Weyl-Kac character formula for affine Lie algebras in Deligne's categories

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 the 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 Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2], and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)].