Séminaire Lotharingien de Combinatoire, B50j (2005), 47 pp.

Adriano Garsia and Nolan Wallach

Some New Applications of Orbit Harmonics

Abstract. We prove a new result in the Theory of Orbit Harmonics and derive from it a new proof of the Cohen-Macauliness of the ring QIm(G) of m-Quasi-Invariants of a Coxeter Group G. Using the non-degeneracy of the fundamental bilinear form on QIm(G), this approach yields also a direct and simple proof that the quotient of QIm(G) by the ideal generated by the homogeneous G-invariants affords a graded version of the left regular representation of G. Originally all of these results were obtained as a combination of some deep work of Etingof-Ginzburg [3], Feigin-Veselov [6] and Felder-Veselov [5]. The arguments here are quite elementary and self contained, except those using the non-degeneracy of the fundamental bilinear form.

Received: November 20, 2004. Accepted: January 14, 2005. Final Version: January 29, 2005.

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