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Tuong Ton-That

Lie group representations and harmonic polynomials of a matrix variable. (English) Zbl 0287.22014

Trans. Am. Math. Soc. 216, 1-46 (1976).
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Cited in 1 Review
Cited in 16 Documents

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
58C99 Calculus on manifolds; nonlinear operators
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References:

[1] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0186.33201
[2] A. Borel and A. Weil, Séminaire Bourbaki 6ième année: 1953/54, Exposé 100 par J.-P. Serre: Répresentations linéaires et espaces homogènes Kähleriens des groupes de Lie compacts, Secrétariat mathématique, Paris, 1959. MR 28 #1087.
[3] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203 – 248. · Zbl 0094.35701 · doi:10.2307/1969996
[4] É. Cartan, Leçons sur la géométrie projective complexe, 2ième éd., Gauthier-Villars, Paris, 1950. MR 12, 849.
[5] C. Chevalley, Theory of Lie groups. Vol. I, Princeton Math. Ser., vol. 8, Princeton Univ. Press. Princeton, N. J., 1946. MR 7, 412. · Zbl 0063.00842
[6] Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. · Zbl 0224.43006
[7] Stephen S. Gelbart, A theory of Stiefel harmonics, Trans. Amer. Math. Soc. 192 (1974), 29 – 50. · Zbl 0292.22016
[8] K. I. Gross and R. A. Kunze, Fourier decompositions of certain representations, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969 – 1970), Dekker, New York, 1972, pp. 119 – 139. Pure and Appl. Math., Vol. 8. · Zbl 0268.43010
[9] -, Bessel functions and representation theory. I (in preparation).
[10] Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87 – 120. · Zbl 0072.01901 · doi:10.2307/2372387
[11] SigurÄ’ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962.
[12] SigurÄ’ur Helgason, Invariants and fundamental functions, Acta Math. 109 (1963), 241 – 258. · Zbl 0112.26303 · doi:10.1007/BF02391814
[13] Carl S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474 – 523. · Zbl 0066.32002 · doi:10.2307/1969810
[14] Kenneth Hoffman and Ray Kunze, Linear algebra, Second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.
[15] Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327 – 404. · Zbl 0124.26802 · doi:10.2307/2373130
[16] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329 – 387. · Zbl 0134.03501 · doi:10.2307/1970237
[17] Daniel A. Levine, Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493 – 522. · Zbl 0196.15803
[18] Hans Maass, Zur Theorie der harmonischen Formen, Math. Ann. 137 (1959), 142 – 149 (German). · Zbl 0083.06002 · doi:10.1007/BF01343242
[19] Hans Maass, Spherical functions and quadratic forms, J. Indian Math. Soc. (N.S.) 20 (1956), 117 – 162. · Zbl 0072.08401
[20] M. A. Naĭmark, Normed rings, 2nd rev. ed., “Nauka”, Moscow, 1968; English transl., Wolters-Noordhoff, Groningen, 1972. · Zbl 0089.10102
[21] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. · Zbl 0232.42007
[22] Robert S. Strichartz, The explicit Fourier decomposition of \?²(\?\?(\?)/\?\?(\?-\?)), Canad. J. Math. 27 (1975), 294 – 310. · Zbl 0275.43009 · doi:10.4153/CJM-1975-036-x
[23] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Ph.D Dissertation, University of California, Irvine, 1974.
[24] Специал\(^{\приме}\)ные функции и теория представлений ��рупп, Издат. ”Наука”, Мосцощ, 1965 (Руссиан). · Zbl 0144.38003
[25] Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations., Trans. Amer. Math. Soc. 158 (1971), 107 – 113. · Zbl 0227.22016
[26] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
[27] D. P. Želobenko, On the theory of representations of complex and real Lie groups, Trudy Moskov. Mat. Obšč. 12 (1963), 53 – 98 (Russian).
[28] D. P. Želobenko, Classical groups. Spectral analysis of finite-dimensional representations, Uspehi Mat. Nauk 17 (1962), no. 1 (103), 27 – 120 (Russian).
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