Moduli space of homologically trivial parabolic (Higgs) bundles on the projective line and applications

We establish an isomorphism between the moduli space of homologically trivial parabolic (Higgs) bundles on ${\mathbb{P}}^1$ and the quiver variety associated to a star-shaped quiver. As applications, we deduce a closed formula for the Littlewood-Richardson coefficients from the Verlinde formula, and solve the nilpotent case of the Deligne-Simpson problem via geometric methods.