On inversion of the Bessel and Gelfand transforms
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- by Masaaki Furusawa and Joseph A. Shalika
- Trans. Amer. Math. Soc. 354 (2002), 837-852
- DOI: https://doi.org/10.1090/S0002-9947-01-02866-5
- Published electronically: July 12, 2001
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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
- Daniel Bump, Solomon Friedberg, and Masaaki Furusawa, Explicit formulas for the Waldspurger and Bessel models, Israel J. Math. 102 (1997), 125β177. MR 1489103, DOI 10.1007/BF02773797
- W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no.Β 3, 387β406. MR 571057
- W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no.Β 2, 207β231. MR 581582
- 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.
- I. G. Macdonald, Spherical functions on a group of $p$-adic type, Publications of the Ramanujan Institute, No. 2, University of Madras, Centre for Advanced Study in Mathematics, Ramanujan Institute, Madras, 1971. MR 0435301
- Mark E. Novodvorsky, New unique models of representations of unitary groups, Compositio Math. 33 (1976), no.Β 3, 289β295. MR 492090
- Yangbo Ye, The fundamental lemma of a relative trace formula for $\textrm {GL}(3)$, Compositio Math. 89 (1993), no.Β 2, 121β162. MR 1255692
- Yangbo Ye, An integral transform and its applications, Math. Ann. 300 (1994), no.Β 3, 405β417. MR 1304430, DOI 10.1007/BF01450494
Bibliographic Information
- Masaaki Furusawa
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3β3β138, Sumiyoshi-ku, OsakaΒ 558β8585, 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
- 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.Β 10640028, the Ministry of Education, Science, Sports and Culture of Japan
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 837-852
- MSC (2000): Primary 22E35; Secondary 11F70, 11F72
- DOI: https://doi.org/10.1090/S0002-9947-01-02866-5
- MathSciNet review: 1862570