# An explicit determination of the $K$-theoretic structure constants of the affine Grassmannian associated to $SL_2$

## TL;DR

This paper explicitly computes the structure constants of the affine Grassmannian associated with SL_2 in both equivariant and non-equivariant K-theory and cohomology, providing inductive formulas and closed forms.

## Contribution

It introduces explicit inductive formulas and closed forms for the structure constants in the K-theoretic and cohomological settings of the affine Grassmannian for SL_2.

## Key findings

- Explicit inductive formulas for K-theoretic structure constants.
- Closed-form expressions for non-equivariant K-theory constants.
- Inductive formulas and closed forms for equivariant cohomology constants.

## Abstract

Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.