On a correspondence between cuspidal

representations of and

Authors:
David Ginzburg, Stephen Rallis and David Soudry

Journal:
J. Amer. Math. Soc. **12** (1999), 849-907

MSC (1991):
Primary 11F27, 11F70, 11F85

Published electronically:
April 26, 1999

MathSciNet review:
1671452

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an irreducible, automorphic, self-dual, 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, self-dual, 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 non-Archimedean field*, Uspehi Mat. Nauk**31**(1976), no. 3(189), 5–70 (Russian). MR**0425030****[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****[F]**Masaaki Furusawa,*On the theta lift from 𝑆𝑂_{2𝑛+1} to ̃𝑆𝑝_{𝑛}*, J. Reine Angew. Math.**466**(1995), 87–110. MR**1353315**, 10.1515/crll.1995.466.87**[G.PS.]**Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis,*Explicit constructions of automorphic 𝐿-functions*, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR**892097****[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**, 10.1155/S107379289700024X**[G.R.S.3]**David Ginzburg, Stephen Rallis, and David Soudry,*Self-dual 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**, 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****[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**, 10.2307/2374103**[J.S.2]**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****[M.V.W]**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****[Sh1]**Freydoon Shahidi,*Twisted endoscopy and reducibility of induced representations for 𝑝-adic groups*, Duke Math. J.**66**(1992), no. 1, 1–41. MR**1159430**, 10.1215/S0012-7094-92-06601-4**[Sh2]**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**, 10.2307/1971524**[Sa]**Gordan Savin,*Local Shimura correspondence*, Math. Ann.**280**(1988), no. 2, 185–190. MR**929534**, 10.1007/BF01456050**[So]**David Soudry,*Rankin-Selberg convolutions for 𝑆𝑂_{2𝑙+1}×𝐺𝐿_{𝑛}: local theory*, Mem. Amer. Math. Soc.**105**(1993), no. 500, vi+100. MR**1169228**, 10.1090/memo/0500

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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.ohio-state.edu

**David Soudry**

Email:
soudry@math.tau.ac.il

DOI:
http://dx.doi.org/10.1090/S0894-0347-99-00300-8

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