zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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×SL(2,), where G is the Coxeter group and SL(2,) is the universal covering of the group SL(2,).

MSC:
33C52Orthogonal polynomials and functions associated with root systems
43A85Analysis on homogeneous spaces
44A15Special transforms (Legendre, Hilbert, etc.)
References:
[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)
[2]Ben Saïd, S.: On a unitary representation of the universal covering group of SL(2,). 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)
[4]Ben Saïd, S., Ørsted, B.: The wave equation for Dunkl operators. To appear in Indag. Math. 16 no.4, (2005)
[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
[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)
[8]Davidson, M., Ólafsson, G., Zhang, G.: Segal-Bargmann transform on Hermitian symmetric spaces. J. Funct. Anal. 204, 157–195 (2003)
[9]Davies, E.B.: Spectral theory and differential operators. Cambridge University Press, 1995
[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)
[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
[12]Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)
[13]Dunkl, C.F.: Reflection groups and othogonal polynomials on the sphere. Math. Z. 197, 33–60 (1988)
[14]Dunkl, C.F.: Integral kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)
[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)
[16]Dunkl, C.F.: Intertwining operators associated to the group S3. Trans. Am. Math. Soc. 347, 3347–3374 (1995)
[17]Dunkl, C.F., de Jeu, M.F., Opdam, E.: Singular polynomials for finite reflection groups. Trans. Am. Math. Soc. 346, 237–256 (1994)
[18]Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. Cambridge Univ. Press, 2001
[19]Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243–348 (2002)
[20]Folland, G.B.: Harmonic analysis in phase space. Princeton University Press, Princeton N. J., 1989
[21]Hall, B.C.: The Segal-Bargmann “Coherent State” transform for compact Lie groups. J. Funct. Anal. 112, 103–151 (1994)
[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
[23]Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. Soc. 3, 821–843 (1980)
[24]Howe, R.: Remarks on classical invariant theory. Trans. Am. Math. Soc. 313, 539–570 (1989)
[25]de Jeu, M.F.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)
[26]Lyakhov, L.N.: Spherical weighted-harmonic functions and singular pseudodifferential operators. Diff. Equa. 21, 693–703 (1985)
[27]Macdonald, I.G.: Some conjectures for root systems. SIAM J. Math. An. 13, 988–1007 (1982)
[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
[29]Olshanetsky, M.A., Perelomov, A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976)
[30]Opdam, E.: Root systems and hypergeometric functions. IV. Compositio Math. 67, 191–209 (1988)
[31]Opdam, E.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compositio Math. 85, 333–373 (1993)
[32]Opdam, E.: Some applications of hypergeometric shift operators. Inv. Math. 98, 1–18 (1989)
[33]Ørsted, B., Zhang, G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43, 551–583 (1994)
[34]Reed, M., Simon, B.: Methods of modern mathematical physics: I. Functional analysis. Academic Press, New York, 1972
[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
[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)
[37]Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519–542 (1998)
[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
[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)
[42]Soltani, F.: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. Pacific J. Math. 214, 379–397 (2004)
[43]Xu, Y.: Orthogonal polynomials for a family of product weight functions on the spheres. Canad. J. Math. 49, 175–192 (1997)
[44]Zhang, G.: Branching coefficients of holomophic representations and Segal-Bargmann transform. J. Funct. Anal. 195, 306–349 (2002)