On the structure of the space of cusp forms for a semisimple group over a number field
HTML articles powered by AMS MathViewer
- by Goran Muić PDF
- Proc. Amer. Math. Soc. 138 (2010), 3147-3158 Request permission
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
- James Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263. MR 2192011
- Armand Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30. MR 202718
- A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 189–207. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. MR 546598
- P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
- R. Godement, The spectral decomposition of cusp-forms, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 225–234. MR 0210828
- Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
- Guy Henniart, La conjecture de Langlands locale pour $\textrm {GL}(3)$, Mém. Soc. Math. France (N.S.) 11-12 (1984), 186 (French, with English summary). MR 743063
- A. Moy, Lecture notes delivered in Beijing (http://www.math.ust.hk/$\sim$amoy/beirun.pdf).
- Allen Moy and Gopal Prasad, Jacquet functors and unrefined minimal $K$-types, Comment. Math. Helv. 71 (1996), no. 1, 98–121. MR 1371680, DOI 10.1007/BF02566411
- Goran Muić, On a construction of certain classes of cuspidal automorphic forms via Poincaré series, Math. Ann. 343 (2009), no. 1, 207–227. MR 2448445, DOI 10.1007/s00208-008-0269-5
- G. Muić, Spectral decomposition of compactly supported Poincaré series and existence of cusp forms, Compositio Math. 146, no. 1 (2010), 1–20.
- G. Muić, On the cusp forms for the congruence subgroups of $SL_2(\mathbb {R})$, Ramanujan J. 21, no. 2 (2010), 223–239.
- G. Muić, On geometric density of Hecke eigenvalues for certain cusp forms, Math. Ann. 347, no. 2 (2010), 479–498.
- Jian-Shu Li and Joachim Schwermer, On the cuspidal cohomology of arithmetic groups, Amer. J. Math. 131 (2009), no. 5, 1431–1464. MR 2559860, DOI 10.1353/ajm.0.0073
- Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
- J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
- Marie-France Vignéras, Correspondances entre representations automorphes de $\textrm {GL}(2)$ sur une extension quadratique de $\textrm {GSp}(4)$ sur $\textbf {Q}$, conjecture locale de Langlands pour $\textrm {GSp}(4)$, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 463–527 (French). MR 853573, DOI 10.1090/conm/053/853573
- Werner Müller, Weyl’s law in the theory of automorphic forms, Groups and analysis, London Math. Soc. Lecture Note Ser., vol. 354, Cambridge Univ. Press, Cambridge, 2008, pp. 133–163. MR 2528465, DOI 10.1017/CBO9780511721410.008
Additional Information
- Goran Muić
- Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
- Email: gmuic@math.hr
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3147-3158
- MSC (2010): Primary 11F70
- DOI: https://doi.org/10.1090/S0002-9939-10-10375-X
- MathSciNet review: 2653939