I study Plancherel measure for Laplace operators on a nilpotent Lie group. Such a measure $\mu$, introduced by Christ, satisfies the following formula $\|K_{F(L)}\|^2_{L^2}= \int_0^{\infty}|F(\lambda)|^2\;d\mu (\lambda),$ where $K_{F(L)}$ is the kernel of the multiplier operator $F(L)$ of a self-adjoint operator $L$ defined on a Lie group. The Plancherel measure gives us a precise description of the $L^2 \to L^{\infty}$ norms of the spectral projectors of the operator $L$. These norms of spectral projectors were investigated in the case of an elliptic operator on a compact manifold by Hörmander and Sogge. I proved the smoothness of the Plancherel measure for a certain class of Laplace operators and showed that this leads to an improvement of Alexopoulos' multiplier theorem.