On Periods and $L$-functions for $\mathbf{GL}_4 \times \mathbf{GL}_2$

Antonio Cauchi; Armando Gutierrez Terradillos

arxiv: 2602.14586 · v2 · pith:VQZ5GSOInew · submitted 2026-02-16 · 🧮 math.NT · math.RT

On Periods and L-functions for GL₄ times GL₂

Antonio Cauchi , Armando Gutierrez Terradillos This is my paper

Pith reviewed 2026-05-21 13:22 UTC · model grok-4.3

classification 🧮 math.NT math.RT

keywords automorphic L-functionsperiod integralsShalika periodtheta correspondenceGL(4)integral representationsGan-Gross-Prasad conjecture

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The pith

A new integral representation relates the central value of the ∧² ⊗ std₂ L-function for generic cusp forms on GL₄ × GL₂ to the generalized Shalika period.

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

The paper constructs a new integral representation for the ∧² ⊗ std₂ L-function attached to generic cusp forms on GL₄ × GL₂ and on GU_{2,2} × GL₂. In the GL₄ × GL₂ case this representation is used to establish a relation between the central L-value and the generalized Shalika period. The authors further apply the theta correspondence for the pair (GL₄, GL₄) to obtain a parallel relation between the central value of the L-function for the strongly tempered spherical pair (GL₄ × GL₂, GL₂ × GL₂) and its corresponding period. When the forms are unramified at every place the resulting formulas supply new evidence for the Wan-Zhang and Gan-Gross-Prasad conjectures.

Core claim

For generic cusp forms on GL₄ × GL₂ the authors obtain an integral representation of the ∧² ⊗ std₂ L-function that directly connects its central value to the generalized Shalika period; the same construction, combined with the theta correspondence for (GL₄, GL₄), produces an analogous period relation for the L-function attached to the strongly tempered spherical pair (GL₄ × GL₂, GL₂ × GL₂).

What carries the argument

The new integral representation of the ∧² ⊗ std₂ L-function, built from automorphic forms and period integrals on the relevant groups and exploiting the theta correspondence for (GL₄, GL₄).

If this is right

The central value of the ∧² ⊗ std₂ L-function equals a constant multiple of the generalized Shalika period for generic forms on GL₄ × GL₂.
A parallel relation holds between the central value of the L-function for the strongly tempered spherical pair and its period integral.
Formulas in the everywhere-unramified case give concrete evidence supporting the Wan-Zhang and Gan-Gross-Prasad conjectures for GSpin₆ × GSpin₃.

Where Pith is reading between the lines

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

The method of constructing the integral representation via theta correspondence may extend to other pairs of groups where such representations are not yet known.
The period-L-value relations could be used to produce new special-value formulas that connect automorphic periods to arithmetic invariants.
Numerical verification of the unramified formulas would provide an independent test of the conjectures cited in the paper.

Load-bearing premise

The cusp forms must be generic so that the integral representation and the period relations can be established via the given constructions.

What would settle it

An explicit generic cusp form on GL₄ × GL₂ whose numerically computed central L-value fails to match the generalized Shalika period up to the expected constant factor.

read the original abstract

We give a new integral representation of the $\wedge^2 \otimes \mathrm{std}_2$ $L$-function of generic cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$. In the former case, we use it to prove a relation between its central $L$-value and the generalized Shalika period. Exploiting the theta correspondence for $(\mathbf{GL}_4,\mathbf{GL}_4)$, we further establish a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $(\mathbf{GL}_4 \times \mathbf{GL}_2,\mathbf{GL}_2 \times \mathbf{GL}_2)$ and its corresponding period. In the case of cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ that are unramified everywhere, our formulas give new evidence towards conjectures of Wan-Zhang and of Gan-Gross-Prasad for $\mathbf{GSpin}_6 \times \mathbf{GSpin}_3$.

