Double Hecke Algebra And Related Quantum Schur Duality

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 HH_r,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.