Hostname: page-component-857557d7f7-nhjpk Total loading time: 0 Render date: 2025-11-21T12:21:11.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Eisenstein Series Arising from Jordan Algebras

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  09 January 2019

Marcela Hanzer  and
Gordan Savin
Marcela Hanzer
Affiliation:
Department of Mathematics, University of Zagreb, Zagreb, Croatia Email: hanmar@math.hr
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Email: savin@math.utah.edu
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 describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Keywords

Eisenstein seriesJordan algebraFourier-Jacobi functor

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 1 , February 2020 , pp. 183 - 201
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author M. H. was supported in part by a Croatian Science Foundation grant no. 9364. Author G. S. was supported in part by an NSF grant DMS-1359774.

References

Aubert, A.-M., Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique . Trans. Amer. Math. Soc. 347(1995), 2179–2189. https://doi.org/10.2307/2154931.Google Scholar
Carter, R., Finite groups of Lie type: conjugacy classes and complex characters . Pure and Applied Mathematics, John Wiley & Sons, New York, 1985.Google Scholar
Gan, W. T. and Savin, G., On minimal representations definitions and properties . Represent. Theory 9(2005), 46–93. https://doi.org/10.1090/S1088-4165-05-00191-3.Google Scholar
Gindikin, S. G. and Karpelevich, F. I., On an integral connected with symmetric Riemann spaces of nonpositive curvature . Translations. Series 2. American Mathematical Society, Providence, RI, 1969, pp. 249–258. https://doi.org/10.1090/trans2/085/12.Google Scholar
Goldfeld, D. and Hundley, J., Automorphic representations and L-functions for the general linear group . Volume I. Cambridge Studies in Advanced Mathematics, 129, Cambridge University Press, Cambridge, 2011. https://doi.org/10.1017/CBO9780511973628.Google Scholar
Hanzer, M., Unitarizability of a certain class of irreducible representations of classical groups . Manuscripta Math. 127(2008), 275–307. https://doi.org/10.1007/s00229-008-0204-9.Google Scholar
Hanzer, M., The unitarizability of the Aubert dual of strongly positive square integrable representations . Israel J. Math. 169(2009), 251–294. https://doi.org/10.1007/s11856-009-0011-3.Google Scholar
Hanzer, M., Non-Siegel Eisenstein series for symplectic groups . Manuscripta Math. 155(2018), 229–302. https://doi.org/10.1007/s00229-017-0927-6.Google Scholar
Hanzer, M. and Muić, G., Degenerate Eisenstein series for Sp (4) . J. Number Theory 146(2015), 310–342. https://doi.org/10.1016/j.jnt.2013.11.002.Google Scholar
Hanzer, M. and Muić, G., On the images and poles of degenerate Eisenstein series for  $\mathit{GL}(n,\mathbb{A})$  and  $\mathit{GL}(n,\mathbb{R})$ . Amer. J. Math. 137(2015), 907–951. https://doi.org/10.1353/ajm.2015.0029.Google Scholar
Ikeda, T., On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series . J. Math. Kyoto Univ. 34(1994), 615–636. https://doi.org/10.1215/kjm/1250518935.Google Scholar
Jacobson, N., Structure and representations of Jordan algebras . American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, RI, 1968.Google Scholar
Kim, H. H., Exceptional modular form of weight 4 on an exceptional domain contained in C 27 . Rev. Mat. Iberoamericana 9(1993), 139–200. https://doi.org/10.4171/RMI/134.Google Scholar
Kobayashi, T. and Savin, G., Global uniqueness of small representations . Math. Z. 281(2015), 215–239. https://doi.org/10.1007/s00209-015-1481-0.Google Scholar
Kudla, S. S. and Rallis, S., On the Weil-Siegel formula . J. Reine Angew. Math. 387(1988), 1–68. https://doi.org/10.1515/crll.1988.391.65.Google Scholar
Kudla, S. S. and Rallis, S., A regularized Siegel-Weil formula: the first term identity . Ann. of Math. (2) 140(1994), 1–80. https://doi.org/10.2307/2118540.Google Scholar
Langlands, R. P., Euler products . Yale Mathematical Monographs, 1, Yale University Press, New Haven, Conn.-London, 1971.Google Scholar
Mœglin, C., Sur certains paquets d’Arthur et involution d’Aubert-Schneider-Stuhler généralisée . Represent. Theory 10(2006), 86–129. https://doi.org/10.1090/S1088-4165-06-00270-6.Google Scholar
Mœglin, C., Vignéras, M.-F., and Waldspurger, J.-L., Correspondances de Howe sur un corps p-adique . Lecture Notes in Mathematics, 1291, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/BFb0082712.Google Scholar
Sahi, S., Unitary representations on the Shilov boundary of a symmetric tube domain . In: Representation theory of groups and algebras , Contemp. Math., 145, American Mathematical Society, Providence, RI, 1993, pp. 275–286. https://doi.org/10.1090/conm/145/1216195.Google Scholar
Sahi, S., Jordan algebras and degenerate principal series . J. Reine Angew. Math. 462(1995), 1–18. https://doi.org/10.1515/crll.1995.462.1.Google Scholar
Weil, A., Sur certains groupes d’opérateurs unitaires . Acta Math. 111(1964), 143–211. https://doi.org/10.1007/BF02391012.Google Scholar
Weissman, M. H., The Fourier-Jacobi map and small representations . Represent. Theory 7(2003), 275–299. https://doi.org/10.1090/S1088-4165-03-00197-3.Google Scholar
Yamana, S., On the Siegel-Weil formula for quaternionic unitary groups . Amer. J. Math. 135(2013), 1383–1432. https://doi.org/10.1353/ajm.2013.0045.Google Scholar