Chapter Two The hypoelliptic Laplacian on X = G/K The purpose of this chapter is to construct the hypoelliptic Laplacian L X , b > b 0 acting on the total space of a vector bundle T X N g over X = G/K. The operator L X will be obtained using general constructions involving b Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant [Ko97]. This chapter is organized as follows. In section 2.1, we introduce a pair (G, K), the symmetric space X = G/K, and the vector bundles T X, N on X. In section 2.2, we construct the canonical flat connection on T X N g. In section 2.3, we define Clifford algebras associated with g. In section 2.4, we obtain flat connections on · (T X N ). In section 2.5, we construct the Casimir operator for G. In section 2.6, we introduce the canonical 3-form g , and its image in the Clifford algebras of g. In section 2.7, we construct the Dirac operator of Kostant.