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

TL;DR

This paper establishes the linkage principle for restricted Verma modules in the critical level category O of affine Kac-Moody algebras, advancing understanding of their structure and block decomposition.

Contribution

It proves the first part of the Feigin-Frenkel conjecture and determines the block decomposition for the restricted category O at the critical level.

Findings

Proved the linkage principle for restricted Verma modules.

Established a version of the BGGH-reciprocity principle.

Determined the block decomposition of the restricted category O.

Abstract

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 BGGH-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.