Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harsih-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations.
We use the coordinated radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spi nis a mixture of quatum spin Calogero-Moser system associated to the restriced root system of G and a one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provid explicit expressions for its first-order quantum Hamiltonians. We also explicitly describe the Schrödinger operator.