Equivariant asymptotics of Szegö kernels under Hamiltonian $${{\\varvec{U}}}(\\mathbf{2})$$U(2)-actions

Abstract

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 components 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 with a weight $$k \\, \\varvec{ \\nu }$$kν, when $$k\\rightarrow +\\infty $$k→+∞. In this paper, part of a series dedicated to this general theme, we consider the case $$G=U(2)$$G=U(2).