  In math.RT/0201073 we constructed an equivalence between the derived category
of equivariant coherent sheaves on the cotangent bundle to the flag variety of
a simple algebraic group and a (quotient of) the category of constructible
sheaves on the affine flag variety of the Langlands dual group. Below we prove
certain properties of this equivalence; provide a similar ``Langlands dual''
description for the category of equivariant coherent sheaves on the nilpotent
cone; and deduce some conjectures by Lusztig and Ostrik.
