  We study a problem concerning parabolic induction in certain p-adic unitary
groups. More precisely, for $E/F$ 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 $\mathrm{GL}_n(E)$. 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.
