Jantzen sum formula for restricted Verma modules over affine Kac-Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already deduced by Arakawa and Fiebig.