Description
I will talk about the quantum variance of Hecke-Maass cusp forms on arithmetic hyperbolic surfaces. This may be
understood as a measure of the correlations between their fluctuations.
On the modular surface, the asymptotic quantum variance had been determined by Luo-Sarnak, Zhao and Sarnak-Zhao. They
showed that it differed in a subtle way (involving central values of $L$-functions) from the classical variance of the
geodesic flow.
I will describe similar results for compact hyperbolic surfaces. The novelty in obtaining these is that previous methods
had required a cusp, through the invocation of the Petersson formula. I will describe a different method based on the
theta correspondence. This will involve some discussion of constructing microlocal lifts via convolution operators,
identities and asymptotics involving integrals of theta functions, the Rallis inner product formula, and the method of
coadjoint orbits.