Existence and Weyl's law for spherical cusp forms

Elon Lindenstrauss*, Akshay Venkatesh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


Let G be a split adjoint semisimple group over 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. This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula.

Original languageAmerican English
Pages (from-to)220-251
Number of pages32
JournalGeometric and Functional Analysis
Issue number1
StatePublished - Apr 2007
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgements. We would like to thank Erez Lapid, Steve Miller and Peter Sarnak for their encouragement of this project. Thanks to Erez Lapid’s suggestions, the first proof of Proposition 3 has been improved. Discussions with Erez also clarified to us the limitations of our techniques. Steve Miller provided detailed and very helpful comments on a draft of this paper. We would like to thank Peter Sarnak and Peter Lax for helpful discussions. This research has been conducted while both E.L. and A.V. were Clay Research Fellows; this generous support from the Clay Mathematics Institute is much appreciated. E.L. was also supported in part by NSF grant DMS-0434403; A.V. was supported in part by NSF grant DMS-0245606.


  • Congruence quotients
  • Cusp forms
  • Trace formulas
  • Weyl law


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