Factors of Automorphy and Other Formal Cohomology Groups for Lie Groups

Abstract

Many rather useful geometric and analytic properties of the automorphic functions associated with a group of analytic transformations of a complex manifold follow from a classification of the factors of automorphy. The limited information now available about the group structures has as yet prevented a complete classification even in many of the most interesting cases. However in all cases of principal interest at present the group is a discrete subgroup of a Lie group of analytic transformations of the manifold. This suggests, as an initial step, the determination of those factors of automorphy for the discrete subgroup which can be extended to factors of automorphy for the full Lie group. All of the factors customarily considered, except those arising in the study of abelian functions, are formed from the Jacobians of the transformations and can be so extended; and the classification theorems to be obtained in the present note show that in these cases, except for a second family of factors of automorphy in E. Cartan's third class of bounded symmetric homogeneous domains, there are no further factors of automorphy which can be so extended. This can be considered as a particular case of a more general problem: that of determining the cohomology groups of a group G with coefficients in a group of functions on a manifold on which G operates as an analytic transformation group. The classes of factors of automorphy correspond to elements of the first cohomology group with coefficients in the multiplicative group of nowhere-vanishing holomorphic functions. In this more general context it is natural to ask in addition for the cohomology groups with other groups of functions, or even groups of differential forms, as coefficients; indeed these other questions arise quite naturally along the way to the original goal. Thus Section 3 is devoted to the determination of the cohomology groups with the group of Co differential forms on the manifold as coefficients; in Section 4 the coefficients are d-closed or Cclosed differential forms, including in particular the additive group of holomorphic functions; and finally in Section 5 the coefficients are the 314