  Let $G$ be a connected real reductive group with maximal compact subgroup $K$
of equal rank, and let $\mathscr M$ be the category of Harish-Chandra modules
for $G$. We relate three differentely defined pairings between two finite
length modules $X$ and $Y$ in $\mathscr M$ : the Euler-Poincar\'e pairing, the
natural pairing between the Dirac indices of $X$ and $Y$, and the elliptic
pairing. (The Dirac index is a virtual finite dimensional representation of
$\widetilde K$, the spin double cover of $K$.) Analogy with the case of Hecke
algebras and a formal (but not rigorous) computation lead us to conjecture that
the first two pairings coincide. In the second part of the paper, we show that
they are both computed as the indices of Fredholm pairs (defined here in an
algebraic sense) of operators acting on the same spaces. We construct index
functions $f_X$ for any finite length Harish-Chandra module $X$. These
functions are very cuspidal in the sense of Labesse, and their orbital
integrals on elliptic elements coincide with the character of $X$. From this we
deduce that the Dirac index pairing coincide with the elliptic pairing. These
results are the archimedean analog of results of Schneider-Stuhler for $p$-adic
groups.
