Schur-Weyl Duality for Deligne Categories

This paper gives an analog to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e., a vector space V with a distinguished nonzero vector {{{1}}}), we give a definition of a complex tensor power of V. This is an {\rm Ind}-object of the Deligne category \underline {{\rm Rep}}(S-{t}) equipped with a natural action of {{\mathfrak {gl}}}(V). This construction allows us to describe a duality between the abelian envelope of the category \underline {{\rm Rep}}(S-{t}) and a localization of the category {{O}}{\mathfrak {p}}-{V} (the parabolic category {{O}} for {{\mathfrak {gl}}}(V) associated with the pair (V, { {{1}}})). In particular, we obtain an exact contravariant functor \widehat {{\rm SW}}-{t} from the category \underline {{\rm Rep}}{\rm ab}(S-{t}) (the abelian envelope of the category \underline {{\rm Rep}}(S-{t})) to a certain quotient of the category {{O}}{\mathfrak {p}}-{V}. This quotient, denoted by \hat {{{O}}}{\mathfrak {p}}-{t, V}, is obtained by taking the full subcategory of {{O}}{\mathfrak {p}}-{V} consisting of modules of degree t, and localizing by the subcategory of finite-dimensional modules. It turns out that the contravariant functor \widehat {{\rm SW}}-{t} makes \hat {{{O}}}{\mathfrak {p}}-{t, V} a Serre quotient of the category \underline {{\rm Rep}}{\rm ab}(S-{t}){\rm op}, and the kernel of \widehat {{\rm SW}}-{t} can be explicitly described.