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2013 | OriginalPaper | Chapter

The Segal–Bargmann Transform on Compact Symmetric Spaces and Their Direct Limits

Authors : Gestur Ólafsson, Keng Wiboonton

Published in: Lie Groups: Structure, Actions, and Representations

Publisher: Springer New York

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Abstract

This article studies the limit of the Segal-Bargman transform on inductive limits of compact symmetric spaces.

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Literature
[1]
go back to reference D.N. Akhiezer and S. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12. D.N. Akhiezer and S. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12.
[2]
go back to reference J-P. Anker and P. Ostellari, The Heat Kernel on Noncompact Symmetric Spaces, in: Lie groups and symmetric spaces: In memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Tranl. (2)210, Amer. Math. Soc. (2004), 27–46. J-P. Anker and P. Ostellari, The Heat Kernel on Noncompact Symmetric Spaces, in: Lie groups and symmetric spaces: In memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Tranl. (2)210, Amer. Math. Soc. (2004), 27–46.
[3]
go back to reference V. Bargmann, On Hilbert spaces of analytic functions and an associated integral transform, Comn. Pure Appl. Math. 14 (1961), 187–214. V. Bargmann, On Hilbert spaces of analytic functions and an associated integral transform, Comn. Pure Appl. Math. 14 (1961), 187–214.
[4]
go back to reference T. Branson, G. Ólafsson and A. Pasquale, The Paley-Wiener Theorem and the local Huygens’ principle for compact symmetric spaces: The even multiplicity case, Indag. Mathem., N.S., 16 (2005), 393–428. T. Branson, G. Ólafsson and A. Pasquale, The Paley-Wiener Theorem and the local Huygens’ principle for compact symmetric spaces: The even multiplicity case, Indag. Mathem., N.S., 16 (2005), 393–428.
[5]
go back to reference J. Faraut, Espaces Hilbertiens invariant de fonctions holomorphes, Semin. Congr., Vol. 7, Soc. Math. de France, Paris, 2003, 101–167. J. Faraut, Espaces Hilbertiens invariant de fonctions holomorphes, Semin. Congr., Vol. 7, Soc. Math. de France, Paris, 2003, 101–167.
[6]
go back to reference  , Analysis on the crown of a Riemannian symmetric space, in: Lie Groups and Symmetric Spaces: In Memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Transl. Ser. 2, Vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 99–110.  , Analysis on the crown of a Riemannian symmetric space, in: Lie Groups and Symmetric Spaces: In Memory of F.I. Karpelevich, S. Gindikin(ed.), Amer. Math. Soc. Transl. Ser. 2, Vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 99–110.
[7]
go back to reference M. Flensted-Jensen, Spherical functions on a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146. M. Flensted-Jensen, Spherical functions on a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146.
[8]
go back to reference  , Discrete series for semisimple symmetric spacdes, Ann of Math. 111 (1980), 253–311.  , Discrete series for semisimple symmetric spacdes, Ann of Math. 111 (1980), 253–311.
[9]
go back to reference G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, 1995.MATH G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, 1995.MATH
[10]
go back to reference R. Goodman and N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, Cambridge, Reprinted with corrections 2003. R. Goodman and N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, Cambridge, Reprinted with corrections 2003.
[11]
go back to reference M. Gordina, Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group, Potential Analysis 12 (2000), 325–357. M. Gordina, Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group, Potential Analysis 12 (2000), 325–357.
[12]
go back to reference B. C. Hall, The Segal-Bargmann transform for compact Lie groups, J. Funct. Anal. 143 (1994), 103–151. B. C. Hall, The Segal-Bargmann transform for compact Lie groups, J. Funct. Anal. 143 (1994), 103–151.
[13]
go back to reference  , Holomorphic methods in analysis and mathematical physics, in: First Summer School in Analysis and Mathematical Physics, Contemp. Math., Vol. 260, Amer. Math. Soc., Providence, RI, 2000, 1–59.  , Holomorphic methods in analysis and mathematical physics, in: First Summer School in Analysis and Mathematical Physics, Contemp. Math., Vol. 260, Amer. Math. Soc., Providence, RI, 2000, 1–59.
[14]
go back to reference  , Harmonic analysis with respect to the heat kernel measure, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 43–78.  , Harmonic analysis with respect to the heat kernel measure, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 43–78.
[15]
go back to reference  , The range of the heat operator, in: Ed.: J. Jorgensen and L. Walling, The Ubiquitous Heat Kernel, 203–231, Contemp. Math., 398, AMS, 2006.  , The range of the heat operator, in: Ed.: J. Jorgensen and L. Walling, The Ubiquitous Heat Kernel, 203–231, Contemp. Math., 398, AMS, 2006.
[16]
go back to reference B. C. Hall and J. Mitchell, The Segal-Bargmann transform for noncompact symmetric spaces of the complex type, J. Funct. Anal. 227 (2005), 338–371. B. C. Hall and J. Mitchell, The Segal-Bargmann transform for noncompact symmetric spaces of the complex type, J. Funct. Anal. 227 (2005), 338–371.
[17]
go back to reference B. C. Hall and A. N. Sengupta, The Segal-Bargmann transform for path-groups, J. Funct. Anal. 152 (1998), 220–254. B. C. Hall and A. N. Sengupta, The Segal-Bargmann transform for path-groups, J. Funct. Anal. 152 (1998), 220–254.
[18]
go back to reference S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Amer. Math. Soc., Providence, RI, 2001.MATH S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Amer. Math. Soc., Providence, RI, 2001.MATH
[19]
go back to reference  , Groups and Geometric Analysis, Amer. Math. Soc., Providence, RI, 2000.  , Groups and Geometric Analysis, Amer. Math. Soc., Providence, RI, 2000.
[20]
go back to reference J. Hilgert and G. Zhang, Segal-Bargmann and Weyl transforms on compact Lie groups, Monatsh. Math. 158 (2009), 285–305. J. Hilgert and G. Zhang, Segal-Bargmann and Weyl transforms on compact Lie groups, Monatsh. Math. 158 (2009), 285–305.
[21]
go back to reference B. Krötz and R. Stanton, Holomorphic extension of representation (II): Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), no. 1, 190–245. B. Krötz and R. Stanton, Holomorphic extension of representation (II): Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), no. 1, 190–245.
[22]
go back to reference B. Krötz, G. Ólafsson and R. Stanton, The Image of the Heat Kernel Transform on Riemannian Symmetric Spaces of the Noncompact Type, Int. Math. Res. Not. 22 (2005), 1307–1329. B. Krötz, G. Ólafsson and R. Stanton, The Image of the Heat Kernel Transform on Riemannian Symmetric Spaces of the Noncompact Type, Int. Math. Res. Not. 22 (2005), 1307–1329.
[23]
go back to reference M. Lassalle, Séries de Laurent des fonctions holomorphes dans la complexification d’un espace symétrique compact, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 167–210. M. Lassalle, Séries de Laurent des fonctions holomorphes dans la complexification d’un espace symétrique compact, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 167–210.
[24]
go back to reference L. Natarajan, E. Rodr\(\acute{i}\)quez-Carrington and J. A. Wolf, The Bott-Borel-Weil Theorem for direct limit Lie groups, Trans. Amer. Math. Soc. 353 (2001), 4583–4622. L. Natarajan, E. Rodr\(\acute{i}\)quez-Carrington and J. A. Wolf, The Bott-Borel-Weil Theorem for direct limit Lie groups, Trans. Amer. Math. Soc. 353 (2001), 4583–4622.
[25]
go back to reference K-H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, 28. Walter de Gruyter & Co., Berlin, 2000 K-H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, 28. Walter de Gruyter & Co., Berlin, 2000
[26]
go back to reference G. Ólafsson, Analytic Continuation in Representation Theory and Harmonic Analysis, in: Global Analysis and Harmonic Analysis, ed. J. P. Bourguignon, T. Branson, and O. Hijazi. S\(\acute{e}\) minares et Congr, Vol 4, (2000), 201–233. The French Math. Soc. G. Ólafsson, Analytic Continuation in Representation Theory and Harmonic Analysis, in: Global Analysis and Harmonic Analysis, ed. J. P. Bourguignon, T. Branson, and O. Hijazi. S\(\acute{e}\) minares et Congr, Vol 4, (2000), 201–233. The French Math. Soc.
[27]
go back to reference G. Ólafsson and H. Schlichtkrull, The Segal-Bargmann transform for the heat equation associated with root systems, Adv. Math. 208 (1) (2007), 422–437. G. Ólafsson and H. Schlichtkrull, The Segal-Bargmann transform for the heat equation associated with root systems, Adv. Math. 208 (1) (2007), 422–437.
[28]
