Autoequivalences of Derived Categories of Moduli Spaces of Vector Bundles

Let $C$ be a smooth projective curve over an algebraically closed field of characteristic zero. For the moduli space $N(r,L)$ of stable vector bundles on $C$ of rank $r$ with fixed determinant $L$, we study the group of exact autoequivalences of its bounded derived category. Combining the Bondal–Orlov reconstruction theorem with the descriptions of $\mathrm{Aut}(N(r,L))$ due to Kouvidakis–Pantev and Newstead, we obtain an explicit description of $\mathrm{Aut}{\mathrm{D}}^{\mathrm{b}}(N(r,L))$. We also construct extension correspondences between moduli spaces of vector bundles of different ranks and use them to define natural Fourier–Mukai type exact functors between their bounded derived categories.