Geometric Schur Duality of Classical Type

Abstract

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial Flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra, $$ \overset{.}{\mathbf{U}} $$ ℐ and $$ \overset{.}{\mathbf{U}} $$ ʅ . We provide a geometric realization of the Schur-type duality of Bao–Wang between such a coideal algebra and Iwahori–Hecke algebra of type B. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra $$ \overset{.}{\mathbf{U}} $$ ℐ are constructed. In an Appendix by three authors, a more subtle 2-step stabilization procedure leading to $$ \overset{.}{\mathbf{U}} $$ ʅ is developed, and then monomial and canonical bases of $$ \overset{.}{\mathbf{U}} $$ ʅ are constructed. It is shown that $$ \overset{.}{\mathbf{U}} $$ ʅ is a subquotient of $$ \overset{.}{\mathbf{U}} $$ ℐ with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.