Non-compact Complex Lie Groups without Non-constant Holomorphic Functions

In this paper we shall consider, on the one hand, a complex Lie group with sufficiently many holomorphic functions and, on the other hand, a complex Lie group whose holomorphic functions are necessarily constant. The former will be called a Stein group and the latter an (H. C.)-group. In the previous paper [3] we considered the complex analytic fibre bundles over Stein manifolds and, among other things, we established a necessary and sufficient condition for a complex Lie group to be a Stein manifold. Using this result, we shall first prove that every connected complex Lie group G contains the smallest closed complex normal subgroup G° such that the factor group G/G° is a Stein group. Next we prove that the subgroup G° is an (H. C.)-group, and so every connected complex Lie group can be obtained by an extension of a Stein group by an (H. C.)-group (Theorem 1 in §2). Using this theorem we can characterize a connected complex Lie group to be holomorphically convex by group theoretical conditions. From this characterization we can show that a connected complex Lie group containing no complex torus is a Stein group if and only if it is holomorphically convex.