Some local-global non-vanishing results for theta lifts from orthogonal groups

Takeda, Shuichiro

doi:10.1090/S0002-9947-09-04787-4

Some local-global non-vanishing results for theta lifts from orthogonal groups
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by Shuichiro Takeda PDF

Trans. Amer. Math. Soc. 361 (2009), 5575-5599 Request permission

Abstract:

We, first, improve a theorem of B. Roberts which characterizes non-vanishing of a global theta lift from $\operatorname {O}(X)$ to $\operatorname {Sp}(n)$ in terms of non-vanishing of local theta lifts. In particular, we will remove all the Archimedean conditions imposed upon his theorem. Secondly, following Roberts, we will apply our theorem to theta lifting of low rank similitude groups. Namely we characterize the non-vanishing condition of a global theta lift from $\operatorname {GO}(4)$ to $\operatorname {GSp}(2)$ in our improved setting. Also we consider non-vanishing conditions of a global theta lift from $\operatorname {GO}(4)$ to $\operatorname {GSp}(1)$ and explicitly compute the lift when it exists.

References

Jeffrey Adams and Dan Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), no. 1, 1–42. MR 1346217, DOI 10.1006/jfan.1995.1099
Jeffrey D. Adler and Dipendra Prasad, On certain multiplicity one theorems, Israel J. Math. 153 (2006), 221–245. MR 2254643, DOI 10.1007/BF02771784
Siegfried Böcherer and Rainer Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J. 121 (1991), 35–96. MR 1096467, DOI 10.1017/S0027763000003391
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $G$, Canad. J. Math. 41 (1989), no. 3, 385–438. MR 1013462, DOI 10.4153/CJM-1989-019-5
James W. Cogdell, Lectures on $L$-functions, converse theorems, and functoriality for $\textrm {GL}_n$, Lectures on automorphic $L$-functions, Fields Inst. Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 2004, pp. 1–96. MR 2071506
Yuval Z. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), no. 3, 295–313 (English, with French summary). MR 984899
Yuval Z. Flicker, On zeroes of the twisted tensor $L$-function, Math. Ann. 297 (1993), no. 2, 199–219. MR 1241802, DOI 10.1007/BF01459497
Yuval Z. Flicker and Dmitrii Zinoviev, On poles of twisted tensor $L$-functions, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 6, 114–116. MR 1344660
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
Michael Harris, David Soudry, and Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to $\textrm {GSp}_4(\textbf {Q})$, Invent. Math. 112 (1993), no. 2, 377–411. MR 1213108, DOI 10.1007/BF01232440
Roger Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on $\textrm {Sp}_{4}$, Duke Math. J. 50 (1983), no. 1, 55–106. MR 700131
Tamotsu Ikeda, On the location of poles of the triple $L$-functions, Compositio Math. 83 (1992), no. 2, 187–237. MR 1174424
Hervé Jacquet, Principal $L$-functions of the linear group, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 63–86. MR 546609
H. Jacquet, : Private communication.
Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR 1880691
Stephen S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229–255. MR 818351, DOI 10.1007/BF01388961
S. Kudla, Notes on the Local Theta Correspondence, unpublished notes, available online.
Stephen S. Kudla and Stephen Rallis, Poles of Eisenstein series and $L$-functions, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 81–110. MR 1159110
Stephen S. Kudla and Stephen Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), no. 1, 1–80. MR 1289491, DOI 10.2307/2118540
Daniel A. Marcus, Number fields, Universitext, Springer-Verlag, New York-Heidelberg, 1977. MR 0457396
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
I. I. Piatetski-Shapiro and S. Rallis, $L$-functions for classical groups, in Lecture Notes in Math. 1254, Springer-Verlag, New York, pp. 1-52.
Dipendra Prasad and Rainer Schulze-Pillot, Generalised form of a conjecture of Jacquet and a local consequence, J. Reine Angew. Math. 616 (2008), 219–236. MR 2369492, DOI 10.1515/CRELLE.2008.023
Brooks Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317. MR 1394579, DOI 10.1007/BF02762709
Brooks Roberts, Tempered representations and the theta correspondence, Canad. J. Math. 50 (1998), no. 5, 1105–1118. MR 1650930, DOI 10.4153/CJM-1998-053-6
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, Nonvanishing of global theta lifts from orthogonal groups, J. Ramanujan Math. Soc. 14 (1999), no. 2, 131–194. MR 1727710
Brooks Roberts, Global $L$-packets for $\textrm {GSp}(2)$ and theta lifts, Doc. Math. 6 (2001), 247–314. MR 1871665

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Some local-global non-vanishing results for theta lifts from orthogonal groups
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