N-point spherical functions and asymptotic boundary KZB equations

Let G be a split real connected Lie group with finite center. In the first part of paper we define and study formal elementary spherical functions. They are power series analogues functions on in which role quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight express terms Harish-Chandra integrate them to regular G. We show that they produce eigenstates for spin versions quantum hyperbolic Calogero–Moser systems. second special subclasses global functions, call N-point Formal arise as limits correlation boundary Wess–Zumino–Witten conformal field theory cylinder when position variables tend infinity. construct compositions equivariant differential intertwiners associated principal representations, Eisenstein integrals. system also common eigenfunctions commuting family first-order operators, asymptotic Knizhnik–Zamolodchikov–Bernard operators. These operators explicitly given folded classical dynamical r-matrices k-matrices.