Spectral asymptotics for arithmetic quotients of ${\rm SL}(n,{\mathbb R})/\rm{SO}(n)$
Erez Lapid and Werner Müller
Source: Duke Math. J. Volume 149, Number 1
(2009), 117-155.
Abstract
In this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space ${\rm SL}(n,{\mathbb R})/\operatorname{SO}(n)$. In particular, we obtain Weyl's law with an estimation on the remainder term. This extends some of the main results of Duistermaat, Kolk, and Varadarajan ([DKV1]) to this setting
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1246453790
Digital Object Identifier: doi:10.1215/00127094-2009-037
Zentralblatt MATH identifier: 05588173
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References
J. G. Arthur, A trace formula for reductive groups, I: Terms associated to classes in $G(\bf Q)$, Duke Math. J. 45 (1978), 911--952.
Mathematical Reviews (MathSciNet): MR0518111
Digital Object Identifier: doi:10.1215/S0012-7094-78-04542-8
Project Euclid: euclid.dmj/1077313104
Zentralblatt MATH: 0499.10032
—, On a family of distributions obtained from Eisenstein series, I: Application of the Paley-Wiener theorem, Amer. J. Math. 104 (1982), 1243--1288.
Mathematical Reviews (MathSciNet): MR0681737
Digital Object Identifier: doi:10.2307/2374061
JSTOR: links.jstor.org
Zentralblatt MATH: 0541.22010
—, On a family of distributions obtained from Eisenstein series, II: Explicit formulas, Amer. J. Math. 104 (1982), 1289--1336.
Mathematical Reviews (MathSciNet): MR0681738
Digital Object Identifier: doi:10.2307/2374062
JSTOR: links.jstor.org
Zentralblatt MATH: 0562.22004
—, A measure on the unipotent variety, Canad. J. Math. 37 (1985), 1237--1274.
Mathematical Reviews (MathSciNet): MR0828844
—, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223--293.
Mathematical Reviews (MathSciNet): MR0932848
Digital Object Identifier: doi:10.1215/S0012-7094-88-05612-8
Project Euclid: euclid.dmj/1077306597
Zentralblatt MATH: 0649.10020
—, The $L\sp 2$-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257--290.
Mathematical Reviews (MathSciNet): MR1001841
Digital Object Identifier: doi:10.1007/BF01389042
Zentralblatt MATH: 0692.22004
—, ``Unipotent automorphic representations: Conjectures'' in Orbites unipotentes et représentations, II, Astérisque 171 --172., Soc. Math. France, Montrouge, 1989, 13--71.
Mathematical Reviews (MathSciNet): MR1021499
Zentralblatt MATH: 0728.22014
G. V. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956), 327--344.
Mathematical Reviews (MathSciNet): MR0080862
Digital Object Identifier: doi:10.1007/BF01473886
A. Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5--30.
Mathematical Reviews (MathSciNet): MR0202718
Digital Object Identifier: doi:10.1007/BF02684289
—, Introduction aux groupes arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg 15, Actualités Sci. Indust. 1341, Hermann, Paris, 1969.
Mathematical Reviews (MathSciNet): MR0244260
P. Deligne, D. Kazhdan, and M.-F. VignéRas, ``Représentations des algèbres centrales simples $p$-adiques'' in Représentations des algèbres centrales simples $p$-adiques, Travaux en Cours, Hermann, Paris, 1984, 33--117.
Mathematical Reviews (MathSciNet): MR0771672
J. J. Duistermaat and V. W. Guillemin, ``The spectrum of positive elliptic operators and periodic geodesics'' in Differential Geometry (Stanford, Calif., 1973), Proc. Sympos. Pure Math. XXVII, Part 2, Amer. Math. Soc., Providence, 1975, 205--209.
Mathematical Reviews (MathSciNet): MR0423438
Zentralblatt MATH: 0322.35071
J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979), 27--93.; Erratum, 54 (1979), 101. $\!$;
Mathematical Reviews (MathSciNet): MR532745
Mathematical Reviews (MathSciNet): MR0549549
Digital Object Identifier: doi:10.1007/BF01389856
—, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math. 49 (1983), 309--398.
Mathematical Reviews (MathSciNet): MR0707179
Zentralblatt MATH: 0524.43008
T. Finis, E. Lapid, and W. MüLler, On the spectral side of Arthur's trace formula, II, preprint, 2007.
R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 150--165.
