On a correspondence between cuspidal representations of and
Authors:
David Ginzburg, Stephen Rallis and David Soudry
Journal:
J. Amer. Math. Soc. 12 (1999), 849907
MSC (1991):
Primary 11F27, 11F70, 11F85
Published electronically:
April 26, 1999
MathSciNet review:
1671452
Fulltext PDF Free Access
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Abstract: Let be an irreducible, automorphic, selfdual, cuspidal representation of , where is the adele ring of a number field . Assume that has a pole at and that . Given a nontrivial character of , we construct a nontrivial space of genuine and globally generic cusp forms on the metaplectic cover of . is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and generic representations of , which lift (``functorially, with respect to ") to . We also present a local counterpart. Let be an irreducible, selfdual, supercuspidal representation of , where is a adic field. Assume that has a pole at . Given a nontrivial character of , we construct an irreducible, supercuspidal (genuine) generic representation of , such that has a pole at , and we prove that is the unique representation of satisfying these properties.
 [B.Z.]
I.
N. Bernšteĭn and A.
V. Zelevinskiĭ, Representations of the group
𝐺𝐿(𝑛,𝐹), where 𝐹 is a local
nonArchimedean field, Uspehi Mat. Nauk 31 (1976),
no. 3(189), 5–70 (Russian). MR 0425030
(54 #12988)
 [D.M.]
Jacques
Dixmier and Paul
Malliavin, Factorisations de fonctions et de vecteurs
indéfiniment différentiables, Bull. Sci. Math. (2)
102 (1978), no. 4, 307–330 (French, with
English summary). MR 517765
(80f:22005)
 [F]
Masaaki
Furusawa, On the theta lift from 𝑆𝑂_{2𝑛+1}
to ̃𝑆𝑝_{𝑛}, J. Reine Angew. Math.
466 (1995), 87–110. MR 1353315
(96g:11052), http://dx.doi.org/10.1515/crll.1995.466.87
 [G.PS.]
Stephen
Gelbart, Ilya
PiatetskiShapiro, and Stephen
Rallis, Explicit constructions of automorphic
𝐿functions, Lecture Notes in Mathematics, vol. 1254,
SpringerVerlag, Berlin, 1987. MR 892097
(89k:11038)
 [G.R.S.1]
D. Ginzburg, S. Rallis and D. Soudry, On explicit lifts of cusp forms from to classical groups, preprint (1997).
 [G.R.S.2]
D.
Ginzburg, S.
Rallis, and D.
Soudry, A new construction of the inverse Shimura
correspondence, Internat. Math. Res. Notices 7
(1997), 349–357. MR 1440573
(98a:11056), http://dx.doi.org/10.1155/S107379289700024X
 [G.R.S.3]
David
Ginzburg, Stephen
Rallis, and David
Soudry, Selfdual automorphic 𝐺𝐿_{𝑛}
modules and construction of a backward lifting from
𝐺𝐿_{𝑛} to classical groups, Internat. Math.
Res. Notices 14 (1997), 687–701. MR 1460389
(98e:22013), http://dx.doi.org/10.1155/S1073792897000457
 [G.R.S.4]
D. Ginzburg, S. Rallis and D. Soudry, functions for symplectic groups, to appear in Bull. de la SMF.
 [J.R.]
Hervé
Jacquet and Stephen
Rallis, Uniqueness of linear periods, Compositio Math.
102 (1996), no. 1, 65–123. MR 1394521
(97k:22025)
 [J.S.1]
H.
Jacquet and J.
A. Shalika, On Euler products and the classification of automorphic
representations. I, Amer. J. Math. 103 (1981),
no. 3, 499–558. MR 618323
(82m:10050a), http://dx.doi.org/10.2307/2374103
 [J.S.2]
Hervé
Jacquet and Joseph
Shalika, Exterior square 𝐿functions, Automorphic
forms, Shimura varieties, and 𝐿functions, Vol.\
II (Ann Arbor, MI,
1988), Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990,
pp. 143–226. MR 1044830
(91g:11050)
 [M.V.W]
Colette
Mœglin, MarieFrance
Vignéras, and JeanLoup
Waldspurger, Correspondances de Howe sur un corps
𝑝adique, Lecture Notes in Mathematics, vol. 1291,
SpringerVerlag, Berlin, 1987 (French). MR 1041060
(91f:11040)
 [Sh1]
Freydoon
Shahidi, Twisted endoscopy and reducibility of induced
representations for 𝑝adic groups, Duke Math. J.
