  Let $M$ be a closed Riemannian manifold carrying an effective and isometric
action of a compact connected Lie group $G$. We derive a refined remainder
estimate in the stationary phase approximation of certain oscillatory integrals
on $T^\ast M \times G$ with singular critical sets that were examined
previously in order to determine the asymptotic distribution of eigenvalues of
an invariant elliptic operator on $M$. As an immediate consequence, we deduce
from this an asymptotic multiplicity formula for families of irreducible
representations in $L^2(M)$. In forthcoming papers, the improved remainder will
be used to prove an equivariant semiclassical Weyl law and a corresponding
equivariant quantum ergodicity theorem.
