Higher rank quantum-classical correspondence

For a compact Riemannian locally symmetric space $\Gamma\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $\Gamma$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.


Introduction
Ruelle resonances for an Anosov flow provide a fundamental spectral invariant that reflects not only many important dynamical properties of the flow but also geometric and topological properties of the underlying manifold.Very recently the concept of resonances was extended to higher rank ‫ޒ‬ n -Anosov actions and led to the notion of Ruelle-Taylor resonances 1 which were shown to be a discrete subset σ RT ⊂ ‫ރ‬ n [Bonthonneau et al. 2020].It was furthermore shown in that same paper that the leading resonances (i.e., those with vanishing real part) are related to mixing properties of the considered Anosov action.In particular, it was shown that if the action is weakly mixing in an arbitrary direction of the abelian group ‫ޒ‬ n , then 0 ∈ ‫ރ‬ n is the only leading resonance.Furthermore, the resonant states at zero give rise to equilibrium measures that share properties of Sinai-Ruelle-Bowen (SRB) measures of Anosov flows.
Apart from the leading resonances the spectrum of Ruelle-Taylor resonances has so far not been studied if n ≥ 2. In particular, when n ≥ 2, it was not known whether there are other resonances than the resonance at zero.Neither was it known whether there is a spectral gap, i.e., whether the real parts of the resonances are bounded away from zero.In this article we shed some light on these questions by examining the Ruelle-Taylor resonances for the class of Weyl chamber flows via harmonic analysis.
Let us briefly introduce the setting: Let G be a real connected noncompact semisimple Lie group with finite center and Iwasawa decomposition G = K AN .Let a be the Lie algebra of A and M the centralizer of A in K .Then A is isomorphic to ‫ޒ‬ n , where n is the real rank of G, and acts on G/M from the right.Hence A also acts on the compact manifold M := \G/M, where ≤ G is a cocompact torsion-free lattice.It can be easily seen that this action is an Anosov action with hyperbolic splitting TM = E 0 ⊕ E s ⊕ E u which can be described explicitly in terms of associated vector bundles (see Section 2A for a general definition of Anosov actions and Proposition 3.1 for the description of the hyperbolic splitting for Weyl chamber flows).Furthermore, if ⊆ a * is the set of restricted roots with simple system and positive system + then the positive Weyl chamber is given by a + = {H ∈ a | α(H ) > 0 ∀α ∈ }.
The Ruelle-Taylor resonances of this Anosov action are defined as follows: For H ∈ a let X H be the vector field on M defined by the right A-action.Then (M) satisfying (X H + λ(H ))u = 0 for all H ∈ a are called resonant states of λ and the dimension of the space of all such distributions is called the multiplicity m(λ) of the resonance λ.It has been shown in [Bonthonneau et al. 2020] that σ RT ⊂ a * ‫ރ‬ is discrete and that all resonances have finite multiplicity.It also follows from that work that the real part of the resonances are located in a certain cone − a * ⊂ a * which is the negative dual cone of the positive Weyl chamber a + (see Section 2B for a precise definition).
In this article we will prove that there is a bijection between a certain subset of the Ruelle-Taylor resonant states and certain joint eigenfunctions of the invariant differential operators on the locally symmetric space \G/K .Before explaining this correspondence in more detail we state two results on the spectrum of Ruelle-Taylor resonances that we can conclude from the correspondence.
The first result says that, for any Weyl chamber flow, there exist infinitely many Ruelle-Taylor resonances by providing a Weyl-lower bound on an appropriate counting function.
Then, for d := dim(G/K ), More generally, let ⊆ a * be open and bounded such that ∂ has finite (n−1)-dimensional Hausdorff measure.Then The second result guarantees a uniform spectral gap.
Theorem 1.2.Let G be a real semisimple Lie group with finite center.Then, for any cocompact torsionfree discrete subgroup ⊂ G, there is a neighborhood G ⊂ a * of 0 such that If G furthermore has Kazhdan's property (T) (e.g., if G is simple of higher rank), then the spectral gap G can be taken uniformly in and only depends on the group G.
Let us now explain in some detail the spectral correspondence that is the key to the above results.
We define the space of first band resonant states as those resonant states that are in addition horocyclically invariant: and we call a Ruelle-Taylor resonance a first band resonance if and only if Res 0 X (λ) ̸ = 0.By working with horocycle operators and vector-valued Ruelle-Taylor resonances we will be able to show that all resonances with real part in a certain neighborhood of zero in a * are always first band resonances (see Proposition 3.7).As the Weyl chamber flow is generated by mutually commuting Hamilton flows, we consider the set of Ruelle-Taylor resonances as a classical spectrum.
Let us briefly describe the quantum side: In the rank 1 case the quantization of the geodesic flow is given by the Laplacian on G/K .In the higher rank case we have to consider the algebra of G-invariant differential operators on G/K which we denote by ‫(ބ‬G/K ).As an abstract algebra this is a polynomial algebra with n algebraically independent operators, among them the Laplace operator.These operators descend to \G/K and we can define the joint eigenspace where χ λ is a character of ‫(ބ‬G/K ) parametrized by λ ∈ a * ‫ރ‬ /W with the Weyl group W .Here χ ρ is the trivial character (see Section 2D).Let σ Q denote the corresponding quantum spectrum {λ ∈ a * ‫ރ‬ | E λ ̸ = {0}}.We have the following correspondence between the classical first band resonant states and the joint quantum eigenspace: ‫ރ‬ be outside the exceptional set Then there is a bijection between the finite-dimensional vector spaces where π * is the push-forward of distributions along the canonical projection π : \G/M → \G/K .
Using this one-to-one correspondence we can then use results about the quantum spectrum to obtain obstructions and existence results on the Ruelle-Taylor resonances.Notably we use results of Duistermaat, Kolk and Varadarajan [Duistermaat et al. 1979] on the spectrum σ Q , but we also deduce refined information on the quantum spectrum.Here we use L p -bounds for spherical functions obtained from asymptotic expansions [van den Ban and Schlichtkrull 1987] and L p -bounds for matrix coefficients based on work by Cowling [1979] andOh [2002].Theorems 1.1 and 1.2 as stated above give only a rough version of the information on the Ruelle-Taylor resonances that we can actually obtain.As the full results require some further notation we refrain from stating them in the introduction and refer to Theorem 5.1.We also refer to Figure 6 for a visualization of the structure of first band resonances for the case G = SL(3, ‫.)ޒ‬ Methods and related results.The key ingredient to the quantum-classical correspondence is that we can in a first step relate the horocyclically invariant first band resonant states with distributional vectors in some principal series representations.Then we can apply the Poisson transform of [Kashiwara et al. 1978] to get a bijection onto the quantum eigenspace E −λ−ρ .The prototype of such a quantum-classical correspondence was first established by Dyatlov, Faure and Guillarmou [Dyatlov et al. 2015] in the case of manifolds of constant curvature or in other words for the rank 1 group G = SO(n, 1).Certain central ideas have, however, already been present for G = SO(2, 1) in the work of Cosentino [2005] and Flaminio and Forni [2003].In the rank 1 setting there exist several generalizations of the quantum classical correspondence of [Dyatlov et al. 2015], for example, to convex cocompact manifolds of constant curvature [Guillarmou et al. 2018;Hadfield 2020], general compact locally symmetric spaces of rank 1 [Guillarmou et al. 2021] and vector bundles [Küster and Weich 2020;2021].
Besides the correspondence between the classical Ruelle resonant states and the quantum Laplace eigenvalues there are several other approaches in the literature establishing exact relations between the Laplace spectrum and the geodesic flow.One approach is to relate the Laplace spectrum to divisors of zeta functions.Such relations have been obtained for rank 1 locally symmetric spaces on various levels of generality by Bunke, Olbrich, Patterson and Perry: G = SO(n, 1) and convex cocompact [Bunke and Olbrich 1997;1999;Patterson and Perry 2001]; G real rank 1 and cocompact [Bunke and Olbrich 1995].
A third approach to an exact quantum-classical correspondence is to relate the Laplace spectrum to a transfer operator which represents a time discretized dynamics of the geodesic flow.This type of correspondence was notably studied for hyperbolic surfaces with cusps; see [Bruggeman and Pohl 2023;Bruggeman et al. 2015;Lewis and Zagier 2001] for results for G = SL(2, ‫)ޒ‬ and discrete subgroups of increasing generality.We refer in particular to the expository article [Pohl and Zagier 2020] and the introduction of [Bruggeman and Pohl 2023] for a current state of the art of these techniques.A very first step towards generalizations of this approach to higher rank has been recently achieved in [Pohl 2020] for the Weyl chamber flow on products of Schottky surfaces by the construction of symbolic dynamics and transfer operators.
Note that in [Dyatlov et al. 2015] not only was the first band of Ruelle resonances related to the Laplace spectrum, but a complete band structure has been established and the higher bands can be related to the Laplace spectrum on divergence-free symmetric tensors.In the present article we do not study the higher bands.This will presumably be a very hard question for general semisimple groups G (note that in [Dyatlov et al. 2015] it was crucial at several points that N ∼ = ‫ޒ‬ n−1 is abelian for G = SO(n, 1)).However, it might be tractable for some concrete groups with simple enough root spaces such as G = SL(3, ‫.)ޒ‬For geodesic flows the phenomenon of such a band structure is quite universal and known in the case of compact locally symmetric spaces of rank 1 [Küster and Weich 2021] and also for geodesic flows on manifolds of pinched negative curvature [Cekić and Guillarmou 2021;Faure and Tsujii 2013;2021].
As mentioned above an important application of Ruelle resonances for Anosov flows are mixing results.More precisely, the existence of a spectral gap in addition with resolvent estimates imply mixing of the flow.For Weyl chamber flows this relation of gaps and mixing rates is not yet established but conjectured to be true.From this perspective, Theorem 1.2 is related to the work of Katok and Spatzier [1994] who showed exponential mixing for the Weyl chamber action in every direction of the closure of the positive Weyl chamber if G has property (T).However it is not known whether their result remains true if the property (T) assumption is dropped.Our result above (Theorem 1.2) ensures a -dependent gap in any case but as mentioned above the precise relation to mixing rates is not yet established.
Finally, Weyl laws for Ruelle resonances of geodesic flows can also be established in variable curvature (or more generally contact Anosov flows) in various settings [Datchev et al. 2014;Faure and Sjöstrand 2011;Faure and Tsujii 2023].In particular, in the very recent article by Faure and Tsujii [2021] the Weyl law also follows because a "first band" of resonances can be related to a quantum operator.The methods in their work are, however, completely different and are based on microlocal analysis rather then global harmonic analysis.

