Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups

Abstract

A connected complex Lie group G is called (linear) reductive, if it is the complexification K r of a maximal compact subgroup K of G. Such a group G is up to a finite (central) covering group theoretically a product (I12") k x S, where S is a semisimple complex Lie group. Reductive groups possess always a unique algebraic structure, so that the multiplication G x G ~ G is a morphism. In [BO] it was shown that a holomorphic function f on a reductive complex Lie group G which is invariant under the right action of a subgroup H, is invariant under the right action of the Zariski closure lq of H. The analogous problem for meromorphic functions and analytic hypersurfaces requires a substantially different approach. In 1983 A.T. Huckleberry and G.A. Margulis [HM] proved the following theorem: Let G be a complex semisimple Lie group and I C G an arbitrary subgroup. Then the following conditions are equivalent: i) 1 is Zariski dense in G, ii) the set ~ ( G ) ~ of l-invariant analytic hypersurfaces in G is empty. Under further assumptions on the group I Ahiezer [Ah] had proven a similar result in 1982. The main purpose of this paper is to generalize the Ahiezer-HuckleberryMargulis theorem. In order to do this it was necessary to consider invariant plurisubharmonic functions and positive semidefinite (1,1)-forms on semisimple groups. The two main theorems of this work are the following. They were conjectured by A.T. Huckleberry. For a subgroup H of a complex Lie group G let Jt~(G) n denote the set of hypersurfaces in G which are invariant under the right action of H on G.