
Journal of Lie Theory 29 (2019), No. 4, 10451070 Copyright Heldermann Verlag 2019 Holomorphic Functions of Exponential Type on Connected Complex Lie Groups Oleg Yu. Aristov Obninsk 249038, Russia aristovoyu@inbox.ru Holomorphic functions of exponential type on a complex Lie group G (introduced by Akbarov) form a locally convex algebra, which is denoted by O_{exp}(G). Our aim is to describe the structure of O_{exp}(G) in the case when G is connected. The following topics are auxiliary for the claimed purpose but of independent interest: (1) a characterization of linear complex Lie group (a result similar to that of Luminet and Valette for real Lie groups); (2) properties of the exponential radical when G is linear; (3) an asymptotic decomposition of a word length function into a sum of three summands (again for linear groups). The main result presents O_{exp}(G) as a complete projective tensor of three factors, corresponding to the length function decomposition. As an application, it is shown that if G is linear then the ArensMichael envelope of O_{exp}(G) is the algebra of all holomorphic functions. Keywords: Complex Lie group, linear group, holomorphic function of exponential type, ArensMichael envelope, submultiplicative weight, length function, exponential radical. MSC: 22E10, 22E30, 32A38, 46F05. [ Fulltextpdf (200 KB)] for subscribers only. 