Euler-Poincar\'e pairing, Dirac index and elliptic pairing for Harish-Chandra modules

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.