Invariant hypercomplex structures and algebraic curves

Abstract: We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{gl}(k,{\mathbb C})$ correspond to algebraic curves $C$ of genus $(k-1)<sup>2$, equipped with a flat projection $\pi:C\to{\mathbb P}<sup>1$ of degree $k$, and an antiholomorphic involution $\sigma:C\to C$ covering the antipodal map on ${\mathbb P}<sup>1$.