On Witten multiple zeta-functions associated with semisimple Lie algebras IV

TL;DR

This paper extends the theory of multi-variable Witten zeta-functions to the G2 root system, defining Bernoulli polynomial analogues, analyzing their generating functions, and establishing functional relations and singularity structures.

Contribution

It introduces G2-type Bernoulli polynomial analogues, studies their generating functions, and derives explicit functional relations for G2 Witten zeta-functions, expanding the understanding of these functions.

Findings

Defined G2-type Bernoulli polynomials and their generating functions

Determined the meromorphic continuation and singularities of G2 zeta-functions

Established explicit functional relations including Witten's volume formulas

Abstract

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.