Euler-poincar 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 M be the category of Harish-Chandra modules for G. We relate three di erently defined pairings between two finite length modules X and Y in M: the Euler-Poincaré pairing, the natural pairing between the Dirac indices of X and Y , and the elliptic pairing of [2]. (The Dirac index I_Dir(X) is a virtual finite-dimensional representation of K , the spin double cover of K.) We construct index functions f_X for any finite length Harish-Chandra module X. Each of these functions is very cuspidal in the sense of Labesse, and its orbital integral on elliptic elements coincides with the character of X. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. Analogy with the case of Hecke algebras studied in [8] and [7] and a formal (but not rigorous) computation led us to conjecture that the first two pairings coincide. 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. Recently, Huang and Sun have established the equality between the Euler-Poincaré and the elliptic pairing, thereby proving directly the analogue of a result of Schneider and Stuhler for p-adic groups [25].