Spherical harmonic polynomials for higher bundles

We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group Spin(n), where important tools are Spin(n)-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators. 0 Introduction Spherical harmonic polynomials or spherical harmonics are polynomial solutions of the Laplace equation φ(x) = ∑ ∂φ/∂xi = 0 on R . These are fundamental and classical objects in mathematics and physics. It is natural that we consider vector-valued spherical harmonic polynomials. For example, the polynomial solutions of the Dirac equation Dφ(x) = 0 on R are studied in Clifford analysis (see [6], [8], and [14]). They are spinor-valued polynomials and called spherical monogenics. We also have other examples in [5], [7], [9], and [12], where we can give spectral information of some basic operators on sphere. Recently, the first-order Spin(n)-equivariant differential operators have been studied like Dirac operator and Rarita-Schwinger operator (see [1]-[5], [10], and [11]). These operators are called higher spin Dirac operators or Stein-Weiss operators. In this paper, we give a method to analyze polynomial sections for natural bundles on R by using higher spin Dirac operators and Clifford homomorphisms. Here, Clifford homomorphism is a natural generalization of Clifford algebra given in [10] and [11]. Let S (resp. H) be the spaces of polynomials (resp. harmonic polynomials) with degree q on the n-dimensional Euclidean space R. We know ∗Department of Mathematical Sciences, Waseda University, 3-4-1 Ohkubo, Shinjukuku, Tokyo, 169-8555, JAPAN. e-mail address : [email protected]