a1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (email: firstname.lastname@example.org)
a2 Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürnberg, Cauerstr. 11, D-91058, Germany (email: email@example.com)
We study the restricted category 𝒪 for an affine Kac–Moody algebra at the critical level. In particular, we prove the first part of the Feigin–Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the Bernstein–Gelfand–Gelfand-reciprocity principle and we determine the block decomposition of the restricted category 𝒪. For the proofs, we need a deformed version of the classical structures, so we mostly work in a relative setting.
(Received April 08 2011)
(Accepted February 01 2012)
(Online publication October 11 2012)
2010 Mathematics Subject Classification
T.A. is partially supported by the JSPS Grant-in-Aid for Scientific Research (B) No. 20340007. P.F. is partially supported by a grant of the Landesstiftung Baden–Württemberg and by the DFG-Schwerpunkt 1388.