Representations of SU(2)in the plane

We present a family of unitary irreducible representations of SU(2) realized in the plane, in terms of the Laguerre polynomials. These functions are similar to the spherical harmonics realized on the sphere. Relations with the space of square integrable functions defined on the plane, L^2(R^2, are analyzed. The realization of representations of Lie groups in spaces with intrinsic symmetry different from the one of the groups is discussed.