On the generic spectrum of a riemannian cover
Annales de l'Institut Fourier, Volume 40 (1990) no. 2, p. 407-442

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.

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.

     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://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, Volume 40 (1990) no. 2, pp. 407-442. doi : 10.5802/aif.1219. http://www.numdam.org/item/AIF_1990__40_2_407_0/

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