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We show that for any irreducible representation of $Sp_{4n}(F_{q})$, thesubspace 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 unipotentrepresentations of $Sp_{4n}(F_{q})$ which have a nonzero$Sp_{2n}(F_{q^{2}})$-invariant and, in particular, we prove that everyirreducible unipotent cuspidal representation has a one-dimensional subspace of$Sp_{2n}(F_{q^{2}})$-invariants. As an application, we give an elementary proofof the fact that the unipotent cuspidal representation is defined over $Q$,which was proved by Lusztig.

$Sp_{2n}(F_{q^{2}})$-Invariants In Irreducible Unipotent Representations of $Sp_{4n}(F_{q})$