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Nonlinear covers of real groups. (English) Zbl 1066.22012

Let \(\mathbb G\) be a semisimple simply connected algebraic group defined over \(\mathbb R\), \(G\) the group of its real points and \(\mathfrak g\) the Lie algebra of \(G\). The author studies the fundamental group \(\pi_1(G)\) and nontrivial covering groups of \(G\) (such a cover is not realizable as a linear Lie group). Let \(\theta\) denote the Cartan involution of \(\mathbb G\) corresponding to \(G\). The root system \(\Delta\) of \(\mathbb G\) with respect to a \(\theta\)-stable Cartan subgroup is considered. Using the natural action of \(\theta\) on \(\Delta\), the author gives uniform proofs of several basic facts concerning \(\pi_1(G)\) and nonlinear covers of \(G\). Some of these have been already proved by different authors using the classification. The first result claims that \(\pi_1(G)\neq 1\) if and only if there exists a long root \(\alpha\in\Delta\), such that \(\theta(\alpha) = \alpha\) and \(\theta(X_{\alpha}) = -X_{\alpha}\), where \(X_{\alpha}\in\mathfrak g(\mathbb C)\) is a root vector of \(\alpha\). The relation with a result of G. Prasad [Adv. Math. 181, 160–164 (2004; Zbl 1037.22040)] is explained. Another result (proved in the “if” direction by B. Kostant) claims that \(\pi_1(G)\neq 1\) if and only if the minimal nilpotent orbit in \(\mathfrak g(\mathbb C)\) is defined over \(\mathbb R\). Now suppose that \(\mathfrak g\) is simple. It is proved that \(\pi_1(G)\) is trivial or is isomorphic to \(\mathbb Z\) or to \(\mathbb Z/2\mathbb Z\). A list of all the groups with \(\pi_1(G) = 1\) is given, and a method to determine \(\pi_1(G)\) from the Kac diagram of \(G\) is described.

MSC:

22E46 Semisimple Lie groups and their representations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E15 General properties and structure of real Lie groups
57T20 Homotopy groups of topological groups and homogeneous spaces

Citations:

Zbl 1037.22040
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