The automorphism group of the moduli space of semi stable vector bundles

Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve $C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms of ${\cal S}{\cal U}(r, L_0)$. The analogue of this result is carried out for the space ${\cal U}(r,d)$ of semi stable vector bundles of rark $r$ and degree $d$. As an application of the technics we use, we give in the appendix at the end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal S}{\cal U}(r, L_0)$ for any rank $r$ and degree $d$.