Equivariant Asymptotics of Szeg\"o kernels under Hamiltonian $U(2)$ actions

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymbol{\nu}$, when $k\rightarrow +\infty$. In this paper, part of a series dedicated to this general theme, we consider the case $G=U(2)$.