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Eisenstein Series Arising from Jordan Algebras

Part of: Lie groups Discontinuous groups and automorphic forms

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: [email protected]
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Email: [email protected]
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Abstract

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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
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