Title: Quantum variance on quaternion algebras, II,
Abstract: A method for determining quantum variance asymptotics on compact quotients
attached to non-split quaternion algebras is developed in general and applied
to "microlocal lifts" in the non-archimedean setting. The results obtained are
in the spirit of recent work of Sarnak--Zhao.
The arguments involve a careful analytic study of the theta correspondence,
the interplay between additive and multiplicative harmonic analysis on
quaternion algebras, the equidistribution of translates of elementary theta
functions, and the Rallis inner product formula.