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

TL;DR

This paper establishes a connection between the Pontryagin product on the equivariant $K$-group of an affine Grassmannian and the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold, providing new insights into equivariant quantum $K$-theory.

Contribution

It constructs an explicit isomorphism linking the equivariant $K$-group of a semi-infinite flag manifold with a localized equivariant quantum $K$-group, offering a new framework for understanding ring structures.

Findings

Pontryagin product coincides with tensor structure

Explicit isomorphism between $K$-groups established

Framework for equivariant quantum $K$-theory and Peterson isomorphism

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.