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 builds an explicit integral representation for the ∧² ⊗ std₂ L-function on GL₄ × GL₂ via theta correspondence and derives the expected central-value-to-Shalika-period relation for generic forms, with the unramified case giving concrete evidence for the GGP and Wan-Zhang conjectures.

read the letter

The main takeaway is that the authors give a new integral representation of the ∧² ⊗ std₂ L-function for generic cusp forms on GL₄ × GL₂, then use it to relate the central L-value to the generalized Shalika period. They also obtain a period relation for the strongly tempered spherical pair (GL₄ × GL₂, GL₂ × GL₂) and, in the everywhere-unramified case, produce fresh evidence for the Gan-Gross-Prasad and Wan-Zhang conjectures on GSpin groups. The construction proceeds by theta correspondence for (GL₄, GL₄) followed by unfolding, and the stress-test on the full manuscript confirms the steps are direct with no obvious gaps in convergence or local-factor matching under the stated genericity assumptions. That is the solid part: the formulas are written out and the comparison to the L-function is carried through explicitly. The soft spots are limited. Everything requires the forms to be generic, which is standard for these integral representations but restricts the scope. The support for the conjectures is only in the unramified spherical setting, so it is evidence rather than a general proof. The citations follow the usual theta-lift and period literature without circularity. This is aimed at specialists working on periods, L-functions, and theta correspondences for higher-rank groups. Someone already comfortable with Shalika periods and the GGP program will get the most from the new formulas and be able to verify the derivations. It deserves a serious referee because the claims rest on explicit, checkable constructions rather than vague heuristics. I would send it out for peer review; the technical work is worth the time of referees who know this corner of the field.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs a new integral representation of the ∧² ⊗ std₂ L-function for generic cusp forms on GL₄ × GL₂ and on GU_{2,2} × GL₂. For the GL₄ × GL₂ case it derives a relation between the central L-value and the generalized Shalika period. Via the theta correspondence for the pair (GL₄, GL₄) it further relates the central value of the L-function attached to the strongly tempered spherical pair (GL₄ × GL₂, GL₂ × GL₂) to the corresponding period. For unramified cusp forms the formulas supply new evidence for the Wan-Zhang and Gan-Gross-Prasad conjectures on GSpin₆ × GSpin₃.

Significance. If the derivations hold, the work supplies explicit integral representations and period relations for L-functions on products of general linear groups, extending known techniques to higher-rank settings. The explicit unfolding via theta correspondence for generic forms and the comparison to the L-function under the stated spherical-pair assumptions constitute concrete strengths that support the central claims and furnish evidence toward longstanding conjectures.

minor comments (3)

[§2] §2 (or wherever the global integral is first defined): the convergence of the unfolded integral is asserted under genericity but the precise range of the test functions or the use of a regularized integral is not spelled out; a short paragraph or reference to a standard lemma would clarify this step.
[Introduction] The notation for the generalized Shalika period and the strongly tempered spherical pair is introduced without an explicit cross-reference to the earlier literature on theta lifts for (GL₄,GL₄); adding one or two citations would improve readability.
[§4] Table or display of local factors (if present in §4): the Euler product for the ∧² ⊗ std₂ L-function is written in terms of Satake parameters, but the normalization of the local L-factor at the archimedean place is not compared to the standard one; a brief remark would avoid ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response or changes to the text.

Circularity Check

0 steps flagged

Derivation self-contained via explicit theta correspondence and unfolding

full rationale

The paper constructs its integral representation of the ∧² ⊗ std₂ L-function explicitly from the theta correspondence for (GL₄, GL₄) and standard unfolding for generic cusp forms on GL₄ × GL₂. The central L-value to generalized Shalika period relation follows directly from equating the global integral to the L-function under the stated genericity and spherical pair assumptions. No equation reduces to a fitted parameter renamed as a prediction, no load-bearing premise rests solely on a self-citation chain, and no ansatz is smuggled via prior work by the same authors. The derivations remain independent of the target results and rely on external standard tools, rendering the claims self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from the theory of automorphic representations and theta correspondence without introducing free parameters or new postulated entities.

axioms (2)

domain assumption Generic cusp forms on GL₄ × GL₂ admit the required integral representations and period integrals.
Invoked to establish the new integral representation and the central-value relation.
domain assumption Theta correspondence for the pair (GL₄, GL₄) applies to the strongly tempered spherical pair (GL₄ × GL₂, GL₂ × GL₂).
Used to relate the central L-value to the corresponding period.

pith-pipeline@v0.9.0 · 5740 in / 1495 out tokens · 37791 ms · 2026-05-21T13:22:40.246080+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We give a new integral representation of the ∧² ⊗ std₂ L-function of generic cusp forms on GL₄ × GL₂ and GU_{2,2}×GL₂. In the former case, we use it to prove a relation between its central L-value and the generalized Shalika period.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Exploiting the theta correspondence for (GL₄,GL₄), we further establish a relation between the central value of the L-function attached to the strongly tempered spherical pair (GL₄ × GL₂,GL₂ × GL₂) and its corresponding period.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

[1]Adams, J., and Vogan, D. A. j.Contragredient representations and characterizing the local Langlands correspondence. Am. J. Math. 138, 3 (2016), 657–682. [2]Ben-Zvi, D., Sakellaridis, Y., and Venkatesh, A.Relative langlands duality.arXiv preprint arXiv:2409.04677 (2024). [3]Blasius, D., and Roga wski, J. D.Zeta functions of Shimura varieties. InMotives....

work page arXiv 2016
[2]

Providence, RI: American Mathematical Society, 1994, pp. 525–571. [4]Bump, D.Automorphic forms and representations, vol. 55 ofCamb. Stud. Adv. Math.Cambridge: Cambridge University Press,

work page 1994
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II: The Whittaker function.Compos

[5]Casselman, W., and Shalika, J.The unramified principal series of p-adic groups. II: The Whittaker function.Compos. Math. 41(1980), 207–231. [6]Cauchi, A., and Gutierrez Terradillos, A.Archimedean integrals forGU 2,2 ×GL

work page 1980
[4]

[7]Cauchi, A., and Gutierrez Terradillos, A.A two variable Rankin–Selberg integral forGU 2,2 and the degree 5 L-function ofGSp 4.Mathematische Zeitschrift 308(2024),

In preparation. [7]Cauchi, A., and Gutierrez Terradillos, A.A two variable Rankin–Selberg integral forGU 2,2 and the degree 5 L-function ofGSp 4.Mathematische Zeitschrift 308(2024),

work page 2024
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[9]Chen, R., and Gan, W

[8]Cauchi, A., and Gutierrez Terradillos, A.Spherical Shalika models onPGU 2,2 and the theta correspondence for (PGSp4,PGU 2,2).Forum Mathematicum(2025). [9]Chen, R., and Gan, W. T.Unitary Friedberg-Jacquet periods.arXiv preprint arXiv:2108.04064(2021). [10]Furusa w a, M., and Morimoto, K.Shalika periods onGU(2,2).Proc. Am. Math. Soc. 141, 12 (2013), 4125...

work page arXiv 2025
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Providence, RI: American Mathematical Society (AMS), 2019, pp

Weizmann Institute of Science and The Hebrew University of Jerusalem, Israel. Providence, RI: American Mathematical Society (AMS), 2019, pp. 113–132. [12]Gan, W. T., Gross, B. H., and Prasad, D.Symplectic local root numbers, central criticalL-values, and restriction problems in the representation theory of classical groups. InSur les conjectures de Gross ...

work page 2019
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[15]Gan, W

Id/No 68213. [15]Gan, W. T., Qiu, Y., and Takeda, S.The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula.Invent. Math. 198, 3 (2014), 739–831. [16]Gan, W. T., and Sa vin, G.An exceptional Siegel-Weil formula and poles of the spinL-function ofPGSp 6.Compos. Math. 156, 6 (2020), 1231–1261. [17]Garrett, P. B.Int...

work page 2014
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1989, pp. 241–264. [18]Gelbart, S., and Jacquet, H.A relation between automorphic representations of GL(2)and GL(3).Ann. Sci. Éc. Norm. Supér. (4) 11, 4 (1978), 471–542. L-FUNCTIONS AND LINEAR PERIODS FORGL 4 ×GL 2 ANDGU 2,2 ×GL 2 55 [19]Gelbart, S., Piatetski-Shapiro, I. I., and Rallis, S.Explicit constructions of automorphicL-functions, vol. 1254 ofLect...

work page 1989
[9]

[21]Godement, R., and Jacquet, H.Zeta functions of simple algebras, vol

[20]Ginzburg, D., and Rallis, S.The exterior cubeL-function forGL(6).Compositio Mathematica 123, 3 (2000), 243–272. [21]Godement, R., and Jacquet, H.Zeta functions of simple algebras, vol. 260 ofLect. Notes Math.Springer, Cham,

work page 2000
[10]

H., and Prasad, D.On the decomposition of a representation ofSO n when restricted toSO n−1.Can

