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

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

1999-07-12

arxiv.org/abs/math/9907061v2

Adv. Math. 156(2000), 44-76

Mathematics

Quantum Algebra

27 pages, LaTeX References added, minor corrections

Scientific paper

10.1006/aima.2000.1951

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3,Z) x Z^3 associated to a non-trivial class in H^3(G,Z).

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