Equivariant Asymptotics of Szeg\"{o} kernels under Hamiltonian $SU(2)$-action

Abstract

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties the equivariant projector associated to a weight $k \, \boldsymbol{ \nu }$, when $\boldsymbol{ \nu }$ is fixed and $k\rightarrow +\infty$.