A reducibility problem for even unitary groups: The depth zero case

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E\mathrm{/}F$ a quadratic extension of p-adic fields the associated unitary group $G=\mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to ${\mathrm{G}\mathrm{L}}_n(E)$. Let $\pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation ${\iota }_P^G\pi$ is reducible.