Spectral Correspondences for Locally Symmetric Spaces

SettingRank 1

Higher rankDiscussion

Spectral Correspondencesfor Locally Symmetric Spaces

Joachim Hilgert

April 12, 2021

AIM-RTNCG This is what I do

J. Hilgert Spectral Correspondences for Locally Symmetric Spaces

SettingRank 1


This is what I do: Commute between locally symmetricspaces and their cotangent bundles

Compare:

Selberg trace formulaeigenvalues! closed geodesics (Duistermaat-Kolk-Varadarajan)

Orbit method (geometric quantization)representations! coadjoint orbits (Kirillov, Kostant)

Microlocal analysisΨDOs! symbols (Sato, Hörmander)

Quantum-classical spectral correspondenceresonant states! resonant states


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Outline

1 Setting

2 Rank 1

3 Higher rank

4 Discussion


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Locally symmetric spaces (quantum)

G non-compact semi-simple real Lie group �nite center

K maximal compact subgroup of G

G/K Riemannian symmetric space (non-compact type)

Γ torsion free discrete subgroup of G

Γ\G/K Riemannian locally symmetric space (non-positivecurvature)

D(G/K ) algebra of G -invariant di�erential operators on G/K

D(G/K ) is commutative and contains the Laplace-Beltramioperator. It descends to Γ\G/K .


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(Co-)tangent bundle of locally symmetric spaces (classical)

T ∗(Γ\G/K ) = Γ\G ×K p∗ ∼= Γ\G ×K p = T (Γ\G/K ).

M centralizer of A in K

A-�ow Γ\G/M × A→ Γ\G/M, (ΓgM, a) 7→ ΓgaM.Weyl chamber �ow � agrees with geodesic �ow in rank 1

G ⊃ MNAN Bruhat decomposition of the open Bruhat cell

Facts:

Γ\G/M ⊆ S(Γ\G/K ) is a generalized energy shell � additionalconserved quantities in higher rank (H '05).

T (Γ\G/M) = Γ\G ×M (n + a + n) is a transversely hyperbolicdecomposition for the A-�ow.


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Γ\G/K compactConvex co-compact hyperbolic spaces

Spectral theory for D(G/K ) and the A-�ow

∆ Laplace-Beltrami operator: D(G/K ) = C[∆].

L2(Γ\G/K ) =⊕

λ∈Spec(∆) L2(Γ\G/K )λ.

A-�ow generated by a single hyperbolic vector �eld X .

1. Approach: via symbolic dynamics & transfer operators

2. Approach: via microlocal analysis (Faure-Sjöstrand '11);get a discrete set of Ruelle resonances λ:

ResX (λ) =

{u ∈ C−∞(Γ\G/M)

∣∣∣ WF(u) ⊆ E∗u

∃j ∈ N : (X + λ)ju = 0

}

E ∗u = Γ\G ×M n∗ unstable part of T ∗(Γ\G/M)


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Quantum-classical spectral correspondence

Ruelle resonances (6∈ R) come in bands: Re(λ) = −ρ− N0‖α0‖First band: Res0X (λ) = {u ∈ ResX (λ) | n · u = 0}L2(Γ\G/K )(λ+ρ)2−ρ2 ∼= Res0X (λ) ∼= (H−∞−(λ+ρ))Γ

(Dyatlov-Faure-Guillarmou '15 for real hyperbolic spaces,

Guillarmou-H-Weich '21)2 SEMYON DYATLOV, FREDERIC FAURE, AND COLIN GUILLARMOU

Re s

Im s

112

Reλ

Imλ

− 12− 3

2− 52

m = 0

m = 1

m = 2

Figure 1. An illustration of Theorem 1, with eigenvalues of the Lapla-

cian on the left and the resonances of geodesic flow, on the right. The

red crosses mark exceptional points where the theorem does not apply.

Remark. We use the Laplace transform (which has poles in the left half-plane) rather

than the Fourier transform as in [Ru, FaSj] to simplify the relation to the parameter

s used for Laplacians on hyperbolic manifolds.

Our main result concerns the case of higher dimensions n+ 1 > 2. The situation is

considerably more involved than in the case of Theorem 1, featuring the spectrum of

the Laplacian on certain tensor bundles. More precisely, for σ ∈ R, denote

MultΔ(σ,m) := dimEigm(σ),

where Eigm(σ), defined in (5.19), is the space of trace-free divergence-free symmetric

sections of ⊗mT ∗M satisfying Δf = σf . Denote by MultR(λ) the geometric multi-

plicity of λ as a Pollicott–Ruelle resonance of the geodesic flow on M (see Theorem 3

and the remarks preceding it for a definition).

