@incollection{AST_2012__346__1_0, author = {Gan, Wee Teck and Gross, Benedict H. and Prasad, Dipendra}, title = {Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups}, booktitle = {Sur les Conjectures de Gross et Prasad. I}, editor = {Gan, W. T. and Gross, B. H. and Prasad, D. and Waldspurger, J.-L.}, series = {Ast\'erisque}, pages = {1--109}, publisher = {Soci\'et\'e math\'ematique de France}, number = {346}, year = {2012}, mrnumber = {3202556}, zbl = {1280.22019}, language = {en}, url = {http://www.numdam.org/item/AST_2012__346__1_0/} }
TY - CHAP AU - Gan, Wee Teck AU - Gross, Benedict H. AU - Prasad, Dipendra TI - Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups BT - Sur les Conjectures de Gross et Prasad. I AU - Collectif ED - Gan, W. T. ED - Gross, B. H. ED - Prasad, D. ED - Waldspurger, J.-L. T3 - Astérisque PY - 2012 SP - 1 EP - 109 IS - 346 PB - Société mathématique de France UR - http://www.numdam.org/item/AST_2012__346__1_0/ LA - en ID - AST_2012__346__1_0 ER -
%0 Book Section %A Gan, Wee Teck %A Gross, Benedict H. %A Prasad, Dipendra %T Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups %B Sur les Conjectures de Gross et Prasad. I %A Collectif %E Gan, W. T. %E Gross, B. H. %E Prasad, D. %E Waldspurger, J.-L. %S Astérisque %D 2012 %P 1-109 %N 346 %I Société mathématique de France %U http://www.numdam.org/item/AST_2012__346__1_0/ %G en %F AST_2012__346__1_0
Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra. Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups, dans Sur les Conjectures de Gross et Prasad. I, Astérisque, no. 346 (2012), pp. 1-109. http://www.numdam.org/item/AST_2012__346__1_0/
[1] Genuine representations of the metaplectic group", Compositio Math. 113, p. 23-66. | DOI | MR | Zbl
and - "[2] Spherical pairs over close local fields", to appear in Comment. Math. Helvetici. | MR | Zbl
, and - "[3] Multiplicity one theorem for ", Selecta Math. 2 (2009), p. 271-294. | DOI | MR | Zbl
and - "[4] Multiplicity one theorems", Annals of Math. 172 (2010), p. 1407-1434. | DOI | MR | Zbl
, , and - "[5] On some problems suggested by the trace formula", in Lie group representations, II (College Park, Md., 1982/1983)", Lecture Notes in Math., vol. 1041, Springer, 1984, p. 1-49. | MR | Zbl
- "[6] Unipotent automorphic representations: conjectures", Astérisque 171-172 (1989), p. 13-71. | Numdam | MR | Zbl
, "[7] Automorphic representations of classical groups", in preparation.
, "[8] The endoscopic classification of representations: orthogonal and symplectic groups", preprint. | MR | Zbl
, "[9] Degenerate principal series for even-orthogonal groups", Represent. Theory 7 (2003), p. 440-480. | DOI | MR | Zbl
and - "[10] Elements of Mathematics, Lie groups and Lie Algebras, Chapters 7-9, Springer, 2005. | MR | Zbl
-[11] Canonical extensions of Harish-Chandra modules to representations of ", Canad. J. Math. 41 (1989), p. 385-438. | DOI | MR | Zbl
- "[12] The unramified principal series of -adic groups. II. The Whittaker function", Compositio Math. 41 (1980), p. 207-231. | EuDML | Numdam | MR | Zbl
and - "[13] Les constantes des équations fonctionnelles des fonctions ", in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, 1973, p. 501-597. Lecture Notes in Math., Vol. 349. | DOI | MR | Zbl
- "[14] Les constantes locales de l'équation fonctionnelle de la fonction d'Artin d'une représentation orthogonale", Invent. Math. 35 (1976), p. 299-316. | DOI | EuDML | MR | Zbl
, "[15] Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques", in Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc, 1979, p. 247-289. | MR | Zbl
, "[16] On the functional equation of the Artin -function for characters of real representations", Invent. Math. 20 (1973), p. 125-138. | DOI | EuDML | MR | Zbl
and - "[17] Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence", to appear in Compositio Math.. | MR | Zbl
and - "[18] The local Langlands conjecture for (4)", International Mathematics Research Notices (2010), p. 2987-3038. | DOI | MR | Zbl
and - "[19] The local Langlands conjecture for (4)", Ann. of Math. 173 (2011), p. 1841-1882. | DOI | MR | Zbl
