In their proof of the Deligne-Langlands conjecture, Kazhdan and Lusztig construct an explicit isomorphism between the equivariant K-group of the Steinberg variety of a complex, reductive, algebraic group G and the extended affine Hecke algebra of the Weyl group of G. Later, Ostrik used their construction to define a "Kazhdan-Lusztig" type basis for the equivariant K-group of the nilpotent cone in the Lie algebra of G.
In these talks, I'll explain known and conjectural connections between (1) Ostrik's construction, (2) the lowest two-sided cell ideal and some special Kazhdan-Lusztig basis elements in the extended, affine Hecke algebra, and (3) G-equivariant vector bundles supported on {0} in the nilpotent cone. In order to keep the talks as accessible as possible, I'll talk only about the group SL_2(C).
Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory II
Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory II