On reducibility of induced representations of odd unitary groups: the depth zero case

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 = U( n, n + 1) contains a parabolic subgroup P with Levi component L isomorphic to GL(n)(E) n ( E ) x 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 t(P)(G) ir is reducible.