Universal subspaces for compact Lie groups

Details Universal subspaces for compact Lie groups Universal subspaces for compact Lie groups

2008-02-13

arxiv.org/abs/0802.1915v3

J. reine angew. Math. 647 (2010), 151-173

Mathematics

Representation Theory

20 pages

Scientific paper

10.1515/CRELLE.2010.076

For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangularization theorem.

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