Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

Abstract

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on of one the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperk\"ahler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this turns out to be the natural $L^2$-metric on a hyperk\"ahler submanifold of the monopole moduli space.