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. In the “ordinary” setting (i.e. when $P$ is minimal), such a representation $\pi$ has a relatively simple structure at $p$, using a theorem of Hida. I will describe a generalization of the latter in the more general $P$-ordinary setting using the theory of types. I will use this structure theorem to analyze and parametrize a $P$-ordinary Hida family $C_\pi$ associated to $\pi$.

Then, I will introduce a $p$-adic family of Eisenstein series that is “compatible” with $C_\pi$. Namely, the Fourier coefficients of the former can be interpolated $p$-adically to induce an “Eisenstein measure” and the family can be paired with $C_\pi$, using an algebraic version of the doubling method, to $p$-adically interpolated special values of $L$-functions.

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.