The non-degeneracy of the bilinear form of m-Quasi-Invariants

We give here a new proof of the non-degeneracy of the fundamental bilinear form for Sn-m-Quasi-Invariants and for m-Quasi-Invariants of classical Weyl groups. We also indicate how our approach can be extended to other Coxeter groups. This bilinear form plays a crucial role in the original proof [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, arXiv: math.QA/0106175 v1, June 2001] that m-Quasi-Invariants are a free module over the invariants as well as in all subsequent proofs [Y. Berest, P. Etingof, V. Ginsburg, Cherednik algebras and differential operators on quasi-invariants, math.QA/0111005; A. Garsia, N. Wallach, Some new applications of orbit harmonics, Sém. Lothar. Combin. 50 (2005), Article B50j]. However, in previous literature this non-degeneracy was stated and used without proof with reference to some deep results of Opdam [E.M. Opdam, Some applications of shift operators, Invent. Math. 98 (1989) 1–18] on shift-differential operators. This result hinges on the validity of a deceptively simple identity on Dunkl operators which, at least in the Sn case, begs for an elementary painless proof. An elementary but by all means not painless proof of this identity can be found in a paper of Dunkl and Hanlon [C. Dunkl, P. Hanlon, Integrals of polynomials associated with tableaux and the Garsia–Haiman conjecture, Math. Z. 228 (1998) 537–567. 71]. Our proof here is not elementary but hopefully it should be painless and informative.