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

TL;DR

This paper explores the relationships between three pairings of Harish-Chandra modules for real reductive groups, establishing their equivalence through index theory and connecting to character theory, thus extending known p-adic results to the archimedean setting.

Contribution

It demonstrates the equivalence of Euler-Poincaré, Dirac index, and elliptic pairings for Harish-Chandra modules using index theory and constructs index functions with specific orbital integral properties.

Findings

The Euler-Poincaré pairing equals the Dirac index pairing.

Both pairings are computed as indices of algebraic Fredholm pairs.

Index functions $f_X$ have orbital integrals matching module characters.

Abstract

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.