Japanese Journal of Mathematics

, Volume 3, Issue 2, pp 215–246 | Cite as

Reflection groups in analysis and applications



This is an overview of the use of reflection groups in analysis, applications in algebra, mathematical physics, and probability.

Keywords and phrases:

Dunkl operators Dunkl transform non-symmetric Jack polynomials Calogero–Sutherland models 

Mathematics Subject Classification (2000):

33 42 (primary) 05 82 (secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Abdelkefi and M. Sifi, Characterization of Besov spaces for the Dunkl operator on the real line, JIPAM. J. Inequal. Pure Appl. Math., 8 (2007), no. 3, article 73, 11 p.Google Scholar
  2. 2.
    T.H. Baker and P.J. Forrester, The Calogero–Sutherland model and generalized classical polynomials, Comm. Math. Phys., 188 (1997), 175–216MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J.J. Betancor, Distributional Dunkl transform and Dunkl convolution operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 221–245MATHMathSciNetGoogle Scholar
  4. 4.
    H.S.M. Coxeter, Regular Polytopes, 3rd ed., Dover Press, New York, 1973.Google Scholar
  5. 5.
    N. Crampé and C.A.S. Young, Sutherland models for complex reflection groups, Nuclear Phys. B, 797 (2008), 499–519, arXiv:0708.2664v2, 29 Nov. 2007MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    F. Dai and Y. Xu, Cesàro means of orthogonal expansions in several variables, Constr. Approx., to appear, arXiv:0705.2477v1, 17 May 2007.Google Scholar
  7. 7.
    C.F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J., 25 (1976), 335–358MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C.F. Dunkl, An addition theorem for Hahn polynomials: the spherical functions, SIAM J. Math. Anal., 9 (1978), 627–637CrossRefMathSciNetGoogle Scholar
  9. 9.
    C.F. Dunkl, Orthogonal polynomials on the sphere with octahedral symmetry, Trans. Amer. Math. Soc., 282 (1984), 555–575MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C.F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z., 197 (1988), 33–60MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311 (1989), 167–183MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C.F. Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry, J. Math. Anal. Appl., 143 (1989), 459–470MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C.F. Dunkl, Operators commuting with Coxeter group actions on polynomials, Invariant Theory and Tableaux, (ed. D. Stanton), IMA Vol. Math. Appl., 19, Springer-Verlag, 1990, pp. 107–117.Google Scholar
  14. 14.
    C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math., 43 (1991), 1213–1227MATHMathSciNetGoogle Scholar
  15. 15.
    C.F.Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123–138.Google Scholar
  16. 16.
    C.F. Dunkl, Intertwining operators associated to the group S 3, Trans. Amer. Math. Soc., 347 (1995), 3347–3374MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    C.F. Dunkl, Orthogonal polynomials of types A and B and related Calogero models, Comm. Math. Phys., 197 (1998), 451–487MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    C.F. Dunkl, Singular polynomials for the symmetric groups, Int. Math. Res. Not., 2004 (2004), 3607–3635MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    C.F. Dunkl, Singular polynomials and modules for the symmetric groups, Int. Math. Res. Not., 2005 (2005), 2409–2436MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    C.F. Dunkl, An intertwining operator for the group B 2, Glas. Math. J., 49 (2007), 291–319MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc., 346 (1994), 237–256MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    C.F. Dunkl and E.M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3), 86 (2003), 70–108MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    C.F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia Math. Appl., 81, Cambridge Univ. Press, Cambridge, 2001Google Scholar
  24. 24.
    P. Etingof and X. Ma, On elliptic Dunkl operators, arXiv:0706.2152v1, 14 Jun. 2007.Google Scholar
  25. 25.
    P.J. Forrester, D.S. McAnally and Y. Nikoyalevsky, On the evaluation formula for Jack polynomials with prescribed symmetry, J. Phys. A, 34 (2001), 8407–8424MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    L. Gallardo and M. Yor, A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab., 34 (2006), 1530–1549MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    V. Ginzburg, N. Guay, E.M. Opdam and R. Rouquier, On the category \({\fancyscript{O}}\) for rational Cherednik algebras, Invent. Math., 154 (2003), 617–651.MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    S. Griffeth, On some orthogonal functions generalizing Jack polynomials, arXiv:0707.0251v2, 20 May 2008.Google Scholar
