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

A. Morimoto
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A. Morimoto: Nagoya University

A chapter in Proceedings of the Conference on Complex Analysis, 1965, pp 256-272 from Springer

Abstract: Abstract 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.

Keywords: Holomorphic Function; Banach Algebra; Discrete Subgroup; Maximal Compact Subgroup; Natural Homomorphism (search for similar items in EconPapers)
Date: 1965
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DOI: 10.1007/978-3-642-48016-4_22

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