Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T21:56:08.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Orbital integrals on $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$

Part of: Lie groups

Published online by Cambridge University Press:  26 February 2021

Hang Xue
Hang Xue*
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, AZ85721, USA
*
e-mail: xuehang@math.arizona.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$ . We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.

Keywords

orbital integralspherical characterssymmetric spaces

MSC classification

Primary: 22E35: Analysis on $p$-adic Lie groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 858 - 886
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work is partially supported by the NSF grant DMS #1901862.

References

Avraham, A. and Gourevitch, D., Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem . Duke Math. J. 149(2009), no. 3, 509–567. http://doi.org/10.1215/00127094-2009-044 With an appendix by the authors and Eitan Sayag.Google Scholar
Friedberg, S. and Jacquet, H., Linear periods . J. Reine Angew. Math. 443(1993), 91–139. http://doi.org/10.1515/crll.1993.443.91 Google Scholar
Godement, R. and Jacquet, H., Zeta functions of simple algebras . Lecture Notes in Mathematics, 260, Springer-Verlag, Berlin, Germany-New York, NY, 1972.Google Scholar
Guo, J., On a generalization of a result of Waldspurger . Can. J. Math. 48(1996), no. 1, 105–142. http://doi.org/10.4153/CJM-1996-005-3 CrossRefGoogle Scholar
Guo, J., Spherical characters on certain $p$ -adic symmetric spaces. MPIM Preprint, 1998. https://www.mpim-bonn.mpg.de/de/preprints?year=&number=&name=guo%2C+J&title=Google Scholar
Hakim, J., Admissible distributions on $p$ -adic symmetric spaces . J. Reine Angew. Math. 455(1994), 1–19. http://doi.org/10.1515/crll.1994.455.1 Google Scholar
Howe, R., The Fourier transform and germs of characters (case of Gln over a p-adic field) . Math. Ann. 208(1974), 305–322. http://doi.org/10.1007/BF01432155 CrossRefGoogle Scholar
Jacquet, H. and Rallis, S., Uniqueness of linear periods . Compos. Math. 102(1996), no. 1, 65–123.Google Scholar
Kottwitz, R. E., Harmonic analysis on reductive p-adic groups and Lie algebras . In: Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., 4, American Mathematical Society, Providence, RI, 2005, pp. 393–522.Google Scholar
Kraft, H. and Procesi, C., Closures of conjugacy classes of matrices are normal . Invent. Math. 53(1979), no. 3, 227–247. http://doi.org/10.1007/BF01389764 CrossRefGoogle Scholar
Laurent, C., Invariant harmonic analysis on the Schwartz space of a reductive p-adic group . In: Harmonic analysis on reductive groups (Brunswick, ME, 1989), Progr. Math., 101, Birkhäuser, Boston, MA, 1991, pp. 101–121, http://doi.org/10.1007/978-1-4612-0455-8_6 Google Scholar
Rader, C. and Rallis, S., Spherical characters on $p$ -adic symmetric spaces . Amer. J. Math. 118(1996), no. 1, 91–178.Google Scholar
Zhang, C., On the smooth transfer for Guo-Jacquet relative trace formulae . Compos. Math. 151(2015), no. 10, 1821–1877.CrossRefGoogle Scholar