Pith. sign in

REVIEW 3 major objections 3 minor 44 references

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper proves that every minimal or next-to-minimal automorphic function on a split simply-laced group is completely determined by its Whittaker coefficients, and it supplies explicit reconstruction sums.

desk verdict The reconstruction theorems are strong and likely correct, but Theorem G's no-cusp-form claim is false as stated for type A1 and needs a serious fix. read the letter →

arxiv 1908.08296 v2 pith:IIVRFMWG submitted 2019-08-21 math.NT hep-thmath.RT

Dmitry Gourevitch , Henrik P. A. Gustafsson , Axel Kleinschmidt , Daniel Persson , Siddhartha Sahi This is my paper · ORCID
classification math.NThep-thmath.RT MSC 11F3011F7022E5520G45
keywords automorphicformsminimalrepresentationnext-to-minimalWhittakercoefficientsFouriernilpotentorbitssimply-lacedgroupsstringtheory
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 that, on a split simply-laced group over a number field, the small automorphic functions—those whose Whittaker support consists of minimal, or next-to-minimal, nilpotent orbits—are completely determined by their Whittaker coefficients, the Fourier coefficients attached to characters of the unipotent radical of a fixed Borel subgroup. This is the analogue, for these small representations, of the classical Piatetski-Shapiro–Shalika recovery of cusp forms on $\mathrm{GL}_n$ from generic Whittaker coefficients. Theorems B and E give explicit infinite sums that reconstruct the full function from those coefficients, and Theorems A and C do the same for maximal parabolic Fourier coefficients. A direct consequence is Theorem G: there are no cusp forms in the minimal or next-to-minimal automorphic spectrum, which settles a question relevant to the $R^4$ and $\partial^4 R^4$ corrections in string theory. Because the formulas are explicit, they turn questions about the structure of these representations into sums that can be evaluated in applications such as string-theory scattering amplitudes.

What carries the argument

The carrying device is the Whittaker pair $(S,\phi)$: a rational semisimple element $S$ and a nilpotent covector $\phi$ with $\mathrm{ad}^*(S)\phi = -2\phi$; it produces the Fourier coefficient $F_{S,\phi}[\eta]$ by integrating $\eta$ against the character $\chi_\phi$ over the unipotent subgroup $N_{S,\phi}$. The main engine is a reduction principle, imported from the companion paper, that computes a Fourier coefficient attached to one Whittaker pair as an integral of the coefficient attached to another, finer pair (Theorem 2.2.6). Iterating this descent along a quasi-abelian enumeration of the simple roots—an ordering in which each successive root is abelian or Heisenberg inside the Levi subgroup built from the previous roots—moves every minimal or next-to-minimal Fourier coefficient down to the standard Whittaker coefficients $W_\phi$ with $\phi$ in the root spaces $g^*_{-\beta}$. Geometric lemmas classify which characters can be conjugated into those root spaces by $\Gamma$, which is what makes the final sums explicit.

What would settle it

Evaluate the right-hand side of the $E_8$ next-to-minimal expansion (1.35) for an explicit next-to-minimal Eisenstein series whose Whittaker coefficients are known, at a generic unramified point; any mismatch with the function itself falsifies Theorem E. A sharper check is to compute the non-abelian integral over $V_{g_8}$ in the $B_8$ and $B_{88}$ terms and compare the result with the known local spherical vector containing a cubic phase; if the integral does not reproduce that phase, the reduction principle fails in the Heisenberg case.

Watch

Extended reading notes

Core claim

The central claim is a reconstruction theorem. Let $G$ be a split simply-laced reductive group with a finite central cover, and let $\eta$ be an automorphic function whose Whittaker support is contained in the minimal, or next-to-minimal, nilpotent orbits. The paper proves that $\eta$ is equal to its constant term plus an explicit sum of terms, each of which is a sum or integral of Whittaker coefficients $W_\phi[\eta]$ with $\phi$ ranging over root spaces that appear in a quasi-abelian enumeration of the simple roots. In the next-to-minimal case the terms $A_i$, $A_{ij}$, $A_{ii}$, $B_n$, $B_{nj}$, $B_{nn}$ appearing in Theorem E are all built from Whittaker coefficients through finite sums, integrals over unipotent groups $V_\gamma$, and shifts by Weyl-group representatives. The same machinery expresses the maximal parabolic Fourier coefficient $F_{S_\alpha,\phi}[\eta]$: for minimal $\eta$ it is a Whittaker coefficient up to conjugation, while for next-to-minimal $\eta$ it is a Whittaker coefficient plus a sum of Whittaker terms, or an integral of a Whittaker coefficient. Consequently the paper derives that no nonzero cuspidal automorphic form belongs to the minimal or next-to-minimal spectrum.

Load-bearing premise

The argument collapses if the reduction principle imported from the companion paper—which expresses a Fourier coefficient for one Whittaker pair as an integral of coefficients for another, finer Whittaker pair—has unstated hypotheses, or if the infinite arithmetic sums over $\Gamma$-quotients and root spaces cannot be rearranged in the space of automorphic functions.

Editorial extensions

If this is right

  • No cuspidal automorphic form can occur in the minimal or next-to-minimal automorphic spectrum (Theorem G).
  • Every maximal parabolic Fourier coefficient of such a form is either zero, a constant term on the Levi subgroup, or an explicit combination of Whittaker coefficients (Theorems A and C).
  • The complete Fourier expansion of $\eta_{\min}$ and $\eta_{\mathrm{ntm}}$ along any quasi-abelian parabolic is explicit, so computations previously done case-by-case for $\mathrm{SL}_n$, $D_5$, and $E_8$ become part of a uniform formula.
  • For the Eisenstein-series cases already known to have collapsing Whittaker coefficients, the reduction implies the corresponding maximal parabolic Fourier coefficients are Eulerian (Remark 1.5.5).
  • The $E_8$ expansions reproduce previously known minimal-orbit and abelian next-to-minimal Fourier expansions and add the missing non-abelian terms.

Reading between the lines

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

  • Editorial inference: these formulas provide a uniqueness principle—two minimal or next-to-minimal automorphic functions with the same Whittaker coefficients are equal, so the space of such functions is parameterized by its Whittaker data.
  • Editorial inference: Theorem G can be read computationally: a cusp form would have vanishing constant terms on every maximal parabolic, and Theorems B and E then force all its Whittaker coefficients to vanish, so testing one coefficient of a candidate residue would suffice to rule it out.
  • Editorial inference: the descent pattern is likely to extend to quasi-split and non-simply-laced groups, where the next-to-minimal orbit class is replaced by several orbits; the $D_5$ example already shows the sums must be adjusted orbit-by-orbit.
  • Editorial inference: because the $E_8$ expansion (1.35) is fully explicit, a numerical implementation on a few Fourier modes could compare the non-abelian $B$ terms with instanton predictions, offering a concrete string-theory test.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. Let G be a finite central extension of the adelic points of a split simply-laced group. The paper defines minimal and next-to-minimal automorphic functions by their Whittaker support, and proves that maximal-parabolic Fourier coefficients of minimal functions reduce to Whittaker coefficients (Theorem A); that minimal functions are reconstructed from Whittaker coefficients along a quasi-abelian enumeration (Theorem B); analogous statements for next-to-minimal functions (Theorems C-F); and that no cuspidal automorphic forms occur in the minimal or next-to-minimal spectrum (Theorem G). The proofs rely on a reduction principle imported from the companion paper [GGK+], on root-system geometry proved here, and on induction over quasi-abelian Levi subgroups. Detailed examples for D5 and E8 and comparisons with [GMV15, KP04, BP17] are provided.

Significance. Conditional on the imported reduction principle, Theorems A-F would be a substantial contribution: they give explicit, parameter-free reconstruction formulas for small automorphic forms and reduce difficult maximal-parabolic Fourier coefficients to Whittaker coefficients. The root-system lemmas (Corollaries 3.0.3-3.0.4, Propositions 4.0.1-4.0.2) are proved carefully, and the worked D5/E8 expansions are consistent with earlier physics literature. However, Theorem G is false as stated: for type A1 the unique nonzero nilpotent orbit is minimal, so all cuspidal automorphic forms are 'minimal' in the paper's sense; the proof of Theorem G contains a false inference about cuspidality and Fourier coefficients. The advertised no-cusp-forms consequence is therefore not merely unproved but contradicted by standard examples.

major comments (3)
  1. [§4.2, proof of Theorem G] The assertion that cuspidality implies 'for any two simple roots ε1, ε2 and any φ ∈ g*_{ε1} ⊕ g*_{ε2}, the Whittaker coefficient Wφ[η′] vanishes identically' is false. Cuspidality only forces the constant term W0[η′] to vanish; nonconstant Fourier coefficients of cuspidal forms are typically nonzero and are the objects that carry cusp forms. For G of type A1, Lemma 2.1.1 says the unique nonzero nilpotent orbit is minimal, and any nonzero cuspidal automorphic form has nonzero Whittaker coefficients for φ ∈ g×_{−α}. Hence Theorem G contradicts the existence of cusp forms on SL2. Moreover, the sums in Theorems B and E contain single-root terms Ai, so even a corrected vanishing statement for two-root coefficients would not make the right-hand side vanish.
  2. [§1.2 and Theorems B/E] The reconstruction identities (1.13) and (1.33) involve infinite sums over Γ-quotients and root spaces, but the paper does not prove convergence or justify rearrangement in C∞(Γ\G). The remark in §1.2 that integrals are 'either compact integrals or represent Fourier expansions of periodic functions' does not cover these infinite sums, especially since automorphic functions here are not required to have moderate growth or finite center action. A rigorous derivation of Theorems B and E needs a convergence statement; without it the equalities are formal.
  3. [§3.3, proof of Theorem C(ii)] After Proposition 3.3.4 identifies the restriction η′ = F_{S,ψ}[η]|_{G′} as minimal or trivial, the proof asserts without further justification that Whittaker coefficients of η′ equal W_{φ+ψ}[η]. This equality is the mechanism by which Theorem B is applied, and it requires a lemma interchanging the integration over N′ with the integration defining η′ and matching the resulting characters. As written, the step is too compressed for (1.19) to be verified.
minor comments (3)
  1. [Equation (1.2)] The displayed complete Fourier expansion contains the same sum over X1 twice; the second term should presumably be over X2, or the formula should be corrected.
  2. [Example 1.4.2] In the displayed expansion, W[ηmin](g) should be W0[ηmin](g) for consistency with (1.13).
  3. [Paragraph after (1.35)] The sentence ending 'defined in §1.4, respectively' is missing a period; the same typo occurs after equation (5.14).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reconstruction formulas are genuine deductions from Fourier analysis and an independently proved reduction principle, not restatements of their inputs.

full rationale

The paper's central theorems express minimal and next-to-minimal automorphic functions as explicit sums of Whittaker coefficients. These are proved by induction using elementary Fourier decomposition, geometric conjugation lemmas proved in the paper, and the reduction principle Theorem 2.2.6 and Heisenberg expansion Proposition 2.2.7 imported from the companion paper [GGK+]. Although [GGK+] shares authors, it is a separate proved result on general automorphic functions whose assumptions do not include the target reconstruction formulas, so it counts as independent support rather than a self-citation that forces the conclusion. The definitions of minimal and next-to-minimal via Whittaker support are used to restrict which Fourier coefficients can appear, not to assume the expansion formulas. No fitted parameter is renamed as a prediction, and no equation is shown to equal its own input by construction. The proof of Theorem G contains a questionable inference from cuspidality to vanishing of nonconstant Whittaker coefficients, especially in rank one, but that is a mathematical correctness concern, not a circularity, so it does not raise the circularity score.

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

No free parameters are fitted. The central claims rest on five imported mathematical inputs, three from the authors' own companion paper; these are stated clearly but not formally verified in this preprint.

assumptions (5)
  • domain assumption Reduction principle for Fourier coefficients, Theorem 2.2.6 of [GGK+]
    Imported from the authors' companion paper; it is the main tool relating Fourier coefficients for different Whittaker pairs. Not proved in this manuscript.
  • domain assumption Heisenberg parabolic expansion, Proposition 2.2.7 of [GGK+]
    Used to handle Heisenberg roots, especially for E8 and type D. Imported from the companion paper.
  • domain assumption Geometric conjugation lemma, Lemma 2.2.8 of [GGK+]
    Used to conjugate next-to-minimal elements into root-space sums. Imported from the companion paper.
  • standard math Classification of quasi-abelian roots and minuscule internal Chevalley modules, using Table 1, [Bou75] and [MS12]
    Used to guarantee existence of quasi-abelian enumerations and transitivity of Weyl group actions on relevant root sets.
  • domain assumption Wave-front set to Whittaker support relation, Lemma 2.0.7, citing [MW87], [Ros95], [Mat87] and [Var14]
    Used to infer minimal or next-to-minimal nature of global functions from local components.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups." pith.science (2026). https://pith.science/paper/IIVRFMWG

@misc{pith2026190808296,
  author       = {Pith},
  title        = {Pith review of: Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIVRFMWG}},
  note         = {Machine review of arXiv:1908.08296}
}
abstract

In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $\eta\in \pi$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whittaker coefficients. A consequence of our results is the non-existence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory.

Figures

Figures reproduced from arXiv: 1908.08296 by the authors.

Figure 1
Figure 1. The various string theory limits associated with different maximal parabolic subgroups Pα. Roots are labeled in the Bourbaki order￾ing. Remark 1.9.1. Theorem G resolves a long-standing question in string theory which concerns the possibility of having contributions from cusp forms in the R4 and ∂ 4R4 amplitudes. The theorem ensures that this can never happen as there are no cusp forms in the minimal or next-to-minim… view at source ↗
Figure 2
Figure 2. Root labels used for D5. 1 10 2 21 6 2 41 2 317 3221 3 3 21 4 3 22 2 515 3 31 4 21 2 5221 5312 5 2 713 73 91 [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 27 canonical work pages

  1. [1]

    Fourier coefficients attached to small automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$

    O. Ahl\' e n, H. P. A. Gustafsson, A. Kleinschmidt, B. Liu, and D. Persson, `` Fourier coefficients attached to small automorphic representations of SL _n( A ) ,'' http://dx.doi.org/10.1016/j.jnt.2018.03.022 J. Number Theory 192 (2018) 80--142 , http://arxiv.org/abs/1707.08937 arXiv:1707.08937 [math.RT]

    work page Pith review arXiv 2018

  2. [2]

    Brylinski and P

    J.-L. Brylinski and P. Deligne, ``Central extensions of reductive groups by K_2 ,'' http://dx.doi.org/10.1007/s10240-001-8192-2 Publ. Math. Inst. Hautes \' E tudes Sci. no. 94, (2001) 5--85

    doi:10.1007/s10240-001-8192-2 2001

  3. [3]

    L. Bao, A. Kleinschmidt, B. E. W. Nilsson, D. Persson, and B. Pioline, `` Instanton corrections to the universal hypermultiplet and automorphic forms on SU(2,1) ,'' http://dx.doi.org/10.4310/CNTP.2010.v4.n1.a5 Commun. Num. Theor. Phys. 4 (2010) 187--266 , http://arxiv.org/abs/0909.4299 arXiv:0909.4299 [hep-th]

    work page Pith review arXiv 2010

  4. [4]

    Bourbaki, \' E l\' e ments de math\' e matique

    N. Bourbaki, \' E l\' e ments de math\' e matique. F asc. XXXVIII : G roupes et alg\`ebres de L ie. C hapitre VII : S ous-alg\`ebres de C artan, \' e l\' e ments r\' e guliers. C hapitre VIII : A lg\`ebres de L ie semi-simples d\' e ploy\' e es . Actualit\' e s Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975

    work page 1975

  5. [5]

    Exact $\nabla^4 R^4$ couplings and helicity supertraces

    G. Bossard and B. Pioline, `` Exact ^4 R^4 couplings and helicity supertraces ,'' http://dx.doi.org/10.1007/JHEP01(2017)050 JHEP 01 (2017) 050 , http://arxiv.org/abs/1610.06693 arXiv:1610.06693 [hep-th]

    work page Pith review arXiv 2017

  6. [6]

    D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple L ie algebras . Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993

    work page 1993

  7. [7]

    Casselman and J

    W. Casselman and J. Shalika, ``The unramified principal series of p -adic groups. II . T he W hittaker function,'' Compositio Math. 41 no. 2, (1980) 207--231

    work page 1980

  8. [8]

    D'Hoker, M

    E. D'Hoker, M. B. Green, and P. Vanhove, ``On the modular structure of the genus-one type II superstring low energy expansion,'' http://dx.doi.org/10.1007/jhep08(2015)041 J. High Energy Phys. no. 8, (2015) 041, front matter+66

    work page doi:10.1007/jhep08(2015)041 2015

Show all 44 references
  1. [9]

    Fleig, H

    P. Fleig, H. P. A. Gustafsson, A. Kleinschmidt, and D. Persson, http://dx.doi.org/10.1017/9781316995860 Eisenstein series and automorphic representations , vol. 176 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2018. http://arxiv.org/abs...

    arXiv

  2. [10]

    Fleig, A

    P. Fleig, A. Kleinschmidt, and D. Persson, `` Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors ,'' http://dx.doi.org/10.4310/CNTP.2014.v8.n1.a2 Commun. Num. Theor. Phys. 08 (2014) 41--100 , http://arxiv.org/abs/1312.3643 arXiv:1312.3643 [hep-th]

    2014 arXiv

  3. [11]

    M. B. Green and M. Gutperle, `` Effects of D instantons ,'' http://dx.doi.org/10.1016/S0550-3213(97)00269-1 Nucl. Phys. B498 (1997) 195--227 , http://arxiv.org/abs/hep-th/9701093 arXiv:hep-th/9701093 [hep-th]

    1997 arXiv

  4. [12]

    M. B. Green and M. Gutperle, `` D Particle bound states and the D instanton measure ,'' http://dx.doi.org/10.1088/1126-6708/1998/01/005 JHEP 01 (1998) 005 , http://arxiv.org/abs/hep-th/9711107 arXiv:hep-th/9711107 [hep-th]

    1998 arXiv

  5. [13]

    Gourevitch, H

    D. Gourevitch, H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, ``A reduction principle for F ourier coefficients of automorphic forms,'' http://arxiv.org/abs/1811.05966v3 arXiv:1811.05966v3 [math.NT]

    arXiv

  6. [14]

    Gomez , D

    R. Gomez , D. Gourevitch , and S. Sahi , `` Whittaker supports for representations of reductive groups ,'' Annales de l'Institut Fourier , http://arxiv.org/abs/1610.00284 arXiv:1610.00284 [math.RT] . To appear

    arXiv

  7. [15]

    Gomez, D

    R. Gomez, D. Gourevitch, and S. Sahi, ``Generalized and degenerate W hittaker models,'' http://dx.doi.org/10.1112/S0010437X16007788 Compos. Math. 153 no. 2, (2017) 223--256

    2017 doi

  8. [16]

    Ginzburg, ``Certain conjectures relating unipotent orbits to automorphic representations,'' http://dx.doi.org/10.1007/BF02777366 Israel J

    D. Ginzburg, ``Certain conjectures relating unipotent orbits to automorphic representations,'' http://dx.doi.org/10.1007/BF02777366 Israel J. Math. 151 (2006) 323--355

    2006 doi

  9. [17]

    D. Ginzburg, ``Towards a classification of global integral constructions and functorial liftings using the small representations method,'' http://dx.doi.org/10.1016/j.aim.2013.12.007 Adv. Math. 254 (2014) 157--186

    2014 doi

  10. [18]

    H. P. A. Gustafsson, A. Kleinschmidt, and D. Persson, `` Small automorphic representations and degenerate Whittaker vectors ,'' http://dx.doi.org/10.1016/j.jnt.2016.02.002 J. Number Theor. 166 (2016) 344--399 , http://arxiv.org/abs/1412.5625 arXiv:1412.5625 [math.NT]

    2016 arXiv

  11. [19]

    M. B. Green, S. D. Miller, and P. Vanhove, `` Small representations, string instantons, and Fourier modes of Eisenstein series ,'' http://dx.doi.org/10.1016/j.jnt.2013.05.018 J. Number Theor. 146 (2015) 187--309 , http://arxiv.org/abs/1111.2983 arXiv:1111.2983 [hep-th]

    2015 arXiv

  12. [20]

    Ginzburg, S

    D. Ginzburg, S. Rallis, and D. Soudry, ``On the automorphic theta representation for simply laced groups,'' http://dx.doi.org/10.1007/BF02773635 Israel J. Math. 100 (1997) 61--116

    1997 doi

  13. [21]

    Ginzburg, S

    D. Ginzburg, S. Rallis, and D. Soudry, http://dx.doi.org/10.1142/9789814304993 The descent map from automorphic representations of GL (n) to classical groups . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011

    2011 doi

  14. [22]

    W. T. Gan and G. Savin, ``On minimal representations definitions and properties,'' http://dx.doi.org/10.1090/S1088-4165-05-00191-3 Represent. Theory 9 (2005) 46--93 (electronic)

    2005 doi

  15. [23]

    Gourevitch and S

    D. Gourevitch and S. Sahi , `` Generalized and degenerate Whittaker quotients and Fourier coefficients ,'' http://arxiv.org/abs/1808.00890 1808.00890 [math.RT]

    arXiv

  16. [24]

    M. B. Green and P. Vanhove, ``Duality and higher derivative terms in M theory,'' http://dx.doi.org/10.1088/1126-6708/2006/01/093 J. High Energy Phys. no. 1, (2006) 093, 37

    2006 doi

  17. [25]

    Hartshorne, Algebraic geometry

    R. Hartshorne, Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52

    1977

  18. [26]

    C. M. Hull and P. K. Townsend, `` Unity of superstring dualities ,'' http://dx.doi.org/10.1016/0550-3213(94)00559-W Nucl. Phys. B438 (1995) 109--137 , http://arxiv.org/abs/hep-th/9410167 arXiv:hep-th/9410167 [hep-th] . [,236(1994)]

    1995 arXiv

  19. [27]

    Jiang, B

    D. Jiang, B. Liu, and G. Savin, ``Raising nilpotent orbits in wave-front sets,'' http://dx.doi.org/10.1090/ert/490 Represent. Theory 20 (2016) 419--450

    2016 doi

  20. [28]

    Kazhdan and A

    D. Kazhdan and A. Polishchuk, ``Minimal representations: spherical vectors and automorphic functionals,'' in Algebraic groups and arithmetic , pp. 127--198. Tata Inst. Fund. Res., Mumbai, 2004. http://arxiv.org/abs/arXiv:math/0209315 arXiv:math/0209315

    2004 arXiv

  21. [29]

    Kobayashi and G

    T. Kobayashi and G. Savin, ``Global uniqueness of small representations,'' http://dx.doi.org/10.1007/s00209-015-1481-0 Math. Z. 281 no. 1-2, (2015) 215--239

    2015 doi

  22. [30]

    I. K. Kostov and P. Vanhove, `` Matrix string partition functions ,'' http://dx.doi.org/10.1016/S0370-2693(98)01381-1 Phys. Lett. B444 (1998) 196--203 , http://arxiv.org/abs/hep-th/9809130 arXiv:hep-th/9809130 [hep-th]

    1998 arXiv

  23. [31]

    Matumoto, ``Whittaker vectors and associated varieties,'' http://dx.doi.org/10.1007/BF01404678 Invent

    H. Matumoto, ``Whittaker vectors and associated varieties,'' http://dx.doi.org/10.1007/BF01404678 Invent. Math. 89 no. 1, (1987) 219--224

    1987 doi

  24. [32]

    G. W. Moore, N. Nekrasov, and S. Shatashvili, `` D particle bound states and generalized instantons ,'' http://dx.doi.org/10.1007/s002200050016 Commun. Math. Phys. 209 (2000) 77--95 , http://arxiv.org/abs/hep-th/9803265 arXiv:hep-th/9803265 [hep-th]

    2000 arXiv

  25. [33]

    S. D. Miller and S. Sahi, ``Fourier coefficients of automorphic forms, character variety orbits, and small representations,'' http://dx.doi.org/10.1016/j.jnt.2012.05.032 J. Number Theory 132 no. 12, (2012) 3070--3108

    2012 doi

  26. [34]

    M glin and J.-L

    C. M glin and J.-L. Waldspurger, ``Mod\`eles de W hittaker d\' e g\' e n\' e r\' e s pour des groupes p -adiques,'' http://dx.doi.org/10.1007/BF01200363 Math. Z. 196 no. 3, (1987) 427--452

    1987 doi

  27. [35]

    M glin and J.-L

    C. M glin and J.-L. Waldspurger, http://dx.doi.org/10.1017/CBO9780511470905 Spectral decomposition and E isenstein series , vol. 113 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1995. Une paraphrase de l'\' E criture [A paraphrase of Scripture]

    1995 doi

  28. [36]

    D. Z . okovi \'c , ``Note on rational points in nilpotent orbits of semisimple groups,'' http://dx.doi.org/10.1016/S0019-3577(97)87565-9 Indag. Math. (N.S.) 9 no. 1, (1998) 31--34

    1998 doi

  29. [37]

    Persson, `` Automorphic instanton partition functions on Calabi-Yau threefolds ,'' http://dx.doi.org/10.1088/1742-6596/346/1/012016 J

    D. Persson, `` Automorphic instanton partition functions on Calabi-Yau threefolds ,'' http://dx.doi.org/10.1088/1742-6596/346/1/012016 J. Phys. Conf. Ser. 346 (2012) 012016 , http://arxiv.org/abs/1103.1014 arXiv:1103.1014 [hep-th]

    2012 arXiv

  30. [38]

    B. Pioline, `` R^4 couplings and automorphic unipotent representations ,'' http://dx.doi.org/10.1007/JHEP03(2010)116 JHEP 03 (2010) 116 , http://arxiv.org/abs/1001.3647 arXiv:1001.3647 [hep-th]

    2010 arXiv

  31. [39]

    Pioline and D

    B. Pioline and D. Persson, `` The automorphic NS5-brane ,'' http://dx.doi.org/10.4310/CNTP.2009.v3.n4.a5 Commun. Num. Theor. Phys. 3 (2009) 697--754 , http://arxiv.org/abs/0902.3274 arXiv:0902.3274 [hep-th]

    2009 arXiv

  32. [40]

    I. I. Piatetski-Shapiro, ``Multiplicity one theorems,'' in Automorphic forms, representations and L -functions ( P roc. S ympos. P ure M ath., O regon S tate U niv., C orvallis, O re., 1977), P art 1 , Proc. Sympos. Pure Math., XXXIII, pp. 209--212. Amer. Math. Soc., Providenc...

    1977

  33. [41]

    Rossmann, ``Picard- L efschetz theory and characters of a semisimple L ie group,'' http://dx.doi.org/10.1007/BF01884312 Invent

    W. Rossmann, ``Picard- L efschetz theory and characters of a semisimple L ie group,'' http://dx.doi.org/10.1007/BF01884312 Invent. Math. 121 no. 3, (1995) 579--611

    1995 doi

  34. [42]

    J. A. Shalika, ``The multiplicity one theorem for GL _ n ,'' http://dx.doi.org/10.2307/1971071 Ann. of Math. (2) 100 (1974) 171--193

    1974 doi

  35. [43]

    Varma, ``On a result of M oeglin and W aldspurger in residual characteristic 2,'' http://dx.doi.org/10.1007/s00209-014-1292-8 Math

    S. Varma, ``On a result of M oeglin and W aldspurger in residual characteristic 2,'' http://dx.doi.org/10.1007/s00209-014-1292-8 Math. Z. 277 no. 3-4, (2014) 1027--1048

    2014 doi

  36. [44]

    M. H. Weissman, ``L-groups and parameters for covering groups,'' http://dx.doi.org/10.24033/ast.1044 Ast\' e risque no. 398, (2018) 33--186 . L-groups and the Langlands program for covering groups

    2018 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.