On the Fourier inversion formula for the full modular group

Ouellette, Keith

doi:10.1090/S1088-4165-2011-00400-3

On the Fourier inversion formula for the full modular group
HTML articles powered by AMS MathViewer

by Keith R. Ouellette

Represent. Theory 15 (2011), 112-125

DOI: https://doi.org/10.1090/S1088-4165-2011-00400-3

Published electronically: February 7, 2011

PDF | Request permission

Abstract:

We offer a new proof of the Fourier inversion and Plancherel formulae for Maass-Eisenstein wave packets. The proof uses truncation, basic analysis, and classical Fourier theory. Brief sketches of the proofs due to Langlands, Lapid, and Casselman are then presented for comparison.

References

James G. Arthur, A (very brief) history of the trace formula, A note on http://www.claymath.org/cw/arthur/pdf/HistoryTraceFormula.pdf, 2007.
Armand Borel, Automorphic forms on $\textrm {SL}_2(\textbf {R})$, Cambridge Tracts in Mathematics, vol. 130, Cambridge University Press, Cambridge, 1997. MR 1482800, DOI 10.1017/CBO9780511896064
Robert S. Doran, Ze-Li Dou, and George T. Gilbert (eds.), Automorphic forms, automorphic representations, and arithmetic, Proceedings of Symposia in Pure Mathematics, vol. 66, American Mathematical Society, Providence, RI, 1999. MR 1703754, DOI 10.1090/pspum/066.2
Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
Erez Lapid, On Arthur’s asymptotic inner product formula of truncated Eisenstein series, to appear in Clay Mathematics Proceedings.
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183 (German). MR 31519, DOI 10.1007/BF01329622
Colette Mœglin and Jean-Loup Waldspurger, Décomposition spectrale et séries d’Eisenstein, Progress in Mathematics, vol. 113, Birkhäuser Verlag, Basel, 1994 (French, with English summary). Une paraphrase de l’Écriture. [A paraphrase of Scripture]. MR 1261867
W. Roelcke, Analytische Fortsetzung der Eisensteinreihen zu den parabolischen Spitzen von Grenzkreisgruppen erster Art, Math. Ann. 132 (1956), 121–129 (German). MR 82562, DOI 10.1007/BF01452322
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Atle Selberg, Discontinuous groups and harmonic analysis, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 177–189. MR 0176097
V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Cambridge Studies in Advanced Mathematics, vol. 16, Cambridge University Press, Cambridge, 1999. Corrected reprint of the 1989 original. MR 1725738