The universal Harish-Chandra $j$-function

Let $F$ be a nonarchimedean local field with residue field of cardinality $q$, let $G$ be the $F$-points of a connected reductive group defined over $F$, let $P$ and $Q$ be two parabolic subgroups with the same Levi factor $M$. We construct intertwining operators $J_{Q|P}$ and the Harish-Chandra $j$-function $j^G$ for finitely generated smooth $A[M]$-modules, where $A$ is any commutative Noetherian algebra over $\mathbb{Z}' :=\mathbb{Z}[\sqrt{q}^{-1}]$. The construction is functorial, compatible with extension of scalars, and generalizes the previously known constructions. We prove a generic Schur's lemma result for parabolic induction, which circumvents the need for generic irreducibility in defining $j^G$. Setting $A=\mathbb{Z}'$ and applying the construction to finitely generated projective generators produces a universal $j$-function that is a rational function with coefficients in the Bernstein center of $M$ over $\mathbb{Z}'$, and which gives the $j$-function of any object via specializing at points of the Bernstein scheme. We conclude by characterizing the local Langlands in families morphism (when it exists) for quasisplit classical groups in terms of an equality of $j$-functions.