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Representations of currents taking values in $PGL(2, Q_q)$

Maria Gabriella Kuhn

Abstract

Let $G=PGL(2,Q_q)$. In this paper we shall investigate the group of measurable currents taking values in $G$. The key observation is that $G$ is acting by automorphisms on a homogeneous tree, which will play the role of the upper half plane in the case of $PSL(2,R)$.

Representations of currents taking values in $PGL(2, Q_q)$

Abstract

Let . In this paper we shall investigate the group of measurable currents taking values in . The key observation is that is acting by automorphisms on a homogeneous tree, which will play the role of the upper half plane in the case of .
Paper Structure (14 sections, 15 theorems, 61 equations)

This paper contains 14 sections, 15 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The tree associated to $PGL(2,F)$
  3. Representations of $G$ and Representations of $\mathcal{A}ut(\mathcal{T})$
  4. The boundary $\Omega$ and the special representation of $\mathcal{A}ut(\mathcal{T})$.
  5. The special representation $\operatorname{sp}_-$ of $\mathcal{A}ut(\mathcal{T})$
  6. The cocycle
  7. Another Realization of $\Xi$.
  8. The case of $SL(2,\Bbb Q_q)$
  9. The case of $PGL(2,\Bbb Q_q)$
  10. The case of $\mathcal{A}ut(\mathcal{T})$
  11. The representation of $\mathcal{G}^0$
  12. Representations of $\mathcal{G}^0$
  13. The representation space for $\mathcal{G}^0$
  14. The group $\mathcal{G}$ of bounded measurable currents

Key Result

Theorem 3.1

Assume that $\beta:G\to H_\pi$ is a nontrivial cocycle and that $(\pi,H_\pi)$ is irreducible. Then $\pi$ cannot be separated from the identity in the Fell topology.

Theorems & Definitions (28)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 18 more