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The real Chevalley involution

Part of: Algebraic number theory: global fields Lie groups

Published online by Cambridge University Press:  27 August 2014

Jeffrey Adams
Jeffrey Adams*
Affiliation:
University of Maryland, College Park, MD 20742-4015, USA email jda@math.umd.edu
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Abstract

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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.

Keywords

Chevalley involutionFrobenius Schur indicator

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 11R39: Langlands-Weil conjectures, nonabelian class field theory
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2127 - 2142
Copyright
© The Author 2014 

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