Segal-Bargmann transforms associated with finite Coxeter groups. (English) Zbl 1109.33015
It is well known that the classical Segal-Bergmann transform maps unitarily from the Schrödinger model to the Fock model intertwining the action of the Heisenberg group. The authors in this paper use the restriction principle, i.e. polarization of a suitable restriction map to construct the Segal-Bergmann transform associated with finite Coxeter groups. A new class of Fock-type spaces have been introduced and studied. The definition and properties of this class of Hilbert spaceces generalize naturally those of the well-known classical Fock spaces. Rösler’s results on the heat-kernel associated with reflection groups [M. Rösler, Commun. Math. Phys. 192, 519–542 (1998; Zbl 0908.33005)] have been used to obtain explicitly the integral representation of the Segal-Bergmann transform. The generalized Segal- Bergmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further the branching decomposition of the generalized Fock spaces under the action of \(G\times\text{SL}( 2,\mathbb R)\), where \(G\) is the Coxeter group and SL\(( 2, \mathbb R)\) is the universal covering of the group SL\(( 2,\mathbb R)\).
Reviewer: Ajay Kumar (Delhi)
MSC:
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 43A85 | Harmonic analysis on homogeneous spaces |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Citations:
Zbl 0908.33005References:
| [1] | Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm. Pure Appl. Math. 14, 187–214 (1961) · Zbl 0107.09102 |
| [2] | Ben Saïd, S.: On a unitary representation of the universal covering group of SL(2,\(\mathbb{R}\)). Preprint |
| [3] | Ben Saïd, S., Ørsted, B.: Bessel functions related to root systems via the trigonometric setting. Int. Math. Res. Not. 9, 551–585 (2005) · Zbl 1065.22010 |
| [4] | Ben Saïd, S., Ørsted, B.: The wave equation for Dunkl operators. To appear in Indag. Math. 16 no.4, (2005) · Zbl 1116.35077 |
| [5] | Ben Saïd, S., Ørsted, B.: On Fock spaces and SL(2)-triples for the Dunkl operators. To appear in Sémin. Congr., Soc. Math. France, Paris, 2005 · Zbl 1180.22013 |
| [6] | Calogero, F.: Solution of the one-dimensional n-body problem. J. Math. Phys. 12, 419–436 (1971) |
| [7] | Cholewinski, F.M.: Generalized Fock spaces and associated operators. SIAM J. Math. Anal. 15, 177–202 (1984) · Zbl 0596.46017 |
| [8] | Davidson, M., Ólafsson, G., Zhang, G.: Segal-Bargmann transform on Hermitian symmetric spaces. J. Funct. Anal. 204, 157–195 (2003) · Zbl 1035.32014 |
| [9] | Davies, E.B.: Spectral theory and differential operators. Cambridge University Press, 1995 · Zbl 0893.47004 |
| [10] | van Dijk, G., Molchanov, M.F.: The Berezin form for rank one para-Hermitian symmetric spaces. J. Math. Pures Appl. 77, 747–799 (1998) · Zbl 0919.43007 |
| [11] | Dunford, N., Schwartz, J.T.: Linear operators: Part II. Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers John Wiley and Sons New York-London, 1963 · Zbl 0128.34803 |
| [12] | Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989) · Zbl 0652.33004 |
| [13] | Dunkl, C.F.: Reflection groups and othogonal polynomials on the sphere. Math. Z. 197, 33–60 (1988) · Zbl 0616.33005 |
| [14] | Dunkl, C.F.: Integral kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991) · Zbl 0827.33010 |
| [15] | Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications. Proceedings, Tampa 1991, Contemp. Math. 138, 123–138 (1992) · Zbl 0789.33008 |
| [16] | Dunkl, C.F.: Intertwining operators associated to the group S3. Trans. Am. Math. Soc. 347, 3347–3374 (1995) · Zbl 0857.22008 |
| [17] | Dunkl, C.F., de Jeu, M.F., Opdam, E.: Singular polynomials for finite reflection groups. Trans. Am. Math. Soc. 346, 237–256 (1994) · Zbl 0829.33010 |
