The CAP representations indexed by Hilbert cusp forms

TL;DR

This paper constructs new CAP representations for various groups by combining classical liftings, extending known theories to higher dimensions, and relating Fourier coefficients of Hilbert cusp forms to quadratic form representation numbers.

Contribution

It introduces novel CAP representations via combined liftings, generalizing Waldspurger and Piatetski-Shapiro theories to higher dimensions.

Findings

Constructed new CAP representations for metaplectic, symplectic, and orthogonal groups.

Established a relation between Fourier coefficients of Hilbert cusp forms and quadratic form representation numbers.

Extended classical liftings to higher-dimensional cases.

Abstract

We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP representations of metaplectic, symplectic and orthogonal groups. These constructions partially generalize the theories of Waldspurger on the Shimura correspondence and of Piatetski-Shapiro on the Saito-Kurokawa lifting to higher dimensions. Applications include a relation between Fourier coefficients of Hilbert cusp forms of weight k+1/2 and a weighted sum of the representation numbers of a quadratic form of rank 2k by a quadratic form of rank 4k.