Classification of the automorphism and isometry groups of Higgs bundle moduli spaces

Let ${\mathcal{M}}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of ${\mathcal{M}}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the group of holomorphic symplectomorphisms, (iii) the group of K"ahler isomorphisms, (iv) the group of automorphisms of the quaternionic structure, (v) the group of hyper-K"ahler isomorphisms. When $n$ and $d$ are coprime we show that ${\mathcal{M}}_{n,d}$ admits an anti-holomorphic isomorphism if and only if the corresponding Riemann surface admits such a map. We then use this to determine the isometry group of ${\mathcal{M}}_{n,d}$.