Maass operators and Eisenstein series

Abstract

Let ~ = G/K be a Hermitian symmetric space of the non-compact type, and let F C G be an arithmetic subgroup. It is obvious that the enveloping algebra U(g 0 of the complexified Lie algebra of G gives rise to operators which transform automorphic forms on N with respect to F to F-automorphic forms of a different type. However, it is in general not easy to write these differential operators explicitly in the coordinates of @. In the case G = Sp(n, IR), when N is the Siegel upper half plane of genus n, Maass constructed certain explicit operators of this type. As a consequence of his construction, he was able to show that his operators transform certain simple Eisenstein series for F to other specific simple Eisenstein series for F. For a summary of his results, and for references, see [6]. In this paper we generalize these results of Maass to arbitrary Hermitian symmetric spaces of the non-compact type. Namely, we show that the enveloping algebra U(g 0 gives rise to differential operators A such that, for certain pairs of canonical automorphy factors J, J ' on G, we have AJ=J': [cf. Theorem (2.4) and Corollary (3.4)]. We do this abstractly, using the transformation properties of the automorphy factors. In particular, we do not give explicit formulas for our differential operators, which we call Maass operators ; it would be interesting to do so in general. We have also restricted our attention to cases in which one of the automorphy factors J, J' is scalar-valued; the same methods probably work more generally. Our interest in these operators arose from their use by Shimura in his study of rationality properties for special values of zeta functions; see [8, 9], and subsequent papers. Katz [4] has also introduced these operators, in another form. The author has extended Shimura's method to the case of Siegel modular forms, in [3], and intends to use the results of the present paper in future work of the same kind. For this reason, a brief and somewhat obtuse discussion of the "rationality" of Maass operators has been included in Sect. 4. In any specific (i.e., coordinatized) example, the results of Sect. 4 can be improved upon.