Szeg\H{o} kernel asymptotics and concentration of Husimi Distributions of eigenfunctions

Let $(M,g)$ be a compact, real analytic Riemannian manifold and $\Pi_\tau D_{\sqrt{\rho}} \Pi_\tau$ be the Toeplitz operator associated to the Reeb vector field on the Grauert tube boundary $\partial M_{\tau}$. We compute scaling asymptotics as $\lambda \to \infty$ for the tempered spectral projection kernels of $\Pi_\tau D_{\sqrt{\rho}} \Pi_\tau$ on $\lambda$ Heisenberg scaled neighborhoods of points of the geodesic flow. We also compute scaling asymptotics for tempered sums of Husimi Distributions of Laplace eigenfunctions on $M$. We show that the leading term of the asymptotics is the Metaplectic representation of the linearization of the geodesic flow on Bargmann--Fock space. As a corollary we obtain sharp $L^2 \to L^{p}$ mapping estimates for the tempered spectral projection kernels of $\Pi_{\tau} D_{\sqrt{\rho} } \Pi_{\tau}$.