Hi, my question concerns the Picard bundles in the special case of an elliptic curve $E$, say over the complex numbers. Let $p$ and $q$ be the projections of $E \times E$.

One defines the $n$-th Picard bundle ($n$ an integer) as follows: take the line bundle associated to the divisor $n$ times O (the point zero weighted with $n$) on $E$, pull it back via $p$, tensor it on $E \times E$ with the Poincaré bundle, and then take the direct image of this under $q$.

Does anybody know whether one can say explicitly what the result of this construction is in the special case of an elliptic curve? (one usually defines the Picard sheaves for general curves and uses the jacobian, see the book of Birkenhake Lange, Complex abelian varieties)

Thanks a lot!