Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T08:09:11.654Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Siegel–Weil formula: The case of singular forms

Part of:Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  04 May 2011

Shunsuke Yamana
Shunsuke Yamana*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan (email: yamana@sci.osaka-cu.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

For the dual pair Sp(n)×O(m) with m≤n, we prove an identity between a special value of a certain Eisenstein series and the regularized integral of a theta function. The proof uses the functional equation of the Eisenstein series and the regularized Siegel–Weil formula for Sp(n)×O(2n+2−m). Analogous results for unitary and orthogonal groups are included.

Keywords

Eisenstein seriesSiegel–Weil formulasingular forms

MSC classification

Secondary:11F27: Theta series; Weil representation; theta correspondences11F70: Representation-theoretic methods; automorphic representations over local and global fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011, pp. 1003 - 1021
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[GT]Gan, W. T. and Takeda, S., The regularized Siegel–Weil formula: the second term identity and non-vanishing of theta lifts from orthogonal groups, J. Reine Angew. Math., to appear.Google Scholar
[How89]Howe, R., Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535–552.CrossRefGoogle Scholar
[HL99]Howe, R. and Lee, S. T., Degenerate principal series representations of GLn(ℂ) and GLn(ℝ), J. Funct. Anal. 166 (1999), 244–309.CrossRefGoogle Scholar
[Ich01]Ichino, A., On the regularized Siegel–Weil formula, J. Reine Angew. Math. 539 (2001), 201–234.Google Scholar
[Ich04]Ichino, A., A regularized Siegel–Weil formula for unitary groups, Math. Z. 247 (2004), 241–277.CrossRefGoogle Scholar
[Ich07]Ichino, A., On the Siegel–Weil formula for unitary groups, Math. Z. 255 (2007), 721–729.CrossRefGoogle Scholar
[Ike96]Ikeda, T., On the residue of the Eisenstein series and the Siegel–Weil formula, Compositio Math. 103 (1996), 183–218.Google Scholar
[JS07]Jiang, D. and Soudry, D., On the genericity of cuspidal automorphic forms of SO(2n+1), II, Compositio Math. 143 (2007), 721–748.CrossRefGoogle Scholar
[Kud97]Kudla, S., Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), 545–646.CrossRefGoogle Scholar
[KR88a]Kudla, S. and Rallis, S., On the Weil–Siegel formula, J. Reine Angew. Math. 387 (1988), 1–68.Google Scholar
[KR88b]Kudla, S. and Rallis, S., On the Weil–Siegel formula II. The isotropic convergent case, J. Reine Angew. Math. 391 (1988), 65–84.Google Scholar
[KR90a]Kudla, S. and Rallis, S., Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), 25–45.CrossRefGoogle Scholar
[KR90b]Kudla, S. and Rallis, S., Poles of Eisenstein series and L-functions, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II, Israel Mathematical Conference Proceedings, vol. 3 (Weizmann, Jerusalem, 1990), 81–110.Google Scholar
[KR92]Kudla, S. and Rallis, S., Ramified degenerate principal series representations for Sp(n), Israel J. Math. 78 (1992), 209–256.CrossRefGoogle Scholar
[KR94]Kudla, S. and Rallis, S., A regularized Siegel–Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), 1–80.CrossRefGoogle Scholar
[KRY06]Kudla, S., Rapoport, M. and Yang, T., Modular forms and special cycles on Shimura curves, Annals of Mathematics Studies, vol. 161 (Princeton University Press, Princeton, NJ, 2006).CrossRefGoogle Scholar
[KS97]Kudla, S. and Sweet, W. J. Jr, Degenerate principal series representations for U(n,n), Israel J. Math. 98 (1997), 253–306.CrossRefGoogle Scholar
[LZ98]Lee, S. T. and Zhu, C.-B., Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. 350 (1998), 5017–5046.CrossRefGoogle Scholar
[LZ08]Lee, S. T. and Zhu, C.-B., Degenerate principal series and local theta correspondence III, J. Algebra 319 (2008), 336–359.CrossRefGoogle Scholar
[Li89]Li, J.-S., Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237–255.CrossRefGoogle Scholar
[Lok06]Loke, H. Y., Howe quotients of unitary characters and unitary lowest weight modules, Represent. Theory 10 (2006), 21–47, with an appendix by S. T. Lee.CrossRefGoogle Scholar
[Moe97]Moeglin, C., Non nullité de certains relêvements par séries théta, J. Lie Theory 7 (1997), 201–229.Google Scholar
[MVW87]Moeglin, C., Vignera, M.-F. and Waldspurger, J.-L., Correspondence de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291 (Springer, New York, 1987).CrossRefGoogle Scholar
[Shi82]Shimura, G., Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), 269–302.CrossRefGoogle Scholar
[Shi99]Shimura, G., Generalized Bessel functions on symmetric spaces, J. Reine Angew. Math. 509 (1999), 35–66.CrossRefGoogle Scholar
[Sie35]Siegel, C. L., Über die analytische Theorie der quadratishen Formen, Ann. of Math. (2) 36 (1935), 527–606.CrossRefGoogle Scholar
[Sie51]Siegel, C. L., Indefinite quadratische Formen und Funktionentheorie I, Math. Ann. 124 (1951), 17–54; II, (1952), 364–387.CrossRefGoogle Scholar
[Swe90]Sweet, W. J. Jr, The metaplectic case of the Siegel–Weil formula, Thesis, University of Maryland (1990).Google Scholar
[Swe95]Sweet, W. J. Jr, Functional equations of p-adic zeta integrals and representations of the metaplectic group, Preprint (1995).Google Scholar
[Tan98]Tan, V., A regularized Siegel–Weil formula on U(2,2) and U(3), Duke Math. J. 94 (1998), 341–378.CrossRefGoogle Scholar
[Tan99]Tan, V., Poles of Siegel Eisenstein series on U(n,n), Canad. J. Math. 51 (1999), 164–175.CrossRefGoogle Scholar
[Wei65]Weil, A., Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87.CrossRefGoogle Scholar
[Yam]Yamana, S., Degenerate principal series representations for quaternionic unitary groups, Israel J. Math., to appear.Google Scholar
[Yam10]Yamana, S., On the Siegel–Weil formula for quaternionic unitary groups, Preprint (2010).Google Scholar