N-point spherical functions and asymptotic boundary KZB equations

Understand

Let $G$ be a split real connected Lie group with finite center.

• In the first part of the paper we define and study formal elementary spherical functions.
• They are formal power series analogues of elementary spherical functions on $G$ in which the role of the quasi-simple admissible $G$-representations is replaced by Verma modules.
• For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of $G$.