Preliminaries
2A. Ruelle-Taylor resonances for higher rank Anosov actions.In this section we recall the main properties of Ruelle-Taylor resonances for higher rank Anosov actions from [Bonthonneau et al. 2020].Let M be a compact Riemannian manifold, let A ≃ ‫ޒ‬ n be an abelian group and let τ : A → Diffeo(M) be a smooth locally free group action.If a := Lie(A) we define the generating map Note that [X H 1 , X H 2 ] = 0 for H i ∈ a.For H ∈ a we denote by ϕ X H t the flow of the vector field X H .The action is called Anosov if there exists H ∈ a and a continuous ϕ X H t -invariant splitting where E 0 := span{X H : H ∈ a} is of dimension n because the action is locally free and there exist C > 0 and ν > 0 such that, for each x ∈ M, where the norm on TM is given by the Riemannian metric on M. Such an H ∈ a is called transversally hyperbolic.We call the set For a fixed positive Weyl chamber W, the set of Ruelle-Taylor resonances can be defined as Here E * u is defined as the annihilator of E 0 ⊕ E u in T * M. The vector space of Ruelle-Taylor resonant states for a resonance λ ∈ σ RT (X) is defined by Remark 2.1.The original definition of Ruelle-Taylor resonances and resonant states is stated via Koszul complexes; see [Bonthonneau et al. 2020, Section 3].More precisely, λ is a resonance if and only if the corresponding Koszul complex is not exact and the resonant states are the cohomologies of this complex.The space of resonant states that we are considering is just the zeroth cohomology.However, it turns out that the Koszul complex is not exact if and only if the zeroth cohomology is nonvanishing, i.e., the two notions coincide; see [Bonthonneau et al. 2020, Theorem 4].
It is known that the resonances have the following properties.
2B. Semisimple Lie groups.In this section we fix the notation for the present article.Let G be a real semisimple connected noncompact Lie group with finite center and Iwasawa decomposition G = K AN .Furthermore, let M := Z K (A) be the centralizer of A in K and G = K AN − be the opposite Iwasawa decomposition.We denote by g, a, n, n − , k and m the corresponding Lie algebras.For g ∈ G, let H (g) be the logarithm of the A-component in the Iwasawa decomposition.We have a K -invariant inner product on g that is induced by the Killing form and the Cartan involution.We have the orthogonal Bruhat decomposition g = a ⊕ m ⊕ α∈ g α into root spaces g α with respect to the a-action via the adjoint action ad.Here ⊆ a * is the set of restricted roots.Denote by W the Weyl group of the root system of restricted roots.Let n be the real rank of G and = {α 1 , . . ., α n } (resp.+ ) the simple (resp.positive) system in determined by the choice of the Iwasawa decomposition.Let m α := dim ‫ޒ‬ g α and ρ := 1 2 α∈ + m α α.Denote by w 0 the longest Weyl group element, i.e., the unique element in W mapping to − .Let a + := {H ∈ a | α(H ) > 0 ∀α ∈ } be the positive Weyl chamber and a * + the corresponding cone in a * via the identification a ↔ a * through the Killing form ⟨ • , • ⟩ restricted to a.We denote by = {α 1 , α 2 , α 1 + α 2 }.As all root spaces are one-dimensional the special element A as the set of diagonal matrices of positive entries with determinant 1, and N as the set of upper triangular matrices with 1's on the diagonal.Then a is the abelian Lie algebra of diagonal matrices and the set of restricted roots is = {ε i − ε j | i ̸ = j}, where ε i (λ) is the i-th diagonal entry of λ.The positive system corresponding to the Iwasawa decomposition is The positive Weyl chamber is and the dual cone is 1 for a visualization in the special case G = SL 3 ‫.)ޒ(‬The Weyl group is the symmetric group S n acting by permutation of the diagonal entries. 2C. Principal series representations.The concept of a principal series representation is an important tool in representation theory of semisimple Lie groups.It can be described using different pictures.We start with the induced picture: Pick λ ∈ a * ‫ރ‬ and (τ, V τ ) an irreducible unitary representation of M. We define where dk is the normalized Haar measure on K .Recall that ρ is the half-sum of positive roots.The group G acts on V τ,λ by the left regular representation.The completion H τ,λ of V τ,λ with respect to the norm is called the induced picture of the (nonunitary) principal series representation with respect to (τ, λ).We also write π τ,λ for this representation.If τ is the trivial representation then we write H λ and π λ and call it the spherical principal series with respect to λ.
Note that for equivalent irreducible unitary representations τ 1 and τ 2 of M the corresponding principal series representations are equivalent as representations as well.In particular, the Weyl group W acts on the unitary dual of M by wτ (m) = τ (w −1 mw), where w ∈ W is given by a representative in the normalizer of A in K , and therefore H λ,wτ is well defined up to equivalence.
A different way to view the principal series representation is the so-called compact picture.Although we don't need this description here, we want to introduce it in order to give a larger overview of these representations.It is given by restricting the function f : G → V τ to K , i.e., a dense subspace is given by with the same norm as above.In this picture the G-action is given by where k K AN is the K -component in the Iwasawa decomposition G = K AN .Furthermore, recall from Section 2B that H (g) ∈ a was defined as the logarithm of the Iwasawa A component.
