Geometric Schur duality of classical type, II

Abstract

We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and two new quantum algebras arising from the geometry of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-step isotropic flag varieties of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This duality is a type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> counterpart of the Schur-Jimbo duality of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Schur-like duality of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper B slash bold upper C"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">B</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf B/\mathbf C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> discovered by Bao-Wang. The new algebras play a role in the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> duality similar to the modified quantum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {gl}(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the modified coideal subalgebras of quantum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {gl}(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper B slash bold upper C"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">B</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf B/\mathbf C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We construct canonical bases for these two algebras.