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

van den Ban, Erik P.

Pac. J. Math. 124, 21-55 (1986).

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

Keywords:

convexity; Iwasawa decomposition; real semisimple Lie group; semisimple symmetric space; Cartan involution; holomorphic continuation