Heat Kernel on Homogeneous Bundles over Symmetric Spaces

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on the two-dimensional sphere S (2) and the hyperbolic plane H (2). We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.