Spectra for Gelfand pairs associated with the Heisenberg group

Chal Benson; Joe Jenkins; Gail Ratcliff; Tefera Worku

Colloquium Mathematicae (1996)

  • Volume: 71, Issue: 2, page 305-328
  • ISSN: 0010-1354

Abstract

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Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group H n . We say that ( K , H n ) is a Gelfand pair when the set L K 1 ( H n ) of integrable K-invariant functions on H n is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for L K 1 ( H n ) can be identified with the set Δ ( K , H n ) of bounded K-spherical functions on H n . In this paper, we study the natural topology on Δ ( K , H n ) given by uniform convergence on compact subsets in H n . We show that Δ ( K , H n ) is a complete metric space and that the ’type 1’ K-spherical functions are dense in Δ ( K , H n ) . Our main result shows that one can embed Δ ( K , H n ) quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ( K ⋉ H n )-invariant differential operators on H n . This viewpoint on the spectrum for Δ ( K , H n ) was previously known for K=U(n) and is referred to as ’the Heisenberg fan’.

How to cite

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Benson, Chal, et al. "Spectra for Gelfand pairs associated with the Heisenberg group." Colloquium Mathematicae 71.2 (1996): 305-328. <http://eudml.org/doc/210444>.

@article{Benson1996,
abstract = {Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L^1_K(H_n)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L^1_K(H_n)$ can be identified with the set $Δ(K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $Δ(K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $Δ(K,H_n)$ is a complete metric space and that the ’type 1’ K-spherical functions are dense in $Δ(K,H_n)$. Our main result shows that one can embed $Δ(K,H_n)$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ($K ⋉ H_n$)-invariant differential operators on $H_n$. This viewpoint on the spectrum for $Δ(K,H_n)$ was previously known for K=U(n) and is referred to as ’the Heisenberg fan’.},
author = {Benson, Chal, Jenkins, Joe, Ratcliff, Gail, Worku, Tefera},
journal = {Colloquium Mathematicae},
keywords = {Gelfand pairs; spherical function; Heisenberg group; left invariant differential operators},
language = {eng},
number = {2},
pages = {305-328},
title = {Spectra for Gelfand pairs associated with the Heisenberg group},
url = {http://eudml.org/doc/210444},
volume = {71},
year = {1996},
}

TY - JOUR
AU - Benson, Chal
AU - Jenkins, Joe
AU - Ratcliff, Gail
AU - Worku, Tefera
TI - Spectra for Gelfand pairs associated with the Heisenberg group
JO - Colloquium Mathematicae
PY - 1996
VL - 71
IS - 2
SP - 305
EP - 328
AB - Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L^1_K(H_n)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L^1_K(H_n)$ can be identified with the set $Δ(K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $Δ(K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $Δ(K,H_n)$ is a complete metric space and that the ’type 1’ K-spherical functions are dense in $Δ(K,H_n)$. Our main result shows that one can embed $Δ(K,H_n)$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ($K ⋉ H_n$)-invariant differential operators on $H_n$. This viewpoint on the spectrum for $Δ(K,H_n)$ was previously known for K=U(n) and is referred to as ’the Heisenberg fan’.
LA - eng
KW - Gelfand pairs; spherical function; Heisenberg group; left invariant differential operators
UR - http://eudml.org/doc/210444
ER -

References

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  1. [1] C. Benson, J. Jenkins and G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85-116. Zbl0704.22006
  2. [2] C. Benson, J. Jenkins and G. Ratcliff, Bounded K-spherical functions on Heisenberg groups, J. Funct. Anal. 105 (1992), 409-443. Zbl0763.22007
  3. [3] C. Benson and G. Ratcliff, A classification for multiplicity free actions, J. Algebra 181 (1996), 152-186. Zbl0869.14021
  4. [4] P. Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup. 14 (1981), 403-432. Zbl0488.60013
  5. [5] G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. Ser. B (7) (1987), 1091-1105. Zbl0632.22005
  6. [6] J. Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 305-317. Zbl0100.32303
  7. [7] J. Faraut, Analyse harmonique et fonctions spéciales, in: J. Faraut et K. Harzallah, Deux courses d'analyse harmonique, Progr. Math. 69, Birkhäuser, Boston, 1987, 1-151. 
  8. [8] G. Folland, Harmonic Analysis in Phase Space, Ann. Math. Stud. 122, Princeton Univ. Press, Princeton, N.J., 1989. Zbl0682.43001
  9. [9] R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer, New York, 1988. Zbl0675.43004
  10. [10] R. Godement, A theory of spherical functions I, Trans. Amer. Math. Soc. 73 (1962), 496-556. Zbl0049.20103
  11. [11] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. 
  12. [12] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565-619. Zbl0733.20019
  13. [13] A. Hulanicki and F. Ricci, A tauberian theorem and tangential convergence of bounded harmonic functions on balls in ℂ n , Invent. Math. 62 (1980), 325-331. Zbl0449.31008
  14. [14] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213. Zbl0431.17007
  15. [15] M. Naimark, Normed Rings, Noordhoff, Groningen, 1964. 
  16. [16] A. l. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer, New York, 1990. 
  17. [17] R. Strichartz, L p harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350-406. Zbl0734.43004
  18. [18] E. G. F. Thomas, An infinitesimal characterization of Gelfand pairs, in: Proc. Conf. in Modern Analysis and Probability in honor of Shizuo Kakutani, New Haven, Conn., 1982, Contemp. Math. 26, Amer. Math. Soc., Providence, R.I., 1984, 379-385. 
  19. [19] Z. Yan, Special functions associated with multiplicity-free representations, preprint. 

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