Positivity in 𝑇-equivariant 𝐾-theory of flag varieties associated to Kac-Moody groups II

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<p>We prove sign-alternation of the structure constants in the basis of the structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties<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>associated to an arbitrary symmetrizable 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>, where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>is any parabolic subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian.</p>

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