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arXiv · 0907.0972

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

Yasushi Komori·Kohji Matsumoto·Hirofumi Tsumura
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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.

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Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura. 2009-07-06. On Witten multiple zeta-functions associated with semisimple Lie algebras IV. https://doi.org/10.1017/s0017089510000613

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