Relativistic Toda lattice of type B and quantum $K$-theory of type C flag variety

Abstract: We introduce a classical integrable system associated with the torus-equivariant quantum $K$-theory of type C flag variety. We prove that its conserved quantities coincide with the generators of the defining ideal of the Borel presentation of the quantum $K$-ring obtained by Kouno and Naito. In particular, the Hamiltonian of the system is naturally regarded as a type B analogue of the relativistic Toda lattice introduced by Ruijsenaars. We also construct Bäcklund transformations describing the discrete time evolution of the system. This construction makes explicit the integrable structure underlying the quantum $K$-theory and provides a framework for further studies of the $K$-theoretic Peterson isomorphism.