2011-02-25

Pavle Pandzic, University of Zagreb and Cornell University

Dirac Operators and Unitarity of Harish-Chandra Modules

I will start by reviewing the well known Dirac inequality of Parhasarathy, which in its simplest form says that a necessary condition for unitarity of a Harish-Chandra module is non-negativity of the square of the corresponding Dirac operator D. I will then explain a sharpening of this inequality concerning the extremal case when $D^2$ has a kernel. The result was conjectured by Vogan and proved by Huang and myself. I will briefly mention a result in the opposite direction, which roughly says that if $D$ is self adjoint with respect to some inner product, then the module is unitary. Finally, I will explain a further sharpening of the Dirac inequality which was conjectured by Salamanca and Vogan, and a possible strategy of proving it (work in progress with Renard)...