Weil Representation and Waldspurger Formula

Abstract

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.