Representations of the D=2 Euclidean and Poincar\'e groups

We present an explicit construction of the unitary irreducible representations of the two-dimensional Euclidean and Poincar\'e groups, together with their Spin double covers, by means of Mackey's theory of induced representations for semidirect products. In dimension D=2, the simplicity of the corresponding little groups allows a complete explicit treatment of momentum orbits, equivariant wavefunctions, and representation operators. For the Euclidean group, the matrix elements of the infinite-dimensional representations are expressed in terms of Bessel functions. For the Poincar\'e group, the richer Lorentzian orbit structure leads to matrix elements involving modified Bessel and Hankel functions and, in some cases, tempered distributions, requiring the use of Rigged Hilbert Spaces. This work illustrates the interplay among induced representations, harmonic analysis on Lie groups, Spin geometry, and special functions in a fully explicit relativistic setting.