Affine Super Schur Duality To the memory of Goro Shimura

Schur duality is an equivalence, for d ≤ n, between the category of finite-dimensional representations over C of the symmetric group S_d on d letters, and the category of finite-dimensional representations over C of GL(n, C) whose irreducible subquotients are subquotients of E^⊗d, E = Cⁿ. The latter are called polynomial representations homogeneous of degree d. It is based on decomposing E^⊗d as a C[S_d ] × GL(n, C)-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex GL(n, C)-modules from the corresponding result for S_d that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor F: M ↦→ M ⊗_C[Sd ] E^⊗d, E now being the super vector space C^m|n, from the category of finite-dimensional C[S_d ⋉ ℤ^d ]-modules, or representations of the affine Weyl, or symmetric, group Sd^a = S_d ⋉ ℤ^d, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra^{U(ŝl(m|n)) that are E⊗d}compatible, namely the subquotients of whose restriction to U(sl(m|n)) are constituents of E^⊗d. Both categories are not semisimple. When d < m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional E^⊗d-compatible representations of the affine superalgebra ŝl(m|n) are tensor products of evaluation representations at distinct points of C^×.