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We consider the gl_N Gaudin model of a tensor power of the standard vectorrepresentation. The geometric Langlands correspondence in the Gaudin modelrelates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebraof functions on a suitable space of N-th order differential operators. In thispaper we introduce a third side of the correspondence: the algebra of functionson the critical set of a master function. We construct isomorphisms of thethird algebra and the first two.  A new object is the Bethe vector averaging maps.

Three sides of the geometric Langlands correspondence for gl_N Gaudin model and Bethe vector averaging maps