Proceedings of the American Mathematical Society

Author(s): Reid C. Huntsinger
Journal: Proc. Amer. Math. Soc. 124 (1996), 2229-2234.
MSC (1991): Primary 22E50
Retrieve article in: PDF
This article is available free of charge

Abstract: Harish-Chandra's formula for the character $\Theta _\pi$ of an irreducible smooth representation $\pi$ of a reductive $p$

-adic group $G$

expresses $\Theta _\pi$ near $1$

as a linear combination of the Fourier transforms of nilpotent $G$

-orbits in the Lie algebra of $G$

. In this note, we prove that if $\pi$ is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.

1.: M. Assem, The Fourier transform and some character formulæ for -adic $\text {SL}_\ell$ , $\ell$ a prime, Amer. J. Math., to appear. CMP 95:05
2.: L. Clozel, Invariant harmonic analysis on the Schwartz space of a reductive -adic group, Harmonic Analysis on Reductive Groups, Bowdoin 1989 (W. Barker & P. Sally, eds.), Progress in Math., vol. 101, Birkhäuser, Boston, 1991, pp. 101--121. MR 93h:22020
3.: Harish-Chandra, Harmonic analysis on reductive -adic groups, Lecture Notes in Math., vol. 162 (Notes by G. Van Dijk), Springer-Verlag, Berlin, Heidelberg and New York, 1970. MR 54:2889
4.: ------, Admissible invariant distributions on reductive -adic groups, Queen's Papers in Pure and Applied Mathematics, vol. 48, Kingston, Ontario, 1978, pp. 281--347. MR 58:28313
5.: ------, The Plancherel formula for reductive -adic groups, Collected Works, vol. IV (ed. V.S. Varadarajan), Springer-Verlag, Berlin, Heidelberg and New York, 1984, pp. 353--367.
6.: R. Herb, Elliptic representations for $\mbox {\rm Sp}(2n)$ and $\mbox {\rm SO}(n)$ , Pacific J. Math. 161 (1993) 347--358. MR 94i:22040
7.: R. Howe, Fourier transform and germs of characters: case of $\text {GL}_n$ over a -adic field, Math. Ann. 208 (1974), 305--322. MR 49:7391
8.: D. Kazhdan, Cuspidal geometry of -adic groups, J. Analyse Math. 47 (1986), 1--36. MR 88g:22017
9.: C. Moeglin and J.L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes -adiques, Math. Z. 196 (1987), 427--452. MR 89f:22024
10.: F. Murnaghan, Asymptotic behavior of supercuspidal characters of -adic $\text {\rm GL}(3)$ and $\text {GL}(4)$ : the generic unramified case, Pacific J. Math. 148 (1991), 107--130. MR 92a:22023
11.: ------, Asymptotic behavior of supercuspidal characters of -adic $\mbox {\rm GSp}(4)$ , Comp. Math. 80 (1991), 15--54. MR 92f:22016
12.: ------, Characters of supercuspidal representations of classical groups, preprint.
13.: Ranga Rao, Orbital integrals on reductive groups, Ann. of Math. 96 (1972), 505--510.
14.: J.D. Rogawski, An application of the building to orbital integrals, Comp. Math. 42 (1981), 417--423. MR 83g:22011
15.: P.J. Sally, Jr., Some remarks on discrete series characters for reductive -adic groups, Representations of Lie Groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math. vol. 14, North-Holland, Amsterdam and New York, 1988, pp. 337--348. MR 91g:22026
16.: J.A. Shalika, A theorem on semisimple -adic groups, Ann. of Math. 95 (1972), 226--242. MR 48:2310
17.: M. Tadic, Notes on representations of non-archimedean $\text {SL}(n)$ , Pacific J. Math. 152 (1992), 375--396. MR 92k:22029
18.: G. Van Dijk, Computation of certain induced characters of -adic groups, Math. Ann. 199 (1972), 229--240.

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 22E50

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS