# Continuity of the Mackey-Higson bijection

## TL;DR

This paper proves that the Mackey-Higson bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan motion group is a continuous map, linking representation topologies.

## Contribution

It establishes the continuity of the Mackey-Higson bijection, connecting the topologies of the tempered dual and the unitary dual in a new way.

## Key findings

- The topology of the tempered dual $ ilde{G}$ is well-understood.
- The topology of the unitary dual $\hat{G}_0$ is characterized.
- The Mackey-Higson bijection $ ilde{G} o \hat{G}_0$ is continuous.

## Abstract

When $G$ is a real reductive group and $G_0$ is its Cartan motion group, the Mackey-Higson bijection is a natural one-to-one correspondence between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. In this short note, we collect some known facts about the topology of the tempered dual $\widetilde{G}$ and that of the unitary dual $\widehat{G_0}$, then verify that the Mackey-Higson bijection $\widetilde{G} \to \widehat{G_0}$ is continuous.