Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 049, 51 pages      arXiv:1103.4593

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Patrick Desrosiers a and Martin Hallnäs b
a) Instituto Matemática y Física, Universidad de Talca, 2 Norte 685, Talca, Chile
b) Department of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU, UK

Received March 22, 2012, in final form July 25, 2012; Published online August 03, 2012

We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain so-called deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.

Key words: symmetric functions; super-symmetric polynomials; (deformed) Calogero-Moser-Sutherland models.

pdf (690 kb)   tex (54 kb)


  1. Atiyah M.F., Macdonald I.G., Introduction to commutative algebra, Addison-Wesley Publishing Co., 1969.
  2. Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
  3. Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system BC, Trans. Amer. Math. Soc. 339 (1993), 581-609.
  4. Bernard D., Gaudin M., Haldane F.D.M., Pasquier V., Yang-Baxter equation in long-range interacting systems, J. Phys. A: Math. Gen. 26 (1993), 5219-5236, hep-th/9301084.
  5. Calogero F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  6. Chalykh O., Feigin M., Veselov A., New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys. 39 (1998), 695-703.
  7. Cherednik I., Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Adv. Math. 106 (1994), 65-95.
  8. Constantine A.G., The distribution of Hotelling's generalized T02, Ann. Math. Statist. 37 (1966), 215-225.
  9. Debiard A., Système différentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type BCp, in Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986), Lecture Notes in Math., Vol. 1296, Springer, Berlin, 1987, 42-124.
  10. Desrosiers P., Dang-Zheng L., Selberg integrals, super hypergeometric functions and applications to β-ensembles of random matrices, arXiv:1109.4659.
  11. Feigin M., Generalized Calogero-Moser systems from rational Cherednik algebras, Selecta Math. (N.S.) 18 (2012), 253-281, arXiv:0809.3487.
  12. Hallnäs M., Langmann E., A unified construction of generalized classical polynomials associated with operators of Calogero-Sutherland type, Constr. Approx. 31 (2010), 309-342, math-ph/0703090.
  13. Heckman G.J., Opdam E.M., Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), 329-352.
  14. Herz C.S., Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474-523.
  15. Heyneman R.G., Sweedler M.E., Affine Hopf algebras. I, J. Algebra 13 (1969), 192-241.
  16. James A.T., Special functions of matrix and single argument in statistics, in Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, 497-520.
  17. James A.T., Constantine A.G., Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. (3) 29 (1974), 174-192.
  18. Kaneko J., Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), 1086-1110.
  19. Kerov S., Okounkov A., Olshanski G., The boundary of the Young graph with Jack edge multiplicities, Int. Math. Res. Not. 1998 (1998), no. 4, 173-199.
  20. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998,
  21. Kohler H., Guhr T., Supersymmetric extensions of Calogero-Moser-Sutherland-like models: construction and some solutions, J. Phys. A: Math. Gen. 38 (2005), 9891-9915, math-ph/0510039.
  22. Korányi A., Hua-type integrals, hypergeometric functions and symmetric polynomials, in International Symposium in Memory of Hua Loo Keng, Vol. II (Beijing, 1988), Springer, Berlin, 1991, 169-180.
  23. Lassalle M., Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), 289-324.
  24. Lassalle M., Polynômes de Hermite généralisés, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 579-582.
  25. Lassalle M., Polynômes de Jacobi généralisés, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 425-428.
  26. Lassalle M., Polynômes de Laguerre généralisés, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 725-728.
  27. Lassalle M., Une formule du binôme généralisée pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 253-256.
  28. Macdonald I.G., Hypergeometric functions, unpublished.
  29. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  30. Moens E.M., Van der Jeugt J., On dimension formulas for gl(m|n) representations, J. Lie Theory 14 (2004), 523-535.
  31. Muirhead R.J., Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1982.
  32. Okounkov A., Olshanski G., Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), 69-78, q-alg/9608020.
  33. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  34. Olshanski G., Laguerre and Meixner orthogonal bases in the algebra of symmetric functions, Int. Math. Res. Not., to appear, arXiv:1103.5848.
  35. Olshanski G., Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes, J. Math. Sci. 174 (2011), 41-57, arXiv:1009.2037.
  36. Opdam E.M., Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1-18.
  37. Opdam E.M., Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, Vol. 8, Mathematical Society of Japan, Tokyo, 2000.
  38. Rains E.M., BCn-symmetric polynomials, Transform. Groups 10 (2005), 63-132, math.QA/0112035.
  39. Reed M., Simon B., Methods of modern mathematical physics. I. Functional analysis, 2nd ed., Academic Press Inc., New York, 1980.
  40. Sergeev A., Superanalogs of the Calogero operators and Jack polynomials, J. Nonlinear Math. Phys. 8 (2001), 59-64, math.RT/0106222.
  41. Sergeev A.N., Veselov A.P., BC Calogero-Moser operator and super Jacobi polynomials, Adv. Math. 222 (2009), 1687-1726, arXiv:0807.3858.
  42. Sergeev A.N., Veselov A.P., Deformed quantum Calogero-Moser problems and Lie superalgebras, Comm. Math. Phys. 245 (2004), 249-278, math-ph/0303025.
  43. Sergeev A.N., Veselov A.P., Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials, Adv. Math. 192 (2005), 341-375, math-ph/0307036.
  44. Sergeev A.N., Veselov A.P., Quantum Calogero-Moser systems: a view from infinity, in XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, 333-337, arXiv:0910.5463.
  45. Stanley R.P., Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115.
  46. van Diejen J.F., Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), 467-497, q-alg/9609032.
  47. van Diejen J.F., Properties of some families of hypergeometric orthogonal polynomials in several variables, Trans. Amer. Math. Soc. 351 (1999), 233-270, q-alg/9604004.
  48. Yan Z.M., A class of generalized hypergeometric functions in several variables, Canad. J. Math. 44 (1992), 1317-1338.

Previous article  Next article   Contents of Volume 8 (2012)