  According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine
Weyl group is identified with the equivariant $K$-group of Steinberg's triple
variety. The $K$-group is equipped with a filtration indexed by closed
$G$-stable subvarieties of the nilpotent variety, where $G$ is the
corresponding reductive algebraic group over $\mathbb{C}$. In this paper we
will show in the case of type $A$ that the filtration is compatible with the
Kazhdan-Lusztig basis of the Hecke algebra.
