$S{p}_{2n}(F_{q^2})$-Invariants In Irreducible Unipotent Representations of $S{p}_{4n}(F_q)$

Abstract

We show that for any irreducible representation of $S{p}_{4n}(F_q)$, the subspace of all its $S{p}_{2n}(F_{q^2})$-invariants is at most one-dimensional. In terms of Lusztig symbols, we give a complete list of irreducible unipotent representations of $S{p}_{4n}(F_q)$ which have a nonzero $S{p}_{2n}(F_{q^2})$-invariant and, in particular, we prove that every irreducible unipotent cuspidal representation has a one-dimensional subspace of $S{p}_{2n}(F_{q^2})$-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.

arXiv comment29 pages