Abstract: I will first discuss the notion of automorphic representations on a unitary group that are $P$-ordinary (at $p$), where $P$ is some parabolic subgroup. I will describe their local structure, as well as the geometry of a $P$-ordinary family $C_\pi$, using the theory of types. Then, I will introduce a $p$-adic family of Eisenstein series (an Eisenstein measure) that is “compatible” with $C_\pi$, using an algebraic version of the doubling method. I will conclude by explaining how this Eisenstein measure corresponds to a $p$-adic $L$-function for $C_\pi$ viewed as an element of a $P$-ordinary Hecke algebra. These results generalize the ones obtained by Eischen-Harris-Li-Skinner in the ordinary setting and are from the speaker’s thesis.