Differential operators preserving relations of automorphy

Abstract

The invariant elements of a?, under this group action, are the automorphic or modular forms associated to the factor of automorphy p for (5, as in [4; 5]. To different factors of automorphy p, p there correspond of course different actions of (5 on a, and thus different families of automorphic forms. There then arises the question of the extent to which these group actions on families of automorphic forms are really distinct. More precisely, one can ask whether there are any linear mappings : a?a? which commute with these group operations, in the sense that 92iP(fIM) = (gf) |PM for all M E (5 and f E a. When the two factors of automorphy are equivalent, in the sense which arises naturally in the classification problem [4], there are trivial such isomorphisms Bq. However, more interestingly, there are cases in which the factors p, p are inequivalent, but in which there are such homomorphisms Bq. In these cases, the resulting maps can lead to quite interesting and nontrivial relations between various classes of automorphic forms. For one-dimensional complex manifolds ., this problem was discussed in [5]; the homomorphisms there led in a natural way to the relations between the Eichler cohomology groups and automorphic forms. In the present paper I shall discuss and classify the linear differential operators which provide such homomorphisms, when (5 is the symplectic group or a general subgroup thereof and . is the Siegel generalized upper half-plane.