The fundamental group of a real flag manifold
Let \(G\) be a real connected semisimple Lie group, \(\mathfrak g\) its Lie algebra, \(\mathfrak p\) a parabolic subalgebra of \(\mathfrak g\). The aim of the paper is to describe the fundamental group \(\pi_1(G/P)\), where \(P\subset G\) is the normalizer of \(\mathfrak p\). As is known, \(\mathfrak p\) is determined by a subset \(F\) of the set \(S\) of restricted simple roots of \(\mathfrak g\). Let \(S^*\subset S\) denote the subset of roots of multiplicity 1. Then \(\pi_1(G/P)\) has the presentation \[ \langle\langle t_{\alpha},\;\alpha\in S^* \parallel t_{\beta}t_{\alpha} = t_{\alpha} t_{\beta}^{\varepsilon(\alpha,\beta)},\;\alpha,\beta\in S^*; t_{\alpha} = e, \;\alpha\in S^*\cap F\rangle\rangle, \] where \(\varepsilon(\alpha,\beta) = (-1)^{(\beta,\alpha^{\vee})}\). The proof makes use of the structure of CW-complexes on \(G/P\) determined by the Bruhat decomposition. The group \(\pi_1(G/P)\) has been already calculated by \textit{A. N. Shchetinin} [Vopr. Teor. Grupp Gomologicheskoj Algebry 2, 175-186 (1979; Zbl 0436.57016)], where a more complicated presentation in terms of roots was obtained, without using any cell decomposition.
scientific article; zbMATH DE number 1251256
scientific article