  Let $G$ be a finite cover of a closed connected transpose-stable subgroup of
$GL(n,\bR)$ with complexified Lie algebra $\frg$. Let $K$ be a maximal compact
subgroup of $G$, and assume that $G$ and $K$ have equal rank. We prove a
translation principle for the Dirac index of virtual $(\frg,K)$-modules. As a
byproduct, to each coherent family of such modules, we attach a polynomial on
the dual of the compact Cartan subalgebra of $\frg$. This ``index polynomial''
generates an irreducible representation of the Weyl group contained in the
coherent continuation representation. We show that the index polynomial is the
exact analogue on the compact Cartan subgroup of King's character polynomial.
The character polynomial was defined in \cite{K1} on the maximally split Cartan
subgroup, and it was shown to be equal to the Goldie rank polynomial up to a
scalar multiple. In the case of representations of Gelfand-Kirillov dimension
at most half the dimension of $G/K$, we also conjecture an explicit
relationship between our index polynomial and the multiplicities of the
irreducible components occuring in the associated cycle of the corresponding
coherent family.