For the example G = PSL 2 ‫,)ޒ(‬ the compact picture allows us to describe this representation explicitly without using the Iwasawa decomposition: since 2D. Invariant differential operators.Let ‫(ބ‬G/K ) be the algebra of G-invariant differential operators on G/K , i.e., differential operators commuting with the left translation by elements g ∈ G. Then we have an algebra isomorphism HC : ‫(ބ‬G/K ) → Poly(a * ) W from ‫(ބ‬G/K ) to the W -invariant complex polynomials on a * which is called the Harish-Chandra homomorphism; see [Helgason 1984, Chapter II Theorem 5.18].For λ ∈ a * ‫ރ‬ , let χ λ be the character of ‫(ބ‬G/K ) defined by χ λ (D) := HC(D)(λ).Obviously, χ λ = χ wλ for w ∈ W . Furthermore, the χ λ exhaust all characters of ‫(ބ‬G/K ); see [Helgason 1984, Chapter III Lemma 3.11].We define the space of joint eigenfunctions We will only work with the subspace of functions of moderate growth Note that E λ and E * λ are G-invariant.
The aim of this section is to establish the following proposition (see Figure 2 for a visualization) that will be needed to obtain a spectral gap in Theorem 4.10.
With this remark, one implication of the proposition is a straightforward consequence of standard estimates for elementary spherical functions: Suppose that Re λ ∈ a * + and Re λ ∈ (1 − 2 p −1 )ρ + − a * .Then we have the following bound on φ λ (see [Knapp 1986, Chapter VII Property 7.15]): where C and d are constants ≥ 0. By the integral formula for G = K A + K (see [Helgason 1984, Chapter I Theorem 5.8]) and the bi-K -invariance of φ λ , we have for a suitable Lebesgue measure on a.Because Re λ ∈ (1 − 2 p −1 )ρ + − a * , we have and we see that the latter is indeed finite by coordinizing a + by x j ↔ α j (H ) with x j > 0. Then d H is a multiple of d x and ρ(H ) = x j ρ j with ρ j > 0. Therefore, φ λ ∈ L p+ε (G).The opposite implication will be proved by combining the proof of [Knapp 1986, Theorem 8.48] with [van den Ban and Schlichtkrull 1987]: according to [van den Ban and Schlichtkrull 1987, Corollary 16.2], the elementary spherical function φ λ has a converging expansion where  1986, Theorem 8.48] shows that Positive definite functions and unitary representations.In this section we recall the correspondence between positive semidefinite elementary spherical functions and irreducible unitary spherical representations.Recall first that a continuous function f : G → ‫ރ‬ is called positive semidefinite if the matrix ( f (x −1 i x j )) i, j for all x 1 , . . ., x k ∈ G is positive semidefinite.If f is positive semidefinite, then f is bounded by f (1) and one has f (x −1 ) = f (x).Moreover, we can define a unitary representation π f associated to f in the following way: if R denotes the right regular representation of G, then π f is the completion of the space spanned by R(x) f with respect to the inner product defined by ⟨R(x) f, R(y) f ⟩ := f (y −1 x), which is positive definite.G acts unitarily on this space by the right regular representation.If f (g) = ⟨π(g)v, v⟩ is a matrix coefficient of a unitary representation π, then f is positive semidefinite and π f is contained in π.
Secondly, recall that a unitary representation is called spherical if it contains a nonzero K -invariant vector.Denote by G sph the subset of the unitary dual consisting of spherical representations.We then have a one-to-one correspondence between positive semidefinite elementary spherical functions and G sph given by φ λ → π φ λ ; see [Helgason 1984, Chapter IV Theorem 3.7].The preimage of an irreducible unitary spherical representation π with normalized K -invariant vector v K is given by g → ⟨π(g)v K , v K ⟩.If the set G sph is endowed with the Fell topology (see [Bekka et al. 2008, Appendix F.2]) and we use the topology of convergence on compact sets on the set of elementary spherical functions, then the above correspondence is a homeomorphism as is easily seen from the definitions.
2H. Associated vector bundles.In order to define the Weyl chamber flow not only on the base manifold but also on vector bundles we recall the definition of the associated vector bundle V τ over a homogeneous space G/M for a unitary finite-dimensional representation (τ, V τ ) of M. Its total space is given by where (gm, v) ∼ (g, τ (m)v) with g ∈ G, m ∈ M and v ∈ V τ .The equivalence classes are denoted by [g, v] and the projection to G/M is [g, v] → g M. A section s of this bundle can be identified with a function s : G → V τ satisfying s(gm) = τ (m) −1 s(g).We will use this identification throughout this article.We also have a G-action on V τ defined by g[g ′ , v] := [gg ′ , v].Therefore, we also have the left regular action on smooth sections of V τ : Identifying s with s, this actions reads gs(g ′ ) = s(g −1 g ′ ).
A special case of an associated vector bundle is the tangent bundle Therefore, we have a canonical connection ∇ on V τ given by where s is a smooth section identified with a s : G → V τ and X is a vector field of G/M identified with X as above.This connection will be used to lift the Weyl chamber flow to V τ .

