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We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a "Miura oper" on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. The Bethe Ansatz equations can be written for an arbitrary Kac-Moody algebra, and we prove an analogue of the last result in this more general setting. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko.

Opers on the Projective Line, Flag Manifolds and Bethe Ansatz