Abstract
The restricted category \(\mathcal {O}\) at the critical level over an affine Kac–Moody algebra is a certain subcategory of the ordinary BGG-category \(\mathcal {O}\). We study a deformed version introduced by Arakawa and Fiebig and calculate the center of the deformed restricted category \(\mathcal {O}\). This complements a result of Fiebig which describes the center of the non-restricted category \(\mathcal {O}\) outside the critical hyperplanes over a symmetrizable Kac–Moody algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arakawa, T., Fiebig, P.: On the restricted Verma modules at the critical level. Trans. Am. Math. Soc. 364(9), 4683–4712 (2012)
Arakawa, T., Fiebig, P.: The linkage principle for restricted critical level representations of affine Kac–Moody algebras. Compos. Math. 148(6), 1787–1810 (2012)
Fiebig, P.: Centers and translation functors for the category \({\cal O}\) over Kac–Moody algebras. Math. Z. 243(4), 689–717 (2003)
Fiebig, P.: On the restricted projective objects in the affine category \({\cal O}\) at the critical level. In: Algebraic Groups and Quantum Groups, Contemp. Math., vol. 565, pp. 55–70. Am. Math. Soc., Providence, RI (2012)
Fiebig, P.: On the subgeneric restricted blocks of affine category \({\cal O}\) at the critical level In: Symmetries Integrable Systems and Representations, vol. 40, pp. 65–84. Springer Proceedings in Mathematics and Statistics (2013)
Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)
Jantzen, J.C.: Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, vol. 750. Springer, Berlin (1979)
Kübel, J.: Jantzen sum formula for restricted Verma modules over affine Kac–Moody algebras at the critical level, preprint (2012). arXiv: 1212.0150
Acknowledgments
I would like to thank my supervisor Peter Fiebig for many inspiring discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the DFG priority program 1388.
Rights and permissions
About this article
Cite this article
Kübel, J. Centers for the restricted category \(\mathcal {O}\) at the critical level over affine Kac–Moody algebras. Math. Z. 276, 1133–1149 (2014). https://doi.org/10.1007/s00209-013-1237-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-013-1237-7