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Transactions of the American Mathematical Society

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Stability of group representations and Haar spectrum

Authors: Robert Azencott and William Parry
Journal: Trans. Amer. Math. Soc. 172 (1972), 317-327
MSC: Primary 22D10; Secondary 28A65
MathSciNet review: 0310128
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Abstract: If $ U$ and $ V$ are commuting unitary representations of locally compact abelian groups $ S$ and $ T$, new representations of $ S$ (perturbations of $ U$) can be obtained from composition with images of $ U$ in $ V$. If most of these representations are equivalent to $ U,U$ is said to be $ V$ stable. We investigate conditions which, together with stability, ensure that $ U$ has (uniform) Haar spectrum. The principal applications are to dynamical systems which possess auxiliary groups with respect to which motion is stable.

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  • [1] W. Parry, Spectral analysis of 𝐺-extensions of dynamical systems, Topology 9 (1970), 217–224. MR 0261581,
  • [2] L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #4841.
  • [3] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 1928; reprint, Dover, New York, 1950.
  • [4] G. W. Mackey, The theory of group representations. Three volumes, Dept. of Matyh., Univ. of Chicago, Chicago, Ill., 1955. Lecture notes (Summer, 1955) prepared by Dr. Fell and Dr. Lowdenslager. MR 0086063
  • [5] O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspehi Matem. Nauk (N.S.) 8 (1953), no. 3(55), 125–126 (Russian). MR 0058883
  • [6] A. I. Plesner and V. A. Rohlin, Spectral theory of linear operators. II, Uspehi Matem. Nauk (N.S.) 1(11) (1946), no. 1, 71–191 (Russian). MR 0021245
  • [7] Paul R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1957) edition. MR 1653399
  • [8] A. Weil, Intégration dans les groupes topologiques et ses applications, 2nd ed., Actualités Sci. Indust., no 869, Hermann, Paris 1951.
  • [9] Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215
  • [10] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480

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Keywords: Representations stable with respect to another, representations very stable with respect to another, spectral multiplicity, uniform multiplicity, Haar spectrum, maximal spectral type, horocycle flow, nilflow, Weyl commutation relation
Article copyright: © Copyright 1972 American Mathematical Society

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