On Bessel distributions for quasi-split groups

Author:
Ehud Moshe Baruch

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2601-2614

MSC (2000):
Primary 22E50; Secondary 11F70

DOI:
https://doi.org/10.1090/S0002-9947-01-02778-7

Published electronically:
March 8, 2001

MathSciNet review:
1828462

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We show that the Bessel distribution attached by Gelfand and Kazhdan and by Shalika to a generic representation of a quasi-split reductive group over a local field is given by a function when it is restricted to the open Bruhat cell. As in the case of the character distribution, this function is real analytic for archimedean fields and locally constant for non-archimidean fields.

**[1]**J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Math. Studies 120, Princeton University Press, 1989. MR**90m:22041****[2]**E.M. Baruch, On Bessel distributions for over a -adic field, J. Number Theory, Vol. 67, No. 2 (1997), 190-202. MR**98k:22042****[3]**E.M. Baruch, Bessel functions for simply laced -adic groups, preprint.**[4]**E.M. Baruch and Z. Mao, Bessel identities and the Waldspurger correspondence, preprint.**[5]**C.J. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of , J. Reine Angew. Math. 375/376(1987), 184-210. MR**88e:22024****[6]**J. Cogdell and I. Piatetski-Shapiro, The Arithmetic and Spectral Analysis of Poincaré Series, Academic Press, 1990. MR**91h:11042****[7]**J. Cogdell and I. Piatetski-Shapiro, Stability of gamma factors for , Manuscripta Math. 95(1998), no. 4, 437-461. MR**99d:11053****[8]**J. Dixmier, Enveloping Algebra, Grad. Stud. in Math., Vol. II, AMS, 1996. Originally published: Amsterdam: North-Holland Pub. Co. 1977. MR**97c:17010**; MR**58:16803b****[9]**S. Gelbart and I. Piatetski-Shapiro, Distinguished representations and modular forms of half integral weight, Inventiones Math. 59 (1980) 145-188. MR**82b:10035****[10]**I.M. Gelfand and M.I. Graev, Construction of irreducible representations of simple algebraic groups over a finite field, Dokl. Akad. Nauk SSSR 147, 3(1962), 529-532; Soviet Math. Dokl. 3(1962), 1646-1649. MR**26:6271****[11]**I.M. Gelfand and D. Kazhdan, Representations of the group where is a local field, Lie groups and their representations, Halsted, New York (1975), 95-118. MR**53:8334****[12]**J. Hakim, Admissible distributions on -adic symmetric spaces, J. Reine Angew. Math. 455(1994), 1-19. MR**96m:22019****[13]**Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119(1965), 457-508. MR**31:4862d****[14]**Harish-Chandra, Admissible invariant distributions on reductive -adic groups, Proceedings of the 1977 annual seminar of the Canadian Mathematical Congress, Queen's papers in Pure and Applied Math., No. 48. Kingston, Ontario. MR**58:28313****[15]**R. Howe and A. Moy, Minimal -types for over a -adic field, Asterisque, 171-172(1989), 257-271. MR**90m:22040****[16]**H. Jacquet, On the nonvanishing of some -functions, Proc. Indian Acad. Sci. Math. Sci. 97(1987), no. 1-3, 117-155. MR**90e:11079****[17]**H. Jacquet, K.F. Lai and S. Rallis, A trace formula for symmetric spaces, Duke Math. J. 70(1993), no. 2, 305-372. MR**94d:11033****[18]**N. N. Lebedev, Special functions and their applications, Dover, 1972. MR**50:2568****[19]**L. Morris, -cuspidal representations, Proc. London Math. Soc. (3) 57(1988), 329-356. MR**89j:22038****[20]**L. Morris, -cuspidal representations of level one, Proc. London Math. Soc. (3) 58(1989), 550-558. MR**90c:22056****[21]**A. Moy and G. Prasad, Jacquet functors and unrefined minimal -types, Comment. Math. Helevtici, 71(1996) 98-121. MR**97c:22021****[22]**C. Rader and S. Rallis, Spherical Characters on -adic symmetric spaces, American J. of Math. Vol. 118, Num. 1 (1996) 91-178. MR**97c:22013****[23]**J.A. Shalika, The multiplicity one theorem for , Ann. of Math. 100(1974) 171-193. MR**50:545****[24]**D. Soudry, The and factors for generic representations of over a local non-Archimedean field . Duke Math. Journal, vol. 51 No. 2, 355-394, 1984. MR**86f:22022**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
22E50,
11F70

Retrieve articles in all journals with MSC (2000): 22E50, 11F70

Additional Information

**Ehud Moshe Baruch**

Affiliation:
Theoretical Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Address at time of publication:
Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064

Email:
baruch@math.ucsc.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02778-7

Keywords:
Bessel distributions,
Bessel functions

Received by editor(s):
November 14, 1998

Received by editor(s) in revised form:
July 7, 1999

Published electronically:
March 8, 2001

Article copyright:
© Copyright 2001
American Mathematical Society