REPRESENTATIVE FUNCTIONS ON COMPLEX ANALYTIC GROUPS

Let G be a complex analytic group. Then the algebra R(G) of representative functions on G is the complex algebra generated by the matrix coordinate functions of the analytic finite dimensional representations of G. Iff is a complex function on G and x E G, let xf be the function on G given by (xf)(y) =f(yx). Then an analytic function f on G is in R(G) if and only if {xf I x E G} spans a finite dimensional vector space [3, Prop. 2.1]. In addition to its C-algebra structure, R(G) has a comultiplication -y: R(G) -? R(G) 0 R(G), induced from the multiplication in G, with which it becomes a Hopf algebra [11, p. 1141] and hence, the algebra of polynomial functions of a pro-affine algebraic group G* [12, Thm. 2.1]. We shall assume throughout that R(G) separates the points of G, or equivalently, that G has a faithfulfinite-dimensional analytic representation. The Hopf algebra R(G) has been investigated by Hochschild and Mostow [312]. If Hom (G, C) = (1), then G is canonically isomorphic to G* so that the Hopf algebra R(G) determines G [5, Thm. 5.2]. If G is algebraic, then R(G) determines G up to isomorphism [11, Thm. 3.2]. But, in general, the Hopf algebra R(G) fails to determine the isomorphism class of G [11, p. 1150]. However, R(G) can still determine many invariants of G of which we only mention here the regular (split) hulls of G viewed throughout as algebraic groups (see Definition 5) [7, Section 4]. We are thus interested in how closely G is determined from the structure of R(G). The solution to this problem in the simply connected case was given by Magid in [16, 17]. One of his results is that if G is simply connected, then R(G) determines the unique reduced regular (split) hull of G and vice versa [16, Coro. 14; 17, Coro. 7]. In this paper, a modified version of Magid's result is given that applies to the general case. In particular, we present necessary and sufficient conditions on faithfully representable complex analytic groups to have isomorphic Hopf algebras of representative functions. The above mentioned result on the simply connected case does not apply to the general case for the following reasons: