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We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed Walgebras.

We prove Drinfeld's conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii algebra, associated to the Langlands dual algebra. The center is identified with a limit of the W-algebra, defined by means of the quantum Drinfeld-Sokolov reduction.

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Introduction.Gaudin's model describes a completely integrable quantum spin chain. Originally 1] it was formulated as a spin model related to the Lie algebra sl 2 . Later it was realized, cf. 2], Sect. 13.2.2 and 3], that one can associate such a model to any semi-simple complex Lie algebra g and a solution of the corresponding classical Yang-Baxter equation 4, 5]. In this work we will focus on the models, corresponding to the rational solutions.Denote by V the nite-dimensional irreducible representation of g of dominant integral highest weight . Let ( ) := ( 1 ; : : : ; N ) be a set of dominant integral weights of g. Consider the tensor product V ( ) := V 1 : : : V N and associate with each factor V i of this tensor product a complex number z i . The hamiltonians of Gaudin's model are mutually commuting operators i = i (z 1 ; : : : ; z N ); i = 1; : : : ; N; acting on the space V ( ) .