The Modified Form Of Mirabolic Quantum Sl₂

As an important research object in algebraic geometry,flag variety is closely related to representation theory,combinatorial mathematics and number theory,and it has important applications in many branches of modern mathematics.It is well known that the geometric realization of Hecke algebra is given by the convolution algebra of double complete flag varieties.Similarly,the geometric realization of quantum Schur algebra is given by the convolution algebra of double partial flag varieties.Through this geometric realization,the canonical basis of the quantum Schur algebra and the geometric realization of the modified form of the quantum group can be obtained.Moreover,Lusztig’s positive conjecture on the modified form of the quantum group can be prove.Therefore,the geometric realization of quantum Schur algebra is of great significance in algebraic representation theory.The geometric realization of Mirabolic quantum group is considered in this dissertation.More precisely,the convolution algebra on double flags and a vector space is considered which is also called Mirabolic quantum Schur algebra.In this sense,the geometric realization of Mirabolic Schur algebra is provided.The modified form of Mirabolic quantum group is obtained by stabilization.