Affinization of category 𝒪 for quantum groups

Mukhin, E.; Young, C.

doi:10.1090/S0002-9947-2014-06039-X

Affinization of category $\mathcal {O}$ for quantum groups
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by E. Mukhin and C. A. S. Young PDF

Trans. Amer. Math. Soc. 366 (2014), 4815-4847 Request permission

Abstract:

Let $\mathfrak {g}$ be a simple Lie algebra. We consider the category $\hat {\mathcal {O}}$ of those modules over the affine quantum group $U_q(\widehat {\mathfrak {g}})$ whose $U_q(\mathfrak {g})$-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category $\hat {\mathcal {O}}$. In particular, we define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.

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