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Background. The Harish-Chandra isomorphism for finite-dimensional semisimple Lie algebras identifies the center of the universal enveloping with invariant polynomials on the Cartan subalgebra. Invariant with respect to shifted action of the Weyl group. The construction of isomorphism is simple - just take value of the central element in irrep of weight lambda.

Question setup consider the loop algebra g((t)). It is known due to Goodman-Wallach, Hayshi, Frenkel-Feigin that universal enveloping algebra (modula some "technical" details like critical level, completion) has huge center which is free polynomial algebra with generators indexed by $C_{k,n}$ k=1...rank(g) , n=$-\infty ... +\infty$. There is also an analogue of the Harish-Chandra isomorphism - we can take the eigenvalues of central elements in the so-called Wakimoto modules.

Question I wonder - the image of this Harish-Chandra is it somehow invariant with respect affine Weyl group ? If yes, what is the action ?

Remark. I suspect that this action should be related with Miura transformation, where G-oper is factorized (d/dz - l_1 ) (d/dz - l_2) = (d/dz - m_1)(d/dz-m_2). So permutation generators I guess sending l_1, l_2 to m_1, m_2. At least if we restrict to finite-dimensional gl_n this indeed coincide with the standard action. (See formula 21 page 9 in our paper).

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