Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the structure of the space of cusp forms for a semisimple group over a number field


Author: Goran Muic
Journal: Proc. Amer. Math. Soc. 138 (2010), 3147-3158
MSC (2010): Primary 11F70
DOI: https://doi.org/10.1090/S0002-9939-10-10375-X
Published electronically: April 9, 2010
MathSciNet review: 2653939
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a semisimple algebraic group defined over a number field $ k$. We study unramified irreducible components of irreducible automorphic cuspidal representations in the space of cusp forms $ \mathcal{A}_{cusp}(G(k)\setminus G(\mathbb{A}))$ using the action of an unramified Hecke algebra on compactly supported cuspidal Poincaré series.


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

  • 1. J. Arthur, An introduction to the trace formula, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005. MR 2192011 (2007d:11058)
  • 2. A. Borel, Some finiteness properties of adele groups over number fields, Publ. Math. Inst. Hautes Études Sci. 16 (1963), 5-30. MR 0202718 (34:2578)
  • 3. A. Borel, H. Jacquet, Automorphic forms and automorphic representations, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, RI, 1979, 189-207. MR 546598 (81m:10055)
  • 4. P. Cartier, Representations of $ p$-adic groups: a survey. Automorphic forms, representations and $ L$-functions, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, RI, 1979, 111-155. MR 546593 (81e:22029)
  • 5. R. Godement, The spectral decomposition of cusp forms, Part 3, Proc. Sympos. Pure Math. IX, Amer. Math. Soc., Providence, RI, 1966, 225-234. MR 0210828 (35:1713)
  • 6. D. Goldfeld, Automorphic forms and $ L$-functions for the group $ \operatorname{GL}(n, \mathbb{R})$, Cambridge Studies in Advanced Mathematics 99, Cambridge Univ. Press, 2006. MR 2254662 (2008d:11046)
  • 7. G. Henniart, La conjecture de Langlands locale pour $ \operatorname{GL}(3)$, Mémories de la Société Mathématique de France, no. 11-12 (1984). MR 743063 (86g:11070)
  • 8. A. Moy, Lecture notes delivered in Beijing (http://www.math.ust.hk/$ \sim$amoy/beirun.pdf).
  • 9. A. Moy, G. Prasad, Jacquet functors and unrefined minimal $ K$-types, Comment. Math. Helv. 71 (1996), 98-121. MR 1371680 (97c:22021)
  • 10. G. Muić, On a construction of certain classes of cuspidal automorphic forms via Poincaré series, Math. Ann. 343 (2009), 207-227. MR 2448445 (2010a:11092)
  • 11. G. Muić, Spectral decomposition of compactly supported Poincaré series and existence of cusp forms, Compositio Math. 146, no. 1 (2010), 1-20.
  • 12. G. Muić, On the cusp forms for the congruence subgroups of $ SL_2(\mathbb{R})$, Ramanujan J. 21, no. 2 (2010), 223-239.
  • 13. G. Muić, On geometric density of Hecke eigenvalues for certain cusp forms, Math. Ann. 347, no. 2 (2010), 479-498.
  • 14. J.-S. Li, J. Schwermer, On the cuspidal cohomology of arithmetic groups, Amer. J. Math. 131 (2009), 1431-1464. MR 2559860
  • 15. F. Shahidi, A proof of Langlands' conjecture on Plancherel measures; complementary series for $ p$-adic groups, Annals of Math. (2) 132 (1990), 273-330. MR 1070599 (91m:11095)
  • 16. J. Tits, Reductive groups over local fields. Automorphic forms, representations and $ L$-functions, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, RI, 1979, 29-69. MR 546588 (80h:20064)
  • 17. M.-F. Vignéras, Correspondances entre representations automorphe de $ \operatorname{GL}(2)$ sur une extension quadratique de $ \operatorname{GSp}(4)$ sur $ \mathbb{Q}$, conjecture locale de Langlands pour $ \operatorname{GSp}(4)$, Contemp. Math. 53, Amer. Math. Soc., Providence, RI, 1986, 463-527. MR 853573 (88e:11118)
  • 18. W. Müller, Weyl's law in the theory of automorphic forms, Groups and Analysis, The Legacy of Hermann Weyl, Cambridge Univ. Press, 2008. MR 2528465

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F70

Retrieve articles in all journals with MSC (2010): 11F70


Additional Information

Goran Muic
Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
Email: gmuic@math.hr

DOI: https://doi.org/10.1090/S0002-9939-10-10375-X
Keywords: Cuspidal automorphic representations, Poincar\' e series
Received by editor(s): September 11, 2009
Received by editor(s) in revised form: December 18, 2009
Published electronically: April 9, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society