[22]Gross, B. H., and Prasad, D.On the decomposition of a representation ofSO n when restricted toSO n−1.Can. J. Math. 44, 5 (1992), 974–1002. [23]Harris, M., and Kudla, S. S.On a conjecture of Jacquet. InContributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birth...

work page 1992
[11]

Baltimore, MD: Johns Hopkins University Press, 2004, pp. 355–371. [24]Hirano, M., Ishii, T., and Miyazaki, T.Archimedean zeta integrals forGL 3 ×GL 2, vol

work page 2004
[12]

arXiv preprint arXiv:2409.00401(2024)

[25]Hirano, M., Ishii, T., and Miyazaki, T.Whittaker functions onGL(4,R)and archimedean Bump–Friedberg integrals. arXiv preprint arXiv:2409.00401(2024). [26]Huang, H., and He, H.Symmetric subgroup actions on isotropic Grassmannians.J. Algebra 337, 1 (2011), 141–168. [27]Ichino, A.On the regularized Siegel-Weil formula.J. Reine Angew. Math. 539(2001), 201–...

work page arXiv 2024
[13]

H., and Krishnamurthy, M.Twisted exterior square lift from GU(2,2) E/F to GL6/F.J

[33]Kim, H. H., and Krishnamurthy, M.Twisted exterior square lift from GU(2,2) E/F to GL6/F.J. Ramanujan Math. Soc. 23, 4 (2008), 381–412. [34]Knapp, A. W.Local Langlands correspondence: the Archimedean case. InProc. Sympos. Pure Math(1994), vol

work page 2008
[14]

S.Seesaw dual reductive pairs

[35]Kudla, S. S.Seesaw dual reductive pairs. InAutomorphic forms of several variables (Katata, 1983), vol. 46 ofProgr. Math.Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. [36]Kudla, S. S.Integrals of Borcherds forms.Compos. Math. 137, 3 (2003), 293–349. [37]Kudla, S. S., and Rallis, S.Poles of Eisenstein series and L-functions. Festschrift in honor of ...

work page 1983
[15]

S., and Rallis, S.A regularized Siegel-Weil formula: The first term identity.Ann

[38]Kudla, S. S., and Rallis, S.A regularized Siegel-Weil formula: The first term identity.Ann. Math. (2) 140, 1 (1994), 1–80. [39]Lee, S. T., and Zhu, C.-B.Degenerate principal series and local theta correspondence. III: The case of complex groups. J. Algebra 319, 1 (2008), 336–359. [40]Mao, Z., W an, C., and Zhang, L.Relative Langlands duality for some ...

work page 1994
[16]

E.Automorphic L-functions for symplectic groupGSp(4)

[42]Novodvorsky, M. E.Automorphic L-functions for symplectic groupGSp(4). InAutomorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part(1979), vol. 2, pp. 87–95. [43]Pitale, A., Saha, A., and Schmidt, R.On the standardL-function forGSp 2n ×GL 1 and algebraicity of symmetric fourthL-values ...

work page arXiv 1977
[17]

[47]Roberts, B.The theta correspondence for similitudes.Israel J. Math. 94(1996), 285–317. [48]Sakellaridis, Y.Spherical functions on spherical varieties.American Journal of Mathematics 135, 5 (2013), 1291–1381. [49]Sakellaridis, Y., and Venkatesh, A.Periods and harmonic analysis on spherical varieties, vol. 396 ofAstérisque. Paris: Société Mathématique d...

work page 1996
[18]

[50]Shimizu, H.Theta series and automorphic forms on GL2.J. Math. Soc. Japan 24(1972), 638–683. [51]Stade, E.Archimedean L-factors onGL(n)×GL(n)and generalized Barnes integrals.Israel Journal of Mathematics 127(2002), 201–219. [52]Sun, B., and Zhu, C.-B.Conservation relations for local theta correspondence.J. Am. Math. Soc. 28, 4 (2015), 939–983. [53]Ton-...

work page 1972
[19]

[56]W atanabe, T.Global theta liftings of general linear groups.J. Math. Sci. Univ. Tokyo 3, 3 (1996), 699–711. [57]Yamana, S.On the Siegel-Weil formula: the case of singular forms.Compos. Math. 147, 4 (2011), 1003–1021. Antonio Cauchi University College Dublin, School of Mathematics and Statistics, Science Centre - South, Belfield, Dublin 4, Dublin, Irel...

