Realization of limit of discrete series using Dirac operators

Question

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see limit of discrete series in Dolbeault cohomology of G/T after you slightly leave the $L^2$-world, so I wonder a similar thing could be done in G/K picture.