[Paper Review] Méthode des orbites et formules du caractère pour les représentations tempérées d'un groupe algébrique réel réductif non connexe

Jean-Yves Ducloux|ArXiv.org|Oct 3, 2000

Advanced Algebra and Geometry3 citations

TL;DR

This paper extends the orbit method to classify tempered unitary representations of non-connected real reductive Lie groups using generalized regular coadjoint orbits and Kirillov-type character formulas. It establishes a parametrization of the tempered dual via 'good parameters' (generalized linear functionals and projective representations), proving that unitary equivalence corresponds exactly to G-conjugacy of these parameters, generalizing results of Adams, Barbasch, and Vogan for connected groups.

ABSTRACT

Let G be a non-connected reductive real Lie group. In this paper, I parametrize the set of irreductible tempered characters of G. Afterwards, I describe these characters by means of some ``Kirillov's formulas'', using the descent method near each elliptic element in G. If G is linear and connected, the parameters that I use are ``final basic'' parameters in the sense of Knapp and Zuckerman.

Motivation & Objective

• To extend the orbit method to non-connected real reductive Lie groups.
• To parametrize the tempered dual of such groups using generalized regular coadjoint orbits.
• To establish a precise correspondence between unitary equivalence of tempered representations and G-conjugacy of 'good parameters'.
• To generalize Kirillov's character formulas to this setting using descent methods near elliptic elements.
• To unify and extend prior results on tempered representations by incorporating limit of discrete series in the induction process.

Proposed method

• Parametrizes tempered representations using pairs $({ ilde{ ho}}, \tau)$, where $\tilde{\rho}$ is a generalized regular linear functional and $\tau$ is a projective representation.
• Applies the descent method near elliptic elements to construct character formulas via Fourier transforms on coadjoint orbits.
• Uses generalized regular coadjoint orbits $\widetilde{\mathfrak{g}}^*_{\text{reg},G}$ to parametrize the tempered dual.
• Relies on the theory of induced representations and translation functors à la Zuckerman to relate representations across parabolic subgroups.
• Employs the framework of Knapp and Zuckerman's 'final basic' parameters in the connected case, extending it to non-connected groups.
• Establishes character formulas through restriction and passage from Levi subgroups $M'$ to $G$, using regularity and integrability conditions.

Experimental results

Research questions

• RQ1 How can the orbit method be extended to classify tempered representations of non-connected real reductive Lie groups?
• RQ2 What is the precise parametrization of the tempered dual in terms of coadjoint orbits and projective representations?
• RQ3 How do Kirillov-type character formulas generalize in this non-connected setting?
• RQ4 What conditions ensure that two representations are unitarily equivalent in this parametrization?
• RQ5 How does the descent method near elliptic elements contribute to character computation?

Key findings

• The tempered dual of a non-connected real reductive group $G$ is parametrized by $G$-orbits of generalized regular coadjoint parameters $({\tilde{\lambda}}, \tau)$, where $\tilde{\lambda}$ is a generalized regular linear functional and $\tau$ is a projective representation.
• Unitary equivalence of tempered representations is equivalent to $G$-conjugacy of their associated 'good parameters' $({\tilde{\lambda}}, \tau)$, providing a precise classification.
• The character of each tempered representation is computed via a generalized Kirillov formula, using Fourier transforms on coadjoint orbits and descent near elliptic elements.
• When $G$ is connected, the parametrization recovers the result of [ABV92, Th. 11.14], with the same unitary equivalence condition.
• The construction generalizes Knapp and Zuckerman's 'final basic' parameters to non-connected groups, using limit of discrete series in the induction process.
• The character formulas are derived via restriction from Levi subgroups $M'$ to $G$, using translation functors and regularity conditions on the parameters.