Real orthogonal representations of algebraic groups

Grosshans, Frank

doi:10.1090/S0002-9947-1971-0281807-7

Real orthogonal representations of algebraic groups
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Trans. Amer. Math. Soc. 160 (1971), 343-352 Request permission

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.

References

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