Alexander Alldridge

Radial parts of superdifferential operators, and applications to invariants

Supergroup actions, slices, and radial parts

Abstract: A theorem of Helgason states that if an action of a Lie group on a manifold admits a slice, then any differential operator admits a `radial’ component on the slice. For actions of Lie supergroups, this is no longer true in general, as the naive slice condition is too weak. We introduce a strong notion of slice that allows for the generalisation of Helgason’s theorem, using the concept of `orbits through generalised points’.

We study in particular the isotropy group action for a Riemannian symmetric superspace. In this setting, the strong slice condition holds if and only if the supermanifold is of `even type’ (i.e. the slice in the naive sense is an ordinary manifold).

To remedy this situation, we consider a weak notion of slices. We prove in general that at least any differential operator locally invariant under the transporter supergroup of the weak slice admits a radial part on the quotient of the weak slice by the transporter supergroup. This quotient is (mildly) singular in general. For the example of Riemannian symmetric superspaces, the transporter supergroup action factors through the Weyl group, which is a reductive Lie group of positive dimension in the cases not of odd type.

We apply these results to the determination of invariant functions on Riemann symmetric superspaces and their flat degenerations.

This work is joint with J. Hilgert and T. Wurzbacher, and with K. Coulembier, respectively.