Stability on the Sato Grassmannian. Applications to the moduli of vector bundles

The action of Sl(r,k[[z]]) on the Sato Grassmannian is studied. Following ideas similar to those of GIT and to those used in the study of vector bundles, the (semi)stable points are introduced. It is shown that any point admits a Harder–Narasimhan filtration and that, if it is semistable, it has a Jordan–Hölder filtration. Finally, theses results are compared with the well-known theory of vector bundles on an algebraic curve.