Spectral Theory and Representations of Nilpotent Groups

We give an estimate of the number $N(\lambda)$ of eigenvalues $<\lambda$ for the image under an irreducible representation of the ``sublaplacian'' on a stratified nilpotent Lie algebra. We also give an estimate for the trace of the heat-kernel associated with this operator. The estimates are formulated in term of geometrical objects related to the representation under consideration. An important particular case is the Schr\"odinger equation with polynomial electrical and magnetical fields.