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

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.