The KdV vacuum operator and its Katz extension

We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra.

As a connection, it has a canonical Katz extension from the disc to the sphere.

We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class.

As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.