Schur-weyl duality for deligne categories II: The limit case

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 1), we constructed in that paper a complex tensor power of V: an Ind-object of the Deligne category Rep(S_ν) which is a Harish-Chandra module for the pair (gl(V), P₁), where P₁ ⊂ GL(V) is the mirabolic subgroup preserving the vector 1. This construction allowed us to obtain an exact contravariant functor SŴ_{ν, V} from the category Rep^ab(S_ν) (the abelian envelope of the category Rep(S_ν) to a certain localization of the parabolic category O associated with the pair (gl(V), P₁). In this paper, we consider the case when V=C^∞. 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^p _v,C ^N with N tending to infinity. The Schur-Weyl functors SW_v,C ^N then give an antiequivalence between this category and the category Rep^ab(S_ν). This duality provides an unexpected tensor structure on the category Ô^p∞ _v,C∞.