A finite Linear Dependence of Discrete Series Multiplicities

Gunja Sachdeva; Kaustabh Mondal

REVIEW 3 major objections 4 minor 17 references

A finite Linear Dependence of Discrete Series Multiplicities

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that a finite sample of discrete series multiplicities along a string of Harish-Chandra parameters determines all multiplicities in that string, with integer coefficients that do not depend on the lattice.

desk verdict The generating-function machinery is a real new idea, but the main theorem as stated overreaches: off-by-one indexing, an unproved integer-coefficient claim, and an unsupported Corollary 1.3 all need fixing. read the letter →

arxiv 2506.00542 v2 pith:BYZP3LP7 submitted 2025-05-31 math.RT math.NT

Kaustabh Mondal , Gunja Sachdeva This is my paper

classification math.RTmath.NT MSC 22E4022E4611F11

keywords discreteseriesrepresentationsmultiplicitiesHarish-Chandraparametersstrongmultiplicityonegeneratingfunctionsellipticelementsholomorphiccuspformsuniformlattices

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

This paper proves that discrete series multiplicities in the square-integrable spectrum of a uniform lattice are determined, on every string of representations (an arithmetic progression of Harish-Chandra parameters sharing one positive root system), by a finite batch of multiplicities. If a finite set of parameters meets every residue class modulo the lattice's maximal elliptic order at least $|\Phi^+|+1$ times, then each multiplicity in the string is an integer linear combination of the sampled ones, with coefficients that do not depend on the lattice. Two uniform lattices whose multiplicities agree on such a finite set therefore agree on the whole string, and even a set of exceptions of density below $1/N_\Gamma$ cannot escape agreement. As an application, equality of cusp-form dimensions on a suitable finite set of weights implies equality for all weights.

What carries the argument

The load-bearing object is the generating function $F_{\lambda_1,\lambda_2,\Gamma}(z)=\sum_{k\ge1} m(\pi_{\lambda_k},\Gamma) z^{k-1}$ for a string $S(\lambda_1,\lambda_2)$. The paper feeds the geometric formula (Thm. 2.7, built from [Wil84] and [HP73]) into this series: each elliptic conjugacy class $[y]$ of $\Gamma$ contributes a rational term whose poles are $N_\Gamma$-th roots of unity, because $e^{w\lambda_2}(y)$ is an $N_\Gamma$-th root of unity. Summing over the finitely many classes gives $F_{\lambda_1,\lambda_2,\Gamma}(z)=p(z)/(1-z^{N_\Gamma})^{|\Phi^+|+1}$ with $\deg p<N_\Gamma(|\Phi^+|+1)$. Expanding the denominator by the binomial series, the coefficient of $z^{mN+j}$ is $\sum_{h=0}^{|\Phi^+|} b_{hN+j}\binom{m-h+|\Phi^+|}{|\Phi^+|}$, and the binomial matrix with distinct $m_i$ is nonsingular; hence $|\Phi^+|+1$ sampled coefficients in each residue class determine the $b$'s, and with them the whole sequence. The coefficients in the resulting linear relations are integers independent of $\Gamma$ because they arise by solving this fixed linear system.

What would settle it

Compute, for an explicit uniform lattice in SL(2,R) with known maximal elliptic order NΓ, the dimensions of cusp forms (equivalently discrete series multiplicities) along a string whose base parameter lies outside D*(G) while the direction lies inside; if the multiplicities fail to satisfy the integer linear relation predicted by a finite sample A meeting each residue class modulo NΓ at least twice (since |Φ+|=1), the theorem's conclusion for the base parameter is false.

