Automorphic forms on $\operatorname {PGSp}(2)$
Author:
Yuval Z. Flicker
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 39-50
MSC (2000):
Primary 11F70; Secondary 22E50, 22E55, 22E45
DOI:
https://doi.org/10.1090/S1079-6762-04-00128-3
Published electronically:
April 23, 2004
MathSciNet review:
2048430
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Abstract: The theory of lifting of automorphic and admissible representations is developed in a new case of great classical interest: Siegel automorphic forms. The self-contragredient representations of PGL(4) are determined as lifts of representations of either symplectic PGSp(2) or orthogonal SO(4) rank two split groups. Our approach to the lifting uses the global tool of the trace formula together with local results such as the fundamental lemma. The lifting is stated in terms of character relations. This permits us to introduce a definition of packets and quasi-packets of representations of the projective symplectic group of similitudes PGSp(2), and analyse the structure of all packets. All representations, not only generic or tempered ones, are studied. Globally we obtain a multiplicity one theorem for the discrete spectrum of the projective symplectic group PGSp(2), a rigidity theorem for packets and quasi-packets, determine all counterexamples to the naive Ramanujan conjecture, and compute the multiplicity of each member in a packet or quasi-packet in the discrete spectrum. The lifting from SO(4) to PGL(4) amounts to establishing a product of two representations of GL(2) with central characters whose product is 1. The rigidity theorem for SO(4) amounts to a strong rigidity statement for a pair of representations of $\operatorname {GL}(2,\mathbb {A})$.
- James Arthur, On some problems suggested by the trace formula, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 1–49. MR 748504, DOI https://doi.org/10.1007/BFb0073144
- 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
- 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
- A. Borel, Automorphic $L$-functions, 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. 27–61. MR 546608
[F1]F1 Y. Flicker, Lifting automorphic forms of PGSp(2) and SO(4) to PGL(4), 2001.
- Yuval Z. Flicker, Regular trace formula and base change lifting, Amer. J. Math. 110 (1988), no. 4, 739–764. MR 955295, DOI https://doi.org/10.2307/2374648
- Yuval Z. Flicker, On the symmetric square: applications of a trace formula, Trans. Amer. Math. Soc. 330 (1992), no. 1, 125–152. MR 1041045, DOI https://doi.org/10.1090/S0002-9947-1992-1041045-0
- Yuval Z. Flicker, Elementary proof of the fundamental lemma for a unitary group, Canad. J. Math. 50 (1998), no. 1, 74–98. MR 1618722, DOI https://doi.org/10.4153/CJM-1998-005-4
- Yuval Z. Flicker, Matching of orbital integrals on ${\rm GL}(4)$ and ${\rm GSp}(2)$, Mem. Amer. Math. Soc. 137 (1999), no. 655, viii+112. MR 1468177, DOI https://doi.org/10.1090/memo/0655
[F6]F6 Y. Flicker, Automorphic forms on SO(4), 2004, preprint.
[F7]F7 Y. Flicker, On Zeta functions of Shimura varieties of PGSp(2), 2003, preprint.
- Yuval Z. Flicker and David A. Kazhdan, Metaplectic correspondence, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 53–110. MR 876160
[FZ]FZ Y. Flicker, D. Zinoviev, Twisted character of a small representation of PGL(4), Moscow Math. J. (2004), in press.
