Schur-Weyl duality for Deligne categories II: the limit case

This paper is a continuation of a previous paper of the author, which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space $V$ (a vector space $V$ with a chosen non-zero vector $\mathbf{1}$), we constructed a complex tensor power of $V$: an $Ind$-object of the Deligne category $\underline{Rep}(S_t)$ which is a Harish-Chandra module for the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$, where $\bar{\mathfrak{P}}_{\mathbf{1}} \subset GL(V)$ is the mirabolic subgroup preserving the vector $\mathbf{1}$. This construction allowed us to obtain an exact contravariant functor $\widehat{SW}_{t, V}$ from the category $\underline{Rep}^{ab}(S_t)$ (the abelian envelope of the category $\underline{Rep}(S_t)$) to a certain localization of the parabolic category $\mathcal{O}$ associated with the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$. In this paper, we consider the case when $V = \mathbb{C}^{\infty}$. We define the appropriate version of the parabolic category $\mathcal{O}$ and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories $\widehat{\mathcal{O}}^{\mathfrak{p}}_{t,\mathbb{C}^N}$ with $N$ tending to infinity. The Schur-Weyl functors $\widehat{SW}_{t, \mathbb{C}^N}$ then give an anti-equivalence between this category and the category $\underline{Rep}^{ab}(S_t)$. This duality provides an unexpected tensor structure on the category $\widehat{\mathcal{O}}^{\mathfrak{p}_{\infty}}_{t, \mathbb{C}^{\infty}}$.