Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



On the Fourier inversion formula for the full modular group

Author: Keith R. Ouellette
Journal: Represent. Theory 15 (2011), 112-125
MSC (2010): Primary 22E45; Secondary 11F72
Published electronically: February 7, 2011
MathSciNet review: 2772585
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. James G. Arthur, A (very brief) history of the trace formula, A note on, 2007.
  • 2. Armand Borel, Automorphic forms on $ {\rm SL}\sb 2({\bf R})$, Cambridge Tracts in Mathematics, vol. 130, Cambridge University Press, Cambridge, 1997. MR 1482800 (98j:11028)
  • 3. William A. Casselman, On the plancherel measure for the continuous spectrum of the modular group, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 19-25. MR 1703754 (2000c:11003)
  • 4. Gerald B. Folland, Real analysis, second 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 (2000c:00001)
  • 5. Tomio Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo, 1973. MR 0429749 (55:2759)
  • 6. 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 (40:2784)
  • 7. Erez Lapid, On Arthur's asymptotic inner product formula of truncated Eisenstein series, to appear in Clay Mathematics Proceedings.
  • 8. Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141-183. MR 0031519 (11,163c)
  • 9. Colette Mœglin and Jean-Loup Waldspurger, Décomposition spectrale et séries d'Eisenstein, Progress in Mathematics, vol. 113, Birkhäuser Verlag, Basel, 1994, Une paraphrase de l'Écriture. [A paraphrase of Scripture]. MR 1261867 (95d:11067)
  • 10. W. Roelcke, Analytische Fortsetzung der Eisensteinreihen zu den parabolischen Spitzen von Grenzkreisgruppen erster Art, Math. Ann. 132 (1956), 121-129. MR 0082562 (18:571b)
  • 11. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York, 1973, McGraw-Hill Series in Higher Mathematics. MR 0365062 (51:1315)
  • 12. Atle 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 0088511 (19:531g)
  • 13. -, Discontinuous groups and harmonic analysis, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 177-189. MR 0176097 (31:372)
  • 14. 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 (2000m:22011)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 22E45, 11F72

Retrieve articles in all journals with MSC (2010): 22E45, 11F72

Additional Information

Keith R. Ouellette
Affiliation: Department of Mathematics, College of the Holy Cross, Worcester, Massachusetts 01610

Received by editor(s): October 21, 2006
Received by editor(s) in revised form: December 10, 2010
Published electronically: February 7, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society