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\(\mathbb{R}\)-regular elements in Zariski-dense subgroups. (English) Zbl 0828.22010
Let \(G\) be a connected semi-simple algebraic group defined over \(\mathbb{R}\) and \(G(\mathbb{R})\) the group of real points of \(G\). Then \(G(\mathbb{R})\) is a real Lie group. Let \(\text{Ad}\) denote the adjoint representation of \(G(\mathbb{R})\) and suppose that \(G(\mathbb{R})\) is non-compact. An element \(g \in G(\mathbb{R})\) is called \(\mathbb{R}\)-regular if the number of eigenvalues (counted with multiplicities) of modulus 1 of \(\text{Ad} (g)\) is minimum possible. The author shows that any subsemigroup of \(G(\mathbb{R})\) which is Zariski dense in \(G\) contains an \(\mathbb{R}\)-regular element. The proof depends on a characterization of \(\mathbb{R}\)-regular elements in terms of exterior powers of the adjoint representation proved by the author and M. S. Raghunathan [Ann. Math., II. Ser. 96, 296-317 (1972; Zbl 0245.22013)].

MSC:
22E15 General properties and structure of real Lie groups
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