Ruelle-Taylor resonances for the Weyl chamber action
We keep the notation from Section 2B.Let be a discrete, torsion-free, cocompact subgroup of G. Then the biquotient M = \G/M is a smooth compact Riemannian manifold where the Riemannian structure is induced by the inner product on g.More precisely, the tangent bundle TM of M is given by the quotient , and the norm of some [g, Y ] with g ∈ G and Y ∈ a ⊕ n ⊕ n − is given by the norm of Y ∈ g.We have a well-defined right A-action on M: Therefore, we have an a-action by smooth vector fields: which we call the Weyl chamber action.
For later use we denote by X : a → Diff 1 (G/M) the corresponding action on G/M.
Proposition 3.1.The A-action on M is Anosov.More precisely, each H ∈ a + is transversally hyperbolic with the splitting . Moreover, for fixed H 0 ∈ a + , the dynamically defined positive Weyl chamber W = {H ∈ a | H is transversally hyperbolic with the same splitting as H 0 } equals a + .Hence the two notions of positive Weyl chambers agree.
Proof.Pick [g, X α ] ∈ \(G × M a ⊕ n ⊕ n − ) and assume that X α is in the root space g α .Then we calculate dϕ X H t Hence we have exponential decay if α ∈ + and exponential growth if α ∈ − + .The general statement is obtained from the observation that g α ⊥ g β ⊥ a for α ̸ = β in .□ 3A.Lifted Weyl chamber action.In order to define horocycle operators we generalize the Weyl chamber action to associated vector bundles.Let (τ, V τ ) be a finite-dimensional unitary representation of M, that is, a complexification of an orthogonal representation.Then we have defined the associated vector bundle The quotient bundle \V τ is the complexification of a Euclidean vector bundle over M, where the Euclidean structure is induced by the inner product on V τ .We identify smooth sections s of this bundle with smooth functions s : G → V τ with s(γ gm) = τ (m −1 )s(g) for all γ ∈ , g ∈ G and m ∈ M.
The canonical connection ∇ descends to a connection ∇ : where s is a smooth section identified as above and X is a vector field of M identified with a smooth function X : G → a ⊕ n ⊕ n − which is left -invariant and right M-equivariant.
Definition 3.2.The lifted Weyl chamber action is defined as follows: where X H is the vector field identified with the constant mapping G → a ⊆ a ⊕ n ⊕ n − , and g → H .
The fact that ∇ is a covariant derivative implies that X τ is an admissible lift of the Weyl chamber action in the sense of (1).
For later use we denote by X τ : a → Diff 1 (G/M, V τ ) the corresponding action on G/M.We can find a nontrivial tube domain in a * ‫ރ‬ which is independent of τ and contains all Ruelle-Taylor resonances for the lifted Weyl chamber action.
Hence it remains to show that ∀t > 0} = 0 for all H ∈ a + .We show the stronger statement that e −t X τ H is unitary.Since M commutes with A, we have a well-defined action of A on \V τ given by ( [g, v])a = [ga, v].This action gives rise to an A-action on sections of the bundle \V τ defined via V τ d g, and it follows that the A-action continued to First band resonances and horocycle operators.In analogy to the rank 1 setting we make the following definition; see [Küster andWeich 2021, Definition 2.11] and[Guillarmou et al. 2021, Definition 3.1] in the scalar case.
Definition 3.4.We call λ ∈ σ RT ( X τ ) a first band resonance and write λ ∈ σ 0 RT ( X τ ) if the vector space of first band resonant states is nontrivial.
The goal of this section is to prove that, in a certain neighborhood of 0 in a * ‫ރ‬ , each Ruelle-Taylor resonance is a first band resonance and Res 0 X τ (λ) = Res X τ (λ).This will be done by introducing so-called horocycle operators as follows.
Recall that TM = \(G × Ad | M a ⊕ n ⊕ n − ) and the bundle \(G × Ad | M n) can be written as and similarly for n − .Therefore, the cotangent bundle . Let us denote the coadjoint action of M on the complexification of g * α by τ α .Note that τ α is unitary with respect to the inner product induced by the Killing form and the Cartan involution.We can now define pr α : (T * M) ‫ރ‬ → \V τ α by fiber-wise restriction to the subbundle \(G × Ad | M g α ).This induces a map denotes the complexification of the canonical connection ∇, then the horocycle operator U α for α ∈ is defined as the composition Note that we have the explicit formula if we again use the identification of sections of some associated vector bundle with left -invariant and right M-equivariant functions indicated by • and the identification V τ ⊗ g * α ≃ Hom(g α , V τ ).We should point out that the space of first band resonant states can be rewritten with the horocycle operators as Res 0 Note that in the case of constant curvature manifolds (i.e., the real hyperbolic case G = PSO(n, 1) of rank 1) there is only one positive root and our definition reduces to the original one due to Dyatlov and Zworski; see [Dyatlov et al. 2015, p. 931].Furthermore, our definition extends the definition of the horocycle operators for arbitrary G of rank 1; see [Küster and Weich 2021].
The horocycle operators fulfill the following important commutation relation.
Lemma 3.6.For all H ∈ a, Proof.Using the formulas ( 5) and ( 6) we obtain and the latter equals . For G = SL 3 ‫)ޒ(‬ the green region depicts the real part of the region where every resonance is a first band resonance; see Proposition 3.7.
We can now prove the main result of this section.