Mathematical Reviews (MathSciNet): MR0289724
Digital Object Identifier: doi:10.2307/1970758
JSTOR: links.jstor.org
M. Goresky, R. Kottwitz, and R. Macpherson, Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J. 89 (1997), 477--554.; Correction, 92 (1998), 665--666. $\!$;
Mathematical Reviews (MathSciNet): MR1470341
Mathematical Reviews (MathSciNet): MR1620542
Digital Object Identifier: doi:10.1215/S0012-7094-97-08922-5
Project Euclid: euclid.dmj/1077241206
Harish-Chandra, Spherical functions on a semisimple Lie group, I, Amer. J. Math. 80 (1958), 241--310.
Mathematical Reviews (MathSciNet): MR0094407
Digital Object Identifier: doi:10.2307/2372786
JSTOR: links.jstor.org
—, Spherical functions on a semisimple Lie group, II, Amer. J. Math. 80 (1958), 553--613.
Mathematical Reviews (MathSciNet): MR0101279
Digital Object Identifier: doi:10.2307/2372772
JSTOR: links.jstor.org
—, Discrete series for semisimple Lie groups, II. Explicit determination of the characters, Acta Math. 116 (1966), 1--111.
Mathematical Reviews (MathSciNet): MR0219666
Digital Object Identifier: doi:10.1007/BF02392813
Zentralblatt MATH: 0199.20102
—, Two theorems on semi-simple Lie groups, Ann. of Math. (2) 83 (1966), 74--128.
Mathematical Reviews (MathSciNet): MR0194556
Digital Object Identifier: doi:10.2307/1970472
JSTOR: links.jstor.org
A. D. Hejhal, The Selberg Trace Formula for $\rm PSL(2,R)$, Vol. I, Lecture Notes in Math. 548, Springer, Berlin, 1976.
Mathematical Reviews (MathSciNet): MR0439755
S. Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297--308.
Mathematical Reviews (MathSciNet): MR0223497
Digital Object Identifier: doi:10.1007/BF01344014
Zentralblatt MATH: 0178.17101
—, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, corrected reprint of the 1984 original, Math. Surveys and Monogr. 83, Amer. Math. Soc., Providence, 2000.
Mathematical Reviews (MathSciNet): MR1790156
—, Differential Geometry, Lie Groups, and Symmetric Spaces, corrected reprint of the 1978 original, Grad. Studies in Math. 34, Amer. Math. Soc., Providence, 2001.
Mathematical Reviews (MathSciNet): MR1834454
L. HöRmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193--218.
Mathematical Reviews (MathSciNet): MR0609014
Digital Object Identifier: doi:10.1007/BF02391913
Zentralblatt MATH: 0164.13201
J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Math. Surveys and Monogr. 43, Amer. Math. Soc., Providence, 1995.
Mathematical Reviews (MathSciNet): MR1343976
Zentralblatt MATH: 0834.20048
M. N. Huxley, ``Scattering matrices for congruence subgroups'' in Modular Forms (Durham, England, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, England, 1984, 141--156.
Mathematical Reviews (MathSciNet): MR0803366
Zentralblatt MATH: 0554.10016
H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of $L$-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55--131.
Mathematical Reviews (MathSciNet): MR1828743
Digital Object Identifier: doi:10.1007/BF02698741
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms, II, Amer. J. Math. 103 (1981), 777--815.
Mathematical Reviews (MathSciNet): MR0623137
Digital Object Identifier: doi:10.2307/2374050
JSTOR: links.jstor.org
Zentralblatt MATH: 0491.10020
—, On Euler products and the classification of automorphic representations, I, Amer. J. Math. 103 (1981), 499--558.
Mathematical Reviews (MathSciNet): MR0618323
Digital Object Identifier: doi:10.2307/2374103
JSTOR: links.jstor.org
Zentralblatt MATH: 0473.12008
J.-P. Labesse and W. MüLler, Weak Weyl's law for congruence subgroups, Asian J. Math. 8 (2004), 733--745.
