In a recent paper, Ben Said, Kobayashi and Orsted obtained a radial deformation of the classical Laplacian and the Dunkl Laplacian. They focus on the construction of a holomorphic semigroup related to their deformation. For special values, this semigroup specializes to generalized Fourier and Hankel transforms and also to the Dunkl transform.
In this talk, I investigate whether there is a Dirac operator underlying this new theory. As it turns out, this can only be achieved by adapting the radial deformation by Ben Said et al. The main advantage of our approach is that we now have a set of first order operators taking over the role of partial derivatives or Dunkl operators, which in principle allow for a more detailed study of this type of deformation.
This is joint work with B. Orsted, P. Somberg and V. Soucek, see arXiv: 0911.4725.