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Szegő kernel asymptotics and concentration of Husimi Distributions of eigenfunctions

We work on the boundary $\partial M_τ$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_τ)$. We compute $λ\to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_τ = \exp tΞ_{\sqrtρ}$ for the spectral projection kernel $Π_{χ, λ}$ associated to $Π_τD_{\sqrtρ} Π_τ$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_τ$ of Laplace eigenfunctions on $M$. Both asymptotic formulae can be expressed in terms of the metaplectic representation of the linearization of the geodesic flow $G^{t}_τ$ on Bargmann--Fock space. As a corollary, we obtain sharp $L^p \to L^{q}$ norm estimates for $Π_{χ, λ}$ and sharp $L^p$ estimates for Husimi distributions.