## Theta correspondences for $\operatorname {GSp}(4)$

HTML articles powered by AMS MathViewer

- by Wee Teck Gan and Shuichiro Takeda PDF
- Represent. Theory
**15**(2011), 670-718 Request permission

## Abstract:

We explicitly determine the theta correspondences for $\operatorname {GSp}_4$ and orthogonal similitude groups associated to various quadratic spaces of rank $4$ and $6$. The results are needed in our proof of the local Langlands correspondence for $\operatorname {GSp}_4$.## References

- Mahdi Asgari and Ralf Schmidt,
*On the adjoint $L$-function of the $p$-adic $\rm GSp(4)$*, J. Number Theory**128**(2008), no. 8, 2340–2358. MR**2394824**, DOI 10.1016/j.jnt.2007.08.012 - Wee Teck Gan and Nadya Gurevich,
*Nontempered A-packets of $G_2$: liftings from $\widetilde \textrm {SL}_2$*, Amer. J. Math.**128**(2006), no. 5, 1105–1185. MR**2262172**, DOI 10.1353/ajm.2006.0040 - W. T. Gan and S. Takeda,
*The local Langlands conjecture for $\operatorname {GSp}(4)$*, to appear in Annals of Math. - Wee Teck Gan and Shuichiro Takeda,
*On Shalika periods and a theorem of Jacquet-Martin*, Amer. J. Math.**132**(2010), no. 2, 475–528. MR**2654780**, DOI 10.1353/ajm.0.0109 - Stephen S. Kudla,
*On the local theta-correspondence*, Invent. Math.**83**(1986), no. 2, 229–255. MR**818351**, DOI 10.1007/BF01388961 - Stephen S. Kudla and Stephen 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., vol. 11, de Gruyter, Berlin, 2005, pp. 273–308. MR**2192827**, DOI 10.1515/9783110892703.273 - Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger,
*Correspondances de Howe sur un corps $p$-adique*, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR**1041060**, DOI 10.1007/BFb0082712 - Goran Muić and Gordan Savin,
*Symplectic-orthogonal theta lifts of generic discrete series*, Duke Math. J.**101**(2000), no. 2, 317–333. MR**1738175**, DOI 10.1215/S0012-7094-00-10128-7 - Dipendra Prasad,
*On the local Howe duality correspondence*, Internat. Math. Res. Notices**11**(1993), 279–287. MR**1248702**, DOI 10.1155/S1073792893000315 - S. Rallis,
*On the Howe duality conjecture*, Compositio Math.**51**(1984), no. 3, 333–399. MR**743016** - Brooks Roberts,
*The theta correspondence for similitudes*, Israel J. Math.**94**(1996), 285–317. MR**1394579**, DOI 10.1007/BF02762709 - Brooks Roberts,
*The non-Archimedean theta correspondence for $\textrm {GSp}(2)$ and $\textrm {GO}(4)$*, Trans. Amer. Math. Soc.**351**(1999), no. 2, 781–811. MR**1458334**, DOI 10.1090/S0002-9947-99-02126-1 - Brooks Roberts and Ralf Schmidt,
*Local newforms for GSp(4)*, Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. MR**2344630**, DOI 10.1007/978-3-540-73324-9 - Paul J. Sally Jr. and Marko Tadić,
*Induced representations and classifications for $\textrm {GSp}(2,F)$ and $\textrm {Sp}(2,F)$*, Mém. Soc. Math. France (N.S.)**52**(1993), 75–133 (English, with English and French summaries). MR**1212952** - J.-L. Waldspurger,
*Un exercice sur $\textrm {GSp}(4,F)$ et les représentations de Weil*, Bull. Soc. Math. France**115**(1987), no. 1, 35–69 (French). MR**897614**, DOI 10.24033/bsmf.2068

## Additional Information

**Wee Teck Gan**- Affiliation: Mathematics Department, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093
- MR Author ID: 621634
- Email: wgan@math.ucsd.edu
**Shuichiro Takeda**- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- MR Author ID: 873141
- Email: stakeda@math.purdue.edu
- Received by editor(s): June 15, 2010
- Received by editor(s) in revised form: March 10, 2011
- Published electronically: November 1, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 670-718 - MSC (2010): Primary 11F27, 11S37, 11S99, 20G99, 22E50
- DOI: https://doi.org/10.1090/S1088-4165-2011-00405-2
- MathSciNet review: 2846304