Representation Theory in Clifford Analysis

This chapter introduces contemporary Clifford analysis as a local function theory of first-order systems of PDEs invariant under various Lie groups. A concept of a symmetry of a system of partial differential equations is the key point of view, it makes it possible to use many efficient tools from the theory of representations of simple Lie groups. A systematic approach is based on a choice of a Klein geometry (a homogeneous space M ≃ G∕P with P being a Lie subgroup of a Lie group G) and on a notion of a homogeneous (invariant) differential operator acting among sections of associated homogeneous vector bundles. The main example is the conformal group G = S p i n ( m + 1 , 1 ) $$G = Spin(m + 1,1)$$ acting on the sphere S m and the Dirac operator. The chapter contains a description of basic properties of solutions of such systems and lists many various examples of the aforementioned scheme. The introductory sections describe the Clifford algebra, its spinor representations, the conformal group of the Euclidean space, the Fegan classification of the conformally invariant first order differential operators, and a series of examples of such operators appearing in the Clifford analysis. They include the Dirac equation for spinor-valued functions, the Hodge and Moisil–Théodoresco systems for differential forms, the Hermitian Clifford analysis, the quaternionic Clifford analysis, the (generalized) Rarita–Schwinger equations, and the massless fields of higher spin. A different point of view to these first-order systems presents them as special solutions of the (twisted) Dirac equation. The second part of this chapter contains a description of basic properties of solutions of the Dirac equation, including the Fischer decomposition of spinor-valued polynomials, the Howe duality, the Taylor and the Laurent series for monogenic functions. The last two sections contains a description of the Gelfand–Tsetlin bases for the spaces of (solid) spherical monogenics and a discussion of possible future direction of research in Clifford analysis.