go back to reference  , Representation theory, Radon transform and the heat equation on a Riemannian symmetric space. Group Representations, Ergodic Theory, and Mathematical Physics; A Tribute to George W. Mackey. in: Contemp. Math., 449 (2008), 315–344.  , Representation theory, Radon transform and the heat equation on a Riemannian symmetric space. Group Representations, Ergodic Theory, and Mathematical Physics; A Tribute to George W. Mackey. in: Contemp. Math., 449 (2008), 315–344.
[29]
go back to reference  , Fourier transforms of spherical distributions on compact symmetric spaces. To appear in Math. Scand.  , Fourier transforms of spherical distributions on compact symmetric spaces. To appear in Math. Scand.
[30]
go back to reference G. Ólafsson and B. Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics (Clausthal, 1995), 3–14, World Sci. Publ., River Edge, NJ, 1996. G. Ólafsson and B. Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics (Clausthal, 1995), 3–14, World Sci. Publ., River Edge, NJ, 1996.
[31]
go back to reference G. Ólafsson and J. A. Wolf, Weyl group invariants and application to spherical harmonic analysis on symmetric spaces. Preprint, arXiv:0901.4765. G. Ólafsson and J. A. Wolf, Weyl group invariants and application to spherical harmonic analysis on symmetric spaces. Preprint, arXiv:0901.4765.
[32]
go back to reference  , Extension of Symmetric Spaces and Restriction of Weyl Groups and Invariant Polynomials, to appear in Contemporary Mathematics.  , Extension of Symmetric Spaces and Restriction of Weyl Groups and Invariant Polynomials, to appear in Contemporary Mathematics.
[33]
go back to reference E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121. E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121.
[34]
go back to reference I. E. Segal, Mathematical Problems of Relatvistic Physics, (Ed. M. Kac) Lectures in Applied Mathematics 2, AMS, 1963. I. E. Segal, Mathematical Problems of Relatvistic Physics, (Ed. M. Kac) Lectures in Applied Mathematics 2, AMS, 1963.
[35]
go back to reference A. R. Sinton, The spherical transform on projective limits of symmetric spaces, J. Lie Theory 17 (2007), no. 4, 869–898. A. R. Sinton, The spherical transform on projective limits of symmetric spaces, J. Lie Theory 17 (2007), no. 4, 869–898.
[36]
go back to reference H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Prog. Math. 49. Birkhäuser, Boston, 1984. H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Prog. Math. 49. Birkhäuser, Boston, 1984.
[37]
go back to reference M. Stenzel, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165 (1999), 44–58. M. Stenzel, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165 (1999), 44–58.
[38]
go back to reference M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka Math. J. 8 (1971), 33–47. M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka Math. J. 8 (1971), 33–47.
[39]
go back to reference S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J. Funct. Anal. 251 (2007), 438–462. S. Thangavelu, Holomorphic Sobolev spaces associated to compact symmetric spaces, J. Funct. Anal. 251 (2007), 438–462.
[40]
go back to reference N. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, 1973. N. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, 1973.
[41]
go back to reference K. Wiboonton, The Segal-Bargmann Transform on Inductive Limits of Compact Symmetric Spaces, Ph.D. Thesis, LSU, 2009. K. Wiboonton, The Segal-Bargmann Transform on Inductive Limits of Compact Symmetric Spaces, Ph.D. Thesis, LSU, 2009.
[43]
go back to reference  , Harmonic analysis on commutative spaces, Math. Surveys & Monographs Vol. 142, Amer. Math. Soc., 2007.  , Harmonic analysis on commutative spaces, Math. Surveys & Monographs Vol. 142, Amer. Math. Soc., 2007.
[44]
go back to reference  , Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces, in: Developments and Trends in Infinite Dimensional Lie Theory, eds. K.-H. Neeb and A. Pianzola, Birkhäuser, to appear in 2009.  , Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces, in: Developments and Trends in Infinite Dimensional Lie Theory, eds. K.-H. Neeb and A. Pianzola, Birkhäuser, to appear in 2009.
Metadata
Title
The Segal–Bargmann Transform on Compact Symmetric Spaces and Their Direct Limits
Authors
Gestur Ólafsson
Keng Wiboonton
Copyright Year
2013
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-7193-6_11

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