Abstract
Automorphic forms of arbitrary real weight can be considered as functions on the universal covering group of SL(2, ℝ). In this situation, we prove an invariant form of the Selberg trace formula for Hecke operators. For this purpose, the Fourier transforms of weightet orbital integrals, obtained by J. Arthur, R. Herb and P. Sally, jr., are explicitly calculated. Our formula does not follow from Arthur's invariant trace formula, since the group has infinite centre, and vector-valued automorphic forms with respect to non-congruence lattices are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akiyama, S.: Selberg trace formula for odd weight.Proc. Japan Acad. 64 (1988), no. 9, Ser. A, 341–344.
Arthur, J.: Some tempered distributions on semisimple Lie groups of real rank one.Ann. Math. 100 (1974), 553–584.
Arthur, J.: The characters of discrete series as orbital integrals.Inv. Math. 32 (1976), 205–261.
Arthur, J.: The trace formula in invariant form.Ann. Math. 114 (1981), 1–74.
Arthur, J.;Herb, R.A.;Sally, jr., P.J.: The Fourier transform of weighted orbital integrals on SL(2, ℝ). In:The Selberg trace formula and related topics. Proceedings of a Summer Research conference held at Bowdoin College in July 22–28, 1984; D. A. Hejhal, P. Sarnak, A. A. Terras (eds.) Contemporary mathematics, vol. 53, Amer. Math. Soc., Providence, RI, 1986, 17–37.
Borel, A.: Density properties of certain subgroups of semisimple groups.Ann. Math. 72 (1960), 179–188.
Colin De Verdière, Y.: Une nouvelle démonstration du prolongement méromorphe de séries d'Eisenstein.C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 361–363.
Colin de Verdière, Y: Pseudo-Laplaciens II.Ann. Inst. Fourier 33 (1983), 87–113.
Elstrodt, J.: Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil I.Math. Annalen 203 (1973), 295–330. Teil II.Math. Zeitschrift 132 (1973), 99–134. Teil III.Math. Annalen 208 (1974), 99–132.
Fischer, J.:An approach to the Selberg trace formula via the Selberg zeta function. Lecture Notes in Math., Vol. 1253, Springer-Verlag, Berlin Heidelberg New York 1987.
Gelfand, I.M.;Graev, M.I.;Piatetski-Shapiro, I.I.:Representation theory and automorphic functions. Saunders, Philadelphia 1969.
Harish-Chandra: Discrete series for semisimple Lie groups. II.Acta Math. 116 (1966), 1–111.
Harish-Chandra: Plancherel formula for the 2×2 real unimodular group.Proc. Nat. Acad. Sci. USA 38 (1952), 337–342.
Harish-Chandra: The characters of semisimple Lie groups.Trans. Amer. Math. Soc. 83 (1956), 98–163.
Harish-Chandra: Harmonic analysis on real reductive groups III. The Maas-Selberg relations and the Plancherel formula.Ann. Math. 104 (1976), 117–201.
Hejhal, D.A.:The Selberg trace formula for PSL(2, ℝ), Vol. 2. Lecture Notes in Math., vol. 1001, Springer-Verlag, Berlin Heidelberg New York 1983.
Hoffmann, W.: The trace formula for Hecke operators over rank one lattices.J. Funct. Anal. 84 (1989), 373–440.
Jacquet, H.;Langlands, R.P.:Automorphic forms on GL(2). Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin Heidelberg New York 1970.
Kubota, T.:Elementary theory of Eisenstein series. Kodansha, Tokyo 1973.
Lang, S.: SL(2, ℝ). Addison-Wesley, Reading 1975.
Molchanov, V.: Plancherel formula for pseudo Riemannian symmetric spaces of the universal covering group of SL(2, ℝ).Siberian Math. J. 25 (1984), 903–917.
Osborne, M.S., Warner, G.:The theory of Eisenstein systems. Academic Press, New York 1981.
Osborne, M.S., Warner, G.: The Selberg trace formula I: Γ-rank one lattices.J. Reine Angew. Math. 324 (1981), 1–113.
Petersson, H.: Zur analytischen Theorie der Grenzkreisgruppen. Teil I.Math. Annalen 115 (1938), 23–67. Teil II.Math. Zeitschrift 44 (1939), 127–155.
Pukanszky, L.: The Plancherel formula for the universal covering group of SL(2, ℝ).Math. Ann. 156 (1964), 96–143.
Raghunathan, M.S.:Discrete subgroups of Lie Groups. Springer-Verlag, Berlin Heidelberg New York 1972.
Roelcke, W.: Das Eigenwertproblem der automorophen Formen in der hyperbolischen Ebene I.Math. Ann. 167 (1966), 292–337. II.Math. Ann. 168 (1967), 261–324.
Sally, P.: Analytic continuation of the irreducible unitary representations of the universal covering group of SL(2, ℝ).Mem. Amer. Math. Soc. 69 (1967), 1–94.
Sally, P.: Intertwining operators and the representations of SL(2, ℝ).J. Funct. Anal. 6 (1960), 441–453.
Varadarajan, V.S.:An introduction to harmonic analysis on semisimple Lie groups. Cambridge studies in advanced mathematics, vol. 16, Cambridge Univ. Press, Cambridge 1989.
Warner, G.: Application of the Selberg trace formula to the Riemann-Roch theorem.J. Differential Geom. 27 (1988), 23–24.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hoffmann, W. An invariant trace formula for the universal covering group of SL(2,ℝ). Ann Glob Anal Geom 12, 19–63 (1994). https://doi.org/10.1007/BF02108286
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02108286