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Lie group representations and harmonic polynomials of a matrix variable


Author: Tuong Ton-That
Journal: Trans. Amer. Math. Soc. 216 (1976), 1-46
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1976-0399366-1
MathSciNet review: 0399366
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Abstract: The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called ``classical separation of variables theorem'' for the symmetric algebra is obtained.

The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations.

The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.


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  • [1] A. Borel, Linear algebraic groups, Math. Lecture Notes Ser., Benjamin, New York and Amsterdam, 1969. MR 40 #4273. MR 0251042 (40:4273)
  • [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] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203-248. MR 19, 681. MR 0089473 (19:681d)
  • [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.
  • [6] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin and New York, 1971. MR 0499948 (58:17690)
  • [7] S. Gelbart, A theory of Stiefel harmonics, Trans. Amer. Math. Soc. 192 (1974), 29-50. MR 0425519 (54:13474)
  • [8] K. I. Gross and R. A. Kunze, Fourier decompositions of certain representations, Symmetric Spaces, Dekker, New York, 1972, pp. 119-139. MR 0427541 (55:572)
  • [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. MR 18, 809. MR 0084104 (18:809d)
  • [11] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [12] -, Invariants and fundamental functions, Acta Math. 109 (1963), 241-258. MR 29 #3581. MR 0166304 (29:3581)
  • [13] C. S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474-523. MR 16, 1107. MR 0069960 (16:1107e)
  • [14] K. Hoffman and R. Kunze, Linear algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 43 #1998. MR 0276251 (43:1998)
  • [15] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404. MR 28 #1252. MR 0158024 (28:1252)
  • [16] -, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 26 #265. MR 0142696 (26:265)
  • [17] D. Levine, Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493-522. MR 0412743 (54:864)
  • [18] H. Maass, Zur Theorie der harmonischen Formen, Math. Ann. 137 (1959), 142-149. MR 22 #12250. MR 0121512 (22:12250)
  • [19] -, Spherical functions and quadratic forms, J. Indian Math. Soc. 20 (1956), 117-162. MR 19, 252. MR 0086837 (19:252a)
  • [20] M. A. Naĭmark, Normed rings, 2nd rev. ed., ``Nauka", Moscow, 1968; English transl., Wolters-Noordhoff, Groningen, 1972.
  • [21] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., no. 32, Princeton Univ. Press, Princeton, N. J., 1971. MR 46 #4102. MR 0304972 (46:4102)
  • [22] R. Strichartz, The explicit Fourier decomposition of $ {L^2}(SO(n)/SO(n - m))$, Canad. J. Math. 27 (1975), 294-310. MR 0380277 (52:1177)
  • [23] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Ph.D Dissertation, University of California, Irvine, 1974.
  • [24] N. Ja. Vilenkin, Special functions and the theory of group representations, ``Nauka", Moscow, 1965; English transl., Transl. Math. Monographs, vol. 22, Amer. Math. Soc., Providence, R. I., 1968. MR 35 #420; 37 #5429. MR 0209523 (35:420)
  • [25] N. R. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107-113. MR 43 #7558. MR 0281844 (43:7558)
  • [26] H. Weyl, The classical groups. Their invariants and representations, 2nd ed., Princeton Univ. Press, Princeton, N. J., 1946. MR 1488158 (98k:01049)
  • [27] D. P. Želobenko, The theory of linear representations of complex and real Lie groups, Trudy Moskov. Mat. Obšč. 12 (1963), 53-98 = Trans. Moscow Math. Soc. 1963, 57-110. MR 29 #2330. MR 0165039 (29:2330)
  • [28] -, Classical groups. Spectral analysis of finite-dimensional representations, Uspehi Mat. Nauk 17 (1962), no. 1 (103), 27-120 = Russian Math. Surveys 17 (1962), no. 1, 1-94. MR 25 #129. MR 0136664 (25:129)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0399366-1
Keywords: Lie group representations, harmonic polynomial functions of a matrix variable, irreducible holomorphic representations of the special orthogonal groups, Borel-Weil-Bott theory, symmetric algebras of polynomial functions, rings of differential operators, G-harmonic polynomials, "generalized separation of variables'' theorem, theory of polynomial invariants, generalized spherical harmonics, Stiefel manifolds, Fourier transforms, generalized Hankel transforms
Article copyright: © Copyright 1976 American Mathematical Society

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