Reduced heat kernels on nilpotent Lie groups

LetU be a basis representation of an irreducible unitary representation of a nilpotent Lie groupG inL 2(R k) and letdU denote the representation of the Lie algebrag obtained by differentiation. Ifb 1,...,b d is a basis ofg andB i =dU(b i ) we consider the operatorsH = - åi,j = 1d cij Bi Bj + åi = 1d ci Bi ,H=-dij=1cijBiBj+di=1ciBi whereC=(c ij ) is a real symmetric strictly positive matrix andc i (Sz j)(x) = òRk dykz (x;y)j(y).(Sz)(x)=Rkdyz(x;y)(y)We prove Gaussian off-diagonal bounds and | kt (x;y) | \leqq a(1 Ùemt) - k \mathord/ \vphantom k 2 2 e - l1 t e - d(x;y)2 (4(1 + e)t) - 1 Unknown control sequence '\leqq' for allt>0 and | kz (x;y) | \leqq ae - l1 \operatornameRe z e - b(| x |a + | y |a ) Unknown control sequence '\leqq' for allz C with Rez 1, for some