Correction to "Factors of Automorphy and Other Formal Cohomology Groups for Lie Groups"

Abstract

It has been brought to my attention that a few words of clarification should be appended to the discussion of the example in Section 6 of my paper Factors of Automorphy *, these Annals, p. 314. The functions jp(T, Z) and >(T, Z) introduced there are factors of automorphy for the matrix group G, which is not the full group of analytic automorphisms in the sense of E. Cartan, since it does not act effectively. The subgroup of G consisting of the matrices of determinant 1, which is isomorphic to the quotient group of G by the subgroup of scalar matrices (representing the transformations leaving all points fixed), is the effective trans orination group involved, and it has an infinite cyclic fundamental group; indeed, all of the Cartan groups proper have infinite cyclic fundamental groups. Hence there can be but a single class of factors of automorphy, the integral powers of the complex Jacobians. The only possibility for more varied factors of automorphy is that there exist an extension of the Cartan group which also acts as a group of analytic transformations in M, but not effectively of course; this is the case in the example under discussion, where a group extension of the Cartan group by the circle group introduces a second non-contractible path and hence a second family of factors of automorphy. The same observations also arise purely formally. The equations defining the two basic factors of automorphy were chosen with the relation at the top of page 325 in mind; a more elegant definition could be given using the relation p( T, Z) >(T, Z) = det T, and thus showing directly that these two factors belong to the same family upon restriction to the subgroup of matrices T of determinant 1.