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.
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Bao, H., Kujawa, J., Li, Y. et al. Geometric Schur Duality of Classical Type. Transformation Groups 23, 329–389 (2018). https://doi.org/10.1007/s00031-017-9447-4
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DOI: https://doi.org/10.1007/s00031-017-9447-4