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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Real orthogonal representations of algebraic groups

Author: Frank Grosshans
Journal: Trans. Amer. Math. Soc. 160 (1971), 343-352
MSC: Primary 20.80
MathSciNet review: 0281807
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Abstract: The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group, $ G$. If $ G$ is split, we construct an extension $ {G^ \ast }$ containing $ G$ and those outer automorphisms of $ G$ fixing the highest weight of the representation. The representation is then extended to $ {G^ \ast }$ and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.

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Keywords: Algebraic groups, representations, invariant symmetric bilinear forms, invariant quaternion hermitian forms
Article copyright: © Copyright 1971 American Mathematical Society