Categorical actions and multiplicities in the Deligne category Rep_(GL_t)

We study the categorical type A action on the Deligne category D_t=Rep_(GL_t) (t∈C) and its “abelian envelope” V_t constructed in [13]. For t∈Z, this action categorifies an action of the Lie algebra sl_Z on the tensor product of the Fock space F with F_t ^∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category V_t the tensor product (in the sense of Losev and Webster, [20]) of categorical sl_Z-modules Pol and Pol_t ^∨. The latter categorify F and F_t ^∨ 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 D_t is abelian semisimple, and the type A action induces a categorical action of sl_Z×sl_Z. This action categorifies the sl_Z×sl_Z-module F⊠F^∨, making D_t the exterior tensor product of the categorical sl_Z-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 V_t.