Quantum $K$-theory of Lagrangian Grassmannian via parabolic Peterson isomorphism

TL;DR

This paper explores the structure of the quantum $K$-ring of the Lagrangian Grassmannian using the parabolic Peterson isomorphism, providing explicit descriptions of the kernel of a key $K$-theoretic map.

Contribution

It explicitly determines the kernel of the $K$-theoretic Peterson map for the Lagrangian Grassmannian, advancing Schubert calculus in quantum $K$-theory.

Findings

Explicit kernel of the Peterson map determined.

Enhanced understanding of Schubert calculus in quantum $K$-theory.

Connections established between affine Grassmannian $K$-homology and quantum $K$-ring.

Abstract

We study Schubert calculus in the torus-equivariant quantum $K$-ring of the Lagrangian Grassmannian $\mathrm{LG}(n)$. Our main tool is the $K$-theoretic Peterson map due to Kato. The map is from the (localized) equivariant $K$-homology ring $K_{*}^{T}(\mathrm{Gr}_{G})$ of the affine Grassmannian $\mathrm{Gr}_{G}$ of the symplectic group $G=\mathrm{Sp}_{2n}(\mathbb{C})$ to the (localized) torus-equivariant quantum $K$-ring $QK_{T}(\mathrm{LG}(n))$. We determine explicitly the kernel of this map.