Abstract: In this survey talk I will describe first the double-coset construction of Iwahori-Hecke algebra of a p-adic reductive group, and then the action of this algebra on the equivariant K-theory of flag varieties, with the goal of explaining a conjectural connection between Kazhdan-Lusztig theory (for affine Hecke algebras) and Schubert calculus (in equivariant K-theory).