Geometric Schur Duality of two parameter quantum group of type A

In this paper, we give an geometric description of the Schur-Weyl duality for two-parameter quantum algebras \(U_{v, t}(gl_n)\), where \(U_{v, t}(gl_n)\) is the deformation of \(U_v(I, \cdot)\), the classic Shur-Weyl duality \((U_{r, s}(gl_n), V^{\otimes d}, H_d(r, s))\) can be seen as a corollary of the Shur-Weyl duality \((U_{v, t}(gl_n), V^{\otimes d}, H_d(v, t))\) by using the galois descend approach. we also establish the Shur-Weyl duality between the algebras \(\widetilde{U_{v, t}(gl_N)^m}\), \(\widehat{U_{v, t}(gl_N)^m}\) and Heck algebra \(H_k(v, t)\).