Petersson Inner Products and Whittaker--Fourier Periods on Even Special Orthogonal and Symplectic Groups

TL;DR

This paper establishes a relation between Whittaker--Fourier coefficients, Petersson inner products, and special values of L-functions for even special orthogonal and symplectic groups, extending known formulas and conjectures.

Contribution

It formulates a new relation connecting Fourier coefficients, inner products, and L-values for these groups, building on prior descent and unfolding techniques.

Findings

Derived formulas relating Fourier coefficients and L-values.

Extended results to odd special orthogonal groups.

Conditional results based on unfolding Whittaker functions.

Abstract

In this article, we would like to formulate a relation between the square norm of Whittaker--Fourier coefficients on even special orthogonal and symplectic groups and Petersson inner products along with the critical value of $L$-functions up to constants. We follow the path of Lapid and Mao to reduce it to the conjectural local identity. Our strategy is based on the work of Ginzburg--Rallis--Soudry on automorphic descent. We present the analogue result for odd special orthogonal groups, which is conditional on unfolding Whittaker functions of descents.