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Visible actions on flag varieties of type B and a generalisation of the Cartan decomposition. (English) Zbl 1347.22012

Summary: We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan 59, 669–691 (2007; Zbl 1124.22003)] 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.

MSC:

22E46 Semisimple Lie groups and their representations
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 1124.22003
Full Text: DOI

References:

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