66 (1992), no. 1, 1–41. MR 1159430
(93b:22034), http://dx.doi.org/10.1215/S0012709492066014
 [Sh2]
Freydoon
Shahidi, A proof of Langlands’ conjecture on Plancherel
measures; complementary series for 𝑝adic groups, Ann. of
Math. (2) 132 (1990), no. 2, 273–330. MR 1070599
(91m:11095), http://dx.doi.org/10.2307/1971524
 [Sa]
Gordan
Savin, Local Shimura correspondence, Math. Ann.
280 (1988), no. 2, 185–190. MR 929534
(89h:22018), http://dx.doi.org/10.1007/BF01456050
 [So]
David
Soudry, RankinSelberg convolutions for
𝑆𝑂_{2𝑙+1}×𝐺𝐿_{𝑛}:
local theory, Mem. Amer. Math. Soc. 105 (1993),
no. 500, vi+100. MR 1169228
(94b:11043), http://dx.doi.org/10.1090/memo/0500
 [B.Z.]
 I.N. Bernstein and A.V. Zelevinsky, Representations of the group , where is a local nonarchimedean field, Russian Math. Surveys, No. 3 p. 168 (1976). MR 54:12988
 [D.M.]
 J. Dixmier and P. Malliavin, Factorizations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math., II. Sér. No. 102, p. 471542 (1978). MR 80f:22005
 [F]
 M. Furusawa, On the theta lift from to , J. reine angew. Math. 466, p. 87110 (1995). MR 96g:11052
 [G.PS.]
 S. Gelbart and I. PiatetskiShapiro, functions for , in Springer Lecture Notes in Math. No. 1254, p. 53136 (1987). MR 89k:11038
 [G.R.S.1]
 D. Ginzburg, S. Rallis and D. Soudry, On explicit lifts of cusp forms from to classical groups, preprint (1997).
 [G.R.S.2]
 D. Ginzburg, S. Rallis and D. Soudry, A new construction of the inverse Shimura correspondence, IMRN, No. 7, p. 349357 (1997). MR 98a:11056
 [G.R.S.3]
 D. Ginzburg, S. Rallis and D. Soudry, Selfdual automorphic modules and construction of a backward lifting from to classical groups, IMRN, No. 14, p. 687701 (1997). MR 98e:22013
 [G.R.S.4]
 D. Ginzburg, S. Rallis and D. Soudry, functions for symplectic groups, to appear in Bull. de la SMF.
 [J.R.]
 H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Math., No. 102, p. 65123 (1997). MR 97k:22025
 [J.S.1]
 H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations I, American J. Math, Vol. 103, No. 3, p. 499558 (1981). MR 82m:10050a
 [J.S.2]
 H. Jacquet and J. Shalika, Exterior square functions, in Automorphic Forms, Shimura Varieties and functions, Vol. II edited by L. Clozel and J.S. Milne, Perspectives in Math., Academic Press, p. 143226 (1990). MR 91g:11050
 [M.V.W]
 C. Moeglin, M.F. Vigneras, J.L. Waldspurger, Correspondences de Howe sur un corps adique, L.N.M. 1291, SpringerVerlag (1987). MR 91f:11040
 [Sh1]
 F. Shahidi, Twisted endescopy and irreducibility of induced representations for adic groups, Duke Math. J. No. 66, p. 141 (1992). MR 93b:22034
 [Sh2]
 F. Shahidi, A proof of Langlands conjecture on Plancherel measures: complementary series for adic groups, Annals of Math. No. 132 p. 273330 (1990). MR 91m:11095
 [Sa]
 G. Savin, Local Shimura correspondence, Math. Ann. 280, p. 185190 (1988). MR 89h:22018
 [So]
 D. Soudry, RankinSelberg convolutions for : Local theory, Memoirs of AMS No. 500, p. 1100 (1993). MR 94b:11043
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Additional Information
David Ginzburg
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email:
ginzburg@math.tau.ac.il
Stephen Rallis
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email:
haar@math.ohiostate.edu
David Soudry
Email:
soudry@math.tau.ac.il
DOI:
http://dx.doi.org/10.1090/S0894034799003008
PII:
S 08940347(99)003008
Received by editor(s):
July 22, 1998
Received by editor(s) in revised form:
March 1, 1999
Published electronically:
April 26, 1999
Additional Notes:
The first and third authors’ research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.
Article copyright:
© Copyright 1999 American Mathematical Society