- David Ginzburg, Stephen Rallis, and David Soudry, Generic automorphic forms on ${\rm SO}(2n+1)$: functorial lift to ${\rm GL}(2n)$, endoscopy, and base change, Internat. Math. Res. Notices 14 (2001), 729–764. MR 1846354, DOI https://doi.org/10.1155/S1073792801000381
- Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR 1702257
- 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, DOI https://doi.org/10.2307/2374103
- David Kazhdan, On lifting, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 209–249. MR 748509, DOI https://doi.org/10.1007/BFb0073149
- Henry H. Kim, Residual spectrum of odd orthogonal groups, Internat. Math. Res. Notices 17 (2001), 873–906. MR 1859343, DOI https://doi.org/10.1155/S1073792801000435
- Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI https://doi.org/10.1215/S0012-7094-84-05129-9
- Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
- Stephen S. Kudla, Stephen Rallis, and David Soudry, On the degree $5$ $L$-function for ${\rm Sp}(2)$, Invent. Math. 107 (1992), no. 3, 483–541. MR 1150600, DOI https://doi.org/10.1007/BF01231900
- A. Borel and H. Jacquet, Automorphic forms and automorphic representations, 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. 189–207. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. MR 546598
- C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de ${\rm GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752
- I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), no. 2, 309–338. MR 689647, DOI https://doi.org/10.1007/BF01389101
- Ilya Piatetski-Shapiro, Work of Waldspurger, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 280–302. MR 748511, DOI https://doi.org/10.1007/BFb0073151
- Brooks Roberts, Global $L$-packets for ${\rm GSp}(2)$ and theta lifts, Doc. Math. 6 (2001), 247–314. MR 1871665
- F. Rodier, Modèle de Whittaker et caractères de représentations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Springer, Berlin, 1975, pp. 151–171. Lecture Notes in Math., Vol. 466 (French). MR 0393355
- François Rodier, Sur les représentations non ramifiées des groupes réductifs $p$-adiques; l’exemple de ${\rm GSp}(4)$, Bull. Soc. Math. France 116 (1988), no. 1, 15–42 (French, with English summary). MR 946277
- Paul J. Sally Jr. and Marko Tadić, Induced representations and classifications for ${\rm GSp}(2,F)$ and ${\rm Sp}(2,F)$, 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 https://doi.org/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 https://doi.org/10.2307/1971524
- Freydoon Shahidi, Langlands’ conjecture on Plancherel measures for $p$-adic groups, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 277–295. MR 1168488
- J.-L. Waldspurger, Un exercice sur ${\rm GSp}(4,F)$ et les représentations de Weil, Bull. Soc. Math. France 115 (1987), no. 1, 35–69 (French). MR 897614
[We]We R. Weissauer, A special case of the fundamental lemma, preprint.
[A]A J. Arthur, On some problems suggested by the trace formula, Lie group representations II, Springer Lecture Notes 1041 (1984), 1–49; Unipotent automorphic representations: conjectures, in: Orbites unipotentes et representations, II, Astérisque 171–172 (1989), 13–71. ;
[BZ1]BZ1 J. Bernstein, A. Zelevinskii, Representations of the group $\operatorname {GL}(n,F)$ where $F$ is a nonarchimedean local field, Russian Math. Surveys 31 (1976), 1–68.
[BZ2]BZ2 J. Bernstein, A. Zelevinsky, Induced representations of reductive $p$-adic groups, Ann. Sci. École Norm. Sup. 10 (1977), 441–472.
[Bo]Bo A. Borel, Automorphic $L$-functions, Proc. Sympos. Pure Math. 33 II, Amer. Math. Soc., Providence, R.I. (1979), 27–63.
[F1]F1 Y. Flicker, Lifting automorphic forms of PGSp(2) and SO(4) to PGL(4), 2001.
[F2]F2 Y. Flicker, Regular trace formula and base-change lifting, Amer. J. Math. 110 (1988), 739–764.
[F3]F3 Y. Flicker, On the symmetric square. IV. Applications of a trace formula, Trans. Amer. Math. Soc. 330 (1992), 125–152; V. Unit elements, Pacific J. Math. 175 (1996), 507–526; VI. Total global comparison; J. Funct. Analyse 122 (1994), 255–278. See also: Automorphic representations of low rank groups, research monograph, 2003. ; ;
[F4]F4 Y. Flicker, I. Elementary proof of a fundamental lemma for a unitary group, Canad. J. Math. 50 (1998), 74–98; II. Packets and liftings for U(3), J. Analyse Math. 50 (1988), 19–63; III. Base change trace identity for U(3), J. Analyse Math. 52 (1989), 39–52; IV. Characters, genericity, and multiplicity one for U(3), preprint. See also: Automorphic representations of the unitary group U(3,$E/F$), research monograph, 2003. ; ;
[F5]F5 Y. Flicker, Matching of orbital integrals on GL(4) and GSp(2), Mem. Amer. Math. Soc. no. 655, vol. 137 (1999), 1–114.
[F6]F6 Y. Flicker, Automorphic forms on SO(4), 2004, preprint.
