Ikeda type lift on ${\rm SO}(3,n+1)$

TL;DR

This paper explicitly constructs Ikeda type lifts on the special orthogonal group SO(3,n+1) over Q using Eisenstein series, resulting in Hecke eigen cusp forms with specific weights derived from elliptic newforms.

Contribution

It introduces explicit construction of Ikeda type lifts on SO(3,n+1) over Q, expanding the understanding of automorphic forms on orthogonal groups.

Findings

Constructed Hecke eigen cusp forms of weight l on SO(3,n+1).

Lifts originate from elliptic newforms with specific weights.

Automorphic representations are cohomological and non-tempered.

Abstract

By using Ikeda's theory for a compatible family of Eisenstein series, we explicitly construct Ikeda type lifts on the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$ with $n\ge 3$ which splits everywhere at finite places. Our lifts are Hecke eigen cusp forms of weight $l$ ($l\ge n+2$, even) and come from elliptic newforms with respect to ${\rm SL}_2(\mathbb{Z})$ which are of weight $l-\frac{n-2}{2}$ when $n$ is even and $2l-n+1$ when $n$ is odd.The corresponding cuspidal automorphic representations are cohomological but non-tempered at any places including the infinity place.