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Mosc. Math. J., 2002, Volume 2, Number 3, Pages 555–566 (Mi mmj63)  

This article is cited in 19 scientific papers (total in 19 papers)

On $m$-quasi-invariants of a Coxeter group

P. Etingofa, V. A. Ginzburgb

a Department of Mathematics, Harvard University
b University of Chicago

Abstract: Let $W$ be a finite Coxeter group in a Euclidean vector space $V$, and let m be a $W$-invariant $\mathbb Z_+$-valued function on the set of reflections in $W$. Chalykh and Veselov introduced an interesting algebra $Q_m$, called the algebra of $m$-quasi-invariants for $W$, such that $\mathbb C[V]_W\subseteq Q_m\subseteq\mathbb C[V]$, $Q_0=\mathbb C[V]$ and $Q_m\supseteq Q_{m'}$ whenever $m\leq m'$. Namely, $Q_m$ is the algebra of quantum integrals of the rational Calogero–Moser system with coupling constant $m$. Feigin and Veselov proposed a number of interesting conjectures concerning the structure of $Q_m$ and verified them for dihedral groups and constant functions $m$. Our objective is to prove some of these conjectures in the general case.

Key words and phrases: Calogero–Moser system, Coxeter groups, $m$-quasi-invariants.


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MSC: 81Rxx, 14-xx
Received: March 2, 2002

Citation: P. Etingof, V. A. Ginzburg, “On $m$-quasi-invariants of a Coxeter group”, Mosc. Math. J., 2:3 (2002), 555–566

Citation in format AMSBIB
\by P.~Etingof, V.~A.~Ginzburg
\paper On $m$-quasi-invariants of a~Coxeter group
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 3
\pages 555--566

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    This publication is cited in the following articles:
    1. G. Felder, A. P. Veselov, “Action of Coxeter groups on $m$-harmonic polynomials and Knizhnik–Zamolodchikov equations”, Mosc. Math. J., 3:4 (2003), 1269–1291  mathnet  crossref  mathscinet  zmath
    2. Berest Yu., Etingof P., Ginzburg V., “Cherednik algebras and differential operators on quasi-invariants”, Duke Math. J., 118:2 (2003), 279–337  crossref  mathscinet  zmath  isi
    3. M. V. Feigin, “Quasi-Invariants of Dihedral Systems”, Math. Notes, 76:5 (2004), 723–737  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Garsia A.M., Wallach N., “$r$-Qsym is free over SYM”, J. Combin. Theory Ser. A, 114:4 (2007), 704–732  crossref  mathscinet  zmath  isi  elib
    5. Bandlow J., Musiker G., “A new characterization for the $m$-quasiinvariants of $S_n$ and explicit basis for two row hook shapes”, J. Combin. Theory Ser. A, 115:8 (2008), 1333–1357  crossref  mathscinet  zmath  isi  elib
    6. Lam T., Pylyavskyy P., “$P$-partition products and fundamental quasi-symmetric function positivity”, Adv. in Appl. Math., 40:3 (2008), 271–294  crossref  mathscinet  zmath  isi
    7. Chalykh O., “Algebro-geometric Schrodinger operators in many dimensions”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 947–971  crossref  mathscinet  zmath  adsnasa  isi
    8. Felder G., Veselov A.P., “Baker-Akhiezer function as iterated residue and Selberg-type integral”, Glasg. Math. J., 51:A (2009), 59–73  crossref  mathscinet  zmath  isi  elib
    9. Tsuchida T., “On Quasiinvariants of $S_n$ of Hook Shape”, Osaka J Math, 47:2 (2010), 461–485  mathscinet  zmath  isi  elib
    10. Berest Yu., Chalykh O., “Quasi-invariants of complex reflection groups”, Compos Math, 147:3 (2011), 965–1002  crossref  mathscinet  zmath  isi  elib
    11. Berest Yu., Samuelson P., “Dunkl Operators and Quasi-Invariants of Complex Reflection Groups”, Mathematical Aspects of Quantization, Contemporary Mathematics, 583, eds. Evens S., Gekhtman M., Hall B., Liu X., Polini C., Amer Mathematical Soc, 2012, 1–23  crossref  mathscinet  zmath  adsnasa  isi
    12. A. B. Zheglov, “On rings of commuting partial differential operators”, St. Petersburg Math. J., 25:5 (2014), 775–814  mathnet  crossref  mathscinet  zmath  isi  elib
    13. Feigin M.V., Hallnaes M.A., Veselov A.P., “Baker-Akhiezer Functions and Generalised Macdonald-Mehta Integrals”, J. Math. Phys., 54:5 (2013), 052106  crossref  mathscinet  zmath  isi  elib
    14. Kurke H., Osipov D., Zheglov A., “Commuting Differential Operators and Higher-Dimensional Algebraic Varieties”, Sel. Math.-New Ser., 20:4 (2014), 1159–1195  crossref  mathscinet  zmath  isi  elib
    15. Feigin M., Johnston D., “a Class of Baker-Akhiezer Arrangements”, Commun. Math. Phys., 328:3 (2014), 1117–1157  crossref  mathscinet  zmath  isi  elib
    16. Correa F., Lechtenfeld O., Plyushchay M., “Nonlinear Supersymmetry in the Quantum Calogero Model”, J. High Energy Phys., 2014, no. 4, 151  crossref  mathscinet  zmath  isi  elib
    17. A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Etingof P., Rains E., “on Cohen-Macaulayness of Algebras Generated By Generalized Power Sums”, Commun. Math. Phys., 347:1 (2016), 163–182  crossref  mathscinet  zmath  isi
    19. Pavel Etingof, “Cherednik and Hecke algebras of varieties with a finite group action”, Mosc. Math. J., 17:4 (2017), 635–666  mathnet  crossref
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