Mathematical Reviews (MathSciNet): MR2127945
Project Euclid: euclid.ajm/1118669697
Zentralblatt MATH: 1162.11344
R. P. Langlands, The dimension of spaces of automorphic forms, Amer. J. Math. 85 (1963), 99--125.
Mathematical Reviews (MathSciNet): MR0156362
Digital Object Identifier: doi:10.2307/2373189
JSTOR: links.jstor.org
Zentralblatt MATH: 0113.25802
E. Lindenstrauss and A. Venkatesh, Existence and Weyl's law for spherical cusp forms, Geom. Funct. Anal. 17 (2007), 220--251.
Mathematical Reviews (MathSciNet): MR2306657
Digital Object Identifier: doi:10.1007/s00039-006-0589-0
Zentralblatt MATH: 1137.22011
W. Luo, Nonvanishing of $L$-values and the Weyl law, Ann. of Math (2) 154 (2001), 477--502.
Mathematical Reviews (MathSciNet): MR1865978
Digital Object Identifier: doi:10.2307/3062104
JSTOR: links.jstor.org
Zentralblatt MATH: 1003.11019
D. S. Miller, On the existence and temperedness of cusp forms for $\rm SL\sb 3(\Bbb Z)$, J. Reine Angew. Math. 533 (2001), 127--169.
Mathematical Reviews (MathSciNet): MR1823867
Digital Object Identifier: doi:10.1515/crll.2001.029
Zentralblatt MATH: 0996.11040
C. Moeglin and J.-L. Waldspurger, Le spectre résiduel de $\rm GL(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), 605--674.
Mathematical Reviews (MathSciNet): MR1026752
W. MüLler, Weyl's law for the cuspidal spectrum of $\rm SL\sb n$, Ann. of Math. (2) 165 (2007), 275--333.
Mathematical Reviews (MathSciNet): MR2276771
Digital Object Identifier: doi:10.4007/annals.2007.165.275
Zentralblatt MATH: 1119.11027
—, ``Weyl's law in the theory of automorphic forms'' in Groups and Analysis: The Legacy of Hermann Weyl (Bielefeld, Germany, 2006), London Math. Soc. Lecture Note Ser. 354, Cambridge Univ. Press, Cambridge, 2008, 133--163.
W. MüLler and B. Speh, Absolute convergence of the spectral side of the Arthur trace formula for $\rm GL\sb n$, with an appendix by E. M. Lapid, Geom. Funct. Anal. 14 (2004), 58--93.
Mathematical Reviews (MathSciNet): MR2053600
Digital Object Identifier: doi:10.1007/s00039-004-0452-0
Zentralblatt MATH: 1083.11031
D. Mumford, Abelian varieties, Tata Inst. Fund. Res. Stud. in Math. 5, Oxford Univ. Press, London, 1970.
Mathematical Reviews (MathSciNet): MR0282985
P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss. 300, Springer, Berlin, 1992.
Mathematical Reviews (MathSciNet): MR1217488
R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), 853--866.
Mathematical Reviews (MathSciNet): MR0812352
Digital Object Identifier: doi:10.1002/cpa.3160380614
Zentralblatt MATH: 0614.10027
—, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), 1--32.
Mathematical Reviews (MathSciNet): MR1127079
Digital Object Identifier: doi:10.2307/2152749
JSTOR: links.jstor.org
Zentralblatt MATH: 0743.30039
A. Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 (1993), 79--105.
Mathematical Reviews (MathSciNet): MR1204788
Digital Object Identifier: doi:10.1007/BF01895514
Zentralblatt MATH: 0785.11034
P. Sarnak, ``On cusp forms'' in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984), Contemp. Math. 53, Amer. Math. Soc., Providence, 1986, 393--407.
Mathematical Reviews (MathSciNet): MR0853570
A. 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.
Mathematical Reviews (MathSciNet): MR0088511
—, Collected Papers, Vol. I, Springer, Berlin, 1989.
Mathematical Reviews (MathSciNet): MR1117906
V. S. Varadarajan, ``The method of stationary phase and applications to geometry and analysis on Lie groups'' in Algebraic and Analytic Methods in Representation Theory (Sønderborg, Denmark, 1994), Perspect. Math. 17, Academic Press, San Diego, 1997, 167--242.
Mathematical Reviews (MathSciNet): MR1415844
Digital Object Identifier: doi:10.1016/B978-012625440-2/50005-7
Zentralblatt MATH: 0903.43004
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