In the first half of the talk I shall review the definition and basic properties of Lusztig's "asymptotic" version of the affine Hecke (studied extensively in his series of papers "Cells in affine Weyl groups" in 1980's) denoted by J. In the 2nd half I will talk about my recent paper with D.Kazhdan which gives a description of the algebra J in terms of representation theory of the corresponding p-adic group. Some open problems will be discussed at the end.