Dirac index and associated cycles for Harish-Chandra modules

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.