Let \((M,g)\) be a closed Riemannian manifold. Its Laplace-Beltrami spectrum \(\mathrm{Spec}_\Lambda(g)\) is the set of eigenvalues (counted with multiplicity) of the Laplace-Beltrami operator \(\Delta_g\), its length spectrum \(\mathrm{Spec}_L(g)\) is the set of lengths of the closed geodesics (or the periods of its geodesic flow \((\Phi_t)_{t\ge0}\) on the unitary cotangent space, also counted with multiplicity). These two spectra are related by the so called Poisson relation: the singular support of the distributional trace \(\mathrm{Tr}\left(e^{it\sqrt{\Delta_g}}\right)\) on \(\mathbb{R}_t\) is included in the symmetric length spectrum \(\pm\mathrm{Spec}_L(g)\) [J. J. Duistermaat and V. W. Guillemin, Invent. Math. 29, 39–79 (1975; Zbl 0307.35071)], see also [Y. Colin de Verdiere, Ann. Inst. Fourier 57, No. 7, 2429–2463 (2007; Zbl 1142.35057)]). The problem of the equality of these two spectra is at the heart of this paper, often formulated as the length spectrum \(\mathrm{Spec}_L(g)\) determination by the spectral spectrum \(\mathrm{Spec}_\Lambda(g)\).
Examples of Riemannian spaces with the equality are given by the compact rank-one symmetric spaces (i.e. \(\mathbb{S}^n\), \(\mathbb{R}P^n\), \(\mathbb{C}P^n\), \(\mathbb{R}H^n\) and \(\mathbb{C}a^2\)), the similar case of Zoll manifolds and, at opposite, by bumpy metrics (which may have a simple length spectrum). This paper gives a large collection of other examples, built in the framework of compact Lie groups and homogenous spaces.
The analysis of the distribution \(\mathrm{Tr}\left(e^{it\sqrt{\Delta_g}}\right)\) singularities is done by Duistermaat-Guillemin under a clean hypothesis: a manifold \(M\) is said to be clean if each period \(\tau\) of its geodesic flow \(\Phi_t\) is clean, meaning that the fixed point set \(\mathrm{Fix}(\Phi_\tau)\) is a union of finitely many closed submanifolds \(Z_1,\dots,Z_r\) such that for each \(u\in Z_j\) the fixed point set \(\mathrm{Fix}(T_u\Phi_\tau)\) is equal to the tangent space \(T_uZ_j\).
The first important result asserts that any compact globally symmetric space is clean. However, the example of the (unclean) Berger metrics on \(\mathrm{SO}(3)\) or \(\mathrm{SU}(2)\) indicates the existence of unclean homogenous spaces.
To look at the spectra of a compact symmetric space \((M=G/K,g)\), the author recalls its structure as detailed by O. Loos [Symmetric spaces. I: General theory. II: Compact spaces and classification. New York-Amsterdam: W. A. Benjamin, Inc. (1969; Zbl 0175.48601)]. There are two types of irreducible symmetric spaces, while in general, such a compact symmetric space \((M,g)\) is of the form \(M=\Gamma\backslash(M_0\times M_1\times\dots \times M_q)\) with metric \(g\) induced by the metric \(G(g_0,c_1,\ldots,c_q)=g_0\times c_1g_1\times\ldots\times c_qg_q\) on the cover \(M_0\times M_1\times\ldots \times M_q\) where \(M_0\) is a compact torus with flat metric \(g_0\) and the other factors \(M_j,j=1,\dots,q\) are simply connected compact irreducible symmetric spaces with metric \(g_j\) induced from the Killing form and real positive constants \(c_j,j=1,\dots,q\); the space \(\Gamma\) is a discrete subgroup of the center of \(M_0\times M_1\times\ldots \times M_q\). Let \(\mathcal{R}_{\mathrm{sym}}(M)=\mathcal{S}^+(\mathrm{dim}M_0)\times \mathbb{R}_+^q\) be the space of symmetric metrics on \(M\) with \(\mathcal{S}^+(d)\) the space of positive real symmetric matrices of order \(d\).
The main result of the paper states the existence of a residual set in \(\mathcal{R}_{\mathrm{sym}}(M)\) such that the compact symmetric space \((M,G(g_0,c_1,\dots,c_q))\) obeys the Poisson relation equality. Moreover the author gives a list of spaces for which the Poisson relation equality holds for any parameter \((g_0,c_1,\dots,c_q)\in\mathcal{R}_{\mathrm{sym}}(M)\), this list containing irreducible \(M\). As corollary, similar results are valid for compact spaces with bi-invariant metrics.
The proof strategy relies on the wave invariants \((\mathrm{Wave}_k(\tau))_{k\in\mathbb{N}}\), which appear in the singularity expansion of the wave trace in the neighborhood of each geodesic period and which has been calculated by Duistermaat-Guillemin for clean manifolds. The classification of symmetric spaces and the structure of their root systems are heavily used to handle this calculations.