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A convexity theorem for semisimple symmetric spaces. (English) Zbl 0809.53058
Pac. J. Math. 162, No. 2, 305-349 (1994); correction ibid. 166, No. 2, 401 (1994).
Let \(\tau\) be an involution of a connected semi-simple Lie group \(G\) with Lie algebra \(\mathfrak g\), and let \(G^ \tau\) (resp. \({\mathfrak h} = {\mathfrak g}^ \tau\)) be the set of fixed points \(g \in G\) (resp. \(X \in {\mathfrak g}\)) of \(\tau\). Using the Cartan involution \(\theta\) of \(G\) satisfying \(\theta \circ \tau = \tau \circ \theta\), and denoting \(K = G^ \tau\), the author first constructs subalgebras \({\mathfrak m}\), \({\mathfrak a}\), \({\mathfrak n}\) (resp. subgroups \(M\), \(A\), \(N\)) of \(\mathfrak g\) (resp. \(G\)) having the following properties: (i) \({\mathfrak g} = {\mathfrak h} + {\mathfrak m} + {\mathfrak a} + {\mathfrak n}\), (ii) \(G^ \tau M_ 0 AN\) is an open subset of \(G\), (iii) \(G^ \tau \cap MAN = G^ \tau \cap M\), and (iv) \({\mathfrak h} \cap ({\mathfrak m} + {\mathfrak a} + {\mathfrak n}) = {\mathfrak h} \cap {\mathfrak m}\). Next, the author shows that for \(g = hman \in G\) with \(h \in G^ \tau\), \(m \in M\), \(a \in A\), \(n \in N\) the element \(\log a \in {\mathfrak a}\) is well-defined. Thus, a map \(L : G^ \tau MAN \to {\mathfrak a}\) can be defined by \(L(g) = \log a\). Now, an open subgroup \(H\) of \(G^ \tau\) is called essentially connected if \(H = H_ 0 Z_{K \cap H}({\mathfrak a})\) holds, where \(Z\) means the centralizer. An element \(a \in A\) is called admissible if \(aH \subset G^ \tau MAN\) holds. The author proves the following convexity theorem: Let \(H \subset G^ \tau\) be essentially connected and let \(a \in A\) be admissible. Then \(L(aH) = \text{conv} (W.\log a) + C(a)\) holds, where \(W\) is the Weyl group derived from the root system \(\Delta\) defined by the maximal Abelian subalgebra \(\mathfrak a\) in \(\{X \in g \mid \tau X = -X\), \(\theta X = -X\}\) and where \(C(a)\) is a certain convex cone in \(\mathfrak a\). The author also succeeds in describing the set \(\log (A_{adm})\), where \(A_{adm}\) denotes the set of all admissible elements \(a \in A\).

53C35 Differential geometry of symmetric spaces
22E46 Semisimple Lie groups and their representations
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