Categorical actions and multiplicities in the Deligne category [formula omitted

Abstract

We study the categorical type A action on the Deligne category Dt=Rep_(GLt) (t∈C) and its “abelian envelope” Vt constructed in [13].For t∈Z, this action categorifies an action of the Lie algebra slZ on the tensor product of the Fock space F with Ft∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category Vt the tensor product (in the sense of Losev and Webster, [20]) of categorical slZ-modules Pol and Polt∨. The latter categorify F and Ft∨ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18].When t∉Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of slZ×slZ. This action categorifies the slZ×slZ-module F⊠F∨, making Dt the exterior tensor product of the categorical slZ-modules Pol, Pol∨.Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt.