Zeta-functions of root systems and Poincar\'e polynomials of Weyl groups

TL;DR

This paper establishes new functional relations among zeta-functions of root systems using signed sums and Bernoulli functions, generalizing previous results and linking these relations to Poincaré polynomials of Weyl groups.

Contribution

It introduces a novel identity connecting zeta-functions and Bernoulli functions of root systems, extending prior work to more general cases and providing criteria for non-vanishing sums.

Findings

Derived explicit functional relations among zeta-functions of root systems.

Established a criterion for the non-vanishing of signed sums using Poincaré polynomials.

Proved a converse theorem relating generating functions for different subsets I.

Abstract

We consider a certain linear combination $S(\mathbf{s},\mathbf{y};I;\Delta)$ of zeta-functions of root systems, where $\Delta$ is a root system of rank $r$ and $I\subset\{1,2,\ldots,r\}$. Showing two different expressions of $S(\mathbf{s},\mathbf{y};I;\Delta)$, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a genralization of the authors' previous result proved in \cite{KMTLondon}, in the case when $I=\emptyset$. We present several explicit examples of such functional relations. A criterion of the non-vanishing of the signed sum, in terms of Poincar{\'e} polynomials of associated Weyl groups, is given. Moreover we prove a certain converse theorem, which implies that the generating function for the case $I=\emptyset$ essentially knows all information on generating functions for general $I$.