  Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine
Kac-Moody algebra. Consider the category of g^-modules at the critical level,
on which the action of the Iwahori subalgebra integrates to algebraic action of
the Iwahori subgroup I. We study the action on this category of the convolution
functors with the "central" sheaves on the affine flag scheme G((t))/I. We show
that each object of our category is an "eigen-module" with respect to these
functors. In order to prove this, we use the fusion product of modules over the
affine Kac-Moody algebra.
