Refined global Gan-Gross-Prasad conjecture for Bessel periods.(English)Zbl 1404.11065
Summary: We formulate a refined version of the global Gan-Gross-Prasad conjecture for general Bessel models, extending the work ofA. Ichino andT. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010;Zbl 1216.11057)] andR. N. Harris [Int. Math. Res. Not. 2014, No. 2, 303–389 (2014;Zbl 1322.11047)] in the co-rank 1 case. It is an explicit formula relating the automorphic period of Bessel type and the central value of certain \(L\)-function. To support such conjecture, we provide two examples for pairs \(\mathrm{SO}_{5}\times\mathrm{SO}_{2}\) and \(\mathrm{SO}_{6}\times\mathrm{SO}_{3}\) (both co-rank 3) in the endoscopic case via theta lifting.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
Bessel periods;Gan-Gross-Prasad conjecture;automorphic periods;central values;\(L\)-functionsCite
References:
| [1] | Aizenbud A., Gourevitch D., Rallis S. and Schiffmann G., Multiplicity one theorems, Ann. of Math. (2) 172 (2010), no. 2, 1407-1434.; Aizenbud, A.; Gourevitch, D.; Rallis, S.; Schiffmann, G., Multiplicity one theorems, Ann. of Math. (2), 172, 2, 1407-1434 (2010) ·Zbl 1202.22012 |
| [2] | Beuzart-Plessis R., La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires, preprint 2012, .; Beuzart-Plessis, R., La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires (2012) ·Zbl 1357.22008 |
| [3] | Böcherer S. and Schulze-Pillot R., The Dirichlet series of Koecher and Maass and modular forms of weight \(\frac{3}{2} \), Math. Z. 209 (1992), no. 2, 273-287.; Böcherer, S.; Schulze-Pillot, R., The Dirichlet series of Koecher and Maass and modular forms of weight \(\frac{3}{2} \), Math. Z., 209, 2, 273-287 (1992) ·Zbl 0773.11031 |
| [4] | Casselman W. and Shalika J., The unramified principal series of p-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207-231.; Casselman, W.; Shalika, J., The unramified principal series of p-adic groups. II. The Whittaker function, Compositio Math., 41, 2, 207-231 (1980) ·Zbl 0472.22005 |
| [5] | Dixmier J. and Malliavin P., Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102 (1978), no. 4, 307-330.; Dixmier, J.; Malliavin, P., Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2), 102, 4, 307-330 (1978) ·Zbl 0392.43013 |
| [6] | Furusawa M., On L-functions for \(\text{GSp}(4)\times\text{GL}(2)\) and their special values, J. reine angew. Math. 438 (1993), 187-218.; Furusawa, M., On L-functions for \(\text{GSp}(4)\times\text{GL}(2)\) and their special values, J. reine angew. Math., 438, 187-218 (1993) ·Zbl 0770.11025 |
| [7] | Furusawa M. and Martin K., On central critical values of the degree four L-functions for \(\text{GSp}(4)\): A simple trace formula, Math. Z. (2014), 10.1007/s00209-013-1248-4.; Furusawa, M.; Martin, K., On central critical values of the degree four L-functions for \(\text{GSp}(4)\): A simple trace formula, Math. Z. (2014) ·Zbl 1310.11056 ·doi:10.1007/s00209-013-1248-4 |
| [8] | Gan W. T., Gross B. H. and Prasad D., Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1-109.; Gan, W. T.; Gross, B. H.; Prasad, D., Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque, 346, 1-109 (2012) ·Zbl 1280.22019 |
| [9] | Gan W. T. and Ichino A., On endoscopy and the refined Gross-Prasad conjecture for \((\text{SO}_5,\text{SO}_4)\), J. Inst. Math. Jussieu 10 (2011), no. 2, 235-324.; Gan, W. T.; Ichino, A., On endoscopy and the refined Gross-Prasad conjecture for \((\text{SO}_5,\text{SO}_4)\), J. Inst. Math. Jussieu, 10, 2, 235-324 (2011) ·Zbl 1241.11058 |
| [10] | Gan W. T., Qiu Y. and Takeda S., The regularized Siegel-Weil formula (the second term identity) and the rallis inner product formula, Invent. Math., to appear.; Gan, W. T.; Qiu, Y.; Takeda, S., The regularized Siegel-Weil formula (the second term identity) and the rallis inner product formula, Invent. Math. ·Zbl 1320.11037 |
| [11] | Ginzburg D., Piatetski-Shapiro I. and Rallis S., L functions for the orthogonal group, Mem. Amer. Math. Soc. 128 (1997), no. 611.; Ginzburg, D.; Piatetski-Shapiro, I.; Rallis, S., L functions for the orthogonal group, Mem. Amer. Math. Soc., 128, 611 (1997) ·Zbl 0884.11022 |
| [12] | Gong Z. and Grenié L., An inequality for local unitary theta correspondence, Ann. Fac. Sci. Toulouse Math. (6) 20 (2011), no. 1, 167-202.; Gong, Z.; Grenié, L., An inequality for local unitary theta correspondence, Ann. Fac. Sci. Toulouse Math. (6), 20, 1, 167-202 (2011) ·Zbl 1254.11049 |
| [13] | Gross B. H., On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287-313.; Gross, B. H., On the motive of a reductive group, Invent. Math., 130, 2, 287-313 (1997) ·Zbl 0904.11014 |
| [14] | Gross B. H. and Prasad D., On the decomposition of a representation of \(\text{SO}_n\) when restricted to \(\text{SO}_{n-1} \), Canad. J. Math. 44 (1992), no. 5, 974-1002.; Gross, B. H.; Prasad, D., On the decomposition of a representation of \(\text{SO}_n\) when restricted to \(\text{SO}_{n-1} \), Canad. J. Math., 44, 5, 974-1002 (1992) ·Zbl 0787.22018 |
| [15] | Gross B. H. and Prasad D., On irreducible representations of \(\text{SO}_{2n+1}\times\text{SO}_{2m} \), Canad. J. Math. 46 (1994), no. 5, 930-950.; Gross, B. H.; Prasad, D., On irreducible representations of \(\text{SO}_{2n+1}\times\text{SO}_{2m} \), Canad. J. Math., 46, 5, 930-950 (1994) ·Zbl 0829.22031 |
| [16] | Harish-Chandra , Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111.; Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math., 116, 1-111 (1966) ·Zbl 0199.20102 |
| [17] | Harish-Chandra , Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104-204.; Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal., 19, 104-204 (1975) ·Zbl 0315.43002 |
| [18] | Harris M., Kudla S. S. and Sweet W. J., Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), no. 4, 941-1004.; Harris, M.; Kudla, S. S.; Sweet, W. J., Theta dichotomy for unitary groups, J. Amer. Math. Soc., 9, 4, 941-1004 (1996) ·Zbl 0870.11026 |
| [19] | Harris R. N., The refined Gross-Prasad conjecture for unitary groups, Int. Math. Res. Not. IMRN (2014), no. 2, 303-389.; Harris, R. N., The refined Gross-Prasad conjecture for unitary groups, Int. Math. Res. Not. IMRN, 2, 303-389 (2014) ·Zbl 1322.11047 |
| [20] | He H., Unitary representations and theta correspondence for type I classical groups, J. Funct. Anal. 199 (2003), no. 1, 92-121.; He, H., Unitary representations and theta correspondence for type I classical groups, J. Funct. Anal., 199, 1, 92-121 (2003) ·Zbl 1021.22008 |
| [21] | Hiraga K. and Saito H., On L-packets for inner forms of \(\text{SL}_n\), Mem. Amer. Math. Soc. 215 (2012), no. 1013.; Hiraga, K.; Saito, H., On L-packets for inner forms of \(\text{SL}_n\), Mem. Amer. Math. Soc., 215, 1013 (2012) ·Zbl 1242.22023 |
| [22] | Howe R., Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535-552.; Howe, R., Transcending classical invariant theory, J. Amer. Math. Soc., 2, 3, 535-552 (1989) ·Zbl 0716.22006 |
| [23] | Ichino A. and Ikeda T., On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal. 19 (2010), no. 5, 1378-1425.; Ichino, A.; Ikeda, T., On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal., 19, 5, 1378-1425 (2010) ·Zbl 1216.11057 |
| [24] | Jacquet H. and Rallis S., On the Gross-Prasad conjecture for unitary groups, On certain L-functions, Clay Math. Proc.13, American Mathematical Society, Providence (2011), 205-264.; Jacquet, H.; Rallis, S., On the Gross-Prasad conjecture for unitary groups, On certain L-functions, Clay Math. Proc.13, 205-264 (2011) ·Zbl 1222.22018 |
| [25] | Jiang D., Sun B. and Zhu C.-B., Uniqueness of Bessel models: The archimedean case, Geom. Funct. Anal. 20 (2010), no. 3, 690-709.; Jiang, D.; Sun, B.; Zhu, C.-B., Uniqueness of Bessel models: The archimedean case, Geom. Funct. Anal., 20, 3, 690-709 (2010) ·Zbl 1200.22008 |
| [26] | Jiang D. and Zhang L., A product of tensor product L-functions of quasi-split classical groups of hermitian type, Geom. Funct. Anal. (2014), 10.1007/s00039-014-0266-7.; Jiang, D.; Zhang, L., A product of tensor product L-functions of quasi-split classical groups of hermitian type, Geom. Funct. Anal. (2014) ·Zbl 1298.11046 ·doi:10.1007/s00039-014-0266-7 |
| [27] | Khoury, Jr. M. J., Multiplicity-one results and explicit formulas for quasi-split p-adic unitary groups, ProQuest LLC, Ph.D. thesis, The Ohio State University, Ann Arbor 2008.; Khoury, Jr., M. J., Multiplicity-one results and explicit formulas for quasi-split p-adic unitary groups (2008) |
| [28] | Kato S., Murase A. and Sugano T., Shintani functions for \(\text{GL}_n\): An explicit formula, unpublished.; Kato, S.; Murase, A.; Sugano, T., Shintani functions for \(\text{GL}_n\): An explicit formula ·Zbl 1037.22034 |
| [29] | Kato S., Murase A. and Sugano T., Whittaker-Shintani functions for orthogonal groups, Tohoku Math. J. (2) 55 (2003), no. 1, 1-64.; Kato, S.; Murase, A.; Sugano, T., Whittaker-Shintani functions for orthogonal groups, Tohoku Math. J. (2), 55, 1, 1-64 (2003) ·Zbl 1037.22034 |
| [30] | Kostant B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 413-455.; Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Éc. Norm. Supér. (4), 6, 413-455 (1973) ·Zbl 0293.22019 |
| [31] | Kudla S. S., On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229-255.; Kudla, S. S., On the local theta-correspondence, Invent. Math., 83, 2, 229-255 (1986) ·Zbl 0583.22010 |
| [32] | Lapid E. and Mao Z., On Whittaker-Fourier coefficients of automorphic forms on \(\widetilde{\text{Sp}}_n\), preprint 2013, .; Lapid, E.; Mao, Z., On Whittaker-Fourier coefficients of automorphic forms on \(#### (2013)\) ·Zbl 1418.11075 |
| [33] | Lapid E. and Mao Z., A conjecture on Whittaker-Fourier coefficients of cusp forms, J. Number Theory (2013), 10.1016/j.jnt.2013.10.003.; Lapid, E.; Mao, Z., A conjecture on Whittaker-Fourier coefficients of cusp forms, J. Number Theory (2013) ·Zbl 1396.11081 ·doi:10.1016/j.jnt.2013.10.003 |
| [34] | Lapid E. and Mao Z., On a new functional equation for local integrals, Automorphic forms and related geometry: Assessing the Legacy of I. I. Piatetski-Shapiro, Contemp. Math. 614, American Mathematical Society, Providence, to appear.; Lapid, E.; Mao, Z., On a new functional equation for local integrals, Automorphic forms and related geometry: Assessing the Legacy of I. I. Piatetski-Shapiro ·Zbl 1306.22008 |
| [35] | Liu Y., Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups, Manuscripta Math. (2014), 10.1007/s00229-014-0666-x.; Liu, Y., Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups, Manuscripta Math. (2014) ·Zbl 1301.11050 ·doi:10.1007/s00229-014-0666-x |
| [36] | Paul A., Howe correspondence for real unitary groups, J. Funct. Anal. 159 (1998), no. 2, 384-431.; Paul, A., Howe correspondence for real unitary groups, J. Funct. Anal., 159, 2, 384-431 (1998) ·Zbl 0924.22012 |
| [37] | Prasad D. and Takloo-Bighash R., Bessel models for \(\text{GSp}(4)\), preprint 2007, .; Prasad, D.; Takloo-Bighash, R., Bessel models for \(\operatorname{GSp} <mml:mo stretchy=''false``>(4 <mml:mo stretchy=''false``>) (2007)\) ·Zbl 1228.11070 |
| [38] | Prasad D. and Takloo-Bighash R., Bessel models for \(\text{GSp}(4)\), J. reine angew. Math. 655 (2011), 189-243.; Prasad, D.; Takloo-Bighash, R., Bessel models for \(\text{GSp}(4)\), J. reine angew. Math., 655, 189-243 (2011) ·Zbl 1228.11070 |
| [39] | Qiu Y., The Bessel period functional on \(\text{SO}(5)\): The nontempered case, preprint 2013, .; Qiu, Y., The Bessel period functional on \(\operatorname{SO} <mml:mo stretchy=''false``>(5 <mml:mo stretchy=''false``>)\): The nontempered case (2013) |
| [40] | Sakellaridis Y. and Venkatesh A., Periods and harmonic analysis on spherical varieties, preprint 2012, .; Sakellaridis, Y.; Venkatesh, A., Periods and harmonic analysis on spherical varieties (2012) ·Zbl 1479.22016 |
| [41] | Sun B., Bounding matrix coefficients for smooth vectors of tempered representations, Proc. Amer. Math. Soc. 137 (2009), no. 1, 353-357.; Sun, B., Bounding matrix coefficients for smooth vectors of tempered representations, Proc. Amer. Math. Soc., 137, 1, 353-357 (2009) ·Zbl 1156.22013 |
| [42] | Sun B. and Zhu C.-B., Multiplicity one theorems: The archimedean case, Ann. of Math. (2) 175 (2012), no. 1, 23-44.; Sun, B.; Zhu, C.-B., Multiplicity one theorems: The archimedean case, Ann. of Math. (2), 175, 1, 23-44 (2012) ·Zbl 1239.22014 |
| [43] | Waldspurger J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173-242.; Waldspurger, J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Math., 54, 2, 173-242 (1985) ·Zbl 0567.10021 |
| [44] | Waldspurger J.-L., Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, \(p\neq 2\), Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Tel Aviv 1989), Israel Math. Conf. Proc. 2, The Weizmann Science Press of Israel, Jerusalem (1990), 267-324.; Waldspurger, J.-L., Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, \(p\neq 2\), Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, 267-324 (1990) ·Zbl 0722.22009 |
| [45] | Waldspurger J.-L., La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235-333.; Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu, 2, 2, 235-333 (2003) ·Zbl 1029.22016 |
| [46] | Waldspurger J.-L., Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: Extension aux représentations tempérées, Astérisque 346 (2012), 171-312.; Waldspurger, J.-L., Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: Extension aux représentations tempérées, Astérisque, 346, 171-312 (2012) ·Zbl 1290.22012 |
| [47] | Waldspurger J.-L., Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann, Astérisque 346 (2012), 313-318.; Waldspurger, J.-L., Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann, Astérisque, 346, 313-318 (2012) ·Zbl 1308.22008 |
| [48] | Wallach N. R., Real reductive groups. II, Pure Appl. Math. 132, Academic Press, Boston 1992.; Wallach, N. R., Real reductive groups. II (1992) ·Zbl 0785.22001 |
| [49] | Yamana S., On the Siegel-Weil formula for quaternionic unitary groups, Amer. J. Math. 135 (2013), no. 5, 1383-1432.; Yamana, S., On the Siegel-Weil formula for quaternionic unitary groups, Amer. J. Math., 135, 5, 1383-1432 (2013) ·Zbl 1356.11025 |
| [50] | Yun Z., The fundamental lemma of Jacquet and Rallis, Duke Math. J. 156 (2011), no. 2, 167-227.; Yun, Z., The fundamental lemma of Jacquet and Rallis, Duke Math. J., 156, 2, 167-227 (2011) ·Zbl 1211.14039 |
| [51] | Zhang C., A note on the local theta correspondence for unitary similitude dual pairs, preprint 2012, .; Zhang, C., A note on the local theta correspondence for unitary similitude dual pairs (2012) |
| [52] | Zhang W., Automorphic period and the central value of Rankin-Selberg L-function, J. Amer. Math. Soc. 27 (2014), 541-612.; Zhang, W., Automorphic period and the central value of Rankin-Selberg L-function, J. Amer. Math. Soc., 27, 541-612 (2014) ·Zbl 1294.11069 |
| [53] | Zhang W., Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups, Ann. of Math., to appear.; Zhang, W., Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups, Ann. of Math. ·Zbl 1322.11048 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