Theorem 2. Let M be a compact hyperbolic manifold of dimension n+1 ≥ 2. Assume

that λ ∈ C \(− n

2− 1

2N0

). Then for λ �∈ −2N, we have (see Figure 2)

MultR(λ) =∑

m≥0

�m/2�∑

�=0

MultΔ

(−

(λ+m+

n

2

)2

+n2

4+m− 2�,m− 2�

)(1.3)

and for λ ∈ −2N, we have

MultR(λ) =∑

m≥0m �=−λ

�m/2�∑

�=0

MultΔ

(−

(λ+m+

n

2

)2

+n2

4+m− 2�,m− 2�

). (1.4)

×: exceptional points ([GHW18], Arends '21)

Tool: Poisson transformations


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Convex co-compact hyperbolic spaces

Extra di�culties:

Ruelle resonances more complicated to establish(Dyatlov-Guillarmou '16)

Spec(∆) no longer discrete

Solution: resonances also for ∆; meromorphic continuation of theresolvent kernel C∞c (Γ\G/K )→ C−∞(Γ\G/K )

Results: similar to the compact case ([GHW18] for surfaces, Had�eld

'20 for higher dimension)

Remark: [DG16] does not work for cusps.


SettingRank 1


Higher rank (compact case)

Extra di�culty: need a suitable spectral theory for commutativefamilies of operators

conceptual: Taylor spectrum (J. Taylor 1970), but now forunbounded operators

technical: extend the Faure-Sjöstrand theory to families(Bonthonneau-Guillarmou-H-Weich '20)

Results: spectral correspondence (H-Weich-Wolf '21)

joint D(G/K )-eigenspaces ! �rst band resonant states


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Higher rank Ruelle resonances 1

M, a: compact manifold, �n. dim. abelian Lie algebra

X : Anosov a-action onM by vector �elds:TM = E0 ⊕ Eu ⊕ Es

dX+λ: C∞(M; Λja∗C)→ C∞(M; Λj+1a∗C)u ⊗ ω 7→ ((X + λ)u) ∧ ω

Koszul complex

C−∞E∗u

(M)⊗ Λa∗CdX+λ−→ C−∞E∗

u

(M)⊗ Λa∗C

Ruelle resonance: λ ∈ a∗C with non-vanishingcohomology


SettingRank 1


Higher rank Ruelle resonances 2

Theorem ([BGHW20])

(i) Ruelle resonances are discrete in a∗C(ii) The cohomology spaces are �nite dimensional

(iii) ∃ scale of anisotropic Sobolev spaces C∞c ⊆ HN ⊆ C−∞ s.th.

kerHN⊗Λa∗C(dX+λ)/ ranHN⊗Λa∗C(dX+λ) ∼=kerC−∞

E∗u

(M)⊗Λa∗C(dX+λ)/ ranC−∞

E∗u

(M)⊗Λa∗C(dX+λ)

for λ in a suitable domain depending on N

(iv) λ Ruelle resonance ⇔ kerC−∞E∗u

(M)(dX+λ) 6= 0

Key: parametrix construction

dX+λQ(λ) + Q(λ)dX+λ = id + K (λ)


SettingRank 1


Applications (rank 1)

Zeros of the Selberg zeta function (Borthwick-Judge-Perry '02,

[GHW18])

Equidistribution of invariant Ruelle densities � convergence toLiouville measure via quantum ergodicity ([GHW21])

New description of Patterson-Sullivan distributions(Anantharaman-Zelditch '07, Hansen-H-Schröder '12)

Betti numbers as resonance multiplicities in negative sectionalcurvature (Küster-Weich '19)


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Applications (higher rank)

Existence of spectral gap and mixing properties for Weylchamber �ows ([BGHW20], [HWW21])

Weyl lower bounds for resonance counting function ([HWW21])

Construction of Sinai-Ruelle-Bowen measures for higher rankactions from leading (λ ∈ ia∗) resonances in 0th cohomology([BGHW20], Bonthonneau-Guillarmou-Weich '21) � just Haarmeasure for loc. sym. spaces

hopefully: progress for Katok-Spatzier rigidity conjecture '94(classi�cation of higher rank Anosov actions)


SettingRank 1


Thank you!


Spectral Correspondences for Locally Symmetric Spaces

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