  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 $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup
$P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times
\mathrm{U}_1(E)$. 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.
