2 00 5 Resolutions of homogeneous bundles on P 2

Homogeneous bundles on P2 = SL(3)/P can be described by representations of the parabolic subgroup P . In 1966 Ramanan proved that if ρ is an irreducible representation of P then the induced bundle Eρ on P 2 is simple and even stable (see [Ram]). Since P is not a reductive group, there is a lot of indecomposable reducible representations of P and to classify homogeneous bundles on P2 and among them the simple ones, the stable ones, etc. by means of the study of the representations of the parabolic subgroup P seems difficult. In this paper our point of view is to consider the minimal free resolution of the bundle. Our aim is to classify homogeneous vector bundles on P2 by means of their minimal resolutions. Precisely we observe that if E is a homogeneous vector bundle on P2 = P(V ) (V complex vector space of dimension 3) there exists a minimal free resolution of E