Fourier-Mukai partners and autoequivalences of moduli spaces of vector bundles

We classify the smooth projective Fourier–Mukai partners and determine the derived autoequivalence group of the moduli space $U_C(r,d)$ of stable bundles of coprime rank $r\ge 2$ and degree $d$ on a smooth complex projective curve, assuming $g(C)\ge 3$ and $\text{Aut}(C)=1$. When the Jacobian $J(C)$ has endomorphism ring $\mathbb{Z}$, the partners are indexed by $(\mathbb{Z}\mathrm{/}r\mathbb{Z}{)}^{\times }\mathrm{/}\{\pm 1\}$. We also prove that two coprime moduli spaces of vector bundles over complex curves of genus at least four are derived equivalent if and only if they are isomorphic.