Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On inversion of the Bessel and Gelfand transforms


Authors: Masaaki Furusawa and Joseph A. Shalika
Journal: Trans. Amer. Math. Soc. 354 (2002), 837-852
MSC (2000): Primary 22E35; Secondary 11F70, 11F72
Published electronically: July 12, 2001
MathSciNet review: 1862570
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of $\text{odd degree}\ge 5$ and the Whittaker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.


References [Enhancements On Off] (What's this?)

  • 1. Daniel Bump, Solomon Friedberg, and Masaaki Furusawa, Explicit formulas for the Waldspurger and Bessel models, Israel J. Math. 102 (1997), 125–177. MR 1489103, 10.1007/BF02773797
  • 2. W. Casselman, The unramified principal series of 𝔭-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
  • 3. W. Casselman and J. Shalika, The unramified principal series of 𝑝-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
  • 4. M. Furusawa and J. A. Shalika, On central critical values for the degree four $L$-functions for ${\mathrm{GSp}}\left(4\right)$: The fundamental lemma, preprint.
  • 5. I. G. Macdonald, Spherical functions on a group of 𝑝-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. MR 0435301
  • 6. Mark E. Novodvorsky, New unique models of representations of unitary groups, Compositio Math. 33 (1976), no. 3, 289–295. MR 0492090
  • 7. Yangbo Ye, The fundamental lemma of a relative trace formula for 𝐺𝐿(3), Compositio Math. 89 (1993), no. 2, 121–162. MR 1255692
  • 8. Yangbo Ye, An integral transform and its applications, Math. Ann. 300 (1994), no. 3, 405–417. MR 1304430, 10.1007/BF01450494

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E35, 11F70, 11F72

Retrieve articles in all journals with MSC (2000): 22E35, 11F70, 11F72


Additional Information

Masaaki Furusawa
Affiliation: Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3–3–138, Sumiyoshi-ku, Osaka 558–8585, Japan
Email: furusawa@sci.osaka-cu.ac.jp

Joseph A. Shalika
Affiliation: Department of Mathematics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218
Email: shalika@math.jhu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02866-5
Received by editor(s): August 7, 2000
Received by editor(s) in revised form: February 26, 2001
Published electronically: July 12, 2001
Additional Notes: The research of Furusawa was partially supported by the Sumitomo Foundation and the Grant-in-Aid for Scientific Research No. 10640028, the Ministry of Education, Science, Sports and Culture of Japan
Article copyright: © Copyright 2001 American Mathematical Society