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GAFA Geometric And Functional Analysis

, Volume 17, Issue 1, pp 220–251 | Cite as

Existence and Weyl’s law for spherical cusp forms

  • Elon Lindenstrauss
  • Akshay Venkatesh
Article

Abstract.

Let G be a split adjoint semisimple group over \({\user2{\mathbb{Q}}} \) and \( K _\infty \subset \mathbf{G} \user2{\mathbb{(R)}} \) a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of \( {\mathbf{G}}({\user2{\mathbb{R}}})/K_{\infty } \). This proves a conjecture of Sarnak for \( \user2{\mathbb{Q}} \) -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula.

Keywords and phrases:

Cusp forms Weyl law trace formulas congruence quotients 

AMS Mathematics Subject Classification:

primary 32N15 secondary 11F03, 11F72, 22E55 

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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  • Elon Lindenstrauss
    • 1
  • Akshay Venkatesh
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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