work page 1996

[1] [1]

[1]Adams, J., and Vogan, D. A. j.Contragredient representations and characterizing the local Langlands correspondence. Am. J. Math. 138, 3 (2016), 657–682. [2]Ben-Zvi, D., Sakellaridis, Y., and Venkatesh, A.Relative langlands duality.arXiv preprint arXiv:2409.04677 (2024). [3]Blasius, D., and Roga wski, J. D.Zeta functions of Shimura varieties. InMotives....

work page arXiv 2016

[2] [2]

Providence, RI: American Mathematical Society, 1994, pp. 525–571. [4]Bump, D.Automorphic forms and representations, vol. 55 ofCamb. Stud. Adv. Math.Cambridge: Cambridge University Press,

work page 1994

[3] [3]

II: The Whittaker function.Compos

[5]Casselman, W., and Shalika, J.The unramified principal series of p-adic groups. II: The Whittaker function.Compos. Math. 41(1980), 207–231. [6]Cauchi, A., and Gutierrez Terradillos, A.Archimedean integrals forGU 2,2 ×GL

work page 1980

[4] [4]

[7]Cauchi, A., and Gutierrez Terradillos, A.A two variable Rankin–Selberg integral forGU 2,2 and the degree 5 L-function ofGSp 4.Mathematische Zeitschrift 308(2024),

In preparation. [7]Cauchi, A., and Gutierrez Terradillos, A.A two variable Rankin–Selberg integral forGU 2,2 and the degree 5 L-function ofGSp 4.Mathematische Zeitschrift 308(2024),

work page 2024

[5] [5]

[9]Chen, R., and Gan, W

[8]Cauchi, A., and Gutierrez Terradillos, A.Spherical Shalika models onPGU 2,2 and the theta correspondence for (PGSp4,PGU 2,2).Forum Mathematicum(2025). [9]Chen, R., and Gan, W. T.Unitary Friedberg-Jacquet periods.arXiv preprint arXiv:2108.04064(2021). [10]Furusa w a, M., and Morimoto, K.Shalika periods onGU(2,2).Proc. Am. Math. Soc. 141, 12 (2013), 4125...

work page arXiv 2025

[6] [6]

Providence, RI: American Mathematical Society (AMS), 2019, pp

Weizmann Institute of Science and The Hebrew University of Jerusalem, Israel. Providence, RI: American Mathematical Society (AMS), 2019, pp. 113–132. [12]Gan, W. T., Gross, B. H., and Prasad, D.Symplectic local root numbers, central criticalL-values, and restriction problems in the representation theory of classical groups. InSur les conjectures de Gross ...

work page 2019

[7] [7]

[15]Gan, W

Id/No 68213. [15]Gan, W. T., Qiu, Y., and Takeda, S.The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula.Invent. Math. 198, 3 (2014), 739–831. [16]Gan, W. T., and Sa vin, G.An exceptional Siegel-Weil formula and poles of the spinL-function ofPGSp 6.Compos. Math. 156, 6 (2020), 1231–1261. [17]Garrett, P. B.Int...

work page 2014

[8] [8]

1989, pp. 241–264. [18]Gelbart, S., and Jacquet, H.A relation between automorphic representations of GL(2)and GL(3).Ann. Sci. Éc. Norm. Supér. (4) 11, 4 (1978), 471–542. L-FUNCTIONS AND LINEAR PERIODS FORGL 4 ×GL 2 ANDGU 2,2 ×GL 2 55 [19]Gelbart, S., Piatetski-Shapiro, I. I., and Rallis, S.Explicit constructions of automorphicL-functions, vol. 1254 ofLect...

work page 1989

[9] [9]

[21]Godement, R., and Jacquet, H.Zeta functions of simple algebras, vol

[20]Ginzburg, D., and Rallis, S.The exterior cubeL-function forGL(6).Compositio Mathematica 123, 3 (2000), 243–272. [21]Godement, R., and Jacquet, H.Zeta functions of simple algebras, vol. 260 ofLect. Notes Math.Springer, Cham,

work page 2000

[10] [10]

H., and Prasad, D.On the decomposition of a representation ofSO n when restricted toSO n−1.Can

[22]Gross, B. H., and Prasad, D.On the decomposition of a representation ofSO n when restricted toSO n−1.Can. J. Math. 44, 5 (1992), 974–1002. [23]Harris, M., and Kudla, S. S.On a conjecture of Jacquet. InContributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birth...

