The Geometric Schur-Weyl Duality For Two Parameter Case Of Type B And C

Schur-Weyl duality relates representation categories of two different algebra,it plays important roles in studying representation theory of quantum algebra,low dimensional topology,representation theory,canonical bases,category theory and so on.Geometric representation theory is using geometric to study representation theory.It is an important branch of representation theory.It has very important applications in the canonical basis theory of quantum groups,Kazhdan-Lusztig theory,Springer theory and geometric Sataka theory.Quantum symmetry pairs is a nontrivial generalization of quantum groups.Quantum symmetry pairs have become a very important research object,and the geometric realization of quantum symmetry pairs can help us study their canonical basis,and the Schur-Weyl duality theory can help us study the relation of its representation category and representation category of the Hecke algebra.Therefore,the geometric realization of the Schur-Weyl duality for quantum symmetry pairs is of great significance.This article mainly study the BLM realization of the Schur-Weyl duality theory for two-parameter quantum symmetry pairs of type B and C.That is,we use the convolution algebra structure of the equivariant C(t)-valued function on the partial flag variety and the complete partial flag variety of type B and C to realize the Schur-algebra and its Schur-Weyl duality,and then we can realize the two-parameter quantum symmetry pairs of typer B and C by using the inverse limit of the Schur-algebra.