Loop structure on equivariant $K$-theory of semi-infinite flag manifolds

Abstract

We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. {\bf 169} (2020)]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold with a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure of equivariant quantum $K$-theory and the Peterson isomorphism.

arXiv comment64pp, v9: final version, typographical corrections/changes from v8, accepted for publication in Ann. of Math

References in corpus (7)

• Jet Schemes of Locally Complete Intersection Canonical Singularities
• Singularities on normal varieties
• K-theory Schubert calculus of the affine Grassmannian
• Reconstruction and Convergence in Quantum $K$-Theory via Difference Equations
• Quantum D-modules and equivariant Floer theory for free loop spaces
• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
• On the quantum K-ring of the flag manifold

Cited by in corpus (6)

• Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
• Frobenius splitting of Schubert varieties of semi-infinite flag manifolds
• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
• Back stable K-theory Schubert calculus
• A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)
• Darboux coordinates on the BFM spaces