0739.11020
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References:

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