Quasi-Invariants of Dihedral Systems
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 776-791.

Voir la notice de l'article provenant de la source Math-Net.Ru

For two-dimensional Coxeter systems with arbitrary multiplicities, a basis of the module of quasi-invariants over the invariants is explicitly constructed. It is proved that the basis thus obtained consists of $m$-harmonic polynomials. Hence this generalizes earlier results of Veselov and the author for systems of constant multiplicity.
@article{MZM_2004_76_5_a12,
     author = {M. V. Feigin},
     title = {Quasi-Invariants of {Dihedral} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {776--791},
     publisher = {mathdoc},
     volume = {76},
     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a12/}
}
TY  - JOUR
AU  - M. V. Feigin
TI  - Quasi-Invariants of Dihedral Systems
JO  - Matematičeskie zametki
PY  - 2004
SP  - 776
EP  - 791
VL  - 76
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a12/
LA  - ru
ID  - MZM_2004_76_5_a12
ER  - 
%0 Journal Article
%A M. V. Feigin
%T Quasi-Invariants of Dihedral Systems
%J Matematičeskie zametki
%D 2004
%P 776-791
%V 76
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a12/
%G ru
%F MZM_2004_76_5_a12
M. V. Feigin. Quasi-Invariants of Dihedral Systems. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 776-791. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a12/

[1] Chalykh O. A., Veselov A. P., “Commutative rings of partial differential operators and Lie algebras”, Comm. Math. Phys., 126 (1990), 597–611 | DOI | MR | Zbl

[2] Calogero F., “Solution of the one-dimensional $n$-body problem with quadratic and/or inversely quadratic pair potential”, J. Math. Phys., 12 (1971), 419–436 | DOI | MR

[3] Moser J., “Three integrable hamiltonian systems connected with isospectral deformations”, Adv. Math., 16 (1975), 197–220 | DOI | MR | Zbl

[4] Olshanetsky M. A., Perelomov A. M., “Quantum integrable systems related to Lie algebras”, Phys. Rep., 94 (1983), 313–404 | DOI | MR

[5] Feigin M., Veselov A. P., “Quasi-invariants of Coxeter groups and $m$-harmonic polynomials”, Intern. Math. Res. Notices, 10 (2002), 521–545 | DOI | MR | Zbl

[6] Heckman G. J., Opdam E. M., “Root systems and hypergeometric functions, I”, Compositio Math., 64 (1987), 329–352 | MR | Zbl

[7] Taniguchi K., On the symmetry of commuting differential operators with singularities along hyperplanes, , 2003 E-print math-ph/0309011

[8] Volchenko K., Kozachko A., Mishachev K., “Koltso kvaziinvariantov dlya grupp diedra”, Vestn. MGU. Ser. 1. Matem., mekh., 1999, no. 1, 48–51 | MR | Zbl

[9] Feigin M., Veselov A. P., Quasi-invariants of Coxeter groups and $m$-harmonic polynomials, , 2001 E-print math-ph/0105014

[10] Etingof P., Ginzburg V., “On $m$-quasi-invariants of a Coxeter group”, Mosc. Math. J., 2:3 (2002), 555–566 | MR | Zbl

[11] Felder G., Veselov A. P., “Action of Coxeter groups on $m$-harmonic polynomials and and Knizhnik–Zamolodchikov equations”, Mosc. Math. J., 3:4 (2003), 1269–1291 | MR | Zbl

[12] Berest Yu., Etingof P., Ginzburg V., Cherednik algebras and differential operators on quasi-invariants, , 2001 E-print math.QA/0111005

[13] Feigin M., Rings of Quantum Integrals for Generalized Calogero–Moser Problems, Ph. D. Thesis, Loughborough University, UK, 2003

[14] Feigin M., Veselov A. P., “Quasi-invariants and quantum integrals of the deformed Calogero–Moser systems”, Intern. Math. Res. Notices, 46 (2003), 2487–2511 | DOI | MR | Zbl

[15] Chalykh O. A., Feigin M. V., Veselov A. P., “Multidimensional Baker–Akhiezer functions and Huygens' principle”, Comm. Math. Phys., 206 (1999), 533–566 | DOI | MR | Zbl