On Hardy and Bergman spaces on complex Ol'shanskiı semigroups

Abstract

Let g denote a finite dimensional real Lie algebra. We call an element X ∈ g elliptic if ad X operates semisimply with purely imaginary spectrum. In the following we assume that g admits a closed convex Inn(g)-invariant cone W ⊆ g with non-empty elliptic interior W 0. If G is a connected Lie group with Lie algebra g, we write S : = G Exp(iW ) for the complex Ol’shanskii semigroup associated to G and W which may be understood as a quotient by π1(G) of the universal covering semigroup of expGC (g + iW ) with GC a simply connected complex Lie group with Lie algebra gC. We note that the interior S 0 : = G Exp(iW 0) of S carries a complex manifold structure and we write Hol(S 0) for the Frechet space of holomorphic functions on S 0. One of the main topics in the complex analysis on complex Ol’shanskii semigroups is the study of Hilbert spaces of holomorphic functions on S 0. Two of them are of particular interest, namely the Hardy space