The purpose of this chapter is to construct semisimple orbital integrals associated with the heat kernel for the hypoelliptic Laplacian . By making b → 0, we show that the corresponding supertrace coincides with the orbital integral associated with the standard elliptic heat kernel. In the next chapters, the evaluation of these elliptic orbital integrals will be obtained by making b → +∞.
This chapter is organized as follows. In section 4.1, we introduce an algebra of invariant kernels q