The First Term Identities for Eisenstein Series

Abstract

In his celebrated paper [We], A. Weil proved the Siegel formula for classical groups, which is called the Siegel Weil formula. The Siegel Weil formula is an identity which relates the value of Siegel Eisenstein series and the average of theta series under certain convergence condition. In order to apply the Siegel Weil formula to the study of special values of automorphic L-functions and automorphic representations, S. Kudla and S. Rallis have extended the Siegel Weil formula beyond the region of convergence, by means of regularization. This extension is expected to work for all classical groups. The regularized Siegel Weil formula by Kudla and Rallis is also called the First Term Identity for Eisenstein series because it is turned out to be an identity relating the first term of the Siegel Eisenstein series and the first term of another Eisenstein series of non-Siegel type, by means of the Laurant expansion of Eisenstein series at certain point. We are interested in characterizing the first terms of Eisenstein series attached to all maximal parabolic subgroups and establishing all possible first term identities for Eisenstein series of the symplectic group. Because our method is not restrictive, it is natural to expect that the similar general first term identities should hold for other groups, at least for classical groups. We shall explain below some more details. Let F be a number field and A=AF the adele ring of F. Let Gn :=Sp(2n) be the symplectic group of rank n, and Pr =M n r N n r be the standard maximal parabolic subgroup of Gn with the Levi part M r isomorphic to GL(r)_Gn&r (m=m(ar , gn&r)). We denote by I r (s) the degenerate principal series representation of Gn(A) induced (normalized induction) from the one dimensional representation |det ar | s of Pr(A). Then, as usual, for any section fs # I r(s), one can form an Eisenstein series as follows: