Pith. sign in

REVIEW 4 major objections 5 minor 106 references

Harmonic analysis on compact standard quotients of non-Riemannian semisimple symmetric spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Standard quotients of non-Riemannian symmetric spaces fall into two spectral types: Type I admits a unique discrete decomposition, Type II forces continuous spectra.

desk verdict A serious, mostly convincing spectral-theory paper whose main Type I theorem survives the largest unshown computation. read the letter →

arxiv 2607.25528 v1 pith:VNSEXHSU submitted 2026-07-28 math.RT math.DGmath.SP

Salah Mehdi , Martin Olbrich This is my paper · ORCID
classification math.RTmath.DGmath.SP MSC 22E4622E4053C3058J5011F72
keywords spectraldecompositioncompactClifford-Kleinformsproperlytransitivetriplessymmetricspacesinvariantdifferentialoperatorsessentialself-adjointnessbranchingrulesanti-deSitterspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a dichotomy in the spectral theory of compact standard quotients $Y=\Gamma\backslash G/H$ of non-Riemannian semisimple symmetric spaces. Because the algebra $D(G/H)$ of invariant differential operators contains no elliptic operator, the usual Riemannian spectral argument fails. The authors show that every standard quotient arises from a properly transitive triple $(G,H,L)$, so $L^2(Y)$ is controlled by the $L$-representations in $L^2(\Gamma\backslash L)$. For Type I triples, $D(G/H)$ acts on each nonzero $L\cap H$-invariant space by a single character; this forces a unique discrete spectral decomposition of $L^2(Y)$, essential self-adjointness of all formally self-adjoint invariant operators, and a complete orthonormal system of smooth joint eigenfunctions. Type II triples instead exhibit genuinely continuous spectral phenomena, demonstrated in detail for a fourfold product of $\mathrm{PSL}(2,\mathbb{R})$.

What carries the argument

The load-bearing object is the properly transitive triple $(G,H,L)$: $G=LH$, $L\cap H$ compact, $L$ reductive, which makes $Y=\Gamma\backslash L/(L\cap H)$. The Type I/Type II dichotomy comes from sphericity of $L_{\max}/(L_{\max}\cap H)$. Casimir transfer formulas express the Laplacian of $G$ as $\alpha\Omega_L+\beta\Omega_{K_L}$ in Type I, with an extra non-compact correction $\Omega_{L\cap\sigma(L)}$ in Type II. For Type I triples, a case-by-case multiplicity-one assertion ensures $D(G/H)$ acts by one character on each nonzero $V_\pi^{L\cap H}$, and a distributional matrix-coefficient map transfers L-spectral data into G-eigendistributions.

What would settle it

For each of the seven Type I triples in the classification, compute $\dim V_\pi^{L\cap H}$ for all principal series representations using the paper's Lemma 6.1. Finding any irreducible admissible $L$-representation with $\dim V_\pi^{L\cap H} \ge 2$ — or, in the exceptional Triple 5, an $\mathrm{Sp}(1)$-module that is not irreducible — would refute Prop. 6.7 and the Type I spectral theorem. For the Type II $\mathrm{PSL}(2,\mathbb{R})$ example, one could compute the joint spectral measure for an irreducible lattice $\Gamma$; if the absolutely continuous part vanished, Theorem 9.2's continuous-spectrum claim would fail.

Watch

Extended reading notes

Core claim

Every standard quotient $Y=\Gamma\backslash G/H$ comes from a properly transitive triple $(G,H,L)$: $L$ acts transitively on $G/H$ and $Y=\Gamma\backslash L/(L\cap H)$, so $L^2(Y)$ is the $L\cap H$-invariant part of $L^2(\Gamma\backslash L)$. For Type I triples, $D(G/H)$ acts on each nonzero $V_\pi^{L\cap H}$ by a single character; hence $L^2(Y)$ has a unique discrete spectral decomposition into joint eigenspaces, with corresponding decompositions of smooth functions and distributions, and every formally self-adjoint invariant operator, Laplacian included, is essentially self-adjoint. Type II triples force continuous spectrum: in the $\mathrm{PSL}(2,\mathbb{R})$ example the unique spectral decomposition contains Lebesgue measure on half-lines, i.e. absolutely continuous spectrum, with embedded

Load-bearing premise

The load-bearing premise is that Prop. 6.7's case-by-case multiplicity-one assertions for spherical triples are complete and correct; the proof states that the computations 'can be done' for Triples 1–7 in Table 1 but does not display them, so an unforeseen exception in any one case would break the single-character action and therefore Theorem 7.3's discrete decomposition.

Editorial extensions

If this is right

  • For any Type I standard quotient, L2(Y) has a complete orthonormal system of smooth joint eigenfunctions of D(G/H).
  • All formally self-adjoint invariant differential operators, in particular the pseudo-Riemannian Laplacian, are essentially self-adjoint on Type I compact quotients, confirming the quantum-completeness conjecture for these spaces.
  • Every integrable discrete series representation for G/H yields an infinite-dimensional family of L2-eigenfunctions on every compact Type I standard quotient.
  • For compact anti-de Sitter quotients, the Laplacian spectrum is explicit: eigenvalues ℓ²−n² appear with infinite multiplicity, a finite set lies in a bounded interval, and an unbounded negative set exists.
  • For the Type II PSL(2,R) example, the spectrum of the Laplacian is all of R, with absolutely continuous spectrum and embedded eigenvalues of infinite multiplicity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the Type I/Type II dichotomy is likely the organizing principle for spectral theory of all compact quotients of non-Riemannian symmetric spaces, not only standard ones; the paper's transitivity result explains why non-standard quotients are harder.
  • Editorial inference: the paper's Conjecture 1.1, if established for irreducible Type II triples, would give a direct-integral analogue of Theorem 7.3; a testable partial check is whether the continuous families of eigendistributions described for Type II exhaust the spectrum as in the PSL(2,R) example.
  • Editorial inference: the explicit rank-one tables could be used to compute Laplacian spectra for compact anti-de Sitter quotients; numerical spectral computations would expose whether the negative essential spectrum accumulates, a question the paper leaves open.
  • Editorial inference: the finite-dimensionality result for non-unitary contributions in rank-one Type I quotients suggests a general principle: only unitary spherical representations of G matter for most of L2(Y), with small corrections from non-unitary ones.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a spectral theory for compact standard quotients Y=Γ\G/H of non-Riemannian semisimple symmetric spaces, organized around properly transitive triples (G,H,L). It proves that cocompact proper actions of reductive subgroups are transitive (Prop. 2.4), imports and uses Onishchik's classification of irreducible triples (Thm. 2.12), computes the embedding of the Casimir operator of G into U(l) (Prop. 3.3), and establishes for Type I triples a unique discrete spectral decomposition of L²(Y) into joint eigenspaces of D(G/H), essential self-adjointness of all formally self-adjoint invariant differential operators (Thm. 7.3), L-admissibility and multiplicity formulas (Thms. 10.9–10.10), a refined spectral decomposition indexed by spherical G-representations (Thm. 11.6), and a detailed spectral analysis for anti-de Sitter quotients (Thm. 8.1). It also analyzes one Type II example with continuous spectrum and embedded eigenvalues (Thm. 9.2) and formulates Conjecture 1.1 for general Type II triples. The central claim is that Type I triples admit a canonical, discrete, unique spectral decomposition.

Significance. If correct, the paper gives a definitive spectral decomposition for all compact standard quotients of Type I and provides the first systematic treatment of the Type II phenomenon, including a fully worked example with absolutely continuous spectrum. The framework of properly transitive triples, the explicit Casimir formulas, and the clear distinction between Type I and Type II are valuable contributions. The paper is commendably explicit about its dependence on Onishchik's external classification and on the overlap with Kassel–Kobayashi's work [54]. The central Theorem 7.3 has an independent route through Cor. 3.4 and Prop. 5.11 that does not rely on the unproved multiplicity-one casework of Prop. 6.7, which is a real strength. The detailed AdS results and the Type II example give concrete, falsifiable spectral predictions.

major comments (4)
  1. [§6, Prop. 6.7] The proof of Prop. 6.7 asserts that the multiplicity-one assertion for Triples 1–7 can be verified by 'explicit computations', but those computations are not displayed anywhere in the paper or the appendices. This is load-bearing: Prop. 6.8, Cor. 7.2, Theorem 11.6, and the π-minimality/branching results (I) all rest on this single-character/multiplicity-one statement. An unrecognized exception in any of the triples would break the discrete, single-character description of the V^{L∩H}_π action. Theorem 7.3 itself survives via Cor. 3.4 and Prop. 5.11, but the paper's representation-theoretic spectral decomposition (K) remains conditional. The authors should either include the missing case-by-case computations or give precise, verifiable references for each of Triples 1–7.
  2. [§6, proof of Prop. 6.8] The final case in the proof, for the triple (SO_e(4,4n), SO_e(3,4n), Sp(1,n)), depends directly on Prop. 6.7's assertion that the Sp(1)-action on V^{Sp(n)}_π is irreducible. The argument identifying the two Sp(1)-actions is plausible, but it cannot be completed if Prop. 6.7 is left as a black box. Since Prop. 6.8 is the key step converting the decomposition (1.8) into the discrete spectral decomposition of Cor. 7.2, this dependency should be made explicit and the missing computation supplied.
  3. [§2, Thm. 2.12 and Table 2] The classification of irreducible properly transitive triples is transferred from Onishchik's work, but the transfer is only summarized. In particular, the Type II determination for Case 12 is sketched via an equivariant-map argument and a case-by-case check is not fully written out. This classification is used in Cor. 3.4 and Prop. 2.11 to reduce the Type I Casimir commutativity to Cases (1)–(4) of Prop. 3.3. The authors should provide a more detailed concordance between their Table 1/Table 2 and Onishchik's tables, and indicate exactly which row of which theorem in [85] yields each case.
  4. [§11, Thm. 11.6] The refined spectral decomposition in Theorem 11.6 is stated as a full Hilbert-space isomorphism with both smooth and distribution decompositions. Its proof depends on the structure of the sets bL_ρ, which in turn uses Prop. 6.7 through the uniqueness of the π-minimal representation (Thm. 10.9). The authors should state explicitly which parts of Theorem 11.6 would fail if the multiplicity-one assertion of Prop. 6.7 had an exception, and which parts are independent of it.
minor comments (5)
  1. [§6, first paragraph] Typos: 'cococompact' should be 'cocompact'.
  2. [§1.5] The statement that Theorem 1.9 of [54] 'is not entirely correct for noncompact quotients' is a serious claim but no counterexample or precise correction is given. A footnote or remark with a reference would be appropriate.
  3. [Table 1] The column 'rank(G/H)' uses the notation rank_R elsewhere; please make the notation consistent.
  4. [§8, Thm. 8.1(e)] The phrase '−∞ is an accumulation point of spec(Δ_Y)' is nonstandard; it would be clearer to state that spec(Δ_Y) is unbounded below and that −∞ is an accumulation point of the spectrum in the extended sense.
  5. [§3, Cor. 3.4] The reduction to Cases (1)–(4) of Prop. 3.3 uses Thm. 2.12 and Prop. 2.11, but the reader must reconstruct the list of Type I irreducible triples. A short explicit sentence enumerating the relevant types would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: self-citations are not load-bearing; the main Type I spectral theorem is derived from stated assumptions and external classification.

full rationale

The derivation chain is largely self-contained. The central Type I result (Thm. 7.3) is obtained from Lemma 7.1, Prop. 6.8, and Prop. 5.11, and the paper explicitly notes that Thm. 7.3 also has a representation-theory-free route via the elliptic operator commuting with all of D(G/H) guaranteed by Cor. 3.4. There is no fitting, no parameter tuned to a predicted quantity, and no normalization that forces the conclusions. The classification of triples is imported from Onishchik's external work, not from the present authors; the overlaps with Kassel-Kobayashi are disclosed and the paper states that (D) and (H) were obtained independently. The only self-references, e.g. the announcement [75] ('Some of these results were previously announced in [75]') and the background citation [17], are not load-bearing: the proofs in the paper do not reduce to those citations. The one genuine weakness is Prop. 6.7, whose proof delegates the multiplicity-one verification to 'explicit computations' that are not displayed; this is an omitted proof / completeness gap, not circularity, because no claim is being assumed in the form of its conclusion. Thus a non-zero score reflects only the presence of minor non-load-bearing self-citations, not a circular derivation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted or chosen ad hoc; all constants are structural. The central claims rest on standard theorems and classifications from the literature, plus explicit but partly delegated case computations. No new physical entities are postulated.

assumptions (7)
  • standard math Onishchik's classification of decompositions of semisimple Lie groups G=LH ([85], Thm 4.1/4.2) is complete and applicable after imposing L∩H compact and H symmetric.
    Theorem 2.12 and Tables 1/2 list irreducible properly transitive triples from this classification; no independent derivation is supplied.
  • standard math Kobayashi's dimension formula for proper and cocompact actions says dim(l_s)+dim(h_s)=dim(s) ([64], Thm 4.7).
    Used in the proof of Prop 2.4 to derive transitivity and in Prop 2.7 for the rank formula.
  • standard math Casselman's subrepresentation theorem and Casselman-Wallach globalization.
    Used in Prop 6.3, Prop 6.7, and Section 10 to reduce to principal series representations.
  • standard math Delorme's embedding theorem for H-spherical representations into P_{σθ}-principal series.
    Used in Prop 10.3 to prove L-admissibility of H-spherical representations.
  • standard math Known branching rules for tensor products of PSL(2,R) representations ([92], [93]).
    Used in Section 9 to compute the Type II example.
  • standard math Selberg's lemma guarantees torsion-free finite index subgroups of lattices.
    Used in Sections 7 and 8 to replace orbifold quotients by manifolds.
  • standard math Howe-Tan [45] and Schlichtkrull [96] results on composition series and unitarity for rank-one principal series.
    Used for Tables 3–6 and the complete rank-one description in Section 11.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Harmonic analysis on compact standard quotients of non-Riemannian semisimple symmetric spaces." pith.science (2026). https://pith.science/paper/VNSEXHSU

@misc{pith2026260725528,
  author       = {Pith},
  title        = {Pith review of: Harmonic analysis on compact standard quotients of non-Riemannian semisimple symmetric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNSEXHSU}},
  note         = {Machine review of arXiv:2607.25528}
}
abstract

Let $Y=\Gamma\backslash G/H$ be a compact standard quotient of a non-Riemannian semisimple symmetric space $X=G/H$. We investigate the spectral decomposition of the algebra ${\bf D}(X)$ of $G$-invariant differential operators on $X$ acting on $L^2(Y)$. The absence of elliptic invariant differential operators makes the spectral theory fundamentally different from the Riemannian case. We first show that standard quotients arise from {\it transitive} actions on $X$ of real reductive subgroups $L$ of $G$ containing the discrete subgroup $\Gamma$. Our approach is based on the geometry of properly transitive triples $(G,H,L)$. We derive explicit formulas expressing the Casimir operator of $G$ in terms of Casimir operators of $L$. Triples fall into two classes: Type I and Type II. For triples of Type I, we prove essential self-adjointness of invariant differential operators and discreteness of the corresponding spectral decomposition. This decomposition is illustrated by a detailed analysis of compact standard quotients of anti-de Sitter spaces. In contrast, Type II triples exhibit genuinely continuous spectral phenomena. A central theme of the paper is the interaction between the representation theories of $G$ and $L$. For Type I triples, we prove $L$-admissibility of $H$-spherical $G$-representations of finite length and establish multiplicity formulas. We show that the resulting correspondence defines a map between irreducible spherical $L$-representations and spherical $G$-representations. For triples of both types, we obtain a representation-theoretic description of eigendistributions via distributional matrix coefficients. As an application, we show that every integrable discrete series representation of $G/H$ contributes an infinite-dimensional family of $L^2$-eigenfunctions on every compact standard quotient of Type I.

Figures

Figures reproduced from arXiv: 2607.25528 by the authors.

Figure 1
Figure 1. Spectrum of ∆Y . Note that the space E0 of solutions of the relativistic wave equation is infinite dimensional. Of course, it would be highly desirable to uncover some structural properties at least of the essential spectrum (i.e. eigenvalues of infinite multiplicity and accumulation points of eigenvalues) in the half-line (−∞, 1−n 2 ), in particular in (−∞, −n 2 ] (contributions of unitary principal series). Howeve… view at source ↗
Figure 2
Figure 2. Expected spectrum for irreducible Γ. Proof. That L 2 (Y ) has a spectral decomposition of the form (9.3) is a direct consequence of Lemma 9.1, the decomposition (9.1), and the branching rules (a)–(d). It remains to verify the more specific claims about the terms involved in the decomposition. For irreducible Γ, we have NΓ(π) = {0} for all non-trivial π of the form (d) (where we allow to switch the factors). Indeed, … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

106 extracted references · 5 linked inside Pith

  1. [54]

    Kassel and T

    F. Kassel and T. Kobayashi,Spectral Analysis on Standard Locally Homogeneous Spaces.Lecture Notes in Mathematics 2367, Springer 2025. (Also available as arXiv:1912.12601.)

    arXiv 2025

  2. [85]

    A. L. Oniˇ sˇ cik,Decompositions of reductive Lie groups.Math. USSR-Sb.9(1969), 515–554

    1969

  3. [1]

    Afentoulidis-Almpanis and G

    S. Afentoulidis-Almpanis and G. Liu,Tensor product decompositions for rank-one spin groups I: Unitary principal series representations.Preprint arXiv:2502.19968. To appear in Israel J. of Math

    arXiv

  4. [2]

    M. W. Baldoni Silva,The unitary dual ofSp(n,1).Trans. Amer. Math. Soc.48(1981), 549–583

    1981

  5. [3]

    van den Ban,Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula.Ark

    E. van den Ban,Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula.Ark. Mat.25(1987), 175–187

    1987

  6. [4]

    van den Ban,The principal series for a reductive symmetric space I.H-fixed distribution vectors.Ann

    E. van den Ban,The principal series for a reductive symmetric space I.H-fixed distribution vectors.Ann. Sci. ´Ecole Morm. Sup.21(1988), 359–412

    1988

  7. [5]

    van den Ban, M

    E. van den Ban, M. Flensted-Jensen, and H. Schlichtkrull,Harmonic analysis on semisimple symmetric spaces: a survey of some general results.In: Representation theory and automorphic forms (Edinburgh 1996), pp. 191–217. Sympos. Pure Math.61, AMS, Providence 1997

    1996

  8. [6]

    Ballmann, H

    W. Ballmann, H. Matthiessen, and S.Mondal,Small eigenvalues of surfaces: old and new.ICCM Not.6(2018), 9–24

    2018

Show all 106 references
  1. [7]

    Ben Said, K

    S. Ben Said, K. Koufany, and G. Zhang,Invariant trilinear forms on spherical principal series of real rank one Lie groups. Internat. J. Math.25(2014), 35 pp

    2014

  2. [8]

    Bergeron and L

    N. Bergeron and L. Clozel,Spectre automorphe des vari´ et´ es hyperboliques et applications topologiques.Ast´ erisque303 (2005), 218 pp

    2005

  3. [9]

    Bergeron and L

    N. Bergeron and L. Clozel,Quelques cons´ equences des travaux d’ Arthur pour le spectre et la topologie des vari´ et´ es hyperboliques.Invent. Math.192(2013), 505–532

    2013

  4. [10]

    Bergeron and L

    N. Bergeron and L. Clozel,Sur le spectre et la topologie des vari´ et´ es hyperboliques de congruence: les cas complexe et quaternionien.Math. Ann.368(2017), 1333–1358

    2017

  5. [11]

    Bernstein,On the support of the Plancherel measure.J

    J. Bernstein,On the support of the Plancherel measure.J. Geom. Phys.5(1987), 663–710

    1987

  6. [12]

    Bernstein and B

    J. Bernstein and B. Kr¨ otz,Smooth globalizations of Harish-Chandra modules.Israel J. Math.199(2014), 45–111

    2014

  7. [13]

    Bien,D-modules and spherical representations

    F. Bien,D-modules and spherical representations. Mathematical Notes 39. Princeton University Press, 1990

    1990

  8. [14]

    Boche´ nski and A

    M. Boche´ nski and A. Tralle,Standard compact Clifford-Klein forms and Lie algebra decompositions.Transform. Groups 30(2025), 1535–1552

    2025

  9. [15]

    Borel,Compact Clifford-Klein forms of symmetric spaces.Topology2(1963), 111–122

    A. Borel,Compact Clifford-Klein forms of symmetric spaces.Topology2(1963), 111–122

    1963

  10. [16]

    Borel and N

    A. Borel and N. Wallach,Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition. Mathematical Surveys 67. American Mathematical Society, Providence, R.I., 2000

    2000

  11. [17]

    Bunke and M

    U. Bunke and M. Olbrich,The spectrum of Kleinian manifolds.J. Funct. Anal.172(2000), 76–164

    2000

  12. [18]

    Buser,Geometry and spectra of compact Riemann surfaces.Progress in Mathematics 106

    P. Buser,Geometry and spectra of compact Riemann surfaces.Progress in Mathematics 106. Birkh¨ auser, 1992

    1992

  13. [19]

    Calabi and L

    E. Calabi and L. Markus,Relativistic space forms.Ann. Math. (2)75(1962), 63–76

    1962

  14. [20]

    Carri` ere,Autour de la conjecture de L

    Y. Carri` ere,Autour de la conjecture de L. Markus sur les vari´ et´ es affines.Invent. Math.95(1989), 615–628

    1989

  15. [21]

    Casselman,Introduction to the Schwartz space ofΓ\G.Canad

    W. Casselman,Introduction to the Schwartz space ofΓ\G.Canad. J. Math.41(1989), 285–320

    1989

  16. [22]

    Casselman,Canonical extensions of Harish-Chandra modules to representations ofG.Canad

    W. Casselman,Canonical extensions of Harish-Chandra modules to representations ofG.Canad. J. Math.41(1989), 385–438

    1989

  17. [23]

    Casselman and M

    W. Casselman and M. S. Osborne,Then-cohomology of representations with infinitesimal character.Compositio Math. 31(1975), 219–227

    1975

  18. [24]

    Colin de Verdi` ere and C

    Y. Colin de Verdi` ere and C. Le Bihan,On essential-selfadjointness of differential operators on closed manifolds.Ann. Fac. Sci. Toulouse Math.31(2022), 1287–1302

    2022

  19. [25]

    D. H. Collingwood,Representations of rank one Lie groups.Pitman, Boston 1985

    1985

  20. [26]

    Delarue, C

    B. Delarue, C. Guillarmou, and D. Monclair,Spectra of Lorentzian quasi-Fuchsian manifolds.Preprint arXiv:2504.21762

  21. [27]

    Deligne,Notes on spinors.Quantum fields and strings: a course for mathematicians, Vol.1, 2, 99–135, Amer

    P. Deligne,Notes on spinors.Quantum fields and strings: a course for mathematicians, Vol.1, 2, 99–135, Amer. Math. Soc., Providence, RI, 1999. HARMONIC ANALYSIS ON STANDARD QUOTIENTS 127

    1999

  22. [28]

    Delorme,Injection de modules sph´ eriques pour les espaces sym´ etriques r´ eductifs dans certaines repr´ esentations induites

    P. Delorme,Injection de modules sph´ eriques pour les espaces sym´ etriques r´ eductifs dans certaines repr´ esentations induites. With an appendix by E. van den Ban and P. Delorme. Lecture Notes in Math.,1243, Noncommutative harmonic analysis and Lie groups (Marseille-Luminy,...

    1985

  23. [29]

    Delorme, F

    P. Delorme, F. Knop, B. Kr¨ otz, and H. Schlichtkrull,Plancherel theory for real spherical spaces: construction of the Bernstein morphisms.J. Amer. Math. Soc.34(2021), 815–908

    2021

  24. [30]

    Dixmier,LesC ∗-alg` ebres et leurs repr´ esentations.Gauthier-Villars & Cie, Paris 1964

    J. Dixmier,LesC ∗-alg` ebres et leurs repr´ esentations.Gauthier-Villars & Cie, Paris 1964

    1964

  25. [31]

    Duistermaat, J.A.C

    J.J. Duistermaat, J.A.C. Kolk, and V.S. Varadarajan,Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math.52(1979), 27–93

    1979

  26. [32]

    Dyatlov and C

    S. Dyatlov and C. Guillarmou,Pollicott-Ruelle resonances for open systems.Ann. Henri Poincar´ e17(2016), 3089–3146

    2016

  27. [33]

    Faraut,Distributions sph´ eriques sur les espaces hyperboliques.J

    J. Faraut,Distributions sph´ eriques sur les espaces hyperboliques.J. Math. Pures App.58(1979), 369–344

    1979

  28. [34]

    Flato, J

    M. Flato, J. Simon, H. Snellman, and D. Sternheimer,Simple facts about analytic vectors and integrability.Ann. Sci. ´Ecole Norm. Sup.5(1972), 423–434

    1972

  29. [35]

    Frahm,Symmetry breaking operators for strongly spherical reductive pairs.Publ

    J. Frahm,Symmetry breaking operators for strongly spherical reductive pairs.Publ. Res. Int. Math. Sci.59(2023), 259–337

    2023

  30. [36]

    I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,Representation theory and automorphic functions.Translated from the Russian by K. A. Hirsch. W. B. Saunders Company, Philadelphia, 1969

    1969

  31. [37]

    Ghys,Flots d’Anosov dont les feuilletages stables sont diff´ erentiables.Ann

    E. Ghys,Flots d’Anosov dont les feuilletages stables sont diff´ erentiables.Ann. Sci. ´Ecole Norm. Sup.20(1987), 251–270

    1987

  32. [38]

    W. M. Goldman,Nonstandard Lorentz space forms.J. Differential Geometry21(1985), 301–308

    1985

  33. [39]

    Goodman and N

    R. Goodman and N. R. Wallach,Representations and invariants of the classical groups.Encyclopedia of Mathematics and its Applications,68, Cambridge University Press, Cambridge 1998

    1998

  34. [40]

    Guillemin and S

    V. Guillemin and S. Sternberg,Geometric asymptotics.Mathematical Surveys14. American Mathematical Society, Prov- idence, R.I., 1977

    1977

  35. [41]

    He,Generalized matrix coefficients for infinite dimensonal unitary representations.J

    H. He,Generalized matrix coefficients for infinite dimensonal unitary representations.J. Ramanujan Math. Soc.29(2014), 253–272

    2014

  36. [42]

    Heckman and H

    G. Heckman and H. Schlichtkrull,Harmonic Analysis and special functions on symmetric spaces.Perspectives in Mathe- matics 14, Academic Press, 1994

    1994

  37. [43]

    Helgason,Groups and geometric analysis.Academic Press, 1984

    S. Helgason,Groups and geometric analysis.Academic Press, 1984

    1984

  38. [44]

    Hotta and R

    R. Hotta and R. Parthasarathy,Multiplicity formulae for discrete series.Invent. Math.26(1974), 133–178

    1974

  39. [45]

    Howe and E

    R. Howe and E. Tan,Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations.Bull. Amer. Math. Soc.28(1993), 1–74

    1993

  40. [46]

    Huang and P

    J.-S. Huang and P. Pandˇ zi´ c,Dirac operators in representation theory.Mathematics: Theory and Apllications. Birkh¨ auser, 2006

    2006

  41. [47]

    J¨ anich,DifferenzierbareG-Mannigfaltigkeiten.Lecture Notes in Mathematics59

    K. J¨ anich,DifferenzierbareG-Mannigfaltigkeiten.Lecture Notes in Mathematics59. Springer-Verlag, Berlin-New York 1968

    1968

  42. [48]

    Kannaka and T

    K. Kannaka and T. Kobayashi,Deformations of standard locally homogeneous spaces.Preprint arXiv:2507.14832

    arXiv

  43. [49]

    Kassel,Deformations of proper actions on reductive homogeneous spaces.Math

    F. Kassel,Deformations of proper actions on reductive homogeneous spaces.Math. Ann.353(2012), 599–632

    2012

  44. [50]

    Kassel and T

    F. Kassel and T. Kobayashi,Stable spectrum for pseudo-Riemannian locally symmetric spaces.C. R. Math. Acad. Sci. Paris349(2011), 29–33

    2011

  45. [51]

    Kassel and T

    F. Kassel and T. Kobayashi,Poincar´ e series for non-Riemannian locally symmetric spaces.Adv. Math.287(2016), 123–236

    2016

  46. [52]

    Kassel and T

    F. Kassel and T. Kobayashi,Invariant differential operators on spherical homogeneous spaces with overgroups.J. Lie Theory29(2019), 663–754

    2019

  47. [53]

    Kassel and T

    F. Kassel and T. Kobayashi,Spectral analysis on pseudo-Riemannian locally symmetric spaces.Proc. Japan Acad. Ser. A Math. Sci.96(2020), 69–74

    2020

  48. [55]

    Kassel and N

    F. Kassel and N. Tholozan,Sharpness of proper and cocompact actions on reductive homogeneous spaces.Preprint arXiv:2410.08179

    arXiv

  49. [56]

    Kazhdan,Some applications of the Weil representation.J

    D. Kazhdan,Some applications of the Weil representation.J. Analyse Math.32(1977), 235–248

    1977

  50. [57]

    Kelmer and P

    D. Kelmer and P. Sarnak,Strong spectral gaps for compact quotients of products ofP SL(2,R).J. Eur. Math. Soc.11 (2009), 283–313

    2009

  51. [58]

    Klingler,Compl´ etude des vari´ etes lorentziennes ` a courbure constante.Math

    B. Klingler,Compl´ etude des vari´ etes lorentziennes ` a courbure constante.Math. Ann.306(1996), 353–370

    1996

  52. [59]

    A. W. Knapp,Representation theory of semisimple groups. An overview based on examples.Princeton University Press, 1986

    1986

  53. [60]

    A. W. Knapp and E. M. Stein,Intertwining operators for semisimple groups.Ann. of Math.93(1971), 489–578. 128 S. MEHDI AND M. OLBRICH

    1971

  54. [61]

    F. Knop, B. Kr¨ otz, T. Pecher, and H. Schlichtkrull,Classification of reductive real spherical pairs I. The simple case. Transform. Groups24(2019), 67–114

    2019

  55. [62]

    F. Knop, B. Kr¨ otz, T. Pecher, and H. Schlichtkrull,Classification of reductive real spherical pairs II. The semisimple case. Transform. Groups24(2019), 467–510

    2019

  56. [63]

    F. Knop, B. Kr¨ otz, and H. Schlichtkrull,The tempered spectrum of a real spherical space.Acta math.218(2017), 319–383

    2017

  57. [64]

    Kobayashi,Proper action on a homogeneous space of reductive type.Math

    T. Kobayashi,Proper action on a homogeneous space of reductive type.Math. Ann.285(1989), 249–263

    1989

  58. [65]

    Kobayashi,Discontinuous groups and Clifford-Klein form of pseudo-Riemannian homgeneous manifolds.In: Algebraic and Analytic Methods in Representation Theory (H

    T. Kobayashi,Discontinuous groups and Clifford-Klein form of pseudo-Riemannian homgeneous manifolds.In: Algebraic and Analytic Methods in Representation Theory (H. Schlichtkrull and B. Orsted, eds.), pp. 99–165. Perspectives in Math 17, Academic Press 1996

    1996

  59. [66]

    Kobayashi and T

    T. Kobayashi and T. Matsuki,Classification of finite-multiplicity symmetric pairs.Transform. Groups19(2014), 457–493

    2014

  60. [67]

    Kobayashi and T

    T. Kobayashi and T. Oshima,Finite multiplicity theorems for induction and restriction.Advances in Math.248(2013), 921–944

    2013

  61. [68]

    Kobayashi and T

    T. Kobayashi and T. Yoshino,Compact Clifford-Klein forms of symmetric spaces - Revisited.Pure Appl. Math. Q.1 (2005), 591–663

    2005

  62. [69]

    Kr¨ otz, and H

    B. Kr¨ otz, and H. Schlichtkrull,Multiplicity bounds and the subrepresentation theorem for real spherical spaces.Trans. Amer. Math. Soc.368(2016), 2749–2762

    2016

  63. [70]

    R. S. Kulkarni,Proper actions and pseudo-Riemannian space forms.Adv. in Math.40(1981), 10–51

    1981

  64. [71]

    R. P. Langlands,Dimensions of spaces of automorphic forms.Proc. Sympos. Pure Math.9(1966), 253–257

    1966

  65. [72]

    R. P. Langlands,On the functional equations satisfied by Eisenstein series.Lecture Notes in Mathematics.544, Springer, 1976

    1976

  66. [73]

    Li,Theta series and construction of automorphic forms.In: Representation theory of groups and algebras, Contemp

    J.-S. Li,Theta series and construction of automorphic forms.In: Representation theory of groups and algebras, Contemp. Math.145(1993), 237–248

    1993

  67. [74]

    R. P. Martin,On the decomposition of tensor products of principal series representations for real-rank one semisimple Lie groups.Trans. Amer. Math. Soc.201(1975), 177–211

    1975

  68. [75]

    Mehdi and M

    S. Mehdi and M. Olbrich,Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations. Adamovi´ c, Dujella et al. (eds.), Lie groups, number theory, and vertex algebras. Representation theory XVI. Dubrovnik,

  69. [76]

    Mehdi and P

    S. Mehdi and P. Pandˇ zi´ c,Representation theoretic embeddings of Dirac operators.Represent. Theory25(2021), 760-779

    2021

  70. [77]

    J. J. Millson,On the first Betti number of a constant negatively curved manifold.Ann. of Math.104(1976), 235–247

    1976

  71. [78]

    Monclair, J.-M

    D. Monclair, J.-M. Schlenker, and N. Tholozan,Gromov-Thurston manifols and anti-de Sitter geometry.Preprint arXiv:2310.12003

    arXiv

  72. [79]

    C. C. Moore,Ergodicity of flows on homogeneous spaces.American J. Math.88(1966), 154–178

    1966

  73. [80]

    G. D. Mostow,Self-adjoint groups.Ann. of Math.62(1955), 44–55

    1955

  74. [81]

    G. D. Mostow,Equivariant embeddings in Euclidean space.Ann. of Math.65(1957), 432–446

    1957

  75. [82]

    Nelson,Analytic vectors.Ann

    E. Nelson,Analytic vectors.Ann. of Math.70(1959), 572–615

    1959

  76. [83]

    Olbrich,Cohomology of convex cocompact groups and invariant distributions on limit sets.Preprint arXiv:0207301

    M. Olbrich,Cohomology of convex cocompact groups and invariant distributions on limit sets.Preprint arXiv:0207301

  77. [84]

    A. L. Oniˇ sˇ cik,Inclusion relations between transitive compact transformation groups.(Russian) Trudy Moskov. Mat. Obˇ sˇ c. 11(1962), 199–242

    1962

  78. [86]

    N. S. Poulsen,OnC ∞-vectors and intertwining bilinear forms for representations of Lie groups.J. Funct. Anal.9(1972), 87–120

    1972

  79. [87]

    R. T. Powers,Selfadjoint algebras of unbounded operators II.Trans. Amer. Math. Soc.187(1974), 261–293

    1974

  80. [88]

    M. S. Raghunathan,Discrete subgroups of Lie groups.Ergebnisse der Mathematik und ihrer Grenzgebiete 68. Springer, Berlin Heidelberg, 1972

    1972

  81. [89]

    J. G. Ratcliffe,Foundations of hyperbolic manifolds.Second edition. Graduate Texts in Mathematics 149. Springer, New York, 2006

    2006

  82. [90]

    Reed and B

    M. Reed and B. Simon,Methods of modern mathematical physics.I. Functional analysis.Second edition. Academic Press, New York - London, 1980

    1980

  83. [91]

    Reed and B

    M. Reed and B. Simon,Methods of modern mathematical physics.II. Fourier analysis, self-adjointness.Academic Press, New York - London, 1975

    1975

  84. [92]

    Repka,Tensor products of unitary representations ofSL 2(R).Bull

    J. Repka,Tensor products of unitary representations ofSL 2(R).Bull. Amer. Math. Soc.82(1976), 930–932

    1976

  85. [93]

    Repka,Tensor products of unitary representations ofSL 2(R).Amer

    J. Repka,Tensor products of unitary representations ofSL 2(R).Amer. J. Math.100(1978), 747–774

    1978

  86. [94]

    Sakellaridis and A

    Y. Sakellaridis and A. Venkatesh,Periods and harmonic analysis on spherical varieties.Asterisque396(2017), 360 pp

    2017

  87. [95]

    H. H. Sch¨ afer,Topological vector spaces.Graduate Texts in Mathematics 3, Springer, Third printing, 1971. HARMONIC ANALYSIS ON STANDARD QUOTIENTS 129

    1971

  88. [96]

    Schlichtkrull,Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms.J

    H. Schlichtkrull,Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms.J. Funct. Anal.70(1987), 194–219

    1987

  89. [97]

    Schlichtkrull, P

    H. Schlichtkrull, P. Trapa, and D. A. Vogan,Laplacians on spheres.Sao Paulo J. Math. Sci.12(2018), 295–358

    2018

  90. [98]

    Shimeno,The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type.J

    N. Shimeno,The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type.J. Funct. Anal.121(1994), 330–388

    1994

  91. [99]

    Tojo,Classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms.Proc

    K. Tojo,Classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms.Proc. Japan Acad. Ser. A Math. Sci.95(2019), 11–15

    2019

  92. [100]

    E. B. Vinberg,Commutative homogeneous spaces and coisotropic symplectic actions.Russian Math. Surveys56(2001), 1–60

    2001

  93. [101]

    N. R. Wallach,On the Selberg trace formula in the case of a compact quotient.Bull. Amer. Math. Soc.82(1976), 171–195

    1976

  94. [102]

    N. R. Wallach,Real reductive groups I.Pure and Applied Mathematics 132. Academic Press, 1988

    1988

  95. [103]

    N. R. Wallach,Real reductive groups II.Pure and Applied Mathematics 132. Academic Press, 1992

    1992

  96. [104]

    Zeghib,On closed anti-de Sitter spacetimes.Math

    A. Zeghib,On closed anti-de Sitter spacetimes.Math. Ann.310(1998), 695–716

    1998

  97. [105]

    D. P. Zhelobenko,Compact Lie groups and their representations.Transl. Math. Monogr. 40, Amer. Math. Soc., Providence 1973. Universit´e de Lorraine, CNRS, IECL, F-57000 Metz, France Email address:salah.mehdi@univ-lorraine.fr F acult´e des Sciences, de la Technologie et de la Co...

    1973

  98. [2019]

    Math.768(2021), 55–63

    Contemp. Math.768(2021), 55–63

    2021

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.