A theory of Stiefel harmonics
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- by Stephen S. Gelbart
- Trans. Amer. Math. Soc. 192 (1974), 29-50
- DOI: https://doi.org/10.1090/S0002-9947-1974-0425519-8
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Abstract:
An explicit theory of special functions is developed for the homogeneous space $SO(n)/SO(n - m)$ generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on $n \times m$ matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of $SO(n)$ and the polynomial representations of $GL(m,{\mathbf {C}})$.References
- S. Bochner, Theta relations with spherical harmonics, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 804–808. MR 48633, DOI 10.1073/pnas.37.12.804 H. Boerner, Representations of groups, North-Holland, Amsterdam, 1963. E. Cartan, Sur la determination d’un système orthogonal complet dans un espace de Riemann symétrique clos, Rend. Circ. Mat. Palermo 53 (1929), 217-252.
- Stephen S. Gelbart, Harmonics on Stiefel manifolds and generalized Hankel transforms, Bull. Amer. Math. Soc. 78 (1972), no. 3, 451–455. MR 480872, DOI 10.1090/S0002-9904-1972-12941-0
- Stephen Gelbart, Holomorphic discrete series for the real symplectic group, Invent. Math. 19 (1973), 49–58. MR 320231, DOI 10.1007/BF01418850
- Sigurđur Helgason, Invariants and fundamental functions, Acta Math. 109 (1963), 241–258. MR 166304, DOI 10.1007/BF02391814
- Carl S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474–523. MR 69960, DOI 10.2307/1969810
- Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- Daniel A. Levine, Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493–522. MR 412743, DOI 10.1090/S0002-9947-1969-0412743-1
- Hans Maass, Spherical functions and quadratic forms, J. Indian Math. Soc. (N.S.) 20 (1956), 117–162. MR 86837
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- E. M. Stein, Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 173–189. MR 0578903 R. Strichartz, The Fourier decomposition of ${L^2}(SO(n)/SO(n - m))$ (to appear).
- André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012
- Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 192 (1974), 29-50
- MSC: Primary 43A85; Secondary 22E45, 33A75
- DOI: https://doi.org/10.1090/S0002-9947-1974-0425519-8
- MathSciNet review: 0425519