Trigonometric Cherednik algebra at critical level and quantum many-body problems

Abstract.

For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra A(k) at critical level in terms of Dunkl type operators. Under this representation the center of A(k) produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of A(k). We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of A(k).