  In this paper, we provide a detailed and systematic study of weak (singular)
Hermite-Einstein structures on homogeneous holomorphic vector bundles over
rational homogeneous varieties. We use standard tools from spectral geometry,
Harmonic analysis, and Cartan's highest weight theory to provide a sufficient
condition in terms of Fourier series and intersection numbers under which an
$L^{2}$-function can be realized as mean curvature of a singular Hermitian
structure on an irreducible homogeneous holomorphic vector bundle. We prove
that the condition provided is necessary and sufficient for functions that
belong to certain interpolation spaces. In the particular case of line bundles
over irreducible Hermitian symmetric spaces of compact type, we describe
explicitly in terms of representation theory the solutions of the underlying
geometric PDE. Also, we establish a sufficient condition in terms of Fourier
coefficients and intersection numbers for solvability and convergence of the
weak Hermite-Einstein flow on irreducible homogeneous holomorphic vector
bundles. As an application of our methods, we describe the first explicit
examples in the literature of solutions to several geometric flows, including
Donaldson's heat flow, Yang-Mills heat flow, and the gradient flow of
Donaldson's Lagrangian, on line bundles over irreducible Hermitian symmetric
spaces of compact type. Additionally, we show that every polynomial central
charge function gives rise to a weak Donaldson's Lagrangian $\mathcal{M}$. In
the particular case of irreducible homogeneous holomorphic vector bundles, we
prove that the gradient flow of $\mathcal{M}$ converges to a Hermitian
structure which is, up to a gauge transformation, a $Z$-critical Hermitian
structure in the large volume limit.
