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

Abstract

We study the categorical type A action on the Deligne category ${\mathcal{D}}_t=\underline{Rep}(G{L}_t)$ (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 $Po{l}_t^{\vee }$. 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 $\mathfrak{F}⊠{\mathfrak{F}}^{\vee }$, making ${\mathcal{D}}_t$ the exterior tensor product of the categorical ${\mathfrak{sl}}_{\mathbb{Z}}$-modules $Pol$, $Po{l}^{\vee }$. Along the way we establish a new relation between the Kazhdan-Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in ${\mathcal{V}}_t$.

ver2: fixed Section 10 and some typos. ver3: fixed support acknowledgement

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Cited by in corpus (1)

• Representations of the oriented skein category