Equivariant asymptotics of Szeg\"o kernels under Hamiltonian $SU(2)\times S^1$-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 and holomorphic manner and that this action linearizes to $A$. Then, there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space $H(X)$. The standard circle action on $H(X)$ commutes with the action of $G$ and thus one has a decompositions labeled by $(k\,\boldsymbol{\nu},\,k)$, where $k\in\mathbb{Z}$ and $\boldsymbol{ \nu }\in \hat{G}$. We consider the local and global asymptotic properties of the corresponding equivariant projector as $k$ goes to infinity. More generally, for a compact connected Lie group, we compute the asymptotics of the dimensions of the corresponding isotypes.