  In this paper we provide, under some mild explicit assumptions, a geometric
description of the category of representations of the centralizer of a regular
unipotent element in a reductive algebraic group in terms of perverse sheaves
on the Langlands dual affine flag variety. This equivalence is suggested and
motivated by the "geometric Langlands" philosophy, and is used in later work to
construct equivalences of categories relating various geometric incarnations of
the affine Hecke algebra of the given reductive group.
