  We revisit the representation theory of the quantum double of the universal
cover of the Lorentz group in 2+1 dimensions, motivated by its role as a
deformed Poincar\'e symmetry and symmetry algebra in (2+1)-dimensional quantum
gravity. We express the unitary irreducible representations in terms of
covariant, infinite-component fields on curved momentum space satisfying
algebraic spin and mass constraints. Adapting and applying the method of group
Fourier transforms, we obtain covariant fields on (2+1)-dimensional Minkowski
space which necessarily depend on an additional internal and circular
dimension. The momentum space constraints turn into differential or
exponentiated differential operators, and the group Fourier transform induces a
star product on Minkowski space and the internal space which is essentially a
version of Rieffel's deformation quantisation via convolution.
