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Reflection groups in analysis and applications. (English) Zbl 1191.33006

Sometimes to discover new results in harmonic analysis and extend classical analysis like Fourier analysis and special functions several variables Lie groups and their representation are studied. The author came in the mid-1980s to realize that it without Lie groups only working with certain finite groups. These are the Coxeter finite reflection groups. The purpose of the paper is to present an overview of how this research area has developed. Some background on finite reflection groups and the development of the differential-difference operators, named Dunkl operators [see C. F. Dunkl, Trans. Am. Math. Soc. 311, No. 1, 167–183 (1989; Zbl 0652.33004)], are presented. There are sections on the interviewing operator, the analogues of the exponential function and the Fourier transform, non-symmetric Jack polynomials, which form an orthogonal basis of polynomials associated with the symmetric groups, and the application of the operators in the solution of the quantum Calogero-Sutherland models.

MSC:

33C52 Orthogonal polynomials and functions associated with root systems
20F55 Reflection and Coxeter groups (group-theoretic aspects)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
81R12 Groups and algebras in quantum theory and relations with integrable systems
05E05 Symmetric functions and generalizations

Citations:

Zbl 0652.33004
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References:

[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. · Zbl 1139.46028
[2] T.H. Baker and P.J. Forrester, The Calogero–Sutherland model and generalized classical polynomials, Comm. Math. Phys., 188 (1997), 175–216 · Zbl 0903.33010 · doi:10.1007/s002200050161
[3] J.J. Betancor, Distributional Dunkl transform and Dunkl convolution operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 221–245 · Zbl 1179.46035
[4] H.S.M. Coxeter, Regular Polytopes, 3rd ed., Dover Press, New York, 1973. · Zbl 0258.05119
[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. 2007 · Zbl 1234.81088 · doi:10.1016/j.nuclphysb.2007.11.028
[6] F. Dai and Y. Xu, Cesàro means of orthogonal expansions in several variables, Constr. Approx., to appear, arXiv:0705.2477v1, 17 May 2007.
[7] C.F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J., 25 (1976), 335–358 · Zbl 0326.33008 · doi:10.1512/iumj.1976.25.25030
[8] C.F. Dunkl, An addition theorem for Hahn polynomials: the spherical functions, SIAM J. Math. Anal., 9 (1978), 627–637 · Zbl 0386.33008 · doi:10.1137/0509043
[9] C.F. Dunkl, Orthogonal polynomials on the sphere with octahedral symmetry, Trans. Amer. Math. Soc., 282 (1984), 555–575 · Zbl 0541.33002 · doi:10.1090/S0002-9947-1984-0732106-7
[10] C.F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z., 197 (1988), 33–60 · Zbl 0616.33005 · doi:10.1007/BF01161629
[11] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311 (1989), 167–183 · Zbl 0652.33004 · doi:10.1090/S0002-9947-1989-0951883-8
[12] C.F. Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry, J. Math. Anal. Appl., 143 (1989), 459–470 · Zbl 0688.33004 · doi:10.1016/0022-247X(89)90052-8
[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.
[14] C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math., 43 (1991), 1213–1227 · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8
[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.
[16] C.F. Dunkl, Intertwining operators associated to the group S 3, Trans. Amer. Math. Soc., 347 (1995), 3347–3374 · Zbl 0857.22008 · doi:10.2307/2155014
[17] C.F. Dunkl, Orthogonal polynomials of types A and B and related Calogero models, Comm. Math. Phys., 197 (1998), 451–487 · Zbl 0936.33006 · doi:10.1007/s002200050460
[18] C.F. Dunkl, Singular polynomials for the symmetric groups, Int. Math. Res. Not., 2004 (2004), 3607–3635 · Zbl 1078.33017 · doi:10.1155/S1073792804140610
[19] C.F. Dunkl, Singular polynomials and modules for the symmetric groups, Int. Math. Res. Not., 2005 (2005), 2409–2436 · Zbl 1102.33012 · doi:10.1155/IMRN.2005.2409
[20] C.F. Dunkl, An intertwining operator for the group B 2, Glas. Math. J., 49 (2007), 291–319 · Zbl 1125.33013 · doi:10.1017/S0017089507003709
