Journal of the Mathematical Society of Japan

Limit formulas of period integrals for a certain symmetric pair II


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Let $(G,H) = (U(p,q),U(p-1,q) \times U(1))$ and $\{\Gamma_n\}$ a tower of congruence uniform lattices in $G$. By the period integrals of automorphic forms on $\Gamma \backslash G$ along $\Gamma_n \cap H\backslash H$ , we introduce a certain discrete measure $d \mu^H_{\Gamma_n}$ on the $H$-spherical unitary dual of $G$. It is shown that the sequence of measures $d \mu^H_{\Gamma_n}$ with growing $n$ converges in a weak sense to the Plancherel measure $d \mu^H$ for the symmetric space $H\backslash G$.

Article information

J. Math. Soc. Japan Volume 63, Number 3 (2011), 1039-1084.

First available in Project Euclid: 1 August 2011

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Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F36

periods of automorphic forms Plancherel measures relative trace formulas


TSUZUKI, Masao. Limit formulas of period integrals for a certain symmetric pair II. Journal of the Mathematical Society of Japan 63 (2011), no. 3, 1039--1084. doi:10.2969/jmsj/06331039.

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  • N. Bergeron, Asymptotique de la norme $L^2$ d'un cycle géodesique dans les revêtments des congruence d'une variété hyperboliques arithmétiques, Math. Z., 241 (2002), 101–125.
  • N. Bergeron and L. Clozel, Spectre automorphe des variétés hyperboliques et applications topologiques, Astérisque, 303 (2005).
  • A. Borel, Introduction aux groupes arithmétique, Hermann, Paris, 1969.
  • A. Borel and Harish-Chandra, Arithmetic groups of algebraic groups, Ann. of Math., 75 (1962), 485–535.
  • D. L. DeGeorge and N. Wallach, Limit formulas for multiplicities in $L^2(\Gamma \bsl G)$, Ann. of Math. (2), 107 (1978), 133–150.
  • D. L. DeGeorge and N. Wallach, Limit formulas for multiplicities in $L^2(\Gamma \backslash G)$. II, Ann. of Math. (2), 109 (1979), 477–495.
  • P. Delorme, Formules limites et formules asymptotiques pour les multiplicités dans $L^2(\Gamma \backslash G)$, Duke Math. J., 53 (1986), 691–731.
  • J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl., 58 (1979), 369–444.
  • G. Heckman and H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspectives in Math., 16, Academic Press, Inc., 1994.
  • T. Oda and M. Tsuzuki, The secondary spherical functions and Green currents associated with certain symmetric pair, Pure Appl. Math. Q., 5 (2009), 977–1028.
  • W. Magnus, F. Oberhettinger and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, in Einzeldarstellungen mit besonderer Berücksichtingung der Anwendungsgebiete Band, 52, Springer-Verlag, New York, 1966.
  • M. Tsuzuki, Limit formulas of period integrals for a certain symmetric pair, J. Func. Anal., 255 (2008), 1139–1190.
  • M. Tsuzuki, Spectral square means for period integrals of wave functions on real hyperbolic spaces, J. Number Theory, 129 (2009), 1387–1438.