On Unitary Representations of GL2n Distinguished by the Symplectic Group

TL;DR

This paper constructs a family of automorphic representations of GL(2n) over p-adic fields that are distinguished by the symplectic group, extending previous local results through automorphic form theory.

Contribution

It introduces a new automorphic approach to classify all irreducible, unitary GL(2n) representations distinguished by the symplectic group, generalizing earlier local results.

Findings

Constructed a family of Sp(2n)-distinguished representations of GL(2n).

Connected automorphic methods with local representation theory.

Provided a classification framework for distinguished unitary representations.

Abstract

We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished by the symplectic group.