Let $G=GU(V)=GU_{n}(q)$ - the general unitary group acting on a vector space $V$, over the finite field of $q^2$ elements.
Let $1 \neq s\in G$ be semisimple and $\lambda \in \mathbb{F}_{q^2}$ an eigenvalue of $s$.
I'm looking for a proof of the following claim:
CLAIM: $E_{\lambda}$ - the eigenspace of $\lambda$, is either totally isotropic or non-degenerate with respect to the unitary (hermitian) form.
Thanks.