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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 $$g=k(g).\exp {\mathcal H}(g).n(g)$$. 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 $${\mathfrak a}\circ$$ of $${\mathfrak p}\cap {\mathfrak q}$$, extend it to a $$\sigma$$- stable maximal abelian subspace $${\mathfrak a}$$ of $${\mathfrak p}$$, and let $$P: {\mathfrak a}\to {\mathfrak a}\circ$$ be the projection given by the direct sum above. Then, for any $$X\in {\mathfrak a}\circ$$, the set $$\{P_{\circ} {\mathcal H}(\exp X.h)$$, $$h\in H\}$$ equals $$C(W\circ.X)+\Gamma$$, where $${\mathcal H}$$ is as above, $$W\circ$$ is the normalizer modulo centralizer of $${\mathfrak a}\circ$$ in $$K\cap H$$, and $$\Gamma$$ is a closed convex polyhedral cone associated to certain roos of ($${\mathfrak g},{\mathfrak a}\circ).$$
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 $$G\times G/diagonal$$. 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.
Reviewer: F.Rouvière

##### 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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