Gamma factors of Selberg zeta functions and functional equation of Ruelle zeta functions (Q1364551)

\textit{M.-F. Vignéras} [Astérisque 61, 235-249 (1979; Zbl 0401.10036)] was the first to see that Barnes multiple gamma functions figure prominently in the functional equation for the Selberg zeta function associated to the modular group acting on the Poincaré upper half plane. This paper considers generalizations to co-compact groups acting on even dimensional locally symmetric spaces \(G/K\) of rank one when the \(K\)-type is non-trivial. As an application, a short proof of the functional equation of the Ruelle zeta function for compact \(2n\)-dimensional real hyperbolic space is given. This uses \textit{D. Fried}'s result that the Ruelle zeta function is a product of generalized Selberg zeta functions [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 491-517 (1986; Zbl 0609.58033) and Invent. Math. 84, 523-540 (1986; Zbl 0621.53035)]. Properties of multiple gamma and sine functions are needed [see \textit{N. Kurokawa}, 1991 Lectures at the Univ. Tokyo and Proc. Japan Acad., Ser. A 68, 256-260 (1992; Zbl 0797.11053)].