Gelfand-Tsetlin Bases for Elliptic Quantum Groups

We study the level-0 representations of the elliptic quantum group Uₐ, ₏ (glN). We give a classification theorem of the finite-dimensional irreducible representations of Uₐ, ₏ (glN) in terms of the theta function analogue of the Drinfeld polynomial for the quantum affine algebra Uq (glN). We also construct the Gelfand-Tsetlin bases for the level-0 Uₐ, ₏ (glN) -modules following the work by Nazarov-Tarasov for the Yangian Y (glN) -modules. This is a construction in terms of the Drinfeld generators. For the case of tensor product of the vector representations, we give another construction of the Gelfand-Tsetlin bases in terms of the L-operators and make a connection between the two constructions. We also compare them with those obtained by the first author by using the Sₙ-action realized by the elliptic dynamical R-matrix on the standard bases. As a byproduct, we obtain an explicit formula for the partition functions of the corresponding 2-dimensional square lattice model in terms of the elliptic weight functions of type A₍-₁.