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The Steinberg module and the cohomology of arithmetic groups. (English) Zbl 0824.20040

For computing homology and cohomology of arithmetic subgroups of \(\text{SL}_ n\) a useful ingredient has been a simplicial complex \(Y\) of dimension equal to \(\nu\), the virtual cohomological dimension of \(\text{SL}_ n(\mathbb{Z})\), on which \(\text{SL}_ n (\mathbb{Z})\) acts cocompactly. The existence of such \(Y\) for a general Chevalley group has not been verified yet. For \(\text{Sp}_ 4 (\mathbb{Z})\) it is an unpublished result of MacPherson and McConnell. In this paper, for a connected semisimple algebraic \(\mathbb{Q}\)-group, the author substitutes the complex \(Y\) for \(S_ G\), the Steinberg representation of \(G(\mathbb{Q})\), and proves Theorem 1: Assume \(G\) is split over \(\mathbb{Z}\) and has no factors of type \(A_ 1\), and let \(K\) be an algebraic closure of \(\mathbb{F}_ p\). Let \(V\) be an irreducible \(G(K)\)-module with a highest weight which is even, positive, and less than \(p\) on each of the simple coroots. Then there is a nonzero \(G(\mathbb{F}_ p)\)-morphism \(H^ \nu(\Gamma (p), \mathbb{Z}) \to V\). Theorem 2: Assume \(G = \text{SL}_ n\). Then \(S_ G\) is cyclic over the integral group algebra of \(G(\mathbb{Z})\). The author computes an explicit generator and points out that this theorem is essentially due to Lee and Szczarba and to Ash and Rudolph. Theorem 3: For any Chevalley group over \(\mathbb{Z}\) with irreducible root system, let \(H := H_ 1(G(\mathbb{Z}, \mathbb{Z}))\). (1) If \(G\) is not of type \(A_ 1\), \(B_ 2\), \(G_ 2\), then \(H=F/2F\), where \(F\) is the fundamental group of \(G\). (2) If \(G\) is of type \(G_ 2\), then \(H = \mathbb{Z}/2\mathbb{Z}\). (3) If \(G\) is of type \(B_ 2\), then \(H = F \otimes \mathbb{Z}/ 2\mathbb{Z}\). (4) If \(G = \text{SL}_ 2\), \(\text{PSL}_ 2\), then \(H = \mathbb{Z}/12\mathbb{Z}\), \((\mathbb{Z}/2\mathbb{Z})^ 3\). This theorem is due to Steinberg. Theorem 4: Assume that \(\Gamma\) is normal in \(G(\mathbb{Z})\) with finite index. Let \(P = LN\) be a parabolic subgroup of \(G\); then under suitable conditions on the highest weight of \(V\) the restriction map \(H^ \nu (\Gamma, V) \to H^ \nu(\Gamma \cap P, V)\) is surjective. The author points out many corollaries.

MSC:

20G10 Cohomology theory for linear algebraic groups
11F75 Cohomology of arithmetic groups
20G05 Representation theory for linear algebraic groups
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