The hypoelliptic Laplacian on X = G/K

Abstract

This chapter constructs the hypoelliptic Laplacian ℒbX > 0 acting on the total space of a vector bundle TX ⊕ N ≃ g over the symmetric space X = G/K. The operator ℒbX is obtained using general constructions involving Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant. The end result is the elliptic Laplacian 𝓛 ^X on X as well as the hypoelliptic Laplacian ℒbX, which is a second order hypoelliptic operator acting on X^. Among other things, this chapter gives a key formula relating 𝓛 ^XℒbX, as well as various formulas involving the operator ℒbX+La.