Abstract:

There are by now several celebrated measure classification results to the effect that a measure is uniform provided it possesses sufficient "invariance" as quantified by stabilizer, entropy, or recurrence. In some applications, part of the challenge is to identify or construct measures to which these hypotheses apply.

I will discuss some examples in which measure classification is used to establish equidistribution on arithmetic homogeneous spaces arising from quaternion algebras. To that end, I will first recall Lindenstrauss's theorem that Hecke-Laplace eigenfunctions on compact arithmetic hyperbolic surfaces equidistribute as their eigenvalues increase. I will then explain a p-adic variant of this result concerning eigenfunctions on the space of non-backtracking paths on certain (p+1)-regular finite graphs. The main new ingredient is the construction of auxiliary measures with the invariance properties required by the measure classification. The construction involves the "p-adic microlocal lifts" mentioned in the title; I will attempt to present the basic idea without assuming prior knowledge of microlocal lifts or representation theory.

There are by now several celebrated measure classification results to the effect that a measure is uniform provided it possesses sufficient "invariance" as quantified by stabilizer, entropy, or recurrence. In some applications, part of the challenge is to identify or construct measures to which these hypotheses apply.

I will discuss some examples in which measure classification is used to establish equidistribution on arithmetic homogeneous spaces arising from quaternion algebras. To that end, I will first recall Lindenstrauss's theorem that Hecke-Laplace eigenfunctions on compact arithmetic hyperbolic surfaces equidistribute as their eigenvalues increase. I will then explain a p-adic variant of this result concerning eigenfunctions on the space of non-backtracking paths on certain (p+1)-regular finite graphs. The main new ingredient is the construction of auxiliary measures with the invariance properties required by the measure classification. The construction involves the "p-adic microlocal lifts" mentioned in the title; I will attempt to present the basic idea without assuming prior knowledge of microlocal lifts or representation theory.

## Date:

Tue, 29/03/2016 - 14:00 to 15:00

## Location:

Manchester building, Hebrew University of Jerusalem, (Room 209)