Olafsson, Spherical representations and mixed symmetric spaces, Represent. Theory 1

Let (g; ) be a real symmetric Lie algebra and g = h + q the corresponding eigenspace decomposition for. We call an element X 2 q hyperbolic if the operator ad X is diagonalizable over R. The existence of "enough " hyperbolic elements in q is impo

Spherical Representations and Mixed Symmetric Spaces

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Spherical Representations and Mixed Symmetric Spaces Bernhard Krotz, Karl-Hermann Neeb and Gestur Olafsson

Introduction Let (g; ) be a real symmetric Lie algebra and g= h+ q the corresponding eigenspace decomposition for . We call an element X 2 q hyperbolic if the operator ad X is diagonalizable over R . The existence of\enough" hyperbolic elements in q is important in many contexts. For Cartan decompositions it is crucial for the restricted root decomposition of semisimple real Lie algebras, and hence for the whole structure theory of these algebras. If (g; ) is a non-compactly causal symmetric (NCC) Lie algebra in the sense of HO96], then q contains open convex cones which are invariant under the group Inng (h) of inner automorphisms of g generated by ead h and which consist entirely of hyperbolic elements. In the last years this class of reductive symmetric Lie algebras and the associated symmetric spaces have become a topic of very active research spreading in more and more areas. For a survey of the state of the art we refer to HO96] and the literature cited there. On the other hand there have been attempts to push this theory further to symmetric Lie algebras which are not necessarily semisimple or reductive. The simplest type (called the complex type) is where g= hC is a complexi cation and is complex conjugation. Among these symmetric Lie algebras those for which h contains an open invariant convex cone W consisting of elliptic elements play a crucial role (cf. Ne94a], Ne96a], Ne96b]). Then iW q= ih is an open cone consisting of hyperbolic elements so that, in the special case of reductive Lie algebras, we obtain on the one hand the non-compactly causal spaces of complex type and, if we allow W= h, also the Riemannian symmetric spaces coming from Cartan involutions of complex semisimple Lie algebras. For the associated symmetric spaces of complex type and the reductive spaces mentioned above one nowadays has a well developed picture of the harmonic analysis (holomorphic representations: Ne94b], Ne95]; spherical functions FHO94], HiNe96]; Hardy spaces HO 91], Kr97]) and the invariant complex analysis (invariant Stein domains and plurisubharmonic functions Ne96b]). The rst step in this program, i.e., the description of an appropriate class of not necessarily reductive symmetric Lie algebras which is general enough to encorporate all the cases mentioned above such as the mixed complex type case, the non-compactly causal spaces, and also the Riemannian symmetric spaces has been carried out in KN96], which we will use as a reference for the basic structure theory and convex geometry of mixed, i.e., non-reductive, symmetric Lie algebras. The next step that we carry out in this paper is the description of the structure of the associated global objects such as complex domains which are curved analogs of tube domains over convex cones. Furthermore we investigate the general represen

tation theory and explain how certain representations can be realized in spaces of holomorphic functions on the aforementioned domains. In Section I we collect the notation and the facts from KN96] we shall need throughout this paper. In Section II we then turn to product decompositions of the corresponding groups.

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