Invariant hypercomplex structures and algebraic curves

We show that U(k)$U(k)$‐invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in gl(k,C)${\mathfrak {g} \mathfrak {l}}(k,{\mathbb {C}})$ correspond to algebraic curves C of genus (k−1)2$(k-1)^2$, equipped with a flat projection π:C→P1$\pi :C\rightarrow {\mathbb {P}}^1$ of degree k, and an antiholomorphic involution σ:C→C$\sigma :C\rightarrow C$ covering the antipodal map on P1${\mathbb {P}}^1$.