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Classification of visible actions on flag varieties. (English) Zbl 1250.32021

Let \(G\) be a simple connected compact Lie group, \(L\), \(H\) be two compact Lie subgroups containing a same maximal connected abelian subgroup \(T\) of \(G\) such that their root systems are subsystems of the root systems of \(G\). Following T. Kobayashi [Transform. Groups 12, No. 4, 671–694 (2007; Zbl 1147.53041)], the author defines the action of \(L\) on the generalized flag manifold \(M=G/H\) (which is also a compact complex manifold and is projective and rational) to be strongly visible if (i) there is a real submanifold \(S\) such that \(O=LS\) is an open subset of \(M\); (ii) there is an anti-holomorphic involution \(\tau\) on \(O\) such that \(S\) is in the fixed point set of \(\tau\).
Now let \(\tau\) be an involution of \(G\) such that \(\tau (t)=t^{-1}\) for all \(t\) in \(T\). Define \(N\) to be the fixed point set of \(G\) with respect to \(\tau\), then \(NH\) defines a real submanifold \(S\) in \(G/H\). The author classifies all the triples \((G, H, L)\) such that \(G=LNH\).
The author calls this a generalized Cartan decomposition of the given matrix group \(G\) with respect to the triple \((H, L, \tau)\). We notice that \(L\) acts on \(G/H\) strongly visible with \(S=NH/H\). This also has something to do with multiplicity-free representations of \(G\) induced from \(G/H\).

MSC:

32M10 Homogeneous complex manifolds
22E46 Semisimple Lie groups and their representations
15A23 Factorization of matrices

Citations:

Zbl 1147.53041
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References:

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