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A convexity theorem for semisimple symmetric spaces. (English) Zbl 0599.22014

In 1973, B. Kostant proved the following convexity theorem. Let \(G=KAN\) be an Iwasawa decomposition of a connected real semisimple Lie group with finite centre, and \({\mathcal H}: G\to {\mathfrak a}\) the corresponding Iwasawa projection onto the Lie algebra of \(A\), defined by \(g=k(g)\cdot \exp {\mathcal H}(g).n(g)\). Then, for any \(X\in {\mathfrak a}\), the set \(\{\mathcal H(\exp X.k), k\in K\}\) equals \(C(W.X)\), the convex hull of the Weyl group orbit of \(X\) in \(\mathfrak a.\)
This result is extended here to a semisimple symmetric space \(G/H\). Here \(G\) is as above, \(\sigma\) is an involution of \(G\), and \(H\) an “essentially connected” open subgroup of the fixed point subgroup of \(\sigma\). Taking for \(K\) a \(\sigma\)-stable maximal compact subgroup of \(G\) gives a Cartan involution \(\theta\) which commutes to \(\sigma\), and a decomposition \[ \mathfrak g = (\mathfrak k\cap \mathfrak h)+(\mathfrak k\cap \mathfrak q)+(\mathfrak p\cap \mathfrak h)+\mathfrak p\cap \mathfrak q) \] into \(\pm 1\) joint eigenspaces of \(\theta\) and \(\sigma\). Fix a maximal abelian subspace \({\mathfrak a}_0\) of \({\mathfrak p}\cap {\mathfrak q}\), extend it to a \(\sigma\)-stable maximal abelian subspace \({\mathfrak a}\) of \({\mathfrak p}\), and let \(P: \mathfrak a\to \mathfrak a_0\) be the projection given by the direct sum above. Then, for any \(X\in\mathfrak a_0\), the set \(\{P_0 {\mathcal H}(\exp X.h)\), \(h\in H\}\) equals \(C(W_0.X)+\Gamma\), where \({\mathcal H}\) is as above, \(W_0\) is the normalizer modulo centralizer of \({\mathfrak a}_0\) in \(K\cap H\), and \(\Gamma\) is a closed convex polyhedral cone associated to certain roots of \((\mathfrak g,\mathfrak a_0).\)
Kostant’s result is the case \(K=H\), \(\theta =\sigma\); the present author considers also the case of a Lie group viewed as a symmetric space \(G\times G/\text{diagonal}\). His proof of the general case goes by induction via centralizers in \(G\); it includes a study of the holomorphic continuation of a certain decomposition of \(G\).

MSC:

22E46 Semisimple Lie groups and their representations
53C35 Differential geometry of symmetric spaces
22E60 Lie algebras of Lie groups
17B20 Simple, semisimple, reductive (super)algebras
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