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The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group

Jean-Yves Ducloux

Abstract

Let G be a possibly disconnected reductive real Lie group. In this paper, I parametrize the set of irreductible tempered characters of G. I then describe these characters using certain ``Kirillov's formulas,'' based on the descent method near each elliptic element in G. If G is linear and connected, the parameters I use are ``final basic'' parameters in the sense of Knapp and Zuckerman.

The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group

Abstract

Let G be a possibly disconnected reductive real Lie group. In this paper, I parametrize the set of irreductible tempered characters of G. I then describe these characters using certain ``Kirillov's formulas,'' based on the descent method near each elliptic element in G. If G is linear and connected, the parameters I use are ``final basic'' parameters in the sense of Knapp and Zuckerman.
Paper Structure (17 sections, 23 theorems)

This paper contains 17 sections, 23 theorems.

Table of Contents

  1. . The “linear form” parameters
  2. The parameters λ ∈ g̃*_{reg,G}
  3. The measures _{G ⋅ λ}
  4. Fixed points of an elliptic element
  5. . The “projective representation” parameters
  6. About special-metalinear and metaplectic groups
  7. The parameters τ ∈ XInd_G(λ)
  8. Integrability condition
  9. . Construction of representations
  10. The case where G is connected
  11. The representations T_{λ,a*+,τ+}G
  12. The injection G ⋅ (λ,τ)↦ T_{λ,τ}G from G \\ XInd_G to Ĝ
  13. . Characters of representations
  14. Character restriction formula
  15. Transition from T_{λ,τ_M'}M' to T_{λ,a*+,τ+}G
  16. ...and 2 more sections

Key Result

Lemma 1..4

We place ourselves in the situation of the previous definition. Let $\,t \!\in\! \left]0,1\right]$. Set $\;\nu_t = \nu + t\epsilon \, \rho_{\mathfrak{g}(\lambda)(\mathrm{i} \rho_{\mathcal{F}^+}),\mathfrak{h}}$, $\mu_t = \mu - 2\mathrm{i} t\rho_{\mathfrak{g}(\nu_+),\mathfrak{h}}$ and $\lambda_t = \mu

Theorems & Definitions (50)

  • Definition 1..1
  • Remark 1..2
  • Definition 1..3
  • Lemma 1..4
  • Lemma 1..5
  • Definition 2..1
  • Proposition 2..2
  • Definition 2..3
  • Remark 2..4
  • Proposition 2..5
  • ...and 40 more