Heat kernels on Lie groups

Let G be a Lie group with a fixed left invariant Haar measure and Λ be a right or left invariant, real, formally negative, elliptic differential operator of second order and without constant term. Then we call the “fundamental solution of the Cauchy problem” for the equation ∂u ∂t = Λu heat kernel. We prove that these heat kernels have most of the useful properties of their special case, the Gauss kernel (2 √ πt ) − n exp ( −∥x∥ 2 4t ) .