On elliptic Dunkl operators (Q841534)

The authors define elliptic Dunkl operators for any finite group \(W\) acting on a compact complex torus \(X\). More precisely, those operators are associated to any topological trivial holomorphic line bundle \(\mathcal{L}\) on \(X\) with trivial stabilizer in \(W\) and to any flat holomorphic connection \(\nabla\) on this bundle. When \(W\) is the Weyl group of a root system and \(X\) is the space of homomorphisms from the root lattice to the elliptic curve, the operators constructed in the present paper coincide with those in [\textit{V. M. Buchstaber, G. Felder} and \textit{A. P. Veselov}, Duke Math. J. 76, No. 3, 885--911 (1994; Zbl 0842.35128)]. The authors prove that the elliptic Dunkl operators commute and they also examine several questions on the representations of associated Hecke algebras, in particular generalizations of the famous Cherednik double affine Hecke algebras. The authors announce a future work where they will construct new quantum integrable systems using their elliptic Dunkl operators.