Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules
Journal de l’École polytechnique - Mathématiques, Volume 3 (2016), pp. 209-229.

Let G be a connected real reductive group with maximal compact subgroup K of equal rank, and let be the category of Harish-Chandra modules for G. We relate three differently defined pairings between two finite length modules X and Y in : 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].

Soit G un groupe réductif réel connexe et soit K un sous-groupe compact maximal que l’on suppose de même rang. Nous relions trois accouplements entre modules de Harish-Chandra de longueur finie X et Y : l’accouplement d’Euler-Poincaré, l’accouplement naturel entre les indices de Dirac de X et Y et l’accouplement elliptique d’Arthur [2] (l’indice de Dirac I Dir (X) est une représentation virtuelle de dimension finie de K ˜, le revêtement Spin à deux feuillets de K). Nous construisons des fonctions indices f X pour tout module de Harish-Chandra de longueur finie X. Chacune de ces fonctions est très cuspidale au sens de Labesse, et son intégrale orbitale coïncide sur les éléments elliptiques avec le caractère de X. De ceci nous déduisons que l’accouplement naturel des indices de Dirac coïncide avec l’accouplement elliptique. Une analogie avec le cas des algèbres de Hecke considéré dans [8] et [7] et un calcul formel (mais non rigoureux) nous ont amenés à conjecturer que les deux premiers accouplements coïncident eux aussi. Nous montrons qu’ils peuvent tout deux être exprimés comme indices de paires de Fredholm (définis ici dans un sens algébrique) d’opérateurs agissant sur les même espaces. Récemment Huang et Sun ont établi l’égalité entre accouplement d’Euler-Poincaré et accouplement elliptique, démontrant ainsi directement l’analogue d’un résultat de Schneider et Stuhler pour les groupes p-adiques [25].

Received:
Accepted:
Published online:
DOI: 10.5802/jep.32
Classification: 22E46,  22E47
Keywords: Harish-Chandra module, elliptic representation, Euler-Poincaré pairing, elliptic pairing, Dirac cohomology
Renard, David 1

1 CMLS, École polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex, France
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Renard, David. Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules. Journal de l’École polytechnique - Mathématiques, Volume 3 (2016), pp. 209-229. doi : 10.5802/jep.32. http://www.numdam.org/articles/10.5802/jep.32/

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