Categorical actions and multiplicities in the Deligne category $\underline{Rep}(GL_t)$

Abstract

We study the categorical type A action on the Deligne category $\mathcal{D}_t=\underline{Rep}(GL_t)$ (here $t \in \mathbb{C}$) and its "abelian envelope" $\mathcal{V}_t$ constructed in arXiv:1511.07699. For $t \in \mathbb{Z}$, this action categorifies an action of the Lie algebra $\mathfrak{sl}_{\mathbb{Z}}$ on the tensor product of the Fock space $\mathfrak{F}$ with $\mathfrak{F}_t^{\vee}$, its restricted dual "shifted" by $t$, as was suggested by I. Losev. In fact, this action makes the category $\mathcal{V}_t$ the tensor product (in the sense of Losev and Webster, arXiv:1303.1336) of categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$ and $Pol_t^{\vee}$. The latter categorify $\mathfrak{F}$ and $\mathfrak{F}_t^{\vee}$ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described by Hong and Yacobi, arXiv:1101.2456 (see also Losev, arXiv:1209.1067). When $t \notin \mathbb Z$, the Deligne category $\mathcal{D}_t$ is abelian semisimple, and the type A action induces a categorical action of $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z}}$. This action categorifies the $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z} }$-module $\mathfrak{F} \boxtimes \mathfrak{F}^{\vee}$, making $\mathcal{D}_t$ the exterior tensor product of the categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$, $Pol^{\vee}$. Along the way we establish a new relation between the Kazhdan-Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in $\mathcal{V}_t$.