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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 Hamiltonianmanner, and that this action linearizes to $A$. Then there is an associatedunitary representation of $G$ on the associated algebro-geometric Hardy space.If the moment map is nowhere vanishing, the isotypical component are all finitedimensional, they are generally not spaces of sections of some power of $A$.One is then led to study the local and global asymptotic properties theisotypical component associated to a weight $k \, \boldsymbol{\nu}$, when$k\rightarrow +\infty$. In this paper, part of a series dedicated to thisgeneral theme, we consider the case $G=U(2)$.

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