Symmetries in quaternionic analysis

This survey-type paper deals with the symmetries related to quaternionic analysis. The main goal is to formulate an $SU$ (2) invariant version of the theory. First, we consider the classical Lie groups related to the algebra of quaternions. After that, we recall the classical Spin(4) invariant case, that is Cauchy-Riemann operators, and recall their basic properties. We define the $SU$ (2) invariant operators called the Coifman-Weiss operators. Then we study their relations with the classical Cauchy-Riemann operators and consider the factorization of the Laplace operator. Using $SU$ (2) invariant harmonic polynomials, we obtain the Fourier series representations for quaternionic valued functions studying in detail the matrix coefficients.