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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
*QI*_{m}(*G*) of *m*-Quasi-Invariants
of a Coxeter Group *G*.
Using the non-degeneracy of the fundamental bilinear form on
*QI*_{m}(*G*),
this approach yields also a direct and simple proof that the
quotient of *QI*_{m}(*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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