Schur–Weyl duality for Deligne categories, II: The limit case

Inna Entova Aizenbud

Vol. 285 (2016), No. 1, 185–224

DOI: 10.2140/pjm.2016.285.185

Abstract

This paper is a continuation of a previous paper by the author (Int. Math. Res. Not. 2015:18 (2015), 8959–9060), 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 nonzero vector $⊮$), we constructed in that paper a complex tensor power of $V$: an $Ind$-object of the Deligne category $\underset{¯}{\left[b\right]Re}\underset{¯}{\left[b\right]p}\left(S_{\nu }\right)$ which is a Harish-Chandra module for the pair $\left(\mathfrak{g}\mathfrak{l}\left(V\right),{\overline{\mathfrak{P}}}_{⊮}\right)$, where ${\overline{\mathfrak{P}}}_{⊮}\subset GL\left(V\right)$ is the mirabolic subgroup preserving the vector $⊮$.

This construction allowed us to obtain an exact contravariant functor ${\widehat{SW}}_{\nu ,V}$ from the category $\underset{¯}{\left[b\right]Re}{\underset{¯}{\left[b\right]p}}^{ab}\left(S_{\nu }\right)$ (the abelian envelope of the category $\underset{¯}{\left[b\right]Re}\underset{¯}{\left[b\right]p}\left(S_{\nu }\right)$) to a certain localization of the parabolic category $O$ associated with the pair $\left(\mathfrak{g}\mathfrak{l}\left(V\right),{\overline{\mathfrak{P}}}_{⊮}\right)$.

In this paper, we consider the case when $V={ℂ}^{\infty }$. We define the appropriate version of the parabolic category $O$ and its localization, and show that the latter is equivalent to a “restricted” inverse limit of categories ${Ô}_{\left[b\right]\nu ,{ℂ}^N}^{\mathfrak{p}}$ with $N$ tending to infinity. The Schur–Weyl functors $\left[t\right]{\widehat{SW}}_{\nu ,{ℂ}^N}$ then give an antiequivalence between this category and the category $\underset{¯}{\left[b\right]Re}{\underset{¯}{\left[b\right]p}}^{ab}\left(S_{\nu }\right)$.

This duality provides an unexpected tensor structure on the category ${Ô}_{\nu ,{ℂ}^{\infty }}^{{\mathfrak{p}}_{\infty }}$.

Keywords

Deligne categories, Schur–Weyl duality, limits of categories, parabolic category $O$

Mathematical Subject Classification 2010

Primary: 17B10, 18D10

Milestones

Received: 17 July 2015

Revised: 8 May 2016

Accepted: 9 May 2016

Published: 27 September 2016