Deformations of the generalised Picard bundle

Let X be a non-singular algebraic curve of genus at least 3 and let M denote the moduli space of stable vector bundles of rank n and fixed determinant of degree d with n and d coprime. For any semistable bundle E over X, we can pull E back to XxM, tensor with a universal bundle and take the direct image W(E) on M. If the degree of E is sufficiently large, this direct image is locally free and we call it a generalised Picard bundle. In this paper we prove an inversion formula allowing us to recover E from W(E) and compute the space of infinitesimal deformations of W(E). We also identify a family of deformations which is locally complete and frequently globally complete as well. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over the Jacobian of X.