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Extended reading notes

Core claim

With $G$ a connected semisimple Lie group with compact Cartan subgroup and $\Gamma$ a uniform lattice, let $S(\lambda_1,\lambda_2)=\{\lambda_1+k\lambda_2:k\in\mathbb{N}\cup\{0\}\}$ be a string of Harish-Chandra parameters with $P^{\lambda_1}=P^{\lambda_2}$, and suppose the base or the direction lies in the sufficiently regular set $D^\star(G)$. Theorem 1.2 asserts that if a finite $A\subset\mathbb{N}\cup\{0\}$ satisfies $|A\cap(j+N_\Gamma\mathbb{Z})|\ge|\Phi^+|+1$ for every residue class $j$ modulo the maximal order $N_\Gamma$ of elliptic elements in $\Gamma$, then for every $\ell\ge0$ the multiplicity $m(\pi_{\lambda_\ell},\Gamma)$ in the string is an integer linear combination $\sum_{k\in A} n(\ell,k)\,m(\pi_{\lambda_k},\Gamma)$ whose coefficients $n(\ell,k)$ are independent of $\Gamma$. The proof shows the multiplicities along a string form a quasi-polynomial of degree $|\Phi^+|$ with period $N_\Gamma$; the condition on $A$ gives $|\Phi^+|+1$ samples in each period, enough to solve for the quasi-polynomial's coefficients, and hence for the whole sequence.

Load-bearing premise

The proof's rationality argument works only for parameters whose multiplicity equals a Lefschetz number, a property the paper verifies for all string members beyond the base when only the direction is regular—but not for the base parameter itself, so the theorem's statement including the base relies on an unproved inclusion.

Editorial extensions

If this is right

If two uniform lattices agree on the multiplicities indexed by a finite set $A$ meeting every residue class modulo $N=\operatorname{lcm}(N_{\Gamma_1},N_{\Gamma_2})$ at least $|\Phi^+|+1$ times, they agree on the whole string (Cor. 1.1).
A density statement replaces the finite set: along a string, two lattices whose multiplicities differ on a set of indices of upper density $<1/N$ must actually agree everywhere on the string (Cor. 1.2).
For parameters in $D^\star(G)$, agreement for all but finitely many discrete series representations forces agreement for all of them (Cor. 1.3), refining the strong multiplicity one theorem.
For $SL(2,\mathbb{R})$, the dimension of every space of holomorphic cusp forms $S_\ell(\Gamma)$ is an integer linear combination of dimensions at finitely many weights, and a comparison of two levels on a finite weight set of size $2N$ decides all weights (Cor. 1.4, Cor. 5.1).
The mechanism extends to multi-directed strings with several directions in the same chamber: finitely many samples per direction determine the whole cone of multiplicities (Thm. 4.3).

Reading between the lines

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

A direct consequence the paper does not spell out: because the generating function is rational with denominator $(1-z^{N_\Gamma})^{|\Phi^+|+1}$, the multiplicity sequence is a quasi-polynomial and therefore satisfies a linear recurrence of order $N_\Gamma(|\Phi^+|+1)$ whose coefficients depend only on the string and on $N_\Gamma$, not on $\Gamma$; such recurrences give an algorithmic way to comput
The lattice-independence of $n(\ell,k)$ suggests these linear relations are intrinsic to the discrete series parameters themselves, so the same finite certificate would likely determine multiplicities for any lattice in $G$ (including non-uniform ones) once a geometric multiplicity formula is available.
For the cusp-form application, because every uniform $SL(2,\mathbb{R})$ lattice arises from a quaternion order, the certificate set of size $2N$ is explicitly computable in classical terms; one could numerically verify the finite-linearity identity for actual dimension formulas and test how far the $|\Phi^+|+1$ sampling threshold is from being sharp.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Take a look at this one if you want to see the finite-determination phenomenon for discrete series multiplicities taken seriously. The core move is new: for a string of Harish-Chandra parameters, the generating function of multiplicities is rational with denominator (1 - z^{N_Gamma})^{|Phi+|+1}, one factor for each elliptic conjugacy class. The proof of Prop 3.1 is credible, and the reduction to Williams and Hotta-Parthasarathy is clean. The cusp-form application in Section 5 is concrete and gives a nice finite-set criterion for equality of dimensions.

The soft spots are in the passage from rationality to the stated theorem. First, there is a real off-by-one: F is defined as sum over k>=1 of m(pi_{lambda_k}) z^{k-1}, so the coefficient of z^r is m(pi_{lambda_{r+1}}), not m(pi_{lambda_r}). Section 4.1 identifies the coefficient of z^{mN+j} with m(pi_{lambda_{mN+j}}), which shifts the data by one. That can be repaired, but the hypothesis on A has to change: the usable data for residue j live in A cap (j+1+NZ), and for j=0 this loses the point a=0. So the theorem as stated, with A allowed to contain 0, does not have the data it needs.

Second, the assertion that the coefficients n(l,k) are integers is not justified. The linear system has a binomial matrix, whose inverse is rational in general; nothing in the proof rules out denominators. For the rational-function class that Prop 3.1 establishes, one can write down a degree-one example where expressing the middle value from two sparse data points requires a half-integer coefficient. The specific example may not arise from actual multiplicities, but the proof claims more than the linear algebra delivers.

Third, the theorem includes the base lambda_0 even though the generating function starts at k=1 and Lemma 2.10 only guarantees lambda_0 in D*(G) when the base is the one in D*. If only the direction is in D*, lambda_0 may fall outside the Williams theorem, so the geometric expression driving everything is unjustified for l=0.

Fourth, Corollary 1.3 is asserted without proof. The multi-string section does not exhaust D*(G), as Prop 4.4 shows, so the 'all but finitely many' statement does not follow from the string results.

The paper is worth a serious referee - the idea is good enough that the gaps are worth fixing - but as submitted it needs correction, not just copyediting.

Referee Report

3 major / 4 minor

Summary. The paper studies multiplicities of discrete series representations in L^2(Γ\G) for a semisimple Lie group G with a compact Cartan subgroup and a uniform lattice Γ. For a "string" of Harish-Chandra parameters λ_k = λ1 + kλ2 sharing a single positive root system, the authors introduce a generating function F(z) = Σ_{k≥1} m(π_{λ_k}) z^{k−1} and prove that it is rational with denominator (1 − z^{N_Γ})^{|Φ+|+1}, using Williams' identification of multiplicities with Lefschetz numbers and the Hotta–Parthasarathy geometric formula. From this they claim a finite linear dependence among the multiplicities of the string, with integer coefficients independent of Γ, along with strong multiplicity-one corollaries and an application to cusp-form dimensions for SL(2,R). The core generating-function idea is promising, but the main theorem as stated contains index-shift errors, an unjustified treatment of the base of the string, and a false integrality claim for the coefficients.

Significance. The central idea of packaging discrete-series multiplicities along a string into a rational generating function is appealing and, if made correct, would give a clean finite-determination statement and a strong multiplicity-one refinement. The rationality proof (Prop. 3.1) is essentially sound, and the paper carefully builds on the theorems of Williams and Hotta–Parthasarathy. The application to cusp-form dimensions is also interesting. However, the main theorem as stated is not established: there are off-by-one indexing errors, an unsupported claim about the base λ0 of the string when only the direction lies in D^*(G), and the integrality of the linear-dependence coefficients is false in a realizable example. The strong multiplicity-one corollary would survive if the integrality claim were weakened to rational coefficients, so the approach is salvageable.

major comments (3)

