On Quantum Unique Ergodicity for Locally Symmetric Spaces
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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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