  Let M be a connected d-dimensional complex projective manifold, and let A be
a holomorphic positive Hermitian line bundle on M, with normalized curvature.
Let G be a compact and connected Lie group of dimension d(G), and let T be a
compact torus T of dimension d(T). Suppose that both G and T act on M in a
holomorphic and Hamiltonian manner, that the actions commute, and linearize to
A. If X is the principal circle-bundle associated to A, then this set-up
determines commuting unitary representations of G and T on the Hardy space H(X)
of X, which may then be decomposed over the irreducible representations of the
two groups. If the moment map for the T-action is nowhere zero, all isotypical
components for the torus are finite-dimensional, and thus provide a collection
of finite-dimensional G-modules. Given a non-zero integral weight n(T) for T,
we consider the isotypical components associated to the multiples kn(T), k that
goes to infinity, and focus on how their structure as G-modules is reflected by
certain local scaling asymptotics on X (and M). More precisely, given a fixed
irreducible character n(G) of G, we study the local scaling asymptotics of the
equivariant Szeg\H{o} projectors associated to n(G) and kn(T), for k that goes
to infinity, investigating their asymptotic concentration along certain loci
defined by the moment maps.