[F7]F7 Y. Flicker, On Zeta functions of Shimura varieties of PGSp(2), 2003, preprint.
[FK]FK Y. Flicker, D. Kazhdan, Metaplectic correspondence, Publ. Math. IHES 64 (1987), 53–110.
[FZ]FZ Y. Flicker, D. Zinoviev, Twisted character of a small representation of PGL(4), Moscow Math. J. (2004), in press.
[GRS]GRS D. Ginzburg, S. Rallis, D. Soudry, On Generic Forms on $\operatorname {SO}(2n+1)$; Functorial Lift to $\operatorname {GL}(2n)$, Endoscopy and Base Change, Internat. Math. Res. Notices 14 (2001), 729–763.
[H]H Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, notes by S. DeBacker and P. Sally, AMS Univ. Lecture Series 16 (1999); see also: Queen’s Papers in Pure and Appl. Math. 48 (1978), 281–346. ;
[JS]JS H. Jacquet, J. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math. 103 (1981), 777–815.
[K]K D. Kazhdan, On lifting, in Lie groups representations II, Springer Lecture Notes 1041 (1984), 209–249.
[Kim]Kim H. Kim, Residual spectrum of odd orthogonal groups, Internat. Math. Res. Notices 17 (2001), 873–906.
[Ko]Ko R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611–650.
[KS]KS R. Kottwitz, D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 pp.
[KRS]KRS S. Kudla, S. Rallis, D. Soudry, On the degree 5 $L$-function for Sp(2), Invent. Math. 107 (1992), 483–541.
[L]L R. Langlands, On the notion of an automorphic representation, Proc. Sympos. Pure Math. 33 I (1979), 203–207.
[MW]MW C. Moeglin, J.-L. Waldspurger, Le spectre résiduel de $\operatorname {GL}(n)$, Ann. Sci. École Norm. Sup. 22 (1989), 605–674.
[PS1]PS1 I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), 309-338; A remark on my paper: “On the Saito-Kurokawa lifting”, Invent. Math. 76 (1984), 75–76. ;
[PS2]PS2 I. Piatetski-Shapiro, Work of Waldspurger, Lie group representations II, Springer Lecture Notes 1041 (1984), 280–302.
[Rb]Rb B. Roberts, Global L-packets for GSp(2) and theta lifts, Doc. Math. 6 (2001), 247–314.
[Ro1]Ro1 F. Rodier, Modèle de Whittaker et caractères de représentations, Noncommutative harmonic analysis, Springer Lecture Notes 466 (1975), 151–171.
[Ro2]Ro2 F. Rodier, Sur les représentations non ramifiées des groupes réductifs $p$-adiques; l’example de GSp(4), Bull. Soc. Math. France 116 (1988), 15–42.
[ST]ST P. Sally, M. Tadic, Induced representations and classifications for $\operatorname {GSp}(2,F)$ and $\operatorname {Sp}(2,F)$, Mem. Soc. Math. France 52 (1993), 75–133.
[Sh1]Sh1 F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–355.
[Sh2]Sh2 F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. 132 (1990), 273–330.
[Sh3]Sh3 F. Shahidi, Langlands’ conjecture on Plancherel measures for $p$-adic groups, in Harmonic analysis on reductive groups, 277–295, Progr. Math., 101, Birkhäuser, Boston, MA, 1991.
[W]W J.-L. Waldspurger, Un exercise sur $\operatorname {GSp}(4,F)$ et les représentations de Weil, Bull. Soc. Math. France 115 (1987), 35–69.
[We]We R. Weissauer, A special case of the fundamental lemma, preprint.
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Additional Information
Yuval Z. Flicker
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174
Email:
flicker@math.ohio-state.edu
Keywords:
Automorphic representations,
symplectic group,
liftings,
twisted endoscopy,
packets,
quasi-packets,
multiplicity one,
rigidity,
functoriality,
twisted trace formula,
character relations
Received by editor(s):
March 4, 2004
Published electronically:
April 23, 2004
Additional Notes:
Partially supported by a Lady Davis Visiting Professorship at the Hebrew University, 2004, and Max-Planck-Institut für Mathematik, Bonn, 2003.
Communicated by:
David Kazhdan
Article copyright:
© Copyright 2004
American Mathematical Society