| [18] | Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. Cambridge Univ. Press, 2001 · Zbl 0964.33001 |
| [19] | Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243–348 (2002) · Zbl 1061.16032 |
| [20] | Folland, G.B.: Harmonic analysis in phase space. Princeton University Press, Princeton N. J., 1989 · Zbl 0682.43001 |
| [21] | Hall, B.C.: The Segal-Bargmann “Coherent State” transform for compact Lie groups. J. Funct. Anal. 112, 103–151 (1994) · Zbl 0838.22004 |
| [22] | Heckman, G.J.: A remark on the Dunkl differential-difference operators. Harmonic analysis on reductive groups, W. Barker, P. Sally (eds.). Progress in Math. 101, Birkhäuser, 1991, pp. 181–191 · Zbl 0749.33005 |
| [23] | Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. Soc. 3, 821–843 (1980) · Zbl 0442.43002 |
| [24] | Howe, R.: Remarks on classical invariant theory. Trans. Am. Math. Soc. 313, 539–570 (1989) · Zbl 0674.15021 |
| [25] | de Jeu, M.F.: The Dunkl transform. Invent. Math. 113, 147–162 (1993) · Zbl 0789.33007 |
| [26] | Lyakhov, L.N.: Spherical weighted-harmonic functions and singular pseudodifferential operators. Diff. Equa. 21, 693–703 (1985) · Zbl 0587.35090 |
| [27] | Macdonald, I.G.: Some conjectures for root systems. SIAM J. Math. An. 13, 988–1007 (1982) · Zbl 0498.17006 |
| [28] | Ólafsson, G., Ørsted, B.: Generalization of the Bargmann transform. Lie theory and its applications in physics (Clausthal, 1995), World Sci. Publishing, River Edge, N.J., 1996, pp. 3–14 · Zbl 0916.22006 |
| [29] | Olshanetsky, M.A., Perelomov, A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976) · Zbl 0342.58017 |
| [30] | Opdam, E.: Root systems and hypergeometric functions. IV. Compositio Math. 67, 191–209 (1988) · Zbl 0669.33008 |
| [31] | Opdam, E.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compositio Math. 85, 333–373 (1993) · Zbl 0778.33009 |
| [32] | Opdam, E.: Some applications of hypergeometric shift operators. Inv. Math. 98, 1–18 (1989) · Zbl 0696.33006 |
| [33] | Ørsted, B., Zhang, G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43, 551–583 (1994) · Zbl 0805.46053 |
| [34] | Reed, M., Simon, B.: Methods of modern mathematical physics: I. Functional analysis. Academic Press, New York, 1972 · Zbl 0242.46001 |
| [35] | Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. Operator Theory: Advances and Applications 73, Basel, Birkhäuser Verlag, 1994, pp. 369–396 · Zbl 0826.33005 |
| [36] | Rösler, M.: Dunkl operators: theory and applications. Orthogonal polynomials and special functions, E.K. Koelink, W. van Assche (eds.), Spring Lecture Notes in Math. 1817, 93–136 (2003) · Zbl 1029.43001 |
| [37] | Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519–542 (1998) · Zbl 0908.33005 |
| [38] | Rösler, M.: Contributions to the theory of Dunkl operators. Habilitations Thesis, TU München, 1999 |
| [39] | Segal, I.E.: Mathematical problems of relativistic physics. Am. Math. Soc., Providence, R. I., 1963 · Zbl 0112.45307 |
| [40] | Sobolev, S.L.: Vvedenie v teoriyu kubaturnykh formul. (Russian) [Introduction to the theory of cubature formulas], Izdat. “Nauka”, Moscow, 1974 |
| [41] | Sifi, M., Soltani, F.: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line. J. Math. Anal. Appl. 270, 92–106 (2002) · Zbl 1012.46033 |
| [42] | Soltani, F.: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. Pacific J. Math. 214, 379–397 (2004) · Zbl 1052.33014 |
| [43] | Xu, Y.: Orthogonal polynomials for a family of product weight functions on the spheres. Canad. J. Math. 49, 175–192 (1997) · Zbl 0872.33008 |
| [44] | Zhang, G.: Branching coefficients of holomophic representations and Segal-Bargmann transform. J. Funct. Anal. 195, 306–349 (2002) · Zbl 1019.22006 |
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