Chapter Eight The case where [ k () , p 0 ] = 0 The purpose of this chapter is to evaluate explicitly the Gaussian integral that appears in the right-hand side of our formula in (6.1.2) for the orbital integrals of the heat kernel, when is nonelliptic and [k () , p 0 ] = 0. Our computations can be easily extended to the more general kernels considered in chapter 6. It is remarkable that the index formulas of chapter 7 play here a key role. This chapter is organized as follows. In section 8.1, we consider the case where G = K. In section 8.2, we compute explicitly the Gaussian integral when is nonelliptic and [k () , p 0 ] = 0. Finally, in section 8.3, the case where G = SL 2 (R) is worked out. We recover the evaluation in [McK72] of the trace of the scalar heat kernel in Selberg's trace formula. We make the same assumptions as in chapters 6 and 7, and we use the corresponding notation.