Generalized Shalika model on $\mathrm{SO}_{4n}(F),$ symplectic linear model on $\mathrm{Sp}_{4n}(F)$ and theta correspondence

TL;DR

This paper demonstrates a transfer of models between representations of special orthogonal and symplectic groups via theta correspondence, confirming a conjecture by D. Jiang.

Contribution

It establishes that representations with a generalized Shalika model in SO(4n) lift to representations with a symplectic linear model in Sp(4n) through theta correspondence.

Findings

Proves the transfer of models between SO(4n) and Sp(4n) representations.

Answers D. Jiang's question on model transfer via theta lift.

Connects models in different classical groups through theta correspondence.

Abstract

We show that if an irreducible admissible representation of $\mathrm{SO}_{4n}(F)$ has a generalized Shalika model, then its small theta lift to $\mathrm{Sp}_{4n}(F)$ has the symplectic linear model, thus answering a question posed by D. Jiang. Here $F$ is a non-archimedean field of characteristic zero.