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GAFA Geometric And Functional Analysis

, Volume 17, Issue 3, pp 960–998 | Cite as

On Quantum Unique Ergodicity for Locally Symmetric Spaces

  • Lior Silberman
  • Akshay Venkatesh
Article

Abstract.

We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally symmetric spaces to Cartan-invariant measures on an appropriate bundle. The construction uses elementary features of the representation theory of semisimple real Lie groups, and can be considered a generalization of Zelditch’s results from the upper half-plane to all locally symmetric spaces of noncompact type. This will be applied in a sequel to settle a version of the quantum unique ergodicity problem on certain locally symmetric spaces.

Keywords and phrases:

Automorphic forms locally symmetric spaces Lie groups quantum chaos quantum unique ergodicity microlocal lift invariant measures 

AMS Mathematics Subject Classification:

Primary 81Q50 Secondary 11F, 37A45, 37D40, 22E45, 35P20 

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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  5. 5.Courant Institute of Mathematical SciencesNew YorkUSA

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