Fock Space Tensor Product Categorifications and Multiplicities in Complex Rank Parabolic Category O

We undertake the study of complex rank analogues of parabolic category O defined using Deligne categories. We regard these categories as a family over an affine space, introduce a stratification on this parameter space, and formulate conjectures on the structural constancy of fibers on each stratum. Using the theory of $\mathfrak{sl}_{\mathbb{Z}}$-categorification, we prove these conjectures for admissible strata. Namely, we axiomatize the notion of multi-Fock tensor product categorifications (MFTPCs), which are interval finite highest weight categories equipped with a compatible action of commuting copies of $\mathfrak{sl}_{\mathbb{Z}}$, categorifying an external tensor product of tensor products of highest and lowest weight Fock space representations. We prove a uniqueness theorem for admissible MFTPCs and show that complex rank parabolic categories O have the structure of MFTPCs. In turn, for suitable choices of parameters, we produce an equivalence of complex rank category O with a stable limit of classical parabolic categories O, resolving our conjecture in the admissible case. These equivalences yield multiplicities of simple objects in Verma modules in terms of stable parabolic Kazhdan--Lusztig polynomials, answering a question posed by Etingof. In particular, for the case of two Levi blocks of non-integral size, we completely describe the structure of the corresponding category O in terms of stable representation theory. As an application, we obtain multiplicities for parabolic analogs of hyperalgebra Verma modules introduced by Haboush in the large rank and large characteristic limit.