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We study the characters of simple modules in the parabolic BGG category ofthe affine Lie algebra in Deligne's category. More specifically, we take thelimit 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 anindecomposable object of Deligne's category $\underline{\mathrm{Rep}}(GL_t)$,$\underline{\mathrm{Rep}}(O_t)$, or $\underline{\mathrm{Rep}}(Sp_t)$, underconditions that the highest weight of $X$ plus the level gives a fundamentalweight, $t$ is transcendental, and the base field $\Bbbk$ has characteristic$0$. We compare our result to the partial result of Etingof, and evaluate thecharacters to the categorical dimensions to get a categorical interpretation ofthe Nekrasov-Okounkov hook length formula.

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