A critical case of Rallis inner product formula
Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n ＋ 1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central va...
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| Published in | Science China. Mathematics Vol. 60; no. 2; pp. 201 - 222 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Science China Press
01.02.2017
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1674-7283 1869-1862 |
| DOI | 10.1007/s11425-015-0770-7 |
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| Abstract | Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n ＋ 1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature. |
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| AbstractList | Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2
n
+1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands
L
-function of π twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature. Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n ＋ 1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature. Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n+1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature. |
| Author | WU ChenYan |
| AuthorAffiliation | Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China |
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| Keywords | Rallis inner product formula 11F27 theta lift regularised Siegel-Weil formula 11F70 function |
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| Notes | regularised Siegel-Weil formula, Rallis inner product formula, theta lift, L-function Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n ＋ 1. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature. 11-5837/O1 ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
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| References | Kudla, Rallis (CR15) 1992; 78 Yamana (CR23) 2011; 147 Weil (CR22) 1965; 113 Yamana (CR24) 2014; 196 Rao (CR20) 1993; 157 Gan, Ichino (CR2) 2014; 195 Gelbart, Piatetski-Shapiro, Rallis (CR5) 1987 Rallis (CR19) 1987 Ichino (CR7) 2001; 539 Zorn (CR25) 2011; 133 Gan, Savin (CR4) 2012; 148 Li (CR17) 1992; 428 Gan (CR1) 2012; 208 Ikeda (CR10) 1996; 103 Kudla, Rallis (CR16) 1994; 140 Ikeda (CR9) 1994; 34 Howe (CR6) 2011 Kudla (CR11) 1996 Ichino (CR8) 2004; 247 Rallis (CR18) 1984; 51 Kudla, Rallis (CR14) 1988; 391 Gan, Qiu, Takeda (CR3) 2014; 198 Kudla, Rallis (CR13) 1988; 387 Kudla (CR12) 1997; 146 Serre (CR21) 1973 S Rallis (770_CR18) 1984; 51 W T Gan (770_CR4) 2012; 148 C Zorn (770_CR25) 2011; 133 W T Gan (770_CR1) 2012; 208 S Rallis (770_CR19) 1987 W T Gan (770_CR3) 2014; 198 R R Rao (770_CR20) 1993; 157 R Howe (770_CR6) 2011 S Kudla (770_CR15) 1992; 78 T Ikeda (770_CR9) 1994; 34 S Yamana (770_CR24) 2014; 196 S Kudla (770_CR13) 1988; 387 A Weil (770_CR22) 1965; 113 S Kudla (770_CR11) 1996 T Ikeda (770_CR10) 1996; 103 S Yamana (770_CR23) 2011; 147 S Kudla (770_CR14) 1988; 391 S Kudla (770_CR16) 1994; 140 S Kudla (770_CR12) 1997; 146 A Ichino (770_CR8) 2004; 247 J S Li (770_CR17) 1992; 428 A Ichino (770_CR7) 2001; 539 W T Gan (770_CR2) 2014; 195 S Gelbart (770_CR5) 1987 J-P Serre (770_CR21) 1973 |
| References_xml | – year: 1987 ident: CR19 publication-title: -Functions and the Oscillator Representation doi: 10.1007/BFb0077894 – volume: 387 start-page: 1 year: 1988 end-page: 68 ident: CR13 article-title: On the Weil-Siegel formula. publication-title: J Reine Angew Math – volume: 247 start-page: 241 year: 2004 end-page: 277 ident: CR8 article-title: A regularized Siegel-Weil formula for unitary groups. publication-title: Math Z doi: 10.1007/s00209-003-0580-5 – start-page: 224 year: 2011 end-page: 331 ident: CR6 article-title: On a notion of rank for unitary representations of the classical groups publication-title: Harmonic Analysis and Group Representation – volume: 140 start-page: 1 year: 1994 end-page: 80 ident: CR16 article-title: A regularized Siegel-Weil formula: the first term identity. publication-title: Ann of Math (2) doi: 10.2307/2118540 – year: 1987 ident: CR5 publication-title: Explicit Constructions of Automorphic L-Functions doi: 10.1007/BFb0078125 – volume: 539 start-page: 201 year: 2001 end-page: 234 ident: CR7 article-title: On the regularized Siegel-Weil formula. publication-title: J Reine Angew Math – volume: 78 start-page: 209 year: 1992 end-page: 256 ident: CR15 article-title: Ramified degenerate principal series representations for Sp( ). publication-title: Israel J Math doi: 10.1007/BF02808058 – volume: 51 start-page: 333 year: 1984 end-page: 399 ident: CR18 article-title: On the Howe duality conjecture. publication-title: Compos Math – year: 1973 ident: CR21 publication-title: A Course in Arithmetic doi: 10.1007/978-1-4684-9884-4 – volume: 208 start-page: 67 year: 2012 end-page: 95 ident: CR1 article-title: Doubling zeta integrals and local factors for metaplectic groups. publication-title: Nagoya Math J doi: 10.1017/S002776300001059X – volume: 428 start-page: 177 year: 1992 end-page: 217 ident: CR17 article-title: Nonvanishing theorems for the cohomology of certain arithmetic quotients. publication-title: J Reine Angew Math – volume: 198 start-page: 739 year: 2014 end-page: 831 ident: CR3 article-title: The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula. publication-title: Invent Math doi: 10.1007/s00222-014-0509-0 – volume: 146 start-page: 545 year: 1997 end-page: 646 ident: CR12 article-title: Central derivatives of Eisenstein series and height pairings. publication-title: Ann of Math (2) doi: 10.2307/2952456 – volume: 133 start-page: 1313 year: 2011 end-page: 1364 ident: CR25 article-title: Theta dichotomy and doubling epsilon factors for ( ). publication-title: Amer J Math doi: 10.1353/ajm.2011.0038 – volume: 103 start-page: 183 year: 1996 end-page: 218 ident: CR10 article-title: On the residue of the Eisenstein series and the Siegel-Weil formula. publication-title: Compos Math – volume: 195 start-page: 509 year: 2014 end-page: 672 ident: CR2 article-title: Formal degrees and local theta correspondence. publication-title: Invent Math doi: 10.1007/s00222-013-0460-5 – volume: 148 start-page: 1655 year: 2012 end-page: 1694 ident: CR4 article-title: Representations of metaplectic groups I: Epsilon dichotomy and local Langlands correspondence. publication-title: Compos Math doi: 10.1112/S0010437X12000486 – volume: 196 start-page: 651 year: 2014 end-page: 732 ident: CR24 article-title: -functions and theta correspondence for classical groups. publication-title: Invent Math doi: 10.1007/s00222-013-0476-x – volume: 113 start-page: 1 year: 1965 end-page: 87 ident: CR22 article-title: Sur la formule de Siegel dans la théorie des groupes classiques. publication-title: Acta Math doi: 10.1007/BF02391774 – volume: 147 start-page: 1003 year: 2011 end-page: 1021 ident: CR23 article-title: On the Siegel-Weil formula: The case of singular forms. publication-title: Compos Math doi: 10.1112/S0010437X11005379 – year: 1996 ident: CR11 publication-title: Notes on the local theta correspondence – volume: 391 start-page: 65 year: 1988 end-page: 84 ident: CR14 article-title: On the Weil-Siegel formula. 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| SubjectTerms | Applications of Mathematics L-函数 Langlands Lifts Mathematics Mathematics and Statistics Weil 产品配方 案例 正交群 积公式 空间相关 |
| Title | A critical case of Rallis inner product formula |
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