The distinction problems for Sp4 and SO3,3
-
Hengfei Lu
Abstract
This paper studies the Prasad conjecture for the special orthogonal group
Funding source: European Research Council
Award Identifier / Grant number: 637912
Funding statement: This research was partially supported by the ERC, StG grant number 637912.
Acknowledgements
This is a certain extension of the author’s Ph.D. thesis. He is grateful to Wee Teck Gan for his guidance and numerous discussions when he was studying at National University of Singapore. He also would like to thank Dipendra Prasad for useful comments. Part of this paper was written down when the author was visiting the Institute for Mathematical Science, NUS in December 2018 where he was invited and partially supported to attend the program: Endoscopy and Beyond. He would like to thank them for their hospitality. He also wants to thank the anonymous referee for a careful reading of the manuscript and numerous suggestions.
Communicated by: Freydoon Shahidi
References
[1] U. K. Anandavardhanan, A. C. Kable and R. Tandon, Distinguished representations and poles of twisted tensor 𝐿-functions, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2875–2883. 10.1090/S0002-9939-04-07424-6Search in Google Scholar
[2]
U. K. Anandavardhanan and D. Prasad,
Distinguished representations for
[3]
U. K. Anandavardhanan and D. Prasad,
Distinguished representations for
[4] H. Atobe and W. T. Gan, On the local Langlands correspondence and Arthur conjecture for even orthogonal groups, Represent. Theory 21 (2017), 354–415. 10.1090/ert/504Search in Google Scholar
[5] R. Beuzart-Plessis, On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad, Invent. Math. 214 (2018), no. 1, 437–521. 10.1007/s00222-018-0807-zSearch in Google Scholar
[6] Y. Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139–172. 10.1515/crll.1991.418.139Search in Google Scholar
[7] Y. Z. Flicker and J. L. Hakim, Quaternionic distinguished representations, Amer. J. Math. 116 (1994), no. 3, 683–736. 10.2307/2374997Search in Google Scholar
[8] W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical 𝐿 values, and restriction problems in the representation theory of classical groups, Sur les conjectures de Gross et Prasad. I, Astérisque 346, Société Mathématique de France, Paris (2012), 1–109. Search in Google Scholar
[9] W. T. Gan and S. Takeda, The local Langlands conjecture for Sp(4), Int. Math. Res. Not. IMRN 2010 (2010), no. 15, 2987–3038. 10.1093/imrn/rnp203Search in Google Scholar
[10]
W. T. Gan and S. Takeda,
The local Langlands conjecture for
[11]
W. T. Gan and S. Takeda,
Theta correspondences for
[12] W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), no. 2, 473–493. 10.1090/jams/839Search in Google Scholar
[13] W. T. Gan and S. Takeda, On the Howe duality conjecture in classical theta correspondence, Advances in the Theory of Automorphic Forms and Their 𝐿-Functions, Contemp. Math. 664, American Mathematical Society, Providence (2016), 105–117. 10.1090/conm/664/13063Search in Google Scholar
[14]
D. Jiang and D. Soudry,
Appendix: On the local descent from
[15] A. C. Kable, Asai 𝐿-functions and Jacquet’s conjecture, Amer. J. Math. 126 (2004), no. 4, 789–820. 10.1353/ajm.2004.0030Search in Google Scholar
[16] S. S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229–255. 10.1007/BF01388961Search in Google Scholar
[17]
S. S. Kudla and S. Rallis,
Ramified degenerate principal series representations for
[18] S. S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, Automorphic Representations, 𝐿-Functions and Applications: Progress and Prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, De Gruyter, Berlin (2005), 273–308. 10.1515/9783110892703.273Search in Google Scholar
[19]
H. Lu,
Theta correspondence and the Prasad conjecture for
[20]
H. Lu,
The Prasad conjectures for
[21]
H. Lu,
The
[22] H. Lu, Some applications of theta correspondence to branching laws, Math. Res. Lett. 27 (2020), no. 1, 243–263. 10.4310/MRL.2020.v27.n1.a12Search in Google Scholar
[23]
H. Lu,
The Prasad conjectures for
[24] N. Matringe, Distinguished representations and exceptional poles of the Asai-𝐿-function, Manuscripta Math. 131 (2010), no. 3–4, 415–426. 10.1007/s00229-009-0327-7Search in Google Scholar
[25]
N. Matringe,
Distinguished generic representations of
[26] D. Prasad, A “relative” local Langlands correspondence, preprint (2015), https://arxiv.org/abs/1512.04347. Search in Google Scholar
[27] B. Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317. 10.1007/BF02762709Search in Google Scholar
[28]
B. Roberts,
Global 𝐿-packets for
[29] B. Sun and C.-B. Zhu, Conservation relations for local theta correspondence, J. Amer. Math. Soc. 28 (2015), no. 4, 939–983. 10.1090/S0894-0347-2014-00817-1Search in Google Scholar
[30]
J.-L. Waldspurger,
Démonstration d’une conjecture de dualité de Howe dans le cas 𝑝-adique,
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case
Articles in the same Issue
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case
