Conjugate and contragredient unitary representations

Definition

Let $(\pi,\mathcal H)$ be a strongly continuous unitary representation of a group $G$ . Its conjugate representation acts on the conjugate Hilbert space $\overline{\mathcal H}$ by

\overline{\pi}(g)\overline{\xi}=\overline{\pi(g)\xi}.

Its contragredient representation acts on the continuous linear dual $\mathcal H'$ by

\pi^\vee(g)\lambda=\lambda\circ\pi(g)^{-1}.

Both actions are strongly continuous and unitary for their natural Hilbert structures. They are canonically unitarily equivalent through the Riesz map $\overline{\xi}\mapsto\langle\,\cdot\,,\xi\rangle$ , with the inner product taken linear in the first variable.

Coefficients

For $\xi,\eta\in\mathcal H$ , the matrix coefficients of the conjugate representation satisfy

c^{\overline{\pi}}_{\overline{\xi},\overline{\eta}}(g) =\overline{c^\pi_{\xi,\eta}(g)}.

Under the Riesz identification, the contragredient action sends the functional represented by $\eta$ to the functional represented by $\pi(g)\eta$ . This calculation is the reason the inverse occurs in the dual-action formula.

Relationship to equivalence

The representation $\pi$ is self-conjugate when it is unitarily equivalent to $\overline{\pi}$ . This condition does not by itself decide whether a compatible real or quaternionic structure exists; those finer alternatives require an antiunitary intertwiner and a condition on its square. Conjugation also reverses scalar phases, which is important even when $\pi$ and $\overline{\pi}$ act on Hilbert spaces of the same dimension.

Conventions and scope

For a general Banach-space representation, the dual action naturally lives on the Banach dual and need not share the Hilbert-space properties used above. In finite-dimensional algebra, “contragredient” is often written using inverse transpose matrices. The Riesz identification is conjugate-linear on $\mathcal H$ but becomes linear when its source is $\overline{\mathcal H}$ Folland, §3.1.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §3.1 on unitary representations and their conjugates.