Sp2n(Fq2)Sp2n(Fq2)-invariants in irreducible unipotent representations of Sp4n(Fq)Sp4n(Fq)

We show that for any irreducible representation of Sp4n(Fq)Sp4n(Fq), the subspace of all its Sp2n(Fq2)Sp2n(Fq2)-invariants is at most one-dimensional. In terms of Lusztig symbols, we give a complete list of irreducible unipotent representations of Sp4n(Fq)Sp4n(Fq) which have a non-zero Sp2n(Fq2)Sp2n(Fq2)-invariant and, in particular, we prove that every irreducible unipotent cuspidal representation has a one-dimensional subspace of Sp2n(Fq2)Sp2n(Fq2)-invariants. As an application, we give an elementary proof of the fact that the unipotent cuspidal representation is defined over QQ, which was proved by Lusztig in [12].