The elliptic gamma function and SL(3, Z) x Z(3)

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. its trigonometric and rational deg enerations are the Jackson q-gamma function and the Euler gamma function, r espectively. The elliptic gamma function appears in Baxter's formula for th e free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this func tion are studied. In particular we show that elliptic gamma functions are g eneralizations of automorphic forms of G = SL(3, Z) x Z(3) associated to a non-trivial class in H-3(G, Z). (C) 2000 Academic Press.