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

Understand

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.