# zbMATH — the first resource for mathematics

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 $$a \in A$$.

##### MSC:
 53C35 Differential geometry of symmetric spaces 22E46 Semisimple Lie groups and their representations
##### Keywords:
convexity theorem
Full Text: