Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect


Jasmin Matz


University of Copenhagen


Tue, 24/11/2020 - 9:00pm


RC-4082, The Red Centre, UNSW


Suppose $M$ is a closed Riemannian manifold with an orthonormal basis $B$
of $L^2(M)$ consisting of Laplace eigenfunctions. A classical result of
Shnirelman and others proves that if the geodesic flow on the cotangent
bundle of $M$ is ergodic, then $M$ is quantum ergodic, in particular, on
average, the probability measures defined by the functions $f$ in $B$ on $M$
tends on average towards the Riemannian measure on $M$ in the high
energy limit (i.e, as the Laplace eigenvalues of $f \to \infty$).
We now want to look at a level aspect of this property, namely, instead
of taking a fixed manifold and high energy eigenfunctions, we take a
sequence of Benjamini-Schramm convergent compact Riemannian manifolds
$M_j$ together with Laplace eigenfunctions $f$ whose eigenvalue varies in
short intervals. This perspective has been recently studied in the
context of graphs by Anantharaman and Le Masson, and for hyperbolic
surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In
my talk I want to discuss joint work with F. Brumley in which we study
this question in higher rank, namely sequences of compact quotients of

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Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)

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