Jordan--Chevalley decomposition for G-bundles on genus one curves

The Jordan--Chevalley decomposition expresses a matrix as a sum of a diagonalizable and a nilpotent matrix that commute. In fact such a decomposition holds both for the Lie algebra and for the group and more generally for any reductive group. One uses extensively this result in representation theory, in the study of the conjugacy classes of matrices and their geometry, etc. In this talk I will explain that there is another situation when one can decompose a "G"-object like this, namely that of semistable G-bundles on elliptic curves: it says that any G-bundle can be written essentialy uniquely as a product of a unipotent bundle and a semisimple bundle. Since G-bundles do not admit a multiplication/addition as is the case for Lie algebras or Lie groups, we need first to make sense of what such a decomposition should mean.

I will present various ways of thinking about the Jordan--Chevalley decomposition and focus on the (geometric) one that admits a generalization to G-bundles. Moreover we will see that degenerating the elliptic curve to a nodal or a cusp leads us back to the conjugacy classes in the Lie group, resp. Lie algebra, and to the corresponding Jordan--Chevalley decomposition, exhibiting thus the trichotomy "rational, trigonometric, elliptic".

Using a tannakian description of semistable G-bundles one can also show that, for non supersingular elliptic curves, the elliptic unipotent cone is isomorphic to the unipotent/nilpotent cone in the group/Lie algebra.

(joint with Sam Gunningham and Penghui Li)