The real Chevalley involution

Publisher: Cambridge University Press

E-ISSN: 1570-5846|150|12|2127-2142

ISSN: 0010-437x

Source: Compositio Mathematica, Vol.150, Iss.12, 2014-12, pp. : 2127-2142

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Abstract

The Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$ , takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$ -conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$ . We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.