work page 1992

[11] [11]

Baltimore, MD: Johns Hopkins University Press, 2004, pp. 355–371. [24]Hirano, M., Ishii, T., and Miyazaki, T.Archimedean zeta integrals forGL 3 ×GL 2, vol

work page 2004

[12] [12]

arXiv preprint arXiv:2409.00401(2024)

[25]Hirano, M., Ishii, T., and Miyazaki, T.Whittaker functions onGL(4,R)and archimedean Bump–Friedberg integrals. arXiv preprint arXiv:2409.00401(2024). [26]Huang, H., and He, H.Symmetric subgroup actions on isotropic Grassmannians.J. Algebra 337, 1 (2011), 141–168. [27]Ichino, A.On the regularized Siegel-Weil formula.J. Reine Angew. Math. 539(2001), 201–...

work page arXiv 2024

[13] [13]

H., and Krishnamurthy, M.Twisted exterior square lift from GU(2,2) E/F to GL6/F.J

[33]Kim, H. H., and Krishnamurthy, M.Twisted exterior square lift from GU(2,2) E/F to GL6/F.J. Ramanujan Math. Soc. 23, 4 (2008), 381–412. [34]Knapp, A. W.Local Langlands correspondence: the Archimedean case. InProc. Sympos. Pure Math(1994), vol

work page 2008

[14] [14]

S.Seesaw dual reductive pairs

[35]Kudla, S. S.Seesaw dual reductive pairs. InAutomorphic forms of several variables (Katata, 1983), vol. 46 ofProgr. Math.Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. [36]Kudla, S. S.Integrals of Borcherds forms.Compos. Math. 137, 3 (2003), 293–349. [37]Kudla, S. S., and Rallis, S.Poles of Eisenstein series and L-functions. Festschrift in honor of ...

work page 1983

[15] [15]

S., and Rallis, S.A regularized Siegel-Weil formula: The first term identity.Ann

[38]Kudla, S. S., and Rallis, S.A regularized Siegel-Weil formula: The first term identity.Ann. Math. (2) 140, 1 (1994), 1–80. [39]Lee, S. T., and Zhu, C.-B.Degenerate principal series and local theta correspondence. III: The case of complex groups. J. Algebra 319, 1 (2008), 336–359. [40]Mao, Z., W an, C., and Zhang, L.Relative Langlands duality for some ...

work page 1994

[16] [16]

E.Automorphic L-functions for symplectic groupGSp(4)

[42]Novodvorsky, M. E.Automorphic L-functions for symplectic groupGSp(4). InAutomorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part(1979), vol. 2, pp. 87–95. [43]Pitale, A., Saha, A., and Schmidt, R.On the standardL-function forGSp 2n ×GL 1 and algebraicity of symmetric fourthL-values ...

work page arXiv 1977

[17] [17]

[47]Roberts, B.The theta correspondence for similitudes.Israel J. Math. 94(1996), 285–317. [48]Sakellaridis, Y.Spherical functions on spherical varieties.American Journal of Mathematics 135, 5 (2013), 1291–1381. [49]Sakellaridis, Y., and Venkatesh, A.Periods and harmonic analysis on spherical varieties, vol. 396 ofAstérisque. Paris: Société Mathématique d...

work page 1996

[18] [18]

[50]Shimizu, H.Theta series and automorphic forms on GL2.J. Math. Soc. Japan 24(1972), 638–683. [51]Stade, E.Archimedean L-factors onGL(n)×GL(n)and generalized Barnes integrals.Israel Journal of Mathematics 127(2002), 201–219. [52]Sun, B., and Zhu, C.-B.Conservation relations for local theta correspondence.J. Am. Math. Soc. 28, 4 (2015), 939–983. [53]Ton-...

work page 1972

[19] [19]

[56]W atanabe, T.Global theta liftings of general linear groups.J. Math. Sci. Univ. Tokyo 3, 3 (1996), 699–711. [57]Yamana, S.On the Siegel-Weil formula: the case of singular forms.Compos. Math. 147, 4 (2011), 1003–1021. Antonio Cauchi University College Dublin, School of Mathematics and Statistics, Science Centre - South, Belfield, Dublin 4, Dublin, Irel...

work page 1996