Vanishing of the leading term in Harish-Chandra’s local character expansion

Huntsinger, Reid

doi:10.1090/S0002-9939-96-03183-8

Vanishing of the leading term
in Harish-Chandra's local character expansion

Author: Reid C. Huntsinger
Journal: Proc. Amer. Math. Soc. 124 (1996), 2229-2234
MSC (1991): Primary 22E50
DOI: https://doi.org/10.1090/S0002-9939-96-03183-8
MathSciNet review: 1307530
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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. Laurent Clozel, Invariant harmonic analysis on the Schwartz space of a reductive 𝑝-adic group, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 101–121. MR 1168480, https://doi.org/10.1007/978-1-4612-0455-8_6
3. Harish-Chandra, Harmonic analysis on reductive 𝑝-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. MR 0414797
4. Harish-Chandra, Admissible invariant distributions on reductive 𝑝-adic groups, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977) Queen’s Univ., Kingston, Ont., 1978, pp. 281–347. Queen’s Papers in Pure Appl. Math., No. 48. MR 0579175
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. Rebecca A. Herb, Elliptic representations for 𝑆𝑝(2𝑛) and 𝑆𝑂(𝑛), Pacific J. Math. 161 (1993), no. 2, 347–358. MR 1242203
7. Roger Howe, The Fourier transform and germs of characters (case of 𝐺𝑙_{𝑛} over a 𝑝-adic field), Math. Ann. 208 (1974), 305–322. MR 342645, https://doi.org/10.1007/BF01432155
8. David Kazhdan, Cuspidal geometry of 𝑝-adic groups, J. Analyse Math. 47 (1986), 1–36. MR 874042, https://doi.org/10.1007/BF02792530
9. C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes 𝑝-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, https://doi.org/10.1007/BF01200363
10. Fiona Murnaghan, Asymptotic behaviour of supercuspidal characters of 𝑝-adic 𝐺𝐿₃ and 𝐺𝐿₄: the generic unramified case, Pacific J. Math. 148 (1991), no. 1, 107–130. MR 1091533
11. Fiona Murnaghan, Asymptotic behaviour of supercuspidal characters of 𝑝-adic 𝐺𝑆𝑝(4), Compositio Math. 80 (1991), no. 1, 15–54. MR 1127058
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. Jonathan D. Rogawski, An application of the building to orbital integrals, Compositio Math. 42 (1980/81), no. 3, 417–423. MR 607380
15. Paul J. Sally Jr., Some remarks on discrete series characters for reductive 𝑝-adic groups, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 337–348. MR 1039842, https://doi.org/10.2969/aspm/01410337
16. J. A. Shalika, A theorem on semi-simple \cal𝑃-adic groups, Ann. of Math. (2) 95 (1972), 226–242. MR 323957, https://doi.org/10.2307/1970797
17. Marko Tadić, Notes on representations of non-Archimedean 𝑆𝐿(𝑛), Pacific J. Math. 152 (1992), no. 2, 375–396. MR 1141803
18. G. Van Dijk, Computation of certain induced characters of -adic groups, Math. Ann. 199 (1972), 229--240.