Representation of the infinite dihedral group

Its irreducible complex representations have dimension at most two; the two-dimensional family pairs eigenvalues $\lambda$ and ${\lambda }^{-1}$ of the rotation generator.

Every nonfaithful finite-dimensional complex representation of the infinite dihedral group is a direct sum of subrepresentations of dimension at most two. Its kernel contains a nonzero power of the rotation, so the representation factors through a finite dihedral group. Maschke's theorem makes that quotient representation semisimple, and all irreducible complex representations of a finite dihedral group have dimension at most two.

Let the rotation and reflection generators act on polynomials of degree at most two byThen $S^2=I$ and $STS=T^{-1}$. The operator $T$ has one size-three unipotent Jordan block, so this three-dimensional representation cannot split into invariant subspaces of dimension at most two.

For $\lambda \ne \pm 1$, the rotation acts by $\mathrm{diag}(\lambda ,{\lambda }^{-1})$ and the reflection interchanges the two eigenlines.