3C.First band resonant states and principal series representation.In this section we identify first band resonant states with certain -invariant vectors in a corresponding principal series representation.The proof follows the line of arguments given in [Küster and Weich 2021, Section 2] in the rank 1 case.This will allow us to apply the Poisson transform and obtain a quantum-classical correspondence.
By analogy to [Küster and Weich 2021, Definition 2.1], we define The following lemma allows us to first study the representation of G in R(λ) and take -invariants afterwards.
Lemma 3.8.The space Res 0 X τ (λ) is isomorphic to the space of -invariants of R(λ), where the isomorphism is defined by considering -invariant sections as sections of the bundle \V τ .
Proof.The only part to observe is that each H and X − ; see [Küster and Weich 2021, Lemma 2.5] for details.□ We will now show that the smooth sections in R(λ) correspond to smooth vectors in the principal series representation for the opposite Iwasawa decomposition.
Lemma 3.9.The smooth sections R The identification is obtained by considering sections s ∈ R(λ) as right M-equivariant functions s : G → V τ .
Proof.The M-equivariance is clear so it remains to show the transformation properties under A and N − .The property (X τ H +λ(H ))s = 0 amounts to (d/dt)| t=0 s(ge t H ) = −λ(H )s(g) for every g ∈ G and H ∈ a. Hence the function ϕ(t) = s(ge t H ) satisfies Therefore, s(ge t H ) = ϕ(t) = e −tλ(H ) s(g).This proves the right A-equivariance.
Hence ϕ is constant.This completes the proof.□ Note that the space W from Lemma 3.9 is already very close to the definition of the induced picture of the principal series representation (see Section 2C).The only difference is that in W we have a right invariance with respect to N − instead of N .This can be easily fixed using a conjugation with the longest Weyl group element and leads to the main result of this section: Proposition 3.10.With the longest Weyl group element w 0 (see Section 2B), we have an isomorphism where (H w 0 τ,w 0 (λ+ρ) ) −∞ denotes the -invariant distributional vectors in the principal series representation π w 0 τ,w 0 (λ+ρ) .
Proof.Pick k 0 ∈ K normalizing a such that the action of Ad(k 0 ) on a is the longest Weyl group element w 0 .We consider the map I s(g) := s(gk 0 ).Then I commutes with the left action by G and one calculates that Hence we have an intertwiner between W and smooth vectors in H w 0 τ,w 0 (λ+ρ) which extends to distributional sections.By Lemma 3.9 we conclude that R(λ) ≃ (H w 0 τ,w 0 (λ+ρ) ) −∞ as G-representations.Taking -invariants and using Lemma 3.8 completes the proof.□ 3D.Quantum-classical correspondence.In the previous section we identified the first band resonant states Res 0 X τ (λ) with -invariant distributional vectors in the principal series (H w 0 τ,w 0 (λ+ρ) ) −∞ .If we restrict ourselves to the scalar case τ = 1, then the Poisson transform P −w 0 (λ+ρ) defines a map from (H w 0 (λ+ρ) ) −∞ to E −w 0 (λ+ρ) , as P −w 0 (λ+ρ) provides a G-equivariant map from (H w 0 (λ+ρ) ) −∞ to E −w 0 (λ+ρ) (see Section 2E).Hence we can identify eigendistributions of the classical motion with quantum states and we call this identification quantum-classical correspondence.More precisely, we have the following result, which immediately gives Theorem 1.3.Proposition 3.11.If λ ∈ a * ‫ރ‬ satisfies 2⟨λ + ρ, α⟩/⟨α, α⟩ ̸ ∈ ‫ގ−‬ >0 for all α ∈ + , then we have a bijection In particular, λ ∈ σ 0 RT ( X) if and only if E −(λ+ρ) ̸ = 0. Furthermore, the isomorphism is given by the push-forward π * of distributions along the canonical projection π : \G/M → \G/K .Proof.In view of Section 2E, the Poisson transform is a bijection from (H w 0 (λ+ρ) ) −∞ → E * −λ−ρ .Restricted to -invariant distributional vectors it is still injective with image E −λ−ρ since is cocompact, and therefore E −λ−ρ = E * −λ−ρ .It remains to show that the isomorphism is the push-forward along the canonical projection.To this end let s ∈ R(λ) be smooth and let π : G/M → G/K be the canonical projection.Then the isomorphism R(λ) → (H w 0 (λ+ρ) ) −∞ carries s to s : G → ‫ރ‬ with s(g) = s(gk 0 ), where k 0 ∈ K is as in the proof of Proposition 3.10.It follows that since K is unimodular.On the other hand, for f ∈ C ∞ c (G/K ), we have s(gk M) dk M f (gK ) dgK if we normalize the Haar measure on M and choose compatible invariant measures on G/K and K /M.Hence π * s = P −w 0 (λ+ρ) s for s ∈ R(λ) ∩ C ∞ (G/M).Using the density of smooth compactly supported functions in R(λ) [Küster and Weich 2021, Corollary 2.9] we obtain the equality for the whole space R(λ).
As before we now restrict to -invariant distributions identified with distributions on \G/M and \G/K to complete the proof.□

Quantum spectrum
In this section we analyze the quantum spectrum of the locally symmetric space \G/K .Recall the definition of the joint eigenspace For the definition of χ λ see Section 2B.Since D ∈ ‫(ބ‬G/K ) is G-invariant, it descends to a differential operator D on the locally symmetric space \G/K .Therefore, the left -invariant functions of E λ (denoted by E λ ) can be identified with joint eigenfunctions on \G/K for each D: This leads to the following definition.
