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

Abstract

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 $1$), 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 $(\mathfrak{gl}(V),{\bar{\mathfrak{P}}}_1)$, where ${\bar{\mathfrak{P}}}_1\subset GL(V)$ is the mirabolic subgroup preserving the vector $1$. 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 $(\mathfrak{gl}(V),{\bar{\mathfrak{P}}}_1)$. 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 ${\hat{\mathcal{O}}}_{t,{\mathbb{C}}^N}^{\mathfrak{p}}$ 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 ${\underline{Rep}}^{ab}(S_t)$. This duality provides an unexpected tensor structure on the category ${\hat{\mathcal{O}}}_{t,{\mathbb{C}}^{\infty }}^{{\mathfrak{p}}_{\infty }}$.

Continuation of arXiv:1403.5509 [math.RT]. 24 pages. Comments welcome

References in corpus (2)

• Stability patterns in representation theory
• On Deligne's category $\uRep^{ab}(S_d)$