On residual automorphic representations and period integrals for symplectic groups

TL;DR

This paper constructs new residual automorphic representations for symplectic groups by lifting cuspidal representations with linear periods, revealing invariant linear forms and generalizing descent methods.

Contribution

It introduces a novel construction of residual automorphic representations for symplectic groups from cuspidal representations of GL_{2n} with linear periods.

Findings

Constructed new irreducible components in the residual spectrum of symplectic groups.

Proved the existence of non-zero invariant linear forms under certain subgroup actions.

Generalized previous descent constructions to broader symplectic group settings.

Abstract

We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of $\mathrm{GL}_{2n}$ with a linear period to an irreducible component of the residual spectrum of the rank $k$ symplectic group $\mathrm{Sp}_k$ for any $k\ge 2n$. We show that this residual representation admits a non-zero $\mathrm{Sp}_n\times \mathrm{Sp}_{k-n}$-invariant linear form. This generalizes a construction of Ginzburg, Rallis and Soudry, the case $k=2n$, that arises in the descent method.