Definition 4.1.The quantum spectrum of \G/K is defined as We now use the quantum-classical correspondence and the Weyl law from [Duistermaat et al. 1979].
Proof of Theorem 1.1.From [Duistermaat et al. 1979, Theorem 8.9] we have, for each set ⊂ a * as in Theorem 1.1, where Vol( \G/K ) is the volume of the compact Riemannian manifold \G/K with Riemannian structure induced by the Killing form and Vol(Ad(K ) ) is the volume of the set Ad(K ) ⊆ Ad(K )a with respect to the Killing form restricted to Ad(K )a.Replacing by \ α∈ + α ⊥ we deduce that since Vol(Ad(K )α ⊥ ) = 0. Therefore, since W acts freely on the Weyl chambers.To complete the proof we observe that σ RT ( X) ⊇ σ 0 RT ( X) and m(λ) ≥ dim(Res 0 The following properties of σ Q were derived by Duistermaat, Kolk and Varadarajan [Duistermaat et al. 1979].We include the proof for the convenience of the reader.Proposition 4.2 (see [Duistermaat et al. 1979, Propositions 2.4 and 3.4, Corollary 3.5]).If λ ∈ σ Q , then the corresponding spherical function φ λ is positive semidefinite.Moreover, there is some w ∈ W such that wλ = − λ and Re λ ∈ conv(Wρ).In particular, ⟨Re λ, Im λ⟩ = 0 and ∥Re λ∥ ≤ ∥ρ∥.
Proof.Pick u ∈ E λ , regarded as a right K -invariant element of L 2 ( \G), normalized such that ⟨u, u⟩ L 2 ( \G) = 1.With the right regular representation R on L 2 ( \G), define (g) := ⟨R(g)u, u⟩.Being a matrix coefficient the function is positive semidefinite.We will show that is the elementary spherical function φ λ .By right K -invariance of u and unitarity of R we get that is K -biinvariant.
(1) = 1 is obvious.Smoothness follows from the fact that u is smooth.Furthermore, by left invariance of D. We conclude that is the elementary spherical function for χ λ , i.e., = φ λ .Since φ λ is positive semidefinite we have φ λ (g) = φ λ (g −1 ) by definition of positive definiteness, and φ λ (g −1 ) = φ | ∥µ∥ ≤ ∥ρ∥} is convex and contains Wρ, the last assertion follows.□ Remark 4.3.In the rank 1 case Proposition 4.2 implies, for λ ∈ σ Q , that λ ∈ a * with ∥λ∥ ≤ ∥ρ∥ or that λ ∈ ia * .In this particular case, this can be obtained not only from Proposition 4.2 but also from the positivity of the Laplacian on \G/K .In the higher rank setting the algebra ‫(ބ‬G/K ) contains more operators; more precisely it is a polynomial algebra in n variables.Using the properties of the Harish-Chandra isomorphism HC one can obtain that − λ ∈ W λ from the self/skew-adjointness of the operators in ‫(ބ‬G/K ).
(3) If Re λ is regular, i.e., ⟨Re λ, α⟩ ̸ = 0 for all α ∈ , we denote by w0 the unique Weyl group element mapping the Weyl chamber containing Re λ to its negative.Then λ ∈ Eig −1 ( w0 ) + i Eig +1 ( w0 ) ⊆ a * ‫ރ‬ , where Eig ±1 denotes the eigenspace for ±1.If −1 is contained in W , then Im λ = 0.In particular, this is true in the rank 1 case but need not hold in general as is seen below.
More specifically, for G = SL 3 ‫,)ޒ(‬ the only Weyl group elements with eigenvalue equal to −1 are the reflections at hyperplanes perpendicular to the roots.Hence λ ∈ σ Q implies Re λ ∈ [−1, 1]α and Im λ ∈ α ⊥ for some α ∈ or λ ∈ ia * .The obstructions for λ to be in σ Q described by Remark 4.4 are less concrete and are visualized in Figure 4.
Let us formulate the condition that φ λ is positive semidefinite in a different way.
Proposition 4.6.The elementary spherical function φ λ is positive semidefinite if and only if the subrepresentation generated by the K -invariant vector in the principal series representation H wλ is unitarizable and irreducible for some w ∈ W . Equivalently, H −w λ has a unitarizable irreducible spherical quotient.
Proof.By Casselman's embedding theorem, π φ λ is a subrepresentation of H τ,ν for some τ ∈ M and ν ∈ a * ‫ރ‬ ; see, e.g., [Knapp 1986, Theorem 8.37].More precisely, the (g, K )-module of K -finite vectors are equivalent.Since the only principal series representations containing K -invariant vectors are the spherical ones, we obtain τ = 1.Since infinitesimally equivalent admissible representations of G have the same set of K -finite matrix coefficients (see [Knapp 1986, Corollary 8.8]), we conclude φ λ = φ ν , i.e., wλ = ν. .Spherical dual in the rank 1 case.The picture on the left describes the real and complex hyperbolic case m 2α ≤ 1.The picture on the right describes the quaternionic case m 2α ≥ 2. In the latter case note that there is a spectral gap separating ρ.
Conversely assume that the subrepresentation generated by the K vector in the principal series representation H wλ is unitarizable and irreducible.Again by the aforementioned result the matrix coefficient φ wλ = φ λ of H wλ is a matrix coefficient of the unitary representation obtained by the unitary structure as well.Hence φ λ is positive semidefinite.Transition to the dual representation implies the second equivalence.□ Remark 4.7.Although the unitary dual is classified for many groups, it is difficult to deduce which elementary spherical functions are positive semidefinite.This is due to the fact that most classifications are not obtained in terms of quotients of the spherical principal series but use different descriptions of admissible representations.However, for rank 1 groups everything is classified (see [Helgason 1984, p. 484]): if α denotes the unique reduced root in + , then φ λ is positive semidefinite if and only if λ ∈ ia * or λ ∈ a * and |⟨λ, α⟩| ≤ ⟨ρ, α⟩ for 2α ̸ ∈ (i.e., in the real hyperbolic case) and |⟨λ, α⟩| ≤ 1 2 m α +1 ⟨α, α⟩ for 2α ∈ or λ = ±ρ.See Figure 5 for a visualization.

4A. Property (T).
In this section we review some facts about Kazhdan's property (T) which will lead to a more precise description of the location of σ Q .Recall that a locally compact group has property (T) if and only if the trivial representation is an isolated point in the unitary dual of the group with respect to the Fell topology; see [Bekka et al. 2008] for a general reference.It is well known that each real simple Lie group of real rank ≥ 2 has property (T); see [Bekka et al. 2008, Theorem 1.6.1].Since the mapping λ → φ λ is continuous and the correspondence between positive semidefinite elementary spherical functions and irreducible unitary spherical representations is a homeomorphism (see Section 2G), we obtain that in some neighborhood of ρ no elementary spherical function is positive semidefinite.We will use a more quantitative description introduced by Oh [2002, Section 7.1].Therefore, we denote by p K (G) the smallest real number such that the K -finite matrix coefficients of π are in L q (G) for any q > p K (G) and nontrivial π ∈ G.

Main Theorem
In this section we present the main theorem of the article and deduce Theorem 1.2 from it.See Figure 6 for a visualization for G = SL 3 ‫.)ޒ(‬Theorem 5.1.Let G be a noncompact real semisimple Lie group with finite center and ≤ G be a discrete, cocompact, torsion-free subgroup.Define Then we have the inclusions σ RT ( X) ∩ (F × ia * ) ⊆ σ 0 RT ( X) and Proof.This is immediate from Propositions 3.7, 3.

2E.
Poisson transform.The representation of G on E * λ can be described via the Poisson transform: If (H τ,λ ) −∞ denotes the distributional vectors in the principal series, then the Poisson transform P

Figure 4 .
Figure4.The situation for SL 3 ‫)ޒ(‬ as obtained from Remark 4.4: if λ ∈ σ Q then Re λ is either equal to zero (blue dot in the left picture) or lies on one of the pink, orange or brown lines depicted on the left.Furthermore, Im λ has to lie in the respective region depicted on the right, i.e., if Re λ = 0, then Im λ can take any value (blue shaded plane); if Re λ lies on the orange line, then Im λ has to lie on the orange line; and so on.
For G = SL n ‫,)ޒ(‬ an element λ ∈ a * ≃ a is regular if and only if the diagonal entries are pairwise distinct.An element λ = diag(λ 1 , . . ., λ n ) ∈ σ Q with Re λ ∈ a * + satisfies Re λ k = − Re λ n+1−k and Im λ k = Im λ n+1−k for all k = 1, . . ., n since the longest Weyl group element is the permutation ( Figure5.Spherical dual in the rank 1 case.The picture on the left describes the real and complex hyperbolic case m 2α ≤ 1.The picture on the right describes the quaternionic case m 2α ≥ 2. In the latter case note that there is a spectral gap separating ρ.

Figure 6 .
Figure 6.Visualization of the real part of a * ‫ރ‬ for G = SL 3 ‫:)ޒ(‬ The pink region is where Ruelle-Taylor resonances can a priori be located in view of the results of [Bonthonneau et al. 2020].The red points and lines depict the region where first band resonances can occur: B ∩ 1 2 conv(Wρ) ∪ Wρ − ρ.The purple shaded region illustrates the real parts in which only first band resonances can occur.Further first band resonances might occur inside the exceptional set A depicted by the black lines.
transversally hyperbolic with the same splitting as H } [van den Ban and Schlichtkrull 1987]t infinity, the Poisson transform produces a joint eigenfunction for a given boundary value; see[van den Ban and Schlichtkrull 1987]for more details.It is important to know for which values of λ ∈ a * ‫ރ‬ the Poisson transform is a bijection.By [van den Ban and Schlichtkrull 1987, Theorem 12.2] we have that P λ : (H −λ ) −∞ 2F.L p -bounds for elementary spherical functions.One can show that in each joint eigenspace E λ there is a unique left K -invariant function which has the value 1 at the identity; see[Helgason 1984, Chapter IV  Corollary 2.3].We denote the corresponding bi-K -invariant function on G by φ λ and call it the elementary spherical function.Therefore, φ λ = φ µ if and only if λ = wµ for some w ∈ W .It is given by the Poisson transform of the constant function with value 1 in the compact picture, i.e., φ λ