[§3, §4.1] The generating function is defined in §3 as F(z) = Σ_{k∈N} m(π_{λ_k}) z^{k−1}, so the coefficient of z^r is m(π_{λ_{r+1}}). In the proof of Theorem 1.2 in §4.1, however, the coefficient of z^{mN+j} is identified with m(π_{λ_{mN+j}}), shifting every index by one. Consequently, the finite set A in Theorem 1.2 should be replaced by A+1 = {a+1 : a ∈ A}, or the generating function should be defined over k∈N∪{0} when the base is in D^*(G). As written, the proof does not establish the claimed linear dependence for the stated set A, and the discrepancy also affects Corollaries 1.1 and 1.2.
[Lemma 2.10, Theorem 1.2] Lemma 2.10 asserts that if either the base or the direction of a string lies in D^*(G), then every member of the string lies in D^*(G). The proof, however, only treats the case that the direction λ2 lies in D^*(G) and concludes for λ1+kλ2 with k≥1; for k=0 it would need ⟨λ1−δ_{λ1}, α⟩ > 0, which is exactly what is not assumed. Thus, when only the direction lies in D^*(G), the base λ0 need not be in D^*(G), and the geometric identity (Thm. 2.7) that underlies Prop. 3.1 is not justified for λ0. The statement of Theorem 1.2, which concludes for λ_ℓ with ℓ=0, is therefore unsupported in this case. The theorem should either restrict to ℓ≥1 when only the direction is in D^*(G), or the proof should include the base in the generating function under the base-in-D^* hypothesis.
[§4.1, after the linear system] The claim that the solution of the linear system yields integer coefficients n(ℓ,k) is not correct. The matrix B_{i,h} = binom(m_i−h+|Φ+|, |Φ+|) is an integer matrix, but its inverse generally has rational entries. The claim is false even for realizable data. Let Γ be a torsion-free uniform lattice in SL(2,R), so NΓ=1 and |Φ+|=1. Take λ1=α and λ2=½α, where α is the positive root; then λ1∈D^*(G), λ_k∈D^*(G) for all k≥0, and the multiplicities along the string are m(π_{λ_k}) = (2+k)V for some positive constant V. For the set A={0,2}, which satisfies |A∩Z|≥2, any relation m(π_{λ_1}) = n_0 m(π_{λ_0}) + n_2 m(π_{λ_2}) that is valid for all volumes V would require 3 = 2n_0 + 4n_2, which has no integer solution. Thus the integrality assertion in Theorem 1.2 is false. The theorem should assert rational coefficients (which still suffice for the corollaries) unless an additional integrality argument is supplied.

minor comments (4)

[Corollary 1.3] Corollary 1.3 is stated as an immediate consequence of Cor. 1.2, but no proof is supplied. Even if the proof is a short application of Cor. 1.2 to the strings S(μ,μ) for μ∈D^*(G), it should be written out, especially because Cor. 1.2 itself inherits the indexing and base issues noted above.
[Lemma 2.10] The statement of Lemma 2.10 overclaims: it asserts that every member of the string lies in D^*(G), but the proof only covers k≥1 when only the direction is in D^*(G). The lemma should be restated to distinguish the cases k=0 and k≥1.
[Throughout] There are several typographical issues: the heading of §2.2 reads "Lefchetz" instead of "Lefschetz"; in the literature review "Schimd" should be "Schmid"; in §4.3, "set quality" should be "set equality"; and the notation for D^*(G) alternates between D⋆(G) and D∗(G).
[§5] In the proof of Prop. 5.2, the assertion that the subrepresentations generated by cusp forms are orthogonal is stated without proof; a reference or a short argument would help. This is not central to the main theorem, but it would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from external geometric formulas and a linear-system solve, not from its conclusion.

full rationale

The derivation chain for Theorem 1.2 is self-contained relative to the cited external results of Williams and Hotta-Parthasarathy. The generating function F_{λ1,λ2,Γ}(z) is defined from the multiplicities themselves, but its rationality is proved using the independent geometric expression m(πλ,Γ)=Σ vol(Γy\Gy)Ψλ(y) for λ∈D*(G), not by assuming the conclusion. The coefficients b_h in the numerator are then obtained by solving a nonsingular linear system whose data are the finitely many multiplicities indexed by A; this is a genuine solve, not a fitted parameter renamed as a prediction, and the claimed Γ-independence would follow from the Γ-independence of the coefficient matrix and the geometric data of elliptic conjugacy classes. The only self-citation is [BMS25], which appears in the literature review and is not load-bearing for any proof step; no uniqueness theorem or ansatz is smuggled in from the authors' prior work. The concerns raised by a skeptical reader about §4.1, such as a possible off-by-one between the coefficient of z^r and m(π_{λ_{r+1}}), or the claim that the inverse of the binomial matrix has integer entries for arbitrary sparse A, are correctness or rigor issues rather than circularity: even if those steps fail, the derivation would not reduce to its own input by definition. No circular step can be exhibited with the paper's own equations, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: the numbers NΓ, |Φ+|, and the binomial coefficients are determined by the group and the lattice. The proof relies on standard external theorems, not on self-citation.

assumptions (5)

standard math Harish-Chandra parametrization: discrete series are in bijection with the set D(G) of regular integral linear forms (Thm 2.1).
Used in Sec 2.1 to label representations by linear forms and to define strings; standard textbook result.
domain assumption Williams' theorem: for λ ∈ D^*(G), m(πλ, Γ) = (-1)^{l(w0)} χ(λ, Γ) (Thm 2.5).
Bridges multiplicities to Lefschetz numbers; cited from [Wil84] and invoked in Sec 3.
domain assumption Hotta-Parthasarathy geometric formula: χ(λ, Γ) equals a finite sum over elliptic conjugacy classes of vol(Γ_y\G_y) Ψ_λ(y) (Thm 2.6, 2.7).
Provides the explicit rational form in Prop 3.1; cited from [HP73].
domain assumption A uniform lattice has only finitely many Γ-conjugacy classes of elliptic elements.
Used in Prop 3.1 to keep the sum over [y] finite; standard result for lattices in semisimple Lie groups.
standard math The binomial matrix C(m_i - h + |Φ+|, |Φ+|) is nonsingular for distinct nonnegative integers m_i.
Follows from shifted binomial coefficients forming a basis of polynomials; used in Sec 4.1 to solve for numerator coefficients.

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Pith. "Pith review of A finite Linear Dependence of Discrete Series Multiplicities." pith.science (2026). https://pith.science/paper/BYZP3LP7

@misc{pith2026250600542,
  author       = {Pith},
  title        = {Pith review of: A finite Linear Dependence of Discrete Series Multiplicities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYZP3LP7}},
  note         = {Machine review of arXiv:2506.00542}
}

abstract

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $\Gamma$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations of $G$. We prove that for any finite subset of $\widehat{G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(\Gamma \backslash G)$ determines all discrete series multiplicities in $L^2(\Gamma \backslash G)$. This allows us to obtain a refinement of the strong multiplicity one result for discrete series representations. As an application, we deduce that for two given levels, the equality of the dimensions of the spaces of cusp forms over a suitable finite set of weights implies the equality of the dimensions of the spaces of cusp forms for all weights.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

On infinitesimal -isospectrality of locally symmetric spaces

Bhagwat C, Mondal K, Sachdeva G. On infinitesimal -isospectrality of locally symmetric spaces. Canadian Mathematical Bulletin. 2025; 68(1): 246-261. doi:10.4153/S0008439524000882

work page doi:10.4153/s0008439524000882 2025
[2]

Bhagwat Chandrasheel, and Rajan C.\,S.\,, On a spectral analog of the strong multiplicity one theorem. Int. Math. Res. Not. IMRN (2011), no.18, 4059--4073

work page 2011
[3]

Automorphic Forms on Adele Groups

Gelbart, Stephen. Automorphic Forms on Adele Groups. Annals of Mathematics Studies, No. 83. Princeton, NJ: Princeton University Press, (1975)

work page 1975
[4]

116 (1966), 1-111

Harish-Chandra, Discrete series for semisimple Lie groups II, Acta Math. 116 (1966), 1-111

work page 1966
[5]

Hotta, R

R. Hotta, R. Parthasarathy, A geometric meaning of the multiplicity of integrable discrete series classes in L^2( G) , Osaka J. Math. 10 (1973), 211-234

work page 1973
[6]

, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, (2001)

Knapp A. , Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, (2001)

work page 2001
[7]

Kelmer, A refinement of strong multiplicity one for spectra of hyperbolic manifolds, Trans

D. Kelmer, A refinement of strong multiplicity one for spectra of hyperbolic manifolds, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5925--5961

work page 2014
[8]

R. P. Langlands, Dimension of spaces of automorphic forms, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) , pp. 253--257

work page 1965

Show all 17 references

[9]

E. A. Lauret and R. J. Miatello, Strong multiplicity one theorems for locally homogeneous spaces of compact-type, Proc. Amer. Math. Soc. 148 (2020), no. 7, 3163--3173

2020
[10]

Osborne, G

S. Osborne, G. Warner, Multiplicities of the integrable discrete series J. Functional Analysis 30 (1978), 287-310

1978
[11]

Parthasarathy, Dirac operator and the discrete series, Annals of Math

R. Parthasarathy, Dirac operator and the discrete series, Annals of Math. 96 (1972), 1-30

1972
[12]

Schmid, On a conjecture of Langlands, Annals of Math

W. Schmid, On a conjecture of Langlands, Annals of Math. 93 (1971), 1-42

1971
[13]

C., Varadarajan, V

Trombi, P. C., Varadarajan, V. S. Asymptotic behaviour of eigen functions on a semisimple Lie group: The discrete spectrum. Acta Math. 129 (1972), 237–280. https://doi.org/10.1007/BF02392217

1972 doi
[14]

Indian Math

André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (NS) 24 (1960), 589–623

1960
[15]

Discrete Series Multiplicities in L^2( G)

Williams, Floyd L. Discrete Series Multiplicities in L^2( G) . American Journal of Mathematics 106, no. 1 (1984), 137–48

1984
[16]

Williams Floyd L., Discrete series multiplicities in L^2( G) II - Proof of Langlands' conjecture, Am. J. Math. 107 (1985) 367-376

1985
[17]

Pitman Res

Williams Floyd L., Lectures on the spectrum of L^2( G) . Pitman Res. Notes Math. Ser., 242 Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, (1991). xiv+348 pp. ISBN:0-582-06863-0

1991

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

On infinitesimal -isospectrality of locally symmetric spaces

Bhagwat C, Mondal K, Sachdeva G. On infinitesimal -isospectrality of locally symmetric spaces. Canadian Mathematical Bulletin. 2025; 68(1): 246-261. doi:10.4153/S0008439524000882

work page doi:10.4153/s0008439524000882 2025

[2] [2]

Bhagwat Chandrasheel, and Rajan C.\,S.\,, On a spectral analog of the strong multiplicity one theorem. Int. Math. Res. Not. IMRN (2011), no.18, 4059--4073

work page 2011

[3] [3]

Automorphic Forms on Adele Groups

Gelbart, Stephen. Automorphic Forms on Adele Groups. Annals of Mathematics Studies, No. 83. Princeton, NJ: Princeton University Press, (1975)

work page 1975

[4] [4]

116 (1966), 1-111

Harish-Chandra, Discrete series for semisimple Lie groups II, Acta Math. 116 (1966), 1-111

work page 1966

[5] [5]

Hotta, R

R. Hotta, R. Parthasarathy, A geometric meaning of the multiplicity of integrable discrete series classes in L^2( G) , Osaka J. Math. 10 (1973), 211-234

work page 1973

[6] [6]

, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, (2001)

Knapp A. , Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, (2001)

work page 2001

[7] [7]

Kelmer, A refinement of strong multiplicity one for spectra of hyperbolic manifolds, Trans

D. Kelmer, A refinement of strong multiplicity one for spectra of hyperbolic manifolds, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5925--5961

work page 2014

[8] [8]

R. P. Langlands, Dimension of spaces of automorphic forms, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) , pp. 253--257

work page 1965