Parabolic Category O in Complex Rank via Fock Space Tensor Product Categorifications

Abstract

We initiate the study of complex rank analogues of parabolic categories $\mathcal{O}$ for general linear Lie algebras defined via Deligne's interpolating categories. We regard these categories as a family varying over an affine parameter space and conjecture that their structure is controlled by a countable locally finite hyperplane arrangement, that is, they are constant along facets. We prove this conjecture on admissible facets using the theory of $\mathfrak{sl}_{\mathbb{Z}}$-categorification. The main technical ingredient is a uniqueness theorem for highest weight categories equipped with a categorical type A action categorifying an ordered tensor product of highest and lowest Fock space representations of $\mathfrak{sl}_{\mathbb{Z}}$. Under some combinatorial conditions on the parameters, this rigidity result allows us to compare complex rank category $\mathcal{O}$ with stable limits of classical parabolic categories $\mathcal{O}$. These equivalences yield character formulas for simple objects in terms of stable limits of parabolic Kazhdan--Lusztig polynomials, answering a problem posed by Etingof. For the case of two Levi blocks of non-integral size, the admissibility assumption is unnecessary, giving a complete description in terms of stable representation theory. As an application, we obtain multiplicities for parabolic analogs of hyperalgebra Verma modules in the large rank and large characteristic limit.