摘要：When I. Schur used representations of the symmetric group Sr to determine polynomial representations of the complex general linear group GLn(C), certain finite-dimensional algebras, known as Schur algebras, played a bridging role between the two. The well-known Schur duality summarizes the relation between the representations of GLn(C) and Sr. Over almost a hundred years, this duality has profoundly influenced representation theory and has evolved in various forms such as the Schur-Weyl duality, Schur-Weyl-Brauer duality, Schur-Weyl-Sergeev duality, and so on. In this talk, I will discuss a latest development, which I call the Schur-Weyl-Hecke duality, by Huanchen Bao and Weiqiang Wang. Based on joint work with Yadi Wu, I will focus on the investigation of the i-quantum groups U ȷ (n) and U ı (n) and their associated q-Schur algebras S ȷ (n, r) and S ı (n, r) of types B and C, respectively. This includes short (element) multiplication formulas, long (element) multiplication formulas, and triangular relations in S ȷ (n, r) and S ı (n, r). We will also give realisations of Beilinson–Lusztig–MacPherson type for both U ȷ (n) and U ı (n) and discuss their Lusztig forms. This allows us to link representations of U ȷ (n) and U ı (n) with those of finite orthogonal and symplectic groups.