Geodesic
Some New Applications of Orbit Harmonics
Séminaire lotharingien de combinatoire, Tome 50 (2003-2005)

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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. 

@article{SLC_2003-2005_50_a9,
     author = {Adriano Garsia and Nolan Wallach},
     title = {Some {New} {Applications} of {Orbit} {Harmonics}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {50},
     year = {2003-2005},
     url = {http://geodesic.mathdoc.fr/item/SLC_2003-2005_50_a9/}
}
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Adriano Garsia; Nolan Wallach. Some New Applications of Orbit Harmonics. Séminaire lotharingien de combinatoire, Tome 50 (2003-2005). http://geodesic.mathdoc.fr/item/SLC_2003-2005_50_a9/