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

TL;DR

This paper develops a framework for complex rank parabolic category O using Deligne categories, proving structural conjectures, establishing equivalences with classical categories, and deriving multiplicity formulas via stable Kazhdan--Lusztig polynomials.

Contribution

It introduces multi-Fock tensor product categorifications, proves their uniqueness, and connects complex rank categories with classical limits, resolving key conjectures in the field.

Findings

Proved conjectures on structural constancy of fibers over stratified parameter space.

Established equivalences between complex rank and classical parabolic categories O.

Derived multiplicities of simple objects using stable Kazhdan--Lusztig polynomials.

Abstract

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.