Weyl-Kac character formula for affine Lie algebra in Deligne's category

TL;DR

This paper extends the Weyl-Kac character formula to affine Lie algebras within Deligne's categories, providing new character formulas for simple modules and linking them to the Nekrasov-Okounkov hook length formula.

Contribution

It introduces a limit-based approach to compute characters of irreducible modules in Deligne's categories, generalizing classical formulas to a categorical setting.

Findings

Derived explicit character formulas for modules in Deligne's categories

Connected categorical characters to Nekrasov-Okounkov hook length formula

Compared results with existing partial formulas by Etingof

Abstract

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of Weyl-Kac formula to compute the character of the irreducible quotient $L(X,k)$ of the parabolic Verma module $M(X,k)$ of level $k$, where $X$ is an indecomposable object of Deligne's category $\underline{\mathrm{Rep}}(GL_t)$, $\underline{\mathrm{Rep}}(O_t)$, or $\underline{\mathrm{Rep}}(Sp_t)$, under conditions that the highest weight of $X$ plus the level gives a fundamental weight, $t$ is transcendental, and the base field $\Bbbk$ has characteristic $0$. We compare our result to the partial result of Etingof, and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula.