Multipliers on complex homogeneous spaces

Abstract

1. Let G be a real Lie group, represented as a transitive group of analytic automorphisms of a simply-connected complex analytic manifold D; if gEG and zED, the action of the transformation representing g on the point z will be denoted by gz. A multiplier for the group G, with respect to its representation as a transformation group on D, is a C?? complex-valued function ,(g; z) on G XD which is holomorphic in z and which satisfies A(g9g2; z) =A(g1; g2Z)A(g2; z) for every gi, g2GG; to exclude the obvious trivial case, we further assume that ,b(g; z) 0. Such functions are sometimes considered in examining the group G and its representations,2 but also arise as the continuous analogs of some structures of interest in the study of automorphic functions;3 our purpose here is to determine the possible multipliers which may arise in connection with the second of the above points of view. We shall always assume here that G is connected. Notice that the set of all multipliers for G forms an abelian group i(G; D) under multiplication. The universal covering group G* of G also acts as a transformation group on D, the action of the transformation representing g*EG* on the point zED being defined by g*z=gz whenever g* covers g; we shall consider firstly the group M(G*; D) of multipliers for G*. Let H* be the isotropy subgroup of G* at some point zo, which point is to be held fixed subsequently, and let K* be the subgroup of G* consisting of all elements represented by the trivial transformation which leaves D pointwise fixed. For our purposes, in particular for Siegel's modular groups, there is no loss of generality in assuming: (i) that there are local CIO mappings z->g* of D into G* such that g*zo =z; (ii) that K* is the center of G*; (iii) that K*G [G*, G*] =e*, that is, the intersection of the center and the commutator subgroup of G* is the trivial subgroup consisting of the identity e* alone; and (iv) that elements of finite order are everywhere dense in the group H*/K*. Whenever f(z) is holomorphic and nowhere vanishing on D,