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Colloquium Mathematicum

1996 | 71 | 2 | 305-328
Tytuł artykułu

Spectra for Gelfand pairs associated with the Heisenberg group

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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'.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
305-328
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-03-01
Twórcy
autor
• Department of Mathematics, and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121, U.S.A.
autor
• Deptartment of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222, U.S.A.
autor
• Department of Mathematics, and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121, U.S.A.
autor
• Deptartment of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222, U.S.A.
Bibliografia
• [1] C. Benson, J. Jenkins and G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85-116.
• [2] C. Benson, J. Jenkins and G. Ratcliff, Bounded K-spherical functions on Heisenberg groups, J. Funct. Anal. 105 (1992), 409-443.
• [3] C. Benson and G. Ratcliff, A classification for multiplicity free actions, J. Algebra 181 (1996), 152-186.
• [4] P. Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup. 14 (1981), 403-432.
• [5] G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. Ser. B (7) (1987), 1091-1105.
• [6] J. Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 305-317.
• [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] G. Folland, Harmonic Analysis in Phase Space, Ann. Math. Stud. 122, Princeton Univ. Press, Princeton, N.J., 1989.
• [9] R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer, New York, 1988.
• [10] R. Godement, A theory of spherical functions I, Trans. Amer. Math. Soc. 73 (1962), 496-556.
• [11] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.
• [12] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565-619.
• [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.
• [14] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213.
• [15] M. Naimark, Normed Rings, Noordhoff, Groningen, 1964.
• [16] A. l. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer, New York, 1990.
• [17] R. Strichartz, $L^p$ harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350-406.
• [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] Z. Yan, Special functions associated with multiplicity-free representations, preprint.
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Bibliografia
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