Picard bundle on the moduli space of torsionfree sheaves

Let Y be an integral nodal projective curve of arithmetic genus g >= 2 with m nodes defined over an algebraically closed field. Let n and d be mutually coprime integers with n >= 2 and d>n(2g-2). Fix a line bundle L of degree d on Y. We prove that the Picard bundle E-L over the 'fixed determinant moduli space' U-L(n,d) is stable with respect to the polarisation theta(L) and its restriction to the moduli space U-L '(n,d), of vector bundles of rank n and determinant L, is stable with respect to any polarisation. There is an embedding of the compactified Jacobian (J) over bar (Y) in the moduli space U-Y(n,d) of rank n and degree d. We show that the restriction of the Picard bundle of rank ng (over U-Y(n,n(2g-1))) to (J) over bar (Y) is stable with respect to any theta divisor theta((J) over bar (Y)).