THE SPINOR HEAT KERNEL IN MAXIMALLY SYMMETRICAL SPACES

The heat kernel K(x, x', t) of the iterated Dirac operator on an N-dimensional simply connected maximally symmetric Riemannian manifold is calculated. On the odd-dimensional hyperbolic spaces K is a Minakshisundaram-DeWitt expansion which terminates to the coefficient a(N-1)/2 and is exact. On the odd spheres the heat kernel may be written as an, image sum of WKB kernels, each term corresponding to a classical path (geodesic). In the even dimensional case the WKB approximation is not exact, but a closed form of K is derived both in terms of (spherical) eigenfunctions and of a "sum over classical paths." The spinor Plancherel measure u(lambda) and zeta-function in the hyperbolic case are also calculated. A simple relation between the analytic structure of mu on H(N) and the degeneracies of the Dirac operator on S(N) is found.