, "[20] Explicit constructions of automorphic -functions, Lecture Notes in Math., vol. 1254, Springer, 1987. | MR | Zbl
, and -[21] On the nonvanishing of the central value of the Rankin-Selberg -functions", J. Amer. Math. Soc. 17 (2004), p. 679-722. | DOI | MR | Zbl
, and - "[22] On the nonvanishing of the central value of the Rankin-Selberg -functions, II", Ohio State Univ. Math. Res. Inst. Publ. 11 (2005), p. 157-191. | MR | Zbl
, and , "[23] Models for certain residual representations of unitary groups", AMS Contemp. Math. 488 (2009), p. 125-146. | DOI | MR | Zbl
, and , "[24] On the decomposition of a representation of when restricted to ", Canad. J. Math. 44 (1992), p. 974-1002. | DOI | MR | Zbl
and, - "[25] On irreducible representations of ", Canad. J. Math. 46 (1994), p. 930-950. | MR | Zbl
and, , "[26] Arithmetic invariants of discrete Langlands parameters", Duke Math. J. 154 (2010), p. 431-508. | DOI | MR | Zbl
and - "[27] Heegner points and derivatives of -series", Invent. Math. 84 (1986), p. 225-320. | DOI | EuDML | MR | Zbl
and - "[28] Theta dichotomy for unitary groups", J. Amer. Math. Soc. 9 (1996), p. 941-1004. | DOI | MR | Zbl
, and - "[29] The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, vol. 151, Princeton Univ. Press, 2001. | MR | Zbl
and -[30] A refined Gross-Prasad conjecture for unitary groups", Ph.D. Thesis, University of California, San Diego, 2011. | MR
- "[31] Une preuve simple des conjectures de Langlands pour sur un corps -adique", Invent. Math. 139 (2000), p. 439-455. | DOI | MR | Zbl
- "[32] Transcending classical invariant theory", J. Amer. Math. Soc. 2 (1989), p. 535-552. | DOI | MR | Zbl
- "[33] On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture", Geom. Funct. Anal. 19 (2010), p. 1378-1425. | DOI | MR | Zbl
and - "[34] Rankin-Selberg convolutions", Amer. J. Math. 105 (1983), p. 367-464. | DOI | MR | Zbl
, and - "[35] On the Gross-Prasad conjecture for unitary groups", Clay Math. Proc. 13 (2011), p. 205-264. | MR | Zbl
and - "[36] Rankin-Selberg convolutions: Archimedean theory", in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, 1990, p. 125-207. | MR | Zbl
and - "[37] Uniqueness of Bessel models: the archimedean case", Geom. and Funct. Anal. 20 (2010), p. 690-709. | DOI | MR | Zbl
, and - "[38] Whittaker-Shintani functions for orthogonal groups", Tohoku Math. J. 55 (2003), p. 1-64. | DOI | MR | Zbl
, and - "[39] Multiplicity-one results and explicit formulas for quasi-split -adic unitary groups", Ph.D. Thesis, The Ohio State University, 2008. | MR
- "[40] The book of involutions, AMS Colloquium Publ., vol. 44, 1998. | DOI | MR | Zbl
, , and -[41] Stable trace formula: elliptic singular terms", Math. Ann. 275 (1986), p. 365-399. | DOI | EuDML | MR | Zbl
- "[42] On the local theta-correspondence", Invent. Math. 83 (1986), p. 229-255. | DOI | EuDML | MR | Zbl
- "[43] Splitting metaplectic covers of dual reductive pairs", Israel J. Math. 87 (1994), p. 361-401. | DOI | MR | Zbl
, "[44] On first occurrence in local theta correspondence", in Automorphic Representations, -functions and Applications: Progress and Prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, 2005, p. 273-308. | MR | Zbl
and - "[45] The Langlands conjecture for of a local field", Ann. of Math. 112 (1980), p. 381-412. | DOI | MR | Zbl
- "[46] -indistinguishability for ", Canad. J. Math. 31 (1979), p. 726-785. | MR | Zbl
and - "[47] Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups", preprint arXiv: 1012.4538. | MR | Zbl
- "[48] Characteristic classes, Princeton Univ. Press, 1974, Annals of Mathematics Studies, No. 76. | MR | Zbl
and -[49] Classification des séries discrètes pour certains groupes classiques -adiques", in Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, p. 209-245. | MR | Zbl
- "[50] Classification et changement de base pour les séries discrètes des groupes unitaires -adiques", Pacific J. Math. 233 (2007), p. 159-204. | DOI | MR | Zbl
, "[51] Correspondances de Howe sur un corps -adique, Springer Lecture Notes, vol. 1291, 1987. | MR | Zbl
, and -[52] Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge Univ. Press, 1995. | MR | Zbl
and -[53] La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général", preprint.
