Induced Representation of Topological groupoids

TL;DR

The paper extends the classical induced representation theory to topological groupoids by constructing $(ind^{G}_{H}(\sigma),\mu)$ for a closed wide subgroupoid $H$ of a locally compact second countable groupoid $G$ with a Haar system, using a full equivariant measure system on $G/H$. It develops the induced representation via a continuous field of Hilbert spaces and a full Hilbert $C_{0}(G^{0})$-module, and establishes fundamental properties including induction in stages, a Mackey-style tensor product theorem for groupoids, and Frobenius reciprocity for compact transitive groupoids. Key contributions are the explicit construction of the induced representation, functorial behavior under equivalence, and the extension of core representation-theoretic results (tensor products and reciprocity) to the groupoid setting. These results pave the way for applying groupoid representation theory to Morita equivalence contexts and groupoid C*-algebras, enabling a robust framework for analyzing representations across groupoid-based models.

Abstract

Let $G$ be a locally compact second countable groupoid with a Haar system. In this article, we introduce the induced representation of $G$ from a continuous unitary representation of a closed wide subgroupoid $H$ with a Haarsystem provided there exists a full equivariant system of measures $μ=\{μ^{u}\}_{u\in G^{0}}$ on $G/H$. We prove some basic properties of induced representation and a theorem on induction in stages. A groupoid version of Mackey's tensor product theorem is also provided. We also prove a groupoid version of Frobenius Reciprocity theorem on compact transitive groupoids.

Theorems & Definitions (37)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof