Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T19:15:51.804Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases

Published online by Cambridge University Press:  20 November 2018

David Ginzburg  and
Erez Lapid
David Ginzburg
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Erez Lapid
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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.

We prove two spectral identities. The first one relates the relative trace formula for the spherical variety $\text{GSpin(4,}\,\text{3)/}{{G}_{2}}$ with a weighted trace formula for $G{{L}_{2}}$. The second relates a spherical variety pertaining to ${{F}_{4}}$ to one of $GSp\left( 6 \right)$. These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.

Keywords

11F7011F7211F3011F67
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1323 - 1340
Copyright
Copyright © Canadian Mathematical Society 2007

References

[Asg02] Asgari, M., Local L-functions for split spinor groups. Canad. J. Math. 54(2002), no. 4, 673–693.Google Scholar
[BG92] Bump, D. and Ginzburg, D., Spin L-functions on symplectic groups. Internat. Math. Res. Notices (1992), no. 8, 153–160.Google Scholar
[BG00] Bump, D. and Ginzburg, D., Spin L-functions on GSp8 and GSp10 . Trans. Amer. Math. Soc. 352(2000), no. 2, 875–899.Google Scholar
[Gin95a] Ginzburg, D., On spin L-functions for orthogonal groups. Duke Math. J. 77(1995), no. 3, 753–798.Google Scholar
[Gin95b] Ginzburg, D., On standard L-functions for E6 and E 7. J. Reine Angew.Math. 465(1995), 101–131.Google Scholar
[GJ00] Ginzburg, D. and Jiang, D., A Siegel-Weil identity for G2 and poles of L-functions. J. Number Theory 82(2000), no. 2, 256–287.Google Scholar
[GR94] Ginzburg, D. and Rallis, S., A tower of Rankin-Selberg integrals. Internat. Math. Res. Notices (1994), no. 5, p. 201 ff. (electronic).Google Scholar
[GR00] Ginzburg, D. and Rallis, S., The exterior cube L-function fo. GL(6). Compositio Math. 123(2000), no. 3, 243–272.Google Scholar
[GRS99] Ginzburg, D., Rallis, S., and Soudry, D., On a correspondence between cuspidal representations of GL2n and Sp2n . J. Amer. Math. Soc. 12(1999), no. 3, 849–907.Google Scholar
[GRS02] Ginzburg, D., Rallis, S., and Soudry, D., Endoscopic representations of Sp2n . J. Inst. Math. Jussieu 1(2002), no. 1, 77–123.Google Scholar
[Jia98] Jiang, D., G 2 -periods and residual representations. J. Reine Angew. Math. (1998), 17–46.Google Scholar
[JLR93] Jacquet, H., Lai, K. F., and Rallis, S., A trace formula for symmetric spaces. Duke Math. J. 70(1993), no. 2, 305–372.Google Scholar
[JLR04] Jacquet, H., Lapid, E., and Rallis, S., A spectral identity for skew symmetric matrices. In: Contributions to Automorphic Forms, Geometry, and Number Theory. Johns Hopkins University Press, Baltimore, MD, 2004, pp. 421–455.Google Scholar
[JS90] Jacquet, H. and Shalika, J., Exterior square L-functions. In: Automorphic Forms, Shimura Varieties, and L-Functions, Vol. II. Academic Press, Boston, MA, 1990, pp. 143–226.Google Scholar
[KR71] Kostant, B. and Rallis, S., Orbits and representations associated with symmetric spaces. Amer. J. Math. 93(1971), 753–809.Google Scholar
[LR04] Lapid, E. and Rallis, S., A spectral identity between symmetric spaces. Israel J. Math. 140(2004), 221–244.Google Scholar
[PSRS92] Piatetski-Shapiro, I., Rallis, S., and Schiffmann, G., Rankin-Selberg integrals for the group G2 . Amer. J. Math. 114(1992), no. 6, 1269–1315.Google Scholar
[Vo97] Vo, S. C., The spin L-function on the symplectic grou. GSp(6). Israel J. Math. 101(1997), 1–71.Google Scholar