Koszul duality for Kac–Moody groups and characters of tilting modules

By Pramod N. Achar, Shotaro Makisumi, Simon Riche, Geordie Williamson. Journal of The American Mathematical Society, 2017

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<p>We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Kazhdan–Lusztig polynomials, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l greater-than h"> <mml:semantics> <mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>></mml:mo> <mml:mi>h</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell > h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l greater-than-or-equal-to 2 h minus 2"> <mml:semantics> <mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> <mml:mi>h</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell \ge 2h-2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.</p>

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9ce0e6d2cb897a37d641f9943166aa1ad0205091