On the generic spectrum of a riemannian cover

Steven Zelditch

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 2, page 407-442
  • ISSN: 0373-0956

Abstract

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Let M be a compact manifold let G be a finite group acting freely on M , and let ℳ G be the (Fréchet) space of G -invariant metric on M . A natural conjecture is that, for a generic metric in ℳ G , all eigenspaces of the Laplacian are irreducible (as orthogonal representations of G ). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim M ≥ dim V for all irreducibles V of G . As an application, we construct isospectral manifolds with simple eigenvalue spectra.

How to cite

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Zelditch, Steven. "On the generic spectrum of a riemannian cover." Annales de l'institut Fourier 40.2 (1990): 407-442. <http://eudml.org/doc/74883>.

@article{Zelditch1990,
abstract = {Let $M$ be a compact manifold let $G$ be a finite group acting freely on $M$, and let $\{\cal M\}_G$ be the (Fréchet) space of $G$-invariant metric on $M$. A natural conjecture is that, for a generic metric in $\{\cal M\}_G$, all eigenspaces of the Laplacian are irreducible (as orthogonal representations of $G$). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim$M\ge $ dim$V$ for all irreducibles $V$ of $G$. As an application, we construct isospectral manifolds with simple eigenvalue spectra.},
author = {Zelditch, Steven},
journal = {Annales de l'institut Fourier},
keywords = {generic spectrum; isospectral manifolds; multiplicity free representation; normal Riemannian cover; Laplacian},
language = {eng},
number = {2},
pages = {407-442},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the generic spectrum of a riemannian cover},
url = {http://eudml.org/doc/74883},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Zelditch, Steven
TI - On the generic spectrum of a riemannian cover
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 2
SP - 407
EP - 442
AB - Let $M$ be a compact manifold let $G$ be a finite group acting freely on $M$, and let ${\cal M}_G$ be the (Fréchet) space of $G$-invariant metric on $M$. A natural conjecture is that, for a generic metric in ${\cal M}_G$, all eigenspaces of the Laplacian are irreducible (as orthogonal representations of $G$). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim$M\ge $ dim$V$ for all irreducibles $V$ of $G$. As an application, we construct isospectral manifolds with simple eigenvalue spectra.
LA - eng
KW - generic spectrum; isospectral manifolds; multiplicity free representation; normal Riemannian cover; Laplacian
UR - http://eudml.org/doc/74883
ER -

References

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  9. [Bro] R. BROOKS, Constructing isospectral manifolds, Am. Math. Monthly, 95 (1988), 823-839. Zbl0673.58046MR89k:58285
  10. [D] H. DONNELLY, G-spaces, the asymptotic splitting of L2(M) into irreducibles, Math. Ann., 237 (1978), 23-40. Zbl0379.53019MR80b:58063
  11. [H] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985. Zbl0601.35001
  12. [K] A. A. KIRILLOV, Elements of the Theory of Representations, Springer-Verlag, 1976. Zbl0342.22001MR54 #447
  13. [PSa] R. PHILLIPS and P. SARNAK, The Weyl theorem and the deformation of discrete groups, Comm. P.A.M., 38 (1985), 853-866. Zbl0614.10027MR87f:11035
  14. [SeT] H. SEIFERT and W. THRELFALL, A textbook of topology, Academic Press, 1980. Zbl0469.55001
  15. [Su] SUNADA T., Riemannian coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186. Zbl0585.58047MR86h:58141
  16. [U] K. UHLENBECK, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. Zbl0355.58017MR57 #4264
  17. [Wig] E. P. WIGNER, Group Theory and its Applications to the Quantum, Mechanics of Atomic Spectra, Academic Press, 1959. Zbl0085.37905
  18. [Z] S. ZELDITCH, Isospectrality in the category of Fourier integral operators I, preprint (1990). Zbl0769.53026

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