In this talk our aim is to explain why there exist hyperkähler metrics on the cotangent bundles and on coadjoint orbits of complex Lie groups. The key observation is that both the cotangent bundle of $G^\mathbb C$ and complex coadjoint orbits can be constructed as hyperkähler quotients in an infinite-dimensional setting: They may be identified with certain moduli spaces of solutions to Nahm's equations, which is a system of non-linear ODEs arising in gauge theory.
In the first half we will describe the hyperkähler quotient construction, which can be viewed as a version of the Marsden-Weinstein symplectic quotient for complex symplectic manifolds. We will then introduce Nahm's equations and explain how their moduli spaces of solutions may be related to the above Lie theoretic objects.