Geometric Schur Duality of Classical Type

Abstract

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,U  andU 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 algebraU  are constructed. In an Appendix by three authors, a more subtle 2-step stabilization procedure leading tȯ U ı is developed, and then monomial and canonical bases ofU ı are constructed. It is shown thatU ı is a subquotient ofU  with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.

Positivity VS Negativity of Canonical Bases

“…Our argument largely follows the line in McGerty's work [18] for n odd (the case for n even needs substantial new work), though we have avoided using the non-degeneracy of the geometric bilinear form of iU (sl n ), which was not available at the outset. Instead, the non-degeneracy of the bilinear form is replaced by arguments involving the stably canonical basis of iU (gl n ) from [2] and the non-degeneracy eventually follows from the almost orthonormality of the canonical basis which we establish.…”

“…Recently the constructions of [1] have been generalized to partial flag varieties of type B and C in [2] (also see [7] for type D). A family of iSchur algebras iS(n, d) was realized geometrically together with canonical (=IC) bases whose structure constants lie in N[v, v −1 ].…”

“…The appearance of the quantum coideal algebra was inspired by [3] where a new approach to Kazhdan-Lusztig theory of type B/C via a new theory of canonical bases arising from quantum coideal algebras was developed. Even though the constructions for n odd and even are quite different with the case of even n being more challenging [3], one can carry out the construction in the even n case by relating to the odd n case via a more subtle two-step stabilization [2].…”

“…As in [3,2], the different behaviors in the cases for n odd and even force us to carry out the studies of the two cases separately in this paper. The case of odd n, indicated by the superscript , is easier and done first, while the remaining case is indicated by the superscript ı.…”

“…In Section 4, we show the stably canonical basis constructed in [2] for the modified quantum coideal algebraU  (gl n ) for n odd does not have positive structure constants.…”