Rankin-Selberg convolution for the Duke-Imamoglu-Ikeda lift

TL;DR

This paper derives a formula expressing the Rankin-Selberg convolution of Duke-Imamoglu-Ikeda lifts in terms of a Dirichlet series, linking automorphic forms and Eisenstein series, with applications to mass equidistribution.

Contribution

It provides a new explicit expression for the Rankin-Selberg convolution of certain Siegel cusp forms in terms of a Dirichlet series, advancing understanding of their L-functions.

Findings

Expressed the Rankin-Selberg convolution in terms of a Dirichlet series.

Connected the convolution to a triple product involving Eisenstein series.

Applied the formula to mass equidistribution under holomorphy assumptions.

Abstract

For two Hecke eigenforms $h_1$ and $h_2$ in the Kohnen plus space of half-integral weight, let $I_n(h_1)$ and $I_n(h_2)$ be the Duke-Imamoglu-Ikeda lift of $h_1$ and $h_2$, respectively, which are Siegel cusp forms with respect to $Sp_n(\ZZ)$. Moreover, let $E_{n/2+1/2}$ be the Cohen Eisenstein series of weight $n/2+1/2$. We then express the Rankin-Selberg convolution $R(s,I_n(h_1),I_n(h_2))$ of $I_n(h_1)$ and $I_n(h_2)$ in terms of a certain Dirichlet series $D(s,h_1,h_2,E_{n/2+1/2})$, which is similar to the triple convolution product of $h_1, h_2$ and $E_{n/2+1/2}$. We apply our formula to mass equidistribution for the Duke-Imamoglu-Ikeda lift assuming the holomorphy of $D(s,h_1,h_1,E_{n/2+1/2})$.