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

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 $K_{\ast }^T({\mathrm{Gr}}_G)$ 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 $Q{K}_T(\mathrm{LG}(n))$. We determine explicitly the kernel of this map.

18 pages