Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3

Abstract

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as sl(2,C)-modules. As finite-dimensional irreducible sl(2,C)-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Guerlebeck. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.