In 1986,Jimbo gave the q-Schur duality on the quantum group,which explained that the images of the representations of the quantum group and the Hecke algebra form double centralizers on V?r,generalized the classical Schur-Weyl duality.This article introduces the double Hecke algebra HHr,it is an infinite dimensional algebra generated by two Hecke algebras.This concept originates from the degenerate double Hecke algebra in the theory of“Schur-Weyl duality related to enhanced reducible algebraic groups”(see literature[1]).This article will study the finite dimensional“natural”representation of the double Hecke algebra(on tensor space).Finally,the article proves that the double Hecke algebra forms a duality with the Levi type quantum group. |