Representative action of the automorphisms of complex analytic groups

Abstract

Let G be a complex analytic group, and denote by W(G) the group of all complex analytic group automorphisms of G. As is well-known, %‘(G) may be identified with a closed complex Lie subgroup of the group W(L) of all Lie algebra automorphisms of the Lie algebra L of G. Here, we shall be concerned with the representability of W(G) on finite-dimensional spaces of holomorphic representative functions on G, where Y’(G) acts naturally by composition. The example of a complex toroid of dimension greater than 1 shows that one should concentrate attention on the identity component W-(G), . Let A denote the Hopf algebra of all holomorphic representative functions on G. In the present context, it is appropriate to assume that A separates the points of G or, equivalently, that G has a faithful finite-dimensional holomorphic representation. This assumption will be in force from now on, and it is usually signalized by saying that G is faithfully representable. In the evident way, A is a right W(G),-module. Let A, denote the unique largest sub-Hopf algebra of A that is stable under the action of W(G)r and locally finite as a %‘“(G),-module. The representation-theoretical significance of B, may be illuminated by considering the semidirect product G . -t’“(G), and observing that a Jinitedimensional holomorphic G-module can be embedded as a G-submodule in a jinite-dimensional holomorphic G . W(G),-module if and only if the associated representative functions belong to A, . Consequently, G . TT(G), is faithfully representable ;f and only ;f A, separetes the points of G. Our main result is an intrinsic characterization of those groups G for which the above condition is satisfied. This depends heavily on the structure theory for faithfully representable analytic groups, and we begin with a recall of the relevant facts. Let G be a faithfully representable complex analytic group. The main structure theorem says that G is a semidirect product S . P, where P is a