[This paper by Agnihotri and Woodward][1] uses a Narasimhan-Seshadri correspondence between parabolic bundles and unitary connections to determine the possible spectrum of a product of two (special) unitary matrices of known spectrum. They start with a triple of unitary matrices with product $1$, N-S relate that to bundles on $\mathbb P^1$ with parabolic structure at three points, classify those bundles as maps of the $\mathbb P^1$ into a Grassmannian, and end up at quantum Schubert calculus of Grassmannians. Maybe not the most obviously natural source of them, but a wonderful application.


  [1]: http://front.math.ucdavis.edu/9712.5013