On the theta correspondence for (𝐺𝑆𝑝(4),𝐺𝑆𝑂(4,2)) and Shalika periods

Morimoto, Kazuki

doi:10.1090/S1088-4165-2014-00451-5

On the theta correspondence for $(\mathrm {GSp}(4), \mathrm {GSO}(4,2))$ and Shalika periods
HTML articles powered by AMS MathViewer

by Kazuki Morimoto

Represent. Theory 18 (2014), 28-87

DOI: https://doi.org/10.1090/S1088-4165-2014-00451-5

Published electronically: April 16, 2014

PDF | Request permission

Abstract:

We consider both local and global theta correspondences for $\mathrm {GSp}_4$ and $\mathrm {GSO}_{4,2}$. Because of the accidental isomorphism $\mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2}$, these correspondences give rise to those between $\mathrm {GSp}_4$ and $\mathrm {GU}_{2,2}$ for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on $\mathrm {GU}(2,2)$ by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.

References

Jeffrey D. Adler and Dipendra Prasad, On certain multiplicity one theorems, Israel J. Math. 153 (2006), 221–245. MR 2254643, DOI 10.1007/BF02771784
Mahdi Asgari and Freydoon Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006), no. 1, 137–190. MR 2219256, DOI 10.1215/S0012-7094-06-13214-3
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
Michel Cognet, Représentation de Weil et changement de base quadratique, Bull. Soc. Math. France 113 (1985), no. 4, 403–457 (French, with English summary). MR 850776, DOI 10.24033/bsmf.2041
Masaaki Furusawa, On the theta lift from $\textrm {SO}_{2n+1}$ to $\widetilde \textrm {Sp}_n$, J. Reine Angew. Math. 466 (1995), 87–110. MR 1353315, DOI 10.1515/crll.1995.466.87
Masaaki Furusawa and Kazuki Morimoto, Shalika periods on $\mathrm {GU}(2,2)$, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4125–4137. MR 3105856, DOI 10.1090/S0002-9939-2013-11690-4
Wee Teck Gan, The Saito-Kurokawa space of $\textrm {PGSp}_4$ and its transfer to inner forms, Eisenstein series and applications, Progr. Math., vol. 258, Birkhäuser Boston, Boston, MA, 2008, pp. 87–123. MR 2402681, DOI 10.1007/978-0-8176-4639-4_{3}
Wee Teck Gan and Atsushi Ichino, On endoscopy and the refined Gross-Prasad conjecture for $(\rm SO_5,SO_4)$, J. Inst. Math. Jussieu 10 (2011), no. 2, 235–324. MR 2787690, DOI 10.1017/S1474748010000198
Wee Teck Gan and Gordan Savin, Real and global lifts from $\rm PGL_3$ to $G_2$, Int. Math. Res. Not. 50 (2003), 2699–2724. MR 2017248, DOI 10.1155/S1073792803130607
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
Wee Teck Gan and Shuichiro Takeda, The local Langlands conjecture for Sp(4), Int. Math. Res. Not. IMRN 15 (2010), 2987–3038. MR 2673717, DOI 10.1093/imrn/rnp203
Wee Teck Gan and Shuichiro Takeda, The local Langlands conjecture for $\textrm {GSp}(4)$, Ann. of Math. (2) 173 (2011), no. 3, 1841–1882. MR 2800725, DOI 10.4007/annals.2011.173.3.12
Wee Teck Gan and Shuichiro Takeda, Theta correspondences for $\textrm {GSp}(4)$, Represent. Theory 15 (2011), 670–718. MR 2846304, DOI 10.1090/S1088-4165-2011-00405-2
David Ginzburg, Stephen Rallis, and David Soudry, Periods, poles of $L$-functions and symplectic-orthogonal theta lifts, J. Reine Angew. Math. 487 (1997), 85–114. MR 1454260, DOI 10.1515/crll.1997.487.85
Michael Harris and Stephen S. Kudla, Arithmetic automorphic forms for the nonholomorphic discrete series of $\textrm {GSp}(2)$, Duke Math. J. 66 (1992), no. 1, 59–121. MR 1159432, DOI 10.1215/S0012-7094-92-06603-8
Hervé Jacquet and Joseph Shalika, Exterior square $L$-functions, Automorphic forms, Shimura varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143–226. MR 1044830
C. David Keys, $L$-indistinguishability and $R$-groups for quasisplit groups: unitary groups in even dimension, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 31–64. MR 892141, DOI 10.24033/asens.1523
A. W. Knapp and E. M. Stein, Irreducibility theorems for the principal series, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Math., Vol. 266, Springer, Berlin, 1972, pp. 197–214. MR 0422512
Kazuko Konno, Induced representations of $\rm U(2,2)$ over a $p$-adic field, J. Reine Angew. Math. 540 (2001), 167–204. MR 1868602, DOI 10.1515/crll.2001.083
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
Stephen S. Kudla, Stephen Rallis, and David Soudry, On the degree $5$ $L$-function for $\textrm {Sp}(2)$, Invent. Math. 107 (1992), no. 3, 483–541. MR 1150600, DOI 10.1007/BF01231900
Jian-Shu Li, Binyong Sun, and Ye Tian, The multiplicity one conjecture for local theta correspondences, Invent. Math. 184 (2011), no. 1, 117–124. MR 2782253, DOI 10.1007/s00222-010-0287-2
Zhengyu Mao and Stephen Rallis, Jacquet modules of the Weil representations and families of relative trace identities, Compos. Math. 140 (2004), no. 4, 855–886. MR 2059223, DOI 10.1112/S0010437X04000399
Nadir Matringe, Distinguished representations and exceptional poles of the Asai-$L$-function, Manuscripta Math. 131 (2010), no. 3-4, 415–426. MR 2592088, DOI 10.1007/s00229-009-0327-7
Goran Muić, A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups, Canad. Math. Bull. 44 (2001), no. 3, 298–312. MR 1847492, DOI 10.4153/CMB-2001-030-4
Hiro-aki Narita, Ameya Pitale, and Ralf Schmidt, Irreducibility criteria for local and global representations, Proc. Amer. Math. Soc. 141 (2013), no. 1, 55–63. MR 2988710, DOI 10.1090/S0002-9939-2012-11438-8
T. Okazaki, $\theta$-correspondence for $\mathrm {PGSp}(4)$ and $\mathrm {PGU}(2,2)$. Automorphic forms, trace formulas and zeta functions, Kyoto 2011, Surikaisekikenkyusho Kokyoroku, 1767(2011) 96–102.
Annegret Paul, On the Howe correspondence for symplectic-orthogonal dual pairs, J. Funct. Anal. 228 (2005), no. 2, 270–310. MR 2175409, DOI 10.1016/j.jfa.2005.03.015
A. Pitale, A. Saha and R. Schmidt, Transfer of Siegel cusp forms of degree 2, to appear in Mem. Amer. Math. Soc.
Brooks Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317. MR 1394579, DOI 10.1007/BF02762709
François Rodier, Whittaker models for admissible representations of reductive $p$-adic split groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 425–430. MR 0354942
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
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
Allan J. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991
Allan J. Silberger, The Knapp-Stein dimension theorem for $p$-adic groups, Proc. Amer. Math. Soc. 68 (1978), no. 2, 243–246. MR 492091, DOI 10.1090/S0002-9939-1978-0492091-5
Allan J. Silberger, Correction: “The Knapp-Stein dimension theorem for $p$-adic groups” [Proc. Amer. Math. Soc. 68 (1978), no. 2, 243–246; MR 58 #11245], Proc. Amer. Math. Soc. 76 (1979), no. 1, 169–170. MR 534411, DOI 10.1090/S0002-9939-1979-0534411-X
David Soudry, A uniqueness theorem for representations of $\textrm {GSO}(6)$ and the strong multiplicity one theorem for generic representations of $\textrm {GSp}(4)$, Israel J. Math. 58 (1987), no. 3, 257–287. MR 917359, DOI 10.1007/BF02771692
B. Sun, and C. B. Zhu, Conservation relations for local theta correspondence. preprint

AMS Website Down

Representation Theory

On the theta correspondence for $(\mathrm {GSp}(4), \mathrm {GSO}(4,2))$ and Shalika periods
HTML articles powered by AMS MathViewer

Abstract: