Zeta-functions of weight lattices of compact connected semisimple Lie groups

TL;DR

This paper introduces zeta-functions for weight lattices of compact connected semisimple Lie groups, extending known formulas for simply-connected groups and establishing new functional relations among these functions.

Contribution

It generalizes zeta-functions to non-simply-connected groups, derives explicit volume formulas, and proves functional relations among these zeta-functions.

Findings

Explicit form of Witten's volume formulas for these zeta-functions

Functional relations among the zeta-functions

Extension of zeta-functions to non-simply-connected groups

Abstract

We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper we consider the general connected (but not necessarily simply-connected) case, prove the explicit form of Witten's volume formulas for these zeta-functions, and further prove functional relations among them which include their volume formulas.