We review the construction of Dirac operators, cohomology and index for Harish-Chandra modules, as well as associated varieties and associated cycles. Then we show how, for certain Harish-Chandra modules, the polynomial giving the dimension of the Dirac index of the corresponding coherent family can be expressed as an integer linear combination of the multiplicities in the characteristic cycle.
This is joint work with S.Mehdi, D.Vogan and R.Zierau.