  This paper studies projections of uniform random elements of (co)adjoint
orbits of compact Lie groups. Such projections generalize several widely
studied ensembles in random matrix theory, including the randomized Horn's
problem, the randomized Schur's problem, and the orbital corners process. In
this general setting, we prove integral formulae for the probability densities,
establish some properties of the densities, and discuss connections to
multiplicity problems in representation theory as well as to known results in
the symplectic geometry literature. As applications, we show a number of
results on marginal problems in quantum information theory and also prove an
integral formula for restriction multiplicities.
