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 $\underset{‾}{Rep}(S_t)$ which is a Harish-Chandra module for the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}{$\mathbf{1}$})$,where$\bar{\mathfrak{P}}{$\mathbf{1}$} \subset GL(V)$isthemirabolicsubgrouppreservingthevector$$\mathbf{1}$$. This construction allowed us to obtain an exact contravariant functor ${\widehat{SW}}_{t,V}$ from the category ${\underset{‾}{Rep}}^{ab}(S_t)$ (the abelian envelope of the category $\underset{‾}{Rep}(S_t)$) to a certain localization of the parabolic category $\mathcal{O}$ associated with the pair $(\mathfrak{g}\mathfrak{l}(V),{\bar{\mathfrak{P}}}_{\mathbf{1}})$. In this paper, we consider the case when $V={\mathbb{C}}^{\mathrm{\infty }}$. 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 ${\widehat{\mathcal{O}}}_{t,{\mathbb{C}}^N}^{\mathfrak{p}}$ 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 ${\underset{‾}{Rep}}^{ab}(S_t)$. This duality provides an unexpected tensor structure on the category $\widehat{$\mathcal{O}$}^{\mathfrak{p}{\infty}}{t, $\mathbb{C}$^{\infty}}$.