Invariant Functionals on the Speh representation

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).