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

TL;DR

This paper investigates the invariants of irreducible unipotent representations of symplectic groups over finite fields, establishing bounds on invariant subspace dimensions and classifying those with nonzero invariants.

Contribution

It provides a complete classification of unipotent representations with nonzero invariants under a subgroup, and offers an elementary proof of their rationality over Q.

Findings

Invariant subspace dimension is at most one for all irreducible representations.

Complete list of unipotent representations with nonzero invariants is given.

Unipotent cuspidal representations are defined over Q.

Abstract

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.