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The real Chevalley involution
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 12 / December 2014
- Published online by Cambridge University Press:
- 27 August 2014, pp. 2127-2142
- Print publication:
- December 2014
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- Article
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- You have access
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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.