and , "[54] Whittaker-Shintani functions on the symplectic group of Fourier-Jacobi type", Compositio Math. 79 (1991), p. 321-349. | EuDML | Numdam | MR | Zbl
and - "[55] Shintani functions and automorphic -functions for ", Tohoku Math. J. 48 (1996), p. 165-202. | DOI | MR | Zbl
and , "[56] Invariant forms for representations of over a local field", Amer. J. Math. 114 (1992), p. 1317-1363. | DOI | MR | Zbl
- "[57] On the local Howe duality correspondence", Internat. Math. Res. Notices (1993), p. 279-287. | DOI | MR | Zbl
, "[58] Theta correspondence for unitary groups", Pacific J. Math. 194 (2000), p. 427-438. | DOI | MR | Zbl
, "[59] On the Howe duality conjecture", Compositio Math. 51 (1984), p. 333-399. | EuDML | Numdam | MR | Zbl
- "[60] Automorphic representations of unitary groups in three variables", Annals of Mathematics Studies 123 (1990). | MR | Zbl
- "[61] Principe de densité pour les groupes réductifs", Compositio Math. 108 (1997), p. 151-184. | DOI | MR | Zbl
- "[62] A course in arithmetic, Grad. Texts in Math., vol. 7, Springer, 1973. | DOI | MR | Zbl
-[63] Linear algebraic groups, second éd., Progress in Math., vol. 9, Birkhäuser, 1998. | MR | Zbl
-[64] Multiplicity one theorems for symplectic groups", preprint arXiv:0903.1417. | Zbl
- "[65] Multiplicity one theorems: the Fourier-Jacobi models", to appear in Amer. J. of Math.. | MR | Zbl
, "[66] Multiplicity one theorems: the Archimedean case", Ann. of Math. 175 (2012), p. 23-44. | DOI | MR | Zbl
and - "[67] A note on special orthogonal groups following Waldspurger", preprint.
and , "[68] Functional equations of p-adic zeta integrals and representations of the metaplectic group", preprint, 1995.
- "[69] Number theoretic background", in Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc, 1979, p. 3-26. | MR | Zbl
- "[70] The local Langlands conjecture", AMS Contemporary Mathematics 145 (1993), p. 305-379. | DOI | MR | Zbl
- "[71] Correspondance de Shimura", J. Math. Pures Appl. 59 (1980), p. 1-132. | MR | Zbl
- "[72] Démonstration d'une conjecture de dualité de Howe dans le cas -adique, , Israel Mathematical Conference Proceedings, vol. 2, Weizmann Science Press of Israel, 1990. | MR | Zbl
,[73] Correspondances de Shimura et quaternions", Forum Math. 3 (1991), p. 219-307. | EuDML | MR | Zbl
, "[74] Une formule intégrale reliée à la conjecture locale de Gross-Prasad", Comp. Math. 146 (2010), p. 1180-1290. | DOI | MR | Zbl
, "[75] Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: extension aux représentations tempérées", this volume, 2012. | Numdam | MR | Zbl
, "[76] Une variante d'un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann", this volume, 2012. | Numdam | MR | Zbl
, "[77] Calcul d'une valeur d'un facteur epsilon par une formule intégrale", to appear in Astérisque. | Numdam | MR
, "[78] La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux", to appear in Astérisque. | Numdam | MR | Zbl
, "[79] On the constant term of a square integrable automorphic form", in Operator algebras and group representations, Vol. II (Neptun, 1980), Monogr. Stud. Math., vol. 18, Pitman, 1984, p. 227-237. | MR | Zbl
- "[80] Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press Inc., 1992. | MR | Zbl
,[81] vectors", in Representations of Lie groups and quantum groups (Trento, 1993), Pitman Res. Notes Math. Ser., vol. 311, Longman Sci. Tech., 1994, p. 205-270. | MR | Zbl
, "[82] Basic number theory, third ed., Die Grund. Math. Wiss., vol. 144, Springer, 1974. | MR | Zbl
-[83] On the Siegel-Weil formula for quaternionic unitary groups", to appear in American J. of Math.. | MR | Zbl
- "[84] Heights of CM points I: Gross-Zagier formula", preprint available at https://wiki.epfl.ch/tan/documents/preprint(83%20pp)%202008%20yuan.pdf.
, and - "[85] Relative trace formula and arithmetic Gross-Prasad conjecture", preprint, 2009.
- "[86] Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups", preprint http://www.math.Columbia.edu/~wzhang/math/online/transfer.pdf, 2011. | MR
, "[87] Theta dichotomy and doubling epsilon factors for ", Amer. J. Math. 133 (2011), p. 1313-1364. | DOI | MR | Zbl
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