Spinor-valued [formula omitted]-spaces on [formula omitted] and representations of [formula omitted

Abstract

We show that the L2-spaces of spinor-valued functions on Sm−1 provide models for the unitary principal series representations of Spin+(1,m) which are characterized by the feature that they contain the fundamental spinor representation(s) of Spin(m). The novelty of our approach is that we use tools characteristic of hypercomplex analysis: the group Spin+(1,m) is realized as a group of Vahlen matrices and the Harish-Chandra modules of such representations are expressed in terms of spherical monogenics which generalize the classical spherical harmonics. By switching to the framework of hypercomplex analysis on the sphere we are able to extend (part of) the well-known results for the (scalar) spherical principal series representations of SO+(1,m) to the spinor setting. In the process we introduce a Clifford algebra-valued Riesz transform on the sphere.