  George Lusztig conjectured that asymptotic affine Hecke algebra of a simply
connected group can be explicitly described in terms of convolution algebras.
Main Theorem of this note (which is a continuation of RT/0010089) is a weak
version of this Conjecture. This version is strong enough to reprove all
previously known results (due to Nanhua Xi) in this direction, for example the
case of type $\tilde A_n$, see QA/0010159.
