Stability of Picard sheaves for vector bundles on curves

For a stable vector bundle E of slope μ(E)>2g−1 on a smooth, projective curve of genus g, we show that the Picard sheaf Eˆ on the Picard variety of the curve is stable. We introduce a homological tool for testing semistability of Picard sheaves. We also obtain the semistability of the general Picard sheaf if μ(E)∈[g−2,g],μ(E)≠g−1.