On Koszul duality for Kac-Moody groups

Abstract

For any Kac-Moody group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with Borel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we give a monoidal equivalence between the derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant mixed complexes on the flag variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper B"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and (a certain completion of) the derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript logical-or"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">G^\vee</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -monodromic mixed complexes on the enhanced flag variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript logical-or Baseline slash upper U Superscript logical-or"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>U</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">G^\vee /U^\vee</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript logical-or"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">G^\vee</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Langlands dual of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also prove variants of this equivalence, one of which is the equivalence between the derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant mixed complexes on the partial flag variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper P"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a certain “Whittaker model” category of mixed complexes on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript logical-or Baseline slash upper B Superscript logical-or"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>B</mml:mi> <mml:mo> ∨ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">G^\vee /B^\vee</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [ <bold>BGS96</bold> ].