Functional Equations of Siegel-Weil Eisenstein Series

Abstract: Functional equation of Eisenstein series is a fundamental and has many applications. The abstract formulation involves intertwining operator, and it is of great interest to make it explicit. In this talk, we talk about an explicit functional equation of Siegel-Weil Eisenstein series associated to a quadratic/hermitian lattice whose localization is a vertex lattice. Instead of using the definition of the intertwining operator to do the calculation directly, we regard the Intertwining operator as a degenerate Whittaker function, and make use of the relation between Whittaker function and local density to inductively compute the formula. Even for the self-dual lattice case, our method is new compared with the classical Gindikin-Karpovich method. As a result, we obtain an elegant functional equation for the corresponding local density polynomials too. This talk is based on a joint work with Chao Li, Yousheng Shi, Tonghai Yang and Baiqing Zhu.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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