ON THE STRUCTURE OF THE AFFINE ASYMPTOTIC HECKE ALGEBRAS

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit description in terms Grothedieck group sheaves on square finite set equivariant under action centralizer nilpotent element reductive group. A weaker form this statement, allowing for possible central extensions stabilizers that action, has been proved by first named author with Ostrik. In present paper we describe an example showing nontrivial do arise, thus above statement is optimal. We also show Lusztig's homomorphism from induces isomorphism cocenters and discuss relation structure cocenter.