Chapter Five Evaluation of supertraces for a model operator In this chapter, given a semisimple element G, we evaluate the supertrace of the heat kernel of a hypoelliptic operator acting over p×g. In section 9.10, this operator will appear in the asymptotics as b + of a rescaled version of the operator L X . b In particular we obtain a function J Y 0 k , Y 0 k k (), which will play a fundamental role in our final formula for the orbital integrals. This chapter is organized as follows. In section 5.1, if Y 0 k k (), we intro- duce the hypoelliptic operator P a,Y 0 k , its heat kernel, and a corresponding supertrace J Y 0 k . In section 5.2, by a conjugation of P a,Y 0 k , we obtain a simpler operator Q a,Y 0 k , where p × p and k have been decoupled. The operator Q a,Y 0 k splits naturally into a scalar part and a matrix part. In section 5.3, we evaluate the trace of the heat kernel of the scalar part of Q a,Y 0 k . In section 5.4, we compute the supertrace of the matrix part of the heat kernel. Finally, in section 5.5, we give an explicit formula for J Y 0 k . 5.1 The operator P a,Y 0 k and the function J Y 0 k Let be a semisimple element of G, which is written in the form given in (4.2.1). Definition 5.1.1. Let z () denote another copy of z (), and let z () denote the corresponding copy of the dual of z (). If · z () , let · be the corresponding element in z () . Also c (z ()) denotes the Clifford algebra of z () , B| z() . We will continue using the notation of chapter 2. In particular, E is taken as in section 2.12. Our operators act on C p × g, · (g ) · z () E . Of course, p × g = p × (p k) . (5.1.1) We denote by y the tautological section of the first copy of p in the right-hand side of (5.1.1), and by Y g = Y p + Y k the tautological section of g = p k. Let dy be the volume form on p, let dY g be the volume form on g = p k.