Geometric Approach to Mirabolic Schur-Weyl Duality of Type A

We commence by constructing the mirabolic quantum Schur algebra, utilizing the convolution algebra defined on the variety of triples of two $n$-step partial flags and a vector. Subsequently, we employ a stabilization procedure to derive the mirabolic quantum ${\mathfrak{g}\mathfrak{l}}_n$. Then we present the geometric approach of the mirabolic Schur-Weyl duality of type $A$.