Unitary Frobenius reciprocity - Imported Knowlpedia Content

Statement

Let $G$ be a compact group, $H\subseteq G$ a closed subgroup, $\pi$ a unitary representation of $G$ , and $\sigma$ a unitary representation of $H$ . Unitary Frobenius reciprocity is the natural linear isomorphism

\operatorname{Hom}_G\!\left(\pi,\operatorname{Ind}_H^G\sigma\right) \cong \operatorname{Hom}_H\!\left(\pi|_H,\sigma\right),

where both sides consist of bounded intertwining operators and $\operatorname{Ind}_H^G\sigma$ is unitary induction. With normalized Haar measure, the correspondence can be chosen compatibly with Hilbert adjoints. In particular, for irreducible finite-dimensional representations it equates the relevant multiplicities.

The intertwiner correspondence

In the equivariant-function model of unitary induction, an intertwiner $T:\pi\to\operatorname{Ind}_H^G\sigma$ is evaluated at the identity to obtain an $H$ -intertwiner $E(T):\pi|_H\to\sigma$ . Conversely, an $H$ -intertwiner $S$ determines

(T_Sv)(x)=S\bigl(\pi(x^{-1})v\bigr).

Compactness ensures that this function is square-integrable and that the construction is bounded. The two operations are inverse Folland, section “The Frobenius Reciprocity Theorem”.

Multiplicity form

If $\pi$ and $\sigma$ are irreducible, compact-group complete reducibility turns the isomorphism into

\operatorname{mult}\!\left(\pi,\operatorname{Ind}_H^G\sigma\right) = \operatorname{mult}\!\left(\sigma,\pi|_H\right).

For $H=\{e\}$ , this recovers that an irreducible $\pi$ occurs in the regular representation of $G$ with multiplicity $\dim\pi$ .

Noncompact warning

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 6, section “The Frobenius Reciprocity Theorem,” for the compact-group Hilbert-space statement.
George W. Mackey, “Induced Representations of Locally Compact Groups II: The Frobenius Reciprocity Theorem,” Annals of Mathematics 58 (1953), 193–221. DOI record. Relevant: analytic reciprocity and the hypotheses needed beyond the finite-group setting.