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, ${\dot{\mathbf{U}}}^{ȷ}$ and ${\dot{\mathbf{U}}}^{ı}$. 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 ${\dot{\mathbf{U}}}^{ȷ}$ are constructed. In an Appendix by three authors, a more subtle $2$-step stabilization procedure leading to ${\dot{\mathbf{U}}}^{ı}$ is developed, and then monomial and canonical bases of ${\dot{\mathbf{U}}}^{ı}$ are constructed. It is shown that ${\dot{\mathbf{U}}}^{ı}$ is a subquotient of ${\dot{\mathbf{U}}}^{ȷ}$ with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.

Final version. To appear in Transformation Groups

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• Positivity vs negativity of canonical bases
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• Quiver varieties and symmetric pairs
• Canonical bases arising from $ı$quantum covering groups
• Mirabolic quantum ${\mathfrak{sl}}_2$