Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T20:44:43.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral decomposition of compactly supported Poincaré series and existence of cusp forms

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 January 2010

Goran Muić
Goran Muić*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we address the issue of existence of cusp forms by using an extension and refinement of a classic method involving (adelic) compactly supported Poincaré series. As a consequence of our adelic approach, we also deal with cusp forms for congruence subgroups.

Keywords

cuspidal automorphic formsPoincaré series

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 1 - 20
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Arthur, J., An introduction to the trace formula, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 1–263.Google Scholar
[2]Bernstein, J., Representations of p-adic groups, Draft of lectures at Harvard University, written by Karl E. Rumelhart (1992).Google Scholar
[3]Bernstein, J. and Zelevinsky, A. V., Representations of the group GL(n,F), where F is a local non-Archimedean field, Uspekhi Mat. Nauk 31 (1976), 5–70 (in Russian).Google Scholar
[4]Bernstein, I. N. and Zelevinsky, A. V., Induced representations of reductive p-adic groups I, Ann. Sci. École Norm. Sup. 10 (1977), 441–472.CrossRefGoogle Scholar
[5]Borel, A., Some finiteness properties of adele groups over number fields, Publ. Math. Inst. Hautes Études Sci. 16 (1963), 5–30.CrossRefGoogle Scholar
[6]Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, in Automorphic Forms, Representations and L-Functions (Oregon State University, Corvallis, OR, 1977), Part 1, Proceedings of Symposia in Pure Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 1979), 189–202.CrossRefGoogle Scholar
[7]Bushnell, C. and Kutzko, P., Smooth representations of reductive p-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), 582–634.CrossRefGoogle Scholar
[8]Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups, Preprint, http://www.math.ubc.ca/cass/research/pdf/p-adic-book.pdf.Google Scholar
[9]Godement, R., The spectral decomposition of cusp forms, in Algebraic groups and discontinuous subgroups (Boulder, Colorado, 1965), Part 3, Proceedings of Symposia in Pure Mathematics, vol. 9 (American Mathematical Society, Providence, RI, 1966), 225–234.CrossRefGoogle Scholar
[10]Goldfeld, D., Automorphic forms and L-functions for the group GL (n,ℝ), Cambridge Studies in Advanced Mathematics, vol. 99 (Cambridge University Press, Cambridge, 2006).Google Scholar
[11]Henniart, G., La conjecture de Langlands locale pour GL(3), Mém. Soc. Math. Fr. (N.S.) 11–12 (1984), 1–186.Google Scholar
[12]Knapp, A. W., Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series, vol. 36 (Princeton University Press, Princeton, NJ, 1986).CrossRefGoogle Scholar
[13]Li, J.-L. and Schwermer, J., On the cuspidal cohomology of arithmetic groups, Amer. J. Math. (to appear).Google Scholar
[14]Moy, A. and Prasad, G., Jacquet functors and unrefined minimal K-types, Comment. Math. Helv. 71 (1996), 98–121.CrossRefGoogle Scholar
[15]Muić, G., On the decomposition of L 2(Γ∖G) in the cocompact case, J. Lie Theory 18 (2008), 937–949.Google Scholar
[16]Muić, G., On a construction of certain classes of cuspidal automorphic forms via Poincaré series, Math. Ann. 343 (2009), 207–227.CrossRefGoogle Scholar
[17]Muić, G., On the cusp forms for the congruence subgroups of SL 2(ℝ), Ramanujan J. (to appear), DOI: 10.1007/s11139-009-9191/-z.CrossRefGoogle Scholar
[18]Müller, W., Weyl’s law in the theory of automorphic forms, in Groups and analysis: the legacy of Hermann Weyl (Cambridge University Press, Cambridge, 2008).Google Scholar
[19]Selberg, A., 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.Google Scholar
[20]Shahidi, F., A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), 273–330.CrossRefGoogle Scholar
[21]Vignéras, M.-F., Correspondances entre representations automorphe de GL(2) sur une extension quadratique de GSp(4) sur ℚ, conjecture locale de Langlands pour GSp(4), Contemp. Math. 53 (1986), 463–527.CrossRefGoogle Scholar