Reduced heat kernels on homogeneous spaces

If S is the semigroup generated by an n-th order strongly elliptic operator on L-p(X; dx) associated with the left regular representation of a unimodu lar Lie group G in the homogeneous space X = G/M, where M is a compact subg roup of G, and k is the reduced heat kernel of S defined by (S(t)phi)(x) = integral(x) k(t)(x; y) phi(y) dy then we prove Gaussian upper bounds for rct and all its derivatives. For reduced heat kernels associated with irreducible unitary representation s on nilpotent Lie groups we prove similar Gaussian bounds.