Induction in stages for unitary representations

Statement

Let $G$ be a second-countable locally compact Hausdorff group, let $H\subseteq K\subseteq G$ be closed subgroups, and let $\sigma$ be a strongly continuous unitary representation of $H$ . Induction in stages is the natural unitary equivalence

\operatorname{Ind}_H^G\sigma \ \simeq\ \operatorname{Ind}_K^G\bigl(\operatorname{Ind}_H^K\sigma\bigr).

Here every induction is the unitary induction constructed from the canonical quasi-invariant measure class on the relevant homogeneous space. The equivalence is canonical up to the harmless choices used to realize those measure classes and intertwines the $G$ -actions.

Geometric mechanism

The quotient map $G/H\to G/K$ has fibers modeled on $K/H$ . Disintegrating a measure in the class on $G/H$ first over $G/K$ and then over $K/H$ identifies an $L^2$ -section over $G/H$ with an $L^2$ -section over $G/K$ whose values are themselves $L^2$ -sections over $K/H$ . The Radon–Nikodym and modular factors combine exactly to give the direct-induction normalization Folland, Chapter 6, “Pseudomeasures and Induction in Stages”.

Consequences

The theorem permits a complicated inducing subgroup to be approached through an intermediate subgroup without changing the resulting representation. In real reductive groups it is the formal reason that induction from a parabolic can be decomposed through a larger parabolic, with the appropriate normalized induction at each step. It also makes the assignment of induction functors coherent for longer chains of closed subgroups.

Finite-group model and analytic content

For finite groups, the equivalence reduces to the associativity isomorphism

\mathbb C[G]\otimes_{\mathbb C[K]} \bigl(\mathbb C[K]\otimes_{\mathbb C[H]}V\bigr) \cong \mathbb C[G]\otimes_{\mathbb C[H]}V.

For locally compact groups, tensor associativity alone does not prove the Hilbert-space statement: quotient measures, completion, and modular corrections are essential parts of the theorem.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 6, section “Pseudomeasures and Induction in Stages.”
George W. Mackey, The Theory of Unitary Group Representations, University of Chicago Press, 1976. Library record. Relevant: Chapter 3 on induced representations and induction in stages.