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

DOI10.1016/J.JALGEBRA.2020.06.027arXiv1906.02902WikidataQ115350451 ScholiaQ115350451MaRDI QIDQ6320129

Aleksei Pakharev

Publication date: 7 June 2019

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 $underlinemathrmRep(G{L}_t)$, $underlinemathrmRep(O_t)$, or $underlinemathrmRep(S{p}_t)$, 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.

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Symmetric monoidal categories (19D23) Monoidal categories, symmetric monoidal categories (18M05)

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