  We study the parabolically induced complex representations of the unitary
group in 5 variables, $ U(5), $ defined over a p-adic field. Let $F$ be a
p-adic field. Let $E : F$ be a field extension of degree two. Let $Gal(E : F )
= \{ 1, \sigma \}.$ We write $ \sigma(x) = \overline{x} \; \forall x \in E. $
Let $ E^* := E \setminus \{ 0 \} $ and let $ E^1 := \{x \in E \mid x
\overline{x} = 1 \}$. $U(5) $ has three proper standard Levi subgroups, the
minimal Levi subgroup $ M_0 \cong E^* \times E^* \times E^1 $ and the two
maximal Levi subgroups $ M_1 \cong GL(2, E) \times E^1 $ and $ M_2 \cong E^*
\times U(3)$. We consider representations induced from the minimal Levi
subgroup $ M_0, $ representations induced from non-cuspidal, not fully-induced
representations of the two maximal Levi subgroups $ M_1 $ and $ M_2, $ and
representations induced from cuspidal representations of $ M_1.$ We describe -
except several particular cases - the unitary dual in terms of
Langlands-quotients.
