ON THE COMPLEX AND CONVEX GEOMETRY OF OLSHANSKII SEMIGROUPS

To a pair of a Lie group G and an open elliptic convex cone W in its L ie algebra one associates a complex semigroup S = GExp(iW) which permi ts an action of G x G by biholomorphic mappings. In the case where W i s a vector space S is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant d omain D subset of or equal to S is Stein is and only if it is of the f orm GExp(D-h), with Dh subset of or equal to iW convex, that each holo morphic function on D extends to the smallest biinvariant Stein domain containing D, and that biinvariant plurisubharmonic functions on D co rrespond to invariant convex functions on D-h.