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Quasi-Invariants of Dihedral Systems

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Abstract

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.

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REFERENCES

  1. O.A. Chalykh and A.P. Veselov, “Commutative rings of partial di.erential operators and Lie algebras,” Comm. Math. Phys., 126 (1990), 597–611.

    Google Scholar 

  2. F. Calogero, “Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials,” J. Math. Phys., 12 (1971), 419–436.

    Google Scholar 

  3. J. Moser, “Three integrable hamiltonian ystems connected with isospectral deformations,”Adv. Math., 16 (1975), 197–220.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  7. K. Taniguchi, “On the symmetry of commuting di.erential operators with singularities along hyperplanes,” in: E-print math-ph/0309011 2003.

  8. K. Volchenko, A. Kozachko, and K. Mishachev, “The ring of quasi-invariants of the dihedral group,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1999), no.1, 48–51.

  9. M. Feigin and A.P. Veselov, “Quasi-invariants of Coxeter groups and m harmonic polynomial,” E-print math-ph/0105014 (2001).

  10. P. Etingof and V. Ginzburg, “On m quasi-invariants of a Coxeter group,” Moscow Math. J., 2 (2002), no.3, 555–566.

    Google Scholar 

  11. G. Felder and A.P. Veselov, “Action of Coxeter groups on m harmonic polynomials and and Knizhnik–Zamolodchikov equations,” Moscow Math. J., 3 (2003), no. 4, 1269–1291.

    Google Scholar 

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

  13. M. Feigin, Rings of Quantum Integrals for Generalized Calogero–Moser Problems Ph.D.Thesis, Lough-borough University, UK, 2003.

    Google Scholar 

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

    Google Scholar 

  15. O.A. Chalykh, M.V. Feigin,and A.P. Veselov, “Multidimensional Baker–Akhiezer functions and Huygens' principle,” Comm. Math. Phys., 206 (1999), no. 3, 533–566.

    Google Scholar 

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Authors and Affiliations

  1. Imperial College, London, Great Britain

    M. V. Feigin

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  1. M. V. Feigin
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Feigin, M.V. Quasi-Invariants of Dihedral Systems. Mathematical Notes 76, 723–737 (2004). https://doi.org/10.1023/B:MATN.0000049671.38147.7e

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