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Commuting tuples in reductive groups and their maximal compact subgroups. (English) Zbl 1306.55007

Let \(G\) be a reductive algebraic group and \(K \subset G\) a maximal compact subgroup. The authors consider the representation spaces \(\operatorname{Hom}({\mathbf Z}^k,K)\) and \(\operatorname{Hom}{(\mathbf Z}^k,G)\) with the topology induced from an embedding into \(K^k\) and \(G^k\), respectively. The goal of this paper is to prove that \(\operatorname{Hom}({\mathbf Z}^k,K)\) is a strong deformation retract of \(\operatorname{Hom}({\mathbf Z}^k,G)\).

MSC:

55P99 Homotopy theory
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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