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Wave Front Sets of Nilpotent Lie Group Representations

Julia Budde, Tobias Weich

Abstract

Let $G$ be a nilpotent, connected, simply connected Lie group with Lie algebra $\mathfrak g$, and $π$ a unitary representation of $G$. The goal is to prove that the wave front set of $π$ coincides with the asymptotic cone of the orbital support of $π$, i.e. $\mathrm{WF}(π)=\mathrm{AC}(\bigcup_{σ\in \mathrm{supp}(π)}\mathcal O_σ)$, where $\mathcal O_σ\subset i\mathfrak g^\ast$ is the coadjoint orbit associated to the irreducible unitary representation $σ\in \hat{G}$ by Kirillov.

Wave Front Sets of Nilpotent Lie Group Representations

Abstract

Let be a nilpotent, connected, simply connected Lie group with Lie algebra , and a unitary representation of . The goal is to prove that the wave front set of coincides with the asymptotic cone of the orbital support of , i.e. , where is the coadjoint orbit associated to the irreducible unitary representation by Kirillov.

Paper Structure

This paper contains 8 sections, 20 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wave Front Sets
  4. Nilpotent Lie Groups
  5. Proof of Theorem \ref{['thm:wf=ac']}
  6. Proof of the Inclusion AC subset WF
  7. Proof of the Inclusion WF subset AC
  8. Example: Restriction of discrete series of $\mathrm{SU}(1,2)$

Key Result

Theorem 1

Let $G$ be a nilpotent, connected, simply connected Lie group and $\pi$ a unitary representation of $G$. Then Where the orbit support def:orbital_support is defined by the Kirillov orbits $\mathcal{O}_\sigma\subset i\mathfrak g^*$ of the unitary irreducible representation $\sigma$.

Figures (3)

  • Figure 1: Orbits of $G_0$ and $G$ in Case II of Theorem \ref{['thm:sigmaOg0']}
  • Figure 2: The choice of $l_m$ and $N_m$.
  • Figure 3: Visualization of the Cartan subalgebra $\mathfrak h$: The compact root is given by $\alpha_3$ and the dashed line represents the reflection hyperplane of the compact Weyl group. The Harish Chandra parameter of the holomorphic discrete series lie in the green region, those of the anti-holomorphic discrete series in the red region and the non-holomorphic ones in the blue region.

Theorems & Definitions (34)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: see howe and harrisheolaf
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Theorem 2: see corgre
  • Theorem 3: see corgre
  • Remark 2.6
  • ...and 24 more