ON REPRESENTATIONS OF THE ELLIPTIC QUANTUM GROUP E(TAU,ETA)(SL(2))

We describe representation theory of the elliptic quantum group E(tau, eta)(sl(2)). It turns out that the representation theory is parallel t o the representation theory of the Yangian Y(sl(2)) and the quantum lo op group U-q(sl(2)). We introduce basic notions of representation theo ry of the elliptic quantum group E(tau,eta)((sl) over tilde(2)) and co nstruct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any ir reducible highest weight module of finite type is isomorphic to a tens or product of evaluation modules and a one-dimensional module. We desc ribe fusion of finite dimensional evaluation modules. In particular, w e show that under certain conditions the tensor product of two evaluat ion modules becomes reducible and contains an evaluation module, in th is case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theo ry becomes special if N-eta = m + l tau, where N, m, l are integers. W e indicate some new features in this case.