# Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules [4pt] of spherical monogenics in dimension 3

## Roman Lávička

Address: Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

E-mail: lavicka@karlin.mff.cuni.cz

Abstract: Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules. As finite-dimensional irreducible ${\mathfrak{sl}}(2,{\mathbb{C}})$-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. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

AMSclassification: primary 30G35; secondary 33C50.

Keywords: spherical monogenics, orthogonal basis, Legendre polynomials, \mathfrak{sl}(2,{\mathbb{C}})-module.