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Affinization of category $ \mathcal{O}$ for quantum groups


Authors: E. Mukhin and C. A. S. Young
Journal: Trans. Amer. Math. Soc. 366 (2014), 4815-4847
MSC (2010): Primary 17B37; Secondary 81R50
DOI: https://doi.org/10.1090/S0002-9947-2014-06039-X
Published electronically: May 5, 2014
MathSciNet review: 3217701
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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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Additional Information

E. Mukhin
Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, LD 270, Indianapolis, Indiana 46202
Email: mukhin@math.iupui.edu

C. A. S. Young
Affiliation: School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, United Kingdom
Email: charlesyoung@cantab.net

DOI: https://doi.org/10.1090/S0002-9947-2014-06039-X
Received by editor(s): April 24, 2012
Received by editor(s) in revised form: November 9, 2012, and November 23, 2012
Published electronically: May 5, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.