  We show that for any irreducible representation of $Sp_{4n}(F_{q})$, the
subspace of all its $Sp_{2n}(F_{q^{2}})$-invariants is at most one-dimensional.
In terms of Lusztig symbols, we give a complete list of irreducible unipotent
representations of $Sp_{4n}(F_{q})$ which have a nonzero
$Sp_{2n}(F_{q^{2}})$-invariant and, in particular, we prove that every
irreducible unipotent cuspidal representation has a one-dimensional subspace of
$Sp_{2n}(F_{q^{2}})$-invariants. As an application, we give an elementary proof
of the fact that the unipotent cuspidal representation is defined over $Q$,
which was proved by Lusztig.
