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VISIBLE ACTIONS ON FLAG VARIETIES OF TYPE B AND A GENERALISATION OF THE CARTAN DECOMPOSITION
Published online by Cambridge University Press: 20 August 2012
Abstract
We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that
$G$ is a connected compact Lie group of type B,
$\sigma $ is a Chevalley–Weyl involution and
$L$,
$H$ are Levi subgroups. First, we prove that
$G=LG^{\sigma }H$ holds if and only if either (I) both
$H$ and
$L$ are maximal and of type A, or (II)
$(G,H)$ is symmetric and
$L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching
$H$ and
$L$. This classification gives a visible action of
$L$ on the generalised flag variety
$G/H$, as well as that of the
$H$-action on
$G/L$ and of the
$G$-action on
$(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’
$B$ with
$\dim B=\mathrm {rank}\, G$ in case I, and
$\dim B=2$ or
$3$ in case II, such that a generalised Cartan decomposition
$G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.
Keywords
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © 2012 Australian Mathematical Publishing Association Inc.
References
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