  We derive the heat kernel for arbitrary tensor fields on S^3 and (Euclidean)
AdS_3 using a group theoretic approach. We use these results to also obtain the
heat kernel on certain quotients of these spaces. In particular, we give a
simple, explicit expression for the one loop determinant for a field of
arbitrary spin s in thermal AdS_3. We apply this to the calculation of the one
loop partition function of N=1 supergravity on AdS_3. We find that the answer
factorizes into left- and right-moving super Virasoro characters built on the
SL(2, C) invariant vacuum, as argued by Maloney and Witten on general grounds.
