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Session S33 - Spectral Geometry

Tuesday, July 13, 13:30 ~ 13:50 UTC-3

Spectra and representation equivalence for compact symmetric spaces of rank one.

Roberto J. Miatello

FaMAF, Universidad Nacional de Córdoba, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be a compact Lie group, $K$ a closed subgroup, and let $M=G/K$ endowed with a $G$-invariant Riemannian metric. Given a finite dimensional representation $(\tau,W_\tau)$ of $K$ let $E_\tau$ be the associated hermitian $G$-homogeneous vector bundle over $M$. There is a self-adjoint, second order, elliptic differential operator $\Delta_\tau$ acting on smooth sections of $E_\tau$, defined by the Casimir element $C$ of $G$.

Given a finite subgroup $\Gamma$ of $G$, the quotient $\Gamma\backslash M$ is a compact good orbifold, with a manifold structure in case $\Gamma$ acts freely on $M$. Then $\Gamma \backslash M$ inherits a Riemannian metric on $\Gamma\backslash M$ and $E_\tau$ naturally induces a vector bundle $E_{\tau,\Gamma}$ over $\Gamma\backslash M$, whose sections are identified with the $\Gamma$-invariant sections of $E_\tau$.

The spectrum of the operator $\Delta_{\tau,\Gamma}$, given by the restriction of $\Delta_\tau$ to the space of $\Gamma$-invariant smooth sections of $E_{\tau}$ -- called the $\tau$-spectrum of $\Gamma\backslash M$ -- can be expressed in Lie theoretical terms. Indeed, if we decompose $L^2(\Gamma \backslash G)_\tau =\sum_{\pi \in \widehat G_\tau} n_\Gamma(\pi)V_\pi$, the multiplicity of $\lambda$ in the spectrum of $\Delta_{\tau,\Gamma}$ has a simple linear expression in terms of the multiplicities $n_\Gamma(\pi)$ with $\pi \in \widehat G_\tau$. In particular, the representation of $G$ in $L^2(\Gamma\backslash G)_\tau$ determines the spectrum of the operator $\Delta_{\tau,\Gamma}$. The converse question arises, whether the $\tau$-spectrum $\Gamma\backslash M$ determines the representation $L^2(\Gamma\backslash G)_\tau$ of $G$.

Jointly with Emilio Lauret we have studied the case of compact symmetric spaces of rank one, giving conditions on $G$, $K$ and $\tau$ so that the $\tau$-spectrum determines the representation of $G$ on $L^2(\Gamma\backslash G)_\tau$. In particular, we show the existence of infinitely many $\tau \in \widehat K$ so that the representation-spectral converse holds.

We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $\tau$ of the representation $\tau_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_C^*M$. We show that for such $\tau$, in most cases $\tau$-isospectrality implies $\tau$-representation equivalence.

Joint work with Emilio A. Lauret (Universidad Nacional del Sur, Bahía Blanca, Argentina).

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