[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), 237–256 · Zbl 0829.33010 · doi:10.2307/2154950
[22] C.F. Dunkl and E.M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3), 86 (2003), 70–108 · Zbl 1042.20025 · doi:10.1112/S0024611502013825
[23] C.F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia Math. Appl., 81, Cambridge Univ. Press, Cambridge, 2001 · Zbl 0964.33001
[24] P. Etingof and X. Ma, On elliptic Dunkl operators, arXiv:0706.2152v1, 14 Jun. 2007. · Zbl 1184.43011
[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), 8407–8424 · Zbl 1004.33010 · doi:10.1088/0305-4470/34/41/302
[26] L. Gallardo and M. Yor, A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab., 34 (2006), 1530–1549 · Zbl 1107.60015 · doi:10.1214/009117906000000133
[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. · Zbl 1071.20005 · doi:10.1007/s00222-003-0313-8
[28] S. Griffeth, On some orthogonal functions generalizing Jack polynomials, arXiv:0707.0251v2, 20 May 2008. · Zbl 1283.05264
[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. · Zbl 0749.33005
[30] K. Hikami, Dunkl operator formalism for quantum many-body problems associated with classical root systems, J. Phys. Soc. Japan, 65 (1996), 394–401 · Zbl 0942.81620 · doi:10.1143/JPSJ.65.394
[31] K. Hikami and M. Wadati, Topics in quantum integrable systems, integrability, topological solitons and beyond, J. Math. Phys., 44 (2003), 3569–3594 · Zbl 1062.81082 · doi:10.1063/1.1588743
[32] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29, Cambridge Univ. Press, Cambridge, 1990 · Zbl 0725.20028
[33] M.F.E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993), 147–162 · Zbl 0789.33007 · doi:10.1007/BF01244305
[34] F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math., 128 (1997), 9–22 · Zbl 0870.05076 · doi:10.1007/s002220050134
[35] T.H. Koornwinder, Yet another proof of the addition formula for Jacobi polynomials, J. Math. Anal. Appl., 61 (1977), 136–141 · Zbl 0406.33006 · doi:10.1016/0022-247X(77)90149-4
[36] S. Lawi, Towards a characterization of Markov processes enjoying the time-inversion property, J. Theoret. Probab., 21 (2008), 144–168 · Zbl 1141.60046 · doi:10.1007/s10959-007-0104-z
[37] I.G. Macdonald, Affine Hecke algebras and Orthogonal Polynomials, Cambridge Tracts in Math., 157, Cambridge Univ. Press, Cambridge, 2003. · Zbl 1024.33001
[38] M. Maslouhi and E.H. Youssfi, Harmonic functions associated to Dunkl operators, Monatsh. Math., 152 (2007), 337–345 · Zbl 1264.31002 · doi:10.1007/s00605-007-0475-3
[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), 77–88 · Zbl 1048.34139 · doi:10.1080/10652460108819335
[40] G.E. Murphy, A new construction of Young’s seminormal representation of the symmetric groups, J. Algebra, 69 (1981), 287–297 · Zbl 0455.20007 · doi:10.1016/0021-8693(81)90205-2
[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. · Zbl 1152.81574
[42] E.M. Opdam, Dunkl operators, Besse functions and the discriminant of a finite Coxeter group, Compositio Math., 85 (1993), 333–373 · Zbl 0778.33009
[43] E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175 (1995), 75–121 · Zbl 0836.43017 · doi:10.1007/BF02392487
[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.
[45] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J., 98 (1999), 445–463 · Zbl 0947.33013 · doi:10.1215/S0012-7094-99-09813-7
[46] M. Rösler and M. de Jeu, Asymptotic analysis for the Dunkl kernel, J. Approx. Theory, 119 (2002), 110–126 · Zbl 1015.43004 · doi:10.1006/jath.2002.3722
[47] M. Rösler and M. Voit, Markov processes related with Dunkl operators, Adv. in Appl. Math., 21 (1998), 575–643 · Zbl 0919.60072 · doi:10.1006/aama.1998.0609
[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. · Zbl 1171.20303
[49] S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math., 97 (2005), 25–55 · Zbl 1131.43006
[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), 349–374 · Zbl 1027.47027 · doi:10.1080/10652460108819358
[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), 2423–2439 · Zbl 0948.39009 · doi:10.1143/JPSJ.65.2423
[52] H. Volkmer, Generalized ellipsoidal and sphero-conal harmonics, SIGMA Symmetry Integrability Geom. Methods Appl., 2 (2006), paper 071, 16 p. · Zbl 1133.35029
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