Invariant Functionals on the Speh representation

TL;DR

This paper investigates invariant functionals on Speh representations of GL(2n,R), providing explicit constructions for even n and showing non-existence for certain subgroups when n is odd, thereby advancing understanding of Klyachko models.

Contribution

It offers explicit formulas for invariant functionals on Speh representations for even n and proves their uniqueness, also establishing non-existence results for odd n, contributing to the theory of Klyachko models.

Findings

Explicit invariant functionals for even n Speh representations.

Uniqueness of the invariant functional up to a constant.

Non-existence of invariant functionals for odd n with respect to U(n).

Abstract

We study Sp(2n,R)-invariant functionals on the spaces of smooth vectors in Speh representations of GL(2n,R). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is invariant under the Siegel parabolic subgroup of Sp(2n,R). For odd n we show that the Speh representations do not admit an invariant functional with respect to the subgroup U(n) of Sp(2n,R) consisting of unitary matrices. Our construction, combined with the argument in [GOSS12], gives a purely local and explicit construction of Klyachko models for all unitary representations of GL(2n,R).