We give a presentation of an elliptic Weyl group W(R) (=the Weyl group for an elliptic root system* 0 R) in terms of the elliptic Dynkin diagram F(R, G) for the elliptic root system. The presentation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups : i) elliptic Coxeter relations attached to the diagram, and ii) a flniteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension W(R, G) of W(R) by an infinite cyclic group, called the hyperbolic extension of W(R). *) an elliptic root system=a 2-extended affine root system (see the introduction and the remark at its end). Contents

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The Gelfand-Tsetlin basis for irreducible c U q (gl(N4-1))-modules of finite dimensions is constructed by means of the lowering operators. § 0. IntroductionIn the previous paper [1], we constructed the Gelfand-Tsetlin basis for irreducible "Ug(^/(]V+l))-modules of finite dimensions, in terms of the lowering operators. In this paper, we will give detailed accounts of our construction of this basis.We give the result when q is a non-zero complex variable and not a root of unity, however from the view of technical arguments in its proof we also have to deal with the case of q to be a transcendental element over C.Let

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