  29. 29.
    G.J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic Analysis on Reductive Groups, Brunswick, ME, 1989, Progr. Math., 101, Birkhäuser Boston, Boston, MA, 1991, pp. 181–191.Google Scholar
  30. 30.
    K. Hikami, Dunkl operator formalism for quantum many-body problems associated with classical root systems, J. Phys. Soc. Japan, 65 (1996), 394–401MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    K. Hikami and M. Wadati, Topics in quantum integrable systems, integrability, topological solitons and beyond, J. Math. Phys., 44 (2003), 3569–3594MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29, Cambridge Univ. Press, Cambridge, 1990Google Scholar
  33. 33.
    M.F.E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993), 147–162MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math., 128 (1997), 9–22MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    T.H. Koornwinder, Yet another proof of the addition formula for Jacobi polynomials, J. Math. Anal. Appl., 61 (1977), 136–141MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    S. Lawi, Towards a characterization of Markov processes enjoying the time-inversion property, J. Theoret. Probab., 21 (2008), 144–168MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    I.G. Macdonald, Affine Hecke algebras and Orthogonal Polynomials, Cambridge Tracts in Math., 157, Cambridge Univ. Press, Cambridge, 2003.Google Scholar
  38. 38.
    M. Maslouhi and E.H. Youssfi, Harmonic functions associated to Dunkl operators, Monatsh. Math., 152 (2007), 337–345MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    M.A. Mourou, Transmutation operators associated with a Dunkl-type differential-difference operator on the real line and certain of their applications, Integral Transform. Spec. Funct., 12 (2001), 77–88MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    G.E. Murphy, A new construction of Young’s seminormal representation of the symmetric groups, J. Algebra, 69 (1981), 287–297MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    S. Odake and R. Sasaki, Exact Heisenberg operator solutions for multiparticle quantum mechanics, J. Math. Phys., 48 (2007), no. 8, 082106, 12 p., arXiv:0706.0768v1, 6 Jun. 2007.Google Scholar
  42. 42.
    E.M. Opdam, Dunkl operators, Besse functions and the discriminant of a finite Coxeter group, Compositio Math., 85 (1993), 333–373MATHMathSciNetGoogle Scholar
  43. 43.
    E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175 (1995), 75–121MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    E.M. Opdam, Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups, (with a preface by T. Oshima), MSJ Mem., 8, Math. Soc. Japan, Tokyo, 2000.Google Scholar
  45. 45.
    M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J., 98 (1999), 445–463MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    M. Rösler and M. de Jeu, Asymptotic analysis for the Dunkl kernel, J. Approx. Theory, 119 (2002), 110–126MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    M. Rösler and M. Voit, Markov processes related with Dunkl operators, Adv. in Appl. Math., 21 (1998), 575–643MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    R. Rouquier, Representations of rational Cherednik algebras, Infinite-Dimensional Aspects of Representation Theory and Applications, Contemp. Math., 392, Amer. Math. Soc., Providence, RI, 2005, pp. 103–131.Google Scholar
  49. 49.
    S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math., 97 (2005), 25–55MathSciNetGoogle Scholar
  50. 50.
    K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform. Spec. Funct., 12 (2001), 349–374MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    H. Ujino and M. Wadati, Rodrigues formula for Hi-Jack symmetric polynomials associated with the quantum Calogero model, J. Phys. Soc. Japan, 65 (1996), 2423–2439MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    H. Volkmer, Generalized ellipsoidal and sphero-conal harmonics, SIGMA Symmetry Integrability Geom. Methods Appl., 2 (2006), paper 071, 16 p.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations