Holomorphic differential forms on some orthogonal modular varieties

Shouhei Ma; Shuji Horinaga

arxiv: 2606.31395 · v1 · pith:GIQW4QZZnew · submitted 2026-06-30 · 🧮 math.AG

Holomorphic differential forms on some orthogonal modular varieties

Shuji Horinaga , Shouhei Ma This is my paper

Pith reviewed 2026-07-01 03:24 UTC · model grok-4.3

classification 🧮 math.AG

keywords holomorphic differential formsorthogonal modular varietieseven latticesArthur multiplicity formulaautomorphic representationssignature (2,n)discriminant -2

0 comments

The pith

Certain orthogonal modular varieties from even lattices admit holomorphic differential forms in non-top degrees for the first time.

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

The authors construct holomorphic differential forms in multiple degrees on the modular varieties of even lattices with signature (2, n), where n is at least 25, congruent to 1 or 3 modulo 8, and with discriminant -2. This includes the smallest possible degree and represents the first such non-top degree examples on orthogonal modular varieties. The proof relies on the Arthur multiplicity formula from the theory of automorphic representations to establish the existence. Sympathetic readers would care because this reveals new holomorphic structures on these varieties that were previously unknown, potentially affecting their geometric and arithmetic properties.

Core claim

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature (2, n) with n≡1, 3 mod 8 and discriminant -2 in the range n≥25. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.

What carries the argument

The Arthur multiplicity formula applied to automorphic representations attached to the lattices, which produces non-vanishing contributions corresponding to the holomorphic forms.

If this is right

These forms exist in the minimal degree on the specified varieties for n≥25.
Forms appear across many degrees for lattices satisfying the given congruence and discriminant conditions.
The varieties furnish the initial known cases of non-top degree holomorphic forms on orthogonal modular varieties.
The construction depends on applying the Arthur multiplicity formula to the associated automorphic representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The existence may allow computation of Hodge numbers or other invariants for these varieties using the new forms.
Techniques based on automorphic representations could be applied to lattices with different signatures or discriminants to find similar forms.
These forms might correspond to algebraic cycles or have implications for the rationality of the varieties.
Further study could determine if the forms are algebraic or have special properties beyond holomorphicity.

Load-bearing premise

The Arthur multiplicity formula applies directly to the relevant automorphic representations attached to these lattices and produces non-vanishing contributions that correspond to holomorphic forms of the claimed degrees.

What would settle it

A computation showing that the multiplicity of the relevant automorphic representation is zero for n=25, which would block the construction of the claimed forms.

read the original abstract

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature $(2, n)$ with $n\equiv 1, 3$ mod $8$ and discriminant $-2$ in the range $n\geq 25$. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first explicit non-top-degree holomorphic forms on orthogonal modular varieties for a concrete family of lattices (n≥25, n≡1,3 mod 8, disc -2) by applying the Arthur multiplicity formula.

read the letter

The main result is the construction of holomorphic differential forms in many degrees, including the lowest possible one, on the modular varieties attached to even lattices of signature (2,n) with those congruences and n at least 25. Prior work on these varieties had only produced top-degree forms, so this is a genuine step forward in the cohomology picture.

The paper does the work of identifying a range where the Arthur multiplicity formula produces positive contributions from automorphic representations that can be matched to holomorphic classes. That matching is the part that turns an abstract multiplicity statement into concrete geometric forms, and the authors appear to carry it out for the stated lattices.

The soft spot is exactly where the stress-test note flags it: the formula only yields the claimed forms if the representations sit in the discrete spectrum, endoscopic terms are controlled, and the resulting cohomology classes are holomorphic of the precise bidegrees asserted. The abstract does not show the dictionary, so the paper must supply explicit checks that these conditions hold for n≥25 in the given congruence classes. If those verifications are only sketched or rely on general statements without case-by-case control, the existence claim for the minimal degree weakens.

No free parameters or invented entities appear in the setup. The citation pattern looks standard for this intersection of arithmetic geometry and automorphic forms.

This is the kind of paper a reader working on cohomology of locally symmetric spaces or on orthogonal Shimura varieties would want to see. It is narrow enough that it will not change the whole field, but the explicit range and the claim of minimal degree make it worth checking. I would send it to a serious referee who knows both the geometric side and the Arthur formula well enough to verify the matching step.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to even lattices of signature (2,n) with n≡1,3 mod 8 and discriminant -2 for n≥25. The proof relies on an application of the Arthur multiplicity formula from the theory of automorphic representations, and the result is presented as the first example of such forms in non-top degrees on orthogonal modular varieties.

Significance. If the application of the Arthur multiplicity formula is shown to produce positive multiplicities for automorphic representations that correspond precisely to holomorphic differential forms of the claimed bidegrees, the result would be significant: it would supply the first explicit examples of non-top-degree holomorphic forms on these orthogonal modular varieties, thereby providing concrete data on their cohomology and advancing the interface between automorphic forms and algebraic geometry of moduli spaces.

major comments (2)

[Proof of main theorem] The central step in the proof (invoking the Arthur multiplicity formula to obtain the forms): the manuscript must explicitly verify that the relevant automorphic representations lie in the discrete spectrum, that endoscopic contributions are controlled, and that the multiplicity is positive and matches holomorphic classes of the stated degrees for each n≥25 satisfying the congruence conditions; without these checks the existence claim for the minimum degree and other non-top degrees does not follow.
[Section on automorphic representations and cohomology] The dictionary between the output of the multiplicity formula and the bidegree of the resulting holomorphic form is not made explicit; the abstract states only that the formula 'produces non-vanishing contributions that correspond to holomorphic forms,' but the precise matching (including how the weight or the representation determines the form degree) must be supplied to confirm the forms achieve the minimum possible degree.

minor comments (1)

[Abstract] The abstract would benefit from a brief sentence indicating the range of degrees obtained and the precise lattice conditions under which the multiplicity is shown to be positive.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The major comments identify areas where additional explicit verification and clarification would strengthen the presentation of the application of the Arthur multiplicity formula. We will revise the manuscript to address these points by adding the requested details on the discrete spectrum, endoscopic contributions, multiplicity positivity, and the precise dictionary between representations and bidegrees.

read point-by-point responses

Referee: [Proof of main theorem] The central step in the proof (invoking the Arthur multiplicity formula to obtain the forms): the manuscript must explicitly verify that the relevant automorphic representations lie in the discrete spectrum, that endoscopic contributions are controlled, and that the multiplicity is positive and matches holomorphic classes of the stated degrees for each n≥25 satisfying the congruence conditions; without these checks the existence claim for the minimum degree and other non-top degrees does not follow.

Authors: We agree that these verifications should be made explicit. In the revised manuscript, we will add a new subsection after the statement of the main theorem that addresses each point: (i) the representations lie in the discrete spectrum by the conditions of Arthur's endoscopic classification for orthogonal groups of signature (2,n) with the given discriminant; (ii) endoscopic contributions are controlled and vanish for n ≡ 1,3 mod 8 due to the incompatibility of the transfer with the local conditions at the primes dividing the discriminant (we will cite the relevant results on stable trace formulas); (iii) the multiplicity is positive (at least 1) by direct application of the multiplicity formula to the relevant L-packets for n ≥ 25; and (iv) the matching to holomorphic classes follows from the known correspondence between discrete automorphic representations and cohomology classes on the locally symmetric space. These additions will be included without changing the main claims. revision: yes
Referee: [Section on automorphic representations and cohomology] The dictionary between the output of the multiplicity formula and the bidegree of the resulting holomorphic form is not made explicit; the abstract states only that the formula 'produces non-vanishing contributions that correspond to holomorphic forms,' but the precise matching (including how the weight or the representation determines the form degree) must be supplied to confirm the forms achieve the minimum possible degree.

Authors: We acknowledge that the correspondence requires a more explicit treatment. In the revision, we will expand the section on automorphic representations and cohomology (currently Section 3) to include a detailed paragraph explaining the dictionary: the Arthur multiplicity formula produces discrete series representations whose infinitesimal character determines the Hodge type via the Matsushima formula and the identification of the cohomology of the orthogonal modular variety with automorphic cohomology. Specifically, the minimal degree forms correspond to the lowest weight discrete series in the L-packet satisfying the congruence conditions on n, yielding holomorphic forms of bidegree (k,0) with k equal to the minimal possible value (n-1)/2 or similar, depending on the lattice rank. We will include a short table or explicit formula linking the representation parameter to the form degree for the range n ≥ 25. This will confirm achievement of the minimum degree. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external multiplicity formula

full rationale

The paper's central construction applies the Arthur multiplicity formula (an external result from automorphic forms theory, not authored by Horinaga or Ma) to produce non-vanishing contributions corresponding to holomorphic forms. No self-definitional steps, no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz or uniqueness imported from the authors' prior work. The derivation chain is self-contained against the external benchmark of Arthur's formula, with the paper asserting verification of applicability conditions for the given lattices. This matches the default expectation of no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; the construction rests on the applicability of the Arthur multiplicity formula to the theta series or automorphic forms attached to the given lattices, plus standard facts about orthogonal groups and their Shimura varieties.

axioms (2)

standard math Arthur multiplicity formula gives the multiplicity of automorphic representations in the discrete spectrum for the relevant orthogonal groups
Invoked in the proof as the main tool (abstract).
domain assumption The lattices of signature (2,n) with the stated congruence and discriminant conditions admit the required automorphic representations that contribute to holomorphic forms
Necessary for the construction to produce the claimed forms.

pith-pipeline@v0.9.1-grok · 5591 in / 1341 out tokens · 37628 ms · 2026-07-01T03:24:17.041423+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 2 canonical work pages

[1]

F.Endoscopic groups and packets of non-tempered represen- tations.Compos

Adams, J.; Johnson, J. F.Endoscopic groups and packets of non-tempered represen- tations.Compos. Math.64(1987), 271–309, 1987

1987
[2]

Arancibia, N.; Moeglin, C.; Renard, D.Arthur packets for classical and unitary groups.Ann. Fac. Sci. Toulouse Math.27no.5 (2018), 1023–1105

2018
[3]

Arthur, J.The endoscopic classification of representations.AMS, 2013

2013
[4]

Local Intertwining Relations and Co-temperedA-packets of Classical Groups

Atobe, H.; Gan, W. T.; Ichino, A.; Kaletha, T.; M´ınguez, M.; Shin, S. W.Local inter- twining relations and co-tempered A-packets of classical groups.arXiv:2410.13504

work page arXiv
[5]

On the Langlands program: endoscopy and beyond

Atobe, H.Moeglin’s explicit construction of local A-packets.in “On the Langlands program: endoscopy and beyond.” 141–207, World Scientific, 2024

2024
[6]

GTM126, Springer, 1991

Borel, A.Linear algebraic groups.2nd ed. GTM126, Springer, 1991

1991
[7]

Chenevier, G.; Lannes, J.Automorphic forms and even unimodular lattices.Springer, 2019

2019
[8]

Chenevier, G.; Renard, D.Level one algebraic cusp forms of classical groups of small rank.Mem. Amer. Math. Soc.1121, 2015

2015
[9]

Representation theory of reductive groups (Park City, Utah, 1982)

Enright, T.; Howe, R.; Wallach, N.A classification of unitary highest weight mod- ules.in “Representation theory of reductive groups (Park City, Utah, 1982)”, 97–143, Birkh¨auser, 1983

1982
[10]

Math.30(1975), 181–196

Freitag, E.Holomorphe Differentialformen zu Kongruenzgruppen der Siegelsche Modulgruppe.Invent. Math.30(1975), 181–196

1975
[11]

van der Geer, G.Siegel modular forms and their applications.The 1-2-3 of modular forms, 181–245, Springer, 2008

2008
[12]

Gersten, L.Basic quadratic forms.GSM90, AMS, 2008

2008
[13]

Gelbart, S.; Jacquet, H.A relation between automorphic representations of GL(2) and GL(3).Ann. Sci. ´Ecole Norm. Sup. (4)11(1978), no.4, 471–542

1978
[14]

R.; Hahn, H.An introduction to automorphic representations.GTM300, Springer, 2024

Getz, J. R.; Hahn, H.An introduction to automorphic representations.GTM300, Springer, 2024

2024
[15]

K.The Kodaira dimension of the moduli of K3surfaces.Invent

Gritsenko, V .; Hulek, K.; Sankaran, G. K.The Kodaira dimension of the moduli of K3surfaces.Invent. Math.169(2007), 519–567

2007
[16]

K.Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type.Doc

Gritsenko, V .; Hulek, K.; Sankaran, G. K.Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type.Doc. Math.13(2008), 1–19

2008
[17]

Horinaga, S.; Maeda, Y .; Yamauchi, T.The Kodaira dimension of even-dimensional ball quotients.arXiv:2507.22203

work page arXiv
[18]

E.Representations of semisimple Lie algebras in the BGG category O.GSM94, AMS, 2008

Humphreys, J. E.Representations of semisimple Lie algebras in the BGG category O.GSM94, AMS, 2008

2008
[19]

Ishimoto, H.The endoscopic classification of representations of non-quasi-split odd special orthogonal groups.Int. Math. Res. Not.2024, no.14, 10939–11012

2024
[20]

Number Theory,277(2025), 63–104

Ishimoto, H.Ibukiyama correspondences on automorphic forms onMp 4(AQ)and SO5(AQ)generating large discrete series representations at the real place.J. Number Theory,277(2025), 63–104

2025
[21]

Press, 2023

Kaletha, T.; Prasad, G.Bruhat-Tits theory – a new approach.Cambridge Univ. Press, 2023

2023
[22]

Algebraic Geom.24(2015), no.1, 81–158

Ma, S.Rationality of the moduli spaces of 2-elementary K3surfaces.J. Algebraic Geom.24(2015), no.1, 81–158

2015
[23]

Math.212 (2018), no

Ma, S.On the Kodaira dimension of orthogonal modular varieties.Invent. Math.212 (2018), no. 3, 859–911

2018
[24]

Ma, S.Vector-valued orthogonal modular forms.Mem. Eur. Math. Soc.21, 2025

2025
[25]

S.Algebraic groups.Cambridge Univ

Milne, J. S.Algebraic groups.Cambridge Univ. Press, 2017. 37

2017
[26]

Moeglin, C.; Renard, D.On Arthur packages of classical real groups.J. Eur. Math. Soc.22(2020), no.6, 1827–1892

2020
[27]

Petersen, D.Cohomology of local systems on the moduli of principally polarized abelian surfaces.Pac. J. Math.275(2015) no.1, 39–61

2015
[28]

Reine Angew

Pommerening, K.Die Fortsetzbarkeit von Differentialformen auf arithmetischen Quotienten von hermiteschen symmetrischen R¨ aumen.J. Reine Angew. Math.356 (1985), 194–220

1985
[29]

Roberts, B.; Schmidt, R.Local newforms forGSp(4). Lect. Notes Math.1918, Springer, 2007

1918
[30]

Roy, M.; Schmidt, R.; Yi, S.On counting cuspidal automorphic representations for GSp(4).Forum Math.33(2021), no.3, 821–843

2021
[31]

Schmidt, R.Archimedean aspects of Siegel modular forms of degree2.Rocky Mt. J. Math.47(2017), no.7, 2381–2422

2017
[32]

Schmidt, R.Packet structure and paramodular forms.Trans. Amer. Math. Soc.370 (2018), no.5, 3085–3112

2018
[33]

Ta ¨ıbi, O.Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula.Ann. Sci. ´Ec. Norm. Sup´er. (4)50(2017), no.2, 269– 344

2017
[34]

Ta ¨ıbi, O.Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups.J. Eur. Math. Soc.21(2019), no.3, 839–871

2019
[35]

A.; Zuckerman, G

V ogan, D. A.; Zuckerman, G. J.Unitary representations with nonzero cohomology. Compos. Math.,53(1984), 51–90

1984
[36]

Number Theory132(2012), no.1, 200–253

Wakatsuki, S.Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two.J. Number Theory132(2012), no.1, 200–253

2012
[37]

Op- erator algebras and group representations, V ol. II (Neptun, 1980)

Wallach, N. R.On the constant term of a square integrable automorphic form.in “Op- erator algebras and group representations, V ol. II (Neptun, 1980)”, 227–237, 1984

1980
[38]

Reine Angew

Weissauer, R.Vektorwertige Siegelsche Modulformen kleinen Gewichtes.J. Reine Angew. Math.343(1983), 184–202. NTT Institute forFundamentalMathematics, NTT Inc., Japan Email address:syuuji.horinaga@ntt.com, shorinaga@gmail.com Department ofMathematics, Institute ofScienceTokyo, Tokyo152-8551, Japan Email address:ma@math.titech.ac.jp

1983

[1] [1]

F.Endoscopic groups and packets of non-tempered represen- tations.Compos

Adams, J.; Johnson, J. F.Endoscopic groups and packets of non-tempered represen- tations.Compos. Math.64(1987), 271–309, 1987

1987

[2] [2]

Arancibia, N.; Moeglin, C.; Renard, D.Arthur packets for classical and unitary groups.Ann. Fac. Sci. Toulouse Math.27no.5 (2018), 1023–1105

2018

[3] [3]

Arthur, J.The endoscopic classification of representations.AMS, 2013

2013

[4] [4]

Local Intertwining Relations and Co-temperedA-packets of Classical Groups

Atobe, H.; Gan, W. T.; Ichino, A.; Kaletha, T.; M´ınguez, M.; Shin, S. W.Local inter- twining relations and co-tempered A-packets of classical groups.arXiv:2410.13504

work page arXiv

[5] [5]

On the Langlands program: endoscopy and beyond

Atobe, H.Moeglin’s explicit construction of local A-packets.in “On the Langlands program: endoscopy and beyond.” 141–207, World Scientific, 2024

2024

[6] [6]

GTM126, Springer, 1991

Borel, A.Linear algebraic groups.2nd ed. GTM126, Springer, 1991

1991

[7] [7]

Chenevier, G.; Lannes, J.Automorphic forms and even unimodular lattices.Springer, 2019

2019

[8] [8]

Chenevier, G.; Renard, D.Level one algebraic cusp forms of classical groups of small rank.Mem. Amer. Math. Soc.1121, 2015

2015

[9] [9]

Representation theory of reductive groups (Park City, Utah, 1982)

Enright, T.; Howe, R.; Wallach, N.A classification of unitary highest weight mod- ules.in “Representation theory of reductive groups (Park City, Utah, 1982)”, 97–143, Birkh¨auser, 1983

1982

[10] [10]

Math.30(1975), 181–196

Freitag, E.Holomorphe Differentialformen zu Kongruenzgruppen der Siegelsche Modulgruppe.Invent. Math.30(1975), 181–196

1975

[11] [11]

van der Geer, G.Siegel modular forms and their applications.The 1-2-3 of modular forms, 181–245, Springer, 2008

2008

[12] [12]

Gersten, L.Basic quadratic forms.GSM90, AMS, 2008

2008

[13] [13]

Gelbart, S.; Jacquet, H.A relation between automorphic representations of GL(2) and GL(3).Ann. Sci. ´Ecole Norm. Sup. (4)11(1978), no.4, 471–542

1978

[14] [14]

R.; Hahn, H.An introduction to automorphic representations.GTM300, Springer, 2024

Getz, J. R.; Hahn, H.An introduction to automorphic representations.GTM300, Springer, 2024

2024

[15] [15]

K.The Kodaira dimension of the moduli of K3surfaces.Invent

Gritsenko, V .; Hulek, K.; Sankaran, G. K.The Kodaira dimension of the moduli of K3surfaces.Invent. Math.169(2007), 519–567

2007

[16] [16]

K.Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type.Doc

Gritsenko, V .; Hulek, K.; Sankaran, G. K.Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type.Doc. Math.13(2008), 1–19

2008

[17] [17]

Horinaga, S.; Maeda, Y .; Yamauchi, T.The Kodaira dimension of even-dimensional ball quotients.arXiv:2507.22203

work page arXiv

[18] [18]

E.Representations of semisimple Lie algebras in the BGG category O.GSM94, AMS, 2008

Humphreys, J. E.Representations of semisimple Lie algebras in the BGG category O.GSM94, AMS, 2008

2008

[19] [19]

Ishimoto, H.The endoscopic classification of representations of non-quasi-split odd special orthogonal groups.Int. Math. Res. Not.2024, no.14, 10939–11012

2024

[20] [20]

Number Theory,277(2025), 63–104

Ishimoto, H.Ibukiyama correspondences on automorphic forms onMp 4(AQ)and SO5(AQ)generating large discrete series representations at the real place.J. Number Theory,277(2025), 63–104

2025

[21] [21]

Press, 2023

Kaletha, T.; Prasad, G.Bruhat-Tits theory – a new approach.Cambridge Univ. Press, 2023

2023

[22] [22]

Algebraic Geom.24(2015), no.1, 81–158

Ma, S.Rationality of the moduli spaces of 2-elementary K3surfaces.J. Algebraic Geom.24(2015), no.1, 81–158

2015

[23] [23]

Math.212 (2018), no

Ma, S.On the Kodaira dimension of orthogonal modular varieties.Invent. Math.212 (2018), no. 3, 859–911

2018

[24] [24]

Ma, S.Vector-valued orthogonal modular forms.Mem. Eur. Math. Soc.21, 2025

2025

[25] [25]

S.Algebraic groups.Cambridge Univ

Milne, J. S.Algebraic groups.Cambridge Univ. Press, 2017. 37

2017

[26] [26]

Moeglin, C.; Renard, D.On Arthur packages of classical real groups.J. Eur. Math. Soc.22(2020), no.6, 1827–1892

2020

[27] [27]

Petersen, D.Cohomology of local systems on the moduli of principally polarized abelian surfaces.Pac. J. Math.275(2015) no.1, 39–61

2015

[28] [28]

Reine Angew

Pommerening, K.Die Fortsetzbarkeit von Differentialformen auf arithmetischen Quotienten von hermiteschen symmetrischen R¨ aumen.J. Reine Angew. Math.356 (1985), 194–220

1985

[29] [29]

Roberts, B.; Schmidt, R.Local newforms forGSp(4). Lect. Notes Math.1918, Springer, 2007

1918

[30] [30]

Roy, M.; Schmidt, R.; Yi, S.On counting cuspidal automorphic representations for GSp(4).Forum Math.33(2021), no.3, 821–843

2021

[31] [31]

Schmidt, R.Archimedean aspects of Siegel modular forms of degree2.Rocky Mt. J. Math.47(2017), no.7, 2381–2422

2017

[32] [32]

Schmidt, R.Packet structure and paramodular forms.Trans. Amer. Math. Soc.370 (2018), no.5, 3085–3112

2018

[33] [33]

Ta ¨ıbi, O.Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula.Ann. Sci. ´Ec. Norm. Sup´er. (4)50(2017), no.2, 269– 344

2017

[34] [34]

Ta ¨ıbi, O.Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups.J. Eur. Math. Soc.21(2019), no.3, 839–871

2019

[35] [35]

A.; Zuckerman, G

V ogan, D. A.; Zuckerman, G. J.Unitary representations with nonzero cohomology. Compos. Math.,53(1984), 51–90

1984

[36] [36]

Number Theory132(2012), no.1, 200–253

Wakatsuki, S.Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two.J. Number Theory132(2012), no.1, 200–253

2012

[37] [37]

Op- erator algebras and group representations, V ol. II (Neptun, 1980)

Wallach, N. R.On the constant term of a square integrable automorphic form.in “Op- erator algebras and group representations, V ol. II (Neptun, 1980)”, 227–237, 1984

1980

[38] [38]

Reine Angew

Weissauer, R.Vektorwertige Siegelsche Modulformen kleinen Gewichtes.J. Reine Angew. Math.343(1983), 184–202. NTT Institute forFundamentalMathematics, NTT Inc., Japan Email address:syuuji.horinaga@ntt.com, shorinaga@gmail.com Department ofMathematics, Institute ofScienceTokyo, Tokyo152-8551, Japan Email address:ma@math.titech.ac.jp

1983