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The Elliptic Gamma Function and SL(3, Z)⋉Z3

✍ Scribed by Giovanni Felder; Alexander Varchenko

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 236 KB
Volume: 156
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.2000.1951

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✦ Synopsis

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 eightvertex 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) _ Z 3 associated to a non-trivial class in H 3 (G, Z).

2000 Academic Press % 0 (z, {)= ` j=0 (1&e 2?i(( j+1) {&z) )(1&e 2?i( j{+z) ).

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