HOMOGENEOUS COMPLEX MANIFOLDS AND REPRESENTATIONS OF SEMISIMPLE LIE GROUPS

Abstract

Let G be a connected semisimple Lie group, K a maximal compact subgroup of G. Assume that rank K = rank G.We keep fixed a Cartan subgroup H of K; H is then also a Cartan subgroup of G. Denote the Lie algebras of G, K, H by go, fo, to, and their complexifications by g, E, b.The adjoint action of b de- termines a rootspace decomposition g = t) ® ( ga, where A is the set of non- zero roots of the pair (g, t).A root a C A is called compact if ga C d, otherwise noncompact.The complexified tangent space of the manifold G/H at eH is naturally isomorphic to 2avEA ha.If A+ C A is a particular system of positive roots, there exists a unique G-invariant complex structure on G/H such that the space of (1,0)-tangent vectors at eH corresponds to Zany+ g ¶ The manifold G/H, endowed with this complex structure, will be denoted by D. Let s =

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