Duke Mathematical Journal

Bounds for multiplicities of automorphic representations

Peter Sarnak and Xiaoxi Xue

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Duke Math. J., Volume 64, Number 1 (1991), 207-227.

First available in Project Euclid: 20 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]


Sarnak, Peter; Xue, Xiaoxi. Bounds for multiplicities of automorphic representations. Duke Math. J. 64 (1991), no. 1, 207--227. doi:10.1215/S0012-7094-91-06410-0. https://projecteuclid.org/euclid.dmj/1077295393

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  • [B] A. Borel, Cohomologie de sous-groupes discrets et représentations de groupes semi-simples, Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan (Orsay, 1974), Soc. Math. France, Paris, 1976, 73–112. Astérisque, No. 32-33.
  • [B-HC] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.
  • [B-W] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J., 1980.
  • [D-R-S] W. Duke, Z. Rudnick, and P. Sarnak, The density of integral points on affine homogeneous varieties, preprint.
  • [D-W]1 David L. de George and Nolan R. Wallach, Limit formulas for multiplicities in $L\sp2(\Gamma \backslash G)$, Ann. Math. (2) 107 (1978), no. 1, 133–150.
  • [D-W]2 D. DeGeorge and N. Wallach, Limit formulas for multiplicities in $L\sp2(\Gamma \backslash G)$. II. The tempered spectrum, Ann. of Math. (2) 109 (1979), no. 3, 477–495.
  • [G-GPS] I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1966 [1977].
  • [Hu] M. Huxley, Exceptional eigenvalues and congruence subgroups, The Selberg trace formula and related topics (Brunswick, Maine, 1984), Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 341–349.
  • [Kat] J. Katznelson thesis, Stanford Univ.
  • [Kn] M. Kneser, Strong approximation, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 187–196.
  • [Kna] A. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986.
  • [Ma] Y. Matsushima, On the first Betti number of compact quotient spaces of higher-dimensional symmetric spaces, Ann. of Math. (2) 75 (1962), 312–330.
  • [Rd] B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 996–1000.
  • [S] P. Sarnak, Diophantine problems and linear groups, to appear in Proc. Internat. Cong. Math., Kyoto, Japan, 1990.
  • [Sc] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985.
  • [Se] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15.
  • [Ti] J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc.9, Providence, R.I., 1966, 1966, pp. 33–62.
  • [X1] X. Xue, On the first Betti numbers of hyperbolic surfaces, Duke Math. J. 64 (1991), no. 1, 85–110.
  • [X2] X. Xue, On the Betti numbers of hyperbolic manifolds, submitted, 1991.