Compositio Mathematica

Research Article

The linkage principle for restricted critical level representations of affine Kac–Moody algebras

Tomoyuki Arakawaa1 and Peter Fiebiga2

a1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (email:

a2 Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürnberg, Cauerstr. 11, D-91058, Germany (email:


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

  • 17B67 (primary);
  • 81R10 (secondary)


  • linkage principle


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.