Some Extensions of the Siegel-weil Formula

In this article I will survey some relatively recent joint work with S.Rallis, in which we extend the classical formula of Siegel and Weil. In the classical case, this formula identifies a special value of a certain Eisenstein series as an integral of a theta function. Our extension identifies the residues of the (normalized) Eisenstein series on Sp(n) as ‘regularized’ integrals of theta functions. Moreover, we obtain an analogous result for the value of the Eisenstein series at its center of symmetry (when n is odd). Both of these identities have applications to special values and poles of Langlands L-functions. Most of our results were announced by Rallis in his lecture [36] at the ICM in Kyoto in August of 1990, and detailed proofs will appear shortly [23]. Thus the present article will be mostly expository. However, some of the results of section III.4 about ‘second term identities’ have not appeared elsewhere. In the last section I have explained how some of our results may be translated into a more classical language.