Abstract
The purpose of this article is to discuss some questions in the harmonic analysis of real and p-adic groups. We shall be particularly concerned with the properties of a certain family of invariant distributions. These distributions arose naturally in a global context, as the terms on the geometric side of the trace formula. However, they are purely local objects, which include the ordinary invariant orbital integrals. One of our aims is to describe how the distributions also arise in a local context. They appear as the terms on the geometric side of a new trace formula, which is simpler than the original one, and is the solution of a natural question in local harmonic analysis. The local trace formula seems to be a promising tool. It might have implications for the difficult local problems which are holding up progress in automorphic forms.
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References
J. Arthur, A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4718–4719.
J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.
J. Arthur, The characters of supercuspidal representations as weighted orbital integrals, Proc. Indian Acad. Sci. 97 (1987), 3–19.
J. Arthur, The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323–383.
J. Arthur, The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501–554.
J. Arthur, Intertwining operators and residues I. Weighted characters, J. Funct. Anal. 84 (1989), 19–84.
J. Arthur, Intertwining operators and residues II. Invariant distributions, Compos. Math. 70 (1989), 51–99
J. Arthur, Unipotent automorphic representations: Conjectures, Astérisque 171-172 (1989), 13–71.
J. Arthur, Towards a local trace formula, Amer. J. Math, (to appear).
J. Arthur, A local trace formula, preprint.
J. Arthur and L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Annals of Math. Studies, Vol. 120, Princeton University Press, Princeton, 1989.
J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math. 47 (1986), 180–192.
L. Clozel, Orbital integrals on p-adic groups: A proof of the Howe conjecture, Ann. of Math. 129 (1989), 237–251.
L. Clozel and P. Delorme, Le théoreme de Paley-Wiener invariant pour les groupes de Lie réductifs II, Ann. Scient. Ec. Norm. Sup. (to appear).
Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. 104 (1976), 117–201, reprinted in Collected Works, Springer-Verlag, Vol. IV, 259-343.
Harish-Chandra, The Plancherel formula for reductive p-adic groups, reproduced in Collected Works, Springer-Verlag, Vol. IV, pp. 353-367.
R. Herb, Discrete series characters and Fourier inversion on semisimple real Lie groups, Trans. Amer. Math. Soc. 277 (1983), 241–262.
D. Kazhdan, Cuspidal geometry on p-adic groups, J. Analyse Math. 47 (1989), 1–36.
R. Kottwitz and J. Rogawski, The distributions in the invariant trace formula are supported on characters, preprint.
R. Langlands, Base Change for GL(2), Annals of Math. Studies 96 (1980), Princeton University Press, Princeton.
R. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.
P. Mischenko, Invariant Tempered Distributions on the Reductive p-Adic Group GL n (F p ), Ph.D. Thesis, University of Toronto, Toronto, 1982.
J. Rogawski, The trace Paley-Wiener theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), 215-229.
J.-L. Waldspurger, Intégrates orbitales spheriques pour GL(N), Asterisque 171—172 (1989), 270–337.
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© 1991 Springer Science+Business Media New York
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Arthur, J. (1991). Some Problems in Local Harmonic Analysis. In: Barker, W.H., Sally, P.J. (eds) Harmonic Analysis on Reductive Groups. Progress in Mathematics, vol 101. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0455-8_3
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DOI: https://doi.org/10.1007/978-1-4612-0455-8_3
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