On the generic spectrum of a riemannian cover
Annales de l'Institut Fourier, Tome 40 (1990) no. 2, p. 407-442
Soient M une variété compacte et G un groupe fini opérant librement sur M, et soit G l’espace (de Fréchet) des métriques G-invariantes sur M. Il est naturel de conjecturer que, pour une métrique générique, tous les espaces propres du laplacien sont irréductibles, en tant que représentations orthogonales de G. (Dans le langage de la physique nous dirions que, génériquement, il n’y a pas de “dégénérescences accidentelles”.) Nous prouvons cette conjecture lorsque dimL dimV pour toutes les représentations irréductibles de G. Comme application, nous construisons des variétés isospectrales à spectres simples.
Let M be a compact manifold let G be a finite group acting freely on M, and let G be the (Fréchet) space of G-invariant metric on M. A natural conjecture is that, for a generic metric in 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 dimM dimV for all irreducibles V of G. As an application, we construct isospectral manifolds with simple eigenvalue spectra.
@article{AIF_1990__40_2_407_0,
     author = {Zelditch, Steven},
     title = {On the generic spectrum of a riemannian cover},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {40},
     number = {2},
     year = {1990},
     pages = {407-442},
     doi = {10.5802/aif.1219},
     zbl = {0722.58044},
     mrnumber = {91g:58294},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_1990__40_2_407_0}
}
Zelditch, Steven. On the generic spectrum of a riemannian cover. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 407-442. doi : 10.5802/aif.1219. http://www.numdam.org/item/AIF_1990__40_2_407_0/

[Al] J. H. Albert, Generic properties of eigenfunctions of elliptic partial differential equations, Trans. Am. Math. Soc., 238 (1978), 341-354. | MR 57 #10743 | Zbl 0379.35023

[Ad] J. F. Adams, Lectures on Lie Groups, Univ. Chicago Press, 1969. | MR 40 #5780 | Zbl 0206.31604

[Ar] V. I. Arnold, Modes and Quasimodes, Fun. Anal. and App., 6 (1972), 44. | MR 45 #6331 | Zbl 0251.70012

[BaUr] S. Bando and H. Urakawa, Generic properties of the eigenvalues of the Laplacian for compact riemannian manifolds, Tôhoku Math. J., 35 (1983), 155-172.

[Be1] G. Besson, On the multiplicity of eigenvalues of the Laplacian, SLN 1339, Springer-Verlag, (1988), 32-53. | MR 90b:58264 | Zbl 0708.53040

[Be2] G. Besson, Propriétés génériques des fonctions propres et multiplicités, preprint (1989). | MR 90k:58226 | Zbl 0697.58056

[B1Wi1] D. D. Bleecker and L. C. Wilson, Splitting the spectrum of a Riemannian manifold, SIAM J. Math. Anal., 11 (5) (1980), 813-818. | MR 81j:58077 | Zbl 0449.58021

[BröT-D] T. Bröcker and T. Tom Dieck, Representations of Compact Lie Groups, Grad. Texts, Springer-Verlag, 98 (1985). | MR 86i:22023 | Zbl 0581.22009

[Bro] R. Brooks, Constructing isospectral manifolds, Am. Math. Monthly, 95 (1988), 823-839. | MR 89k:58285 | Zbl 0673.58046

[D] H. Donnelly, G-spaces, the asymptotic splitting of L2(M) into irreducibles, Math. Ann., 237 (1978), 23-40. | MR 80b:58063 | Zbl 0379.53019

[H] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985. | Zbl 0601.35001

[K] A. A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, 1976. | MR 54 #447 | Zbl 0342.22001

[PSa] R. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. P.A.M., 38 (1985), 853-866. | MR 87f:11035 | Zbl 0614.10027

[SeT] H. Seifert and W. Threlfall, A textbook of topology, Academic Press, 1980.

[Su] Sunada T., Riemannian coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186. | MR 86h:58141 | Zbl 0585.58047

[U] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. | MR 57 #4264 | Zbl 0355.58017

[Wig] E. P. Wigner, Group Theory and its Applications to the Quantum, Mechanics of Atomic Spectra, Academic Press, 1959. | Zbl 0085.37905

[Z] S. Zelditch, Isospectrality in the category of Fourier integral operators I, preprint (1990). | Zbl 0769.53026