
Journal of Lie Theory 12 (2002), No. 2, 495502 Copyright Heldermann Verlag 2002 On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces Nazih Nahlus Dept. of Mathematics, American University, Beirut, Lebanon, New York, NY 100226297, U.S.A. Let L be a closed analytic subgroup of a faithfully representable complex analytic group G, let R(G) be the algebra of complex analytic representative functions on G, and let G_{0} be the universal algebraic subgroup (or algebraic kernel) of G. In this paper, we show many characterizations of the property that the homogenous space G/L is (representationally) separable, i.e, R(G)^{L} separates the points of G/L. This yield new characterizations for the observability of L in G and new characterizations for the existence of a quasiaffine structure on G/L. For example, G/L is separable if and only if the intersection of G_{0} and L is an observable algebraic subgroup of G_{0}. Moreover, L is observable in G if and only if G/L is separable and L_{0} is equal to the intersection of G_{0} and L. Similarly, we discuss a weaker separability of G/L and the existence of a representative algebraic structure on it. [ Fulltextpdf (147 KB)] 