Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series

TL;DR

This paper studies the analytic properties of a Dirichlet series linked to Fourier-Jacobi coefficients of cusp forms on orthogonal groups, revealing integral representations, meromorphic continuation, and functional equations.

Contribution

It introduces a new integral representation for the Dirichlet series and establishes its meromorphic continuation and functional equation, connecting orthogonal Eisenstein series with Epstein zeta functions.

Findings

Derived integral representation for the Dirichlet series

Proved meromorphic continuation to the complex plane

Established a functional equation for the series in the case of the $E_8$ lattice

Abstract

We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature $(2,n+2)$. Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one $1$-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally $4 \mid n$, we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree $2$. We obtain, in this way, the meromorphic continuation of the Dirichlet series to $\mathbb{C}$ as a corollary. In the case of the $E_8$ lattice, we are able to further deduce a precise functional equation for the Dirichlet series.