A finite-dimensional irreducible representation of a noncompact group

The product group

G=\mathbb R\times\operatorname{SU}(2)

is noncompact, nonabelian, and type I, yet has the two-dimensional irreducible unitary representation

\pi:G\longrightarrow U(\mathbb C^2),\qquad \pi(t,k)=e^{it}k.

Here $\operatorname{SU}(2)$ is the special unitary group in its defining representation.

Verification

Scalar multiplication commutes with $k$ , so $\pi(t,k)\pi(s,h)=\pi(t+s,kh)$ . The operators are unitary and depend continuously on their parameters. Any invariant subspace must be invariant under $\operatorname{SU}(2)$ , which acts transitively on the unit sphere of $\mathbb C^2$ ; a nonzero such subspace is therefore all of $\mathbb C^2$ .

The real factor makes $G$ noncompact, and the special-unitary factor makes it nonabelian. Compact and abelian groups, and their direct product here, are type I.

Simplest counterexample

Every topological group has its one-dimensional trivial unitary representation $g\mapsto1$ , which is irreducible. Noncompactness by itself therefore cannot force all irreducible unitary representations to be infinite-dimensional. The displayed two-dimensional example also addresses nontriviality and noncommutativity.

References

Anton Deitmar and Gerrit van Dijk, Trace class groups. Examples 1.5 (first bullet), Theorem 1.7, and Proposition 1.10(a) verify the type I claim via trace-class groups; irreducibility is checked directly above.