  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).
