Doubled Hecke Algebras and Related Quantum Schur Duality

This article introduces the double Hecke algebra, which 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 reductive algebraic groups. We will study the finite dimensional natural representation of the double Hecke algebra on tensor space and prove that the double Hecke algebra forms a duality with the Levi type quantum group.