On the generic spectrum of a riemannian cover

Zelditch, Steven

Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 407-442.

Soient $M$ une variété compacte et $G$ un groupe fini opérant librement sur $M$, et soit ${ℳ}_G$ 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 dim$L\ge$ dim$V$ 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 ${ℳ}_G$ 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$M\ge$ dim$V$ for all irreducibles $V$ of $G$. As an application, we construct isospectral manifolds with simple eigenvalue spectra.

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