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

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