  The starting point of our analysis is an old idea of writing an eigenfunction
expansion for a heat kernel considered in the case of a hypoelliptic heat
kernel on a nilpotent Lie group $G$. One of the ingredients of this approach is
the generalized Fourier transform. The formula one gets using this approach is
explicit as long as we can find all unitary irreducible representations of $G$.
In the current paper we consider an $n$-step nilpotent Lie group $G_{n}$ as an
illustration of this technique. First we apply Kirillov's orbit method to
describe these representations for $G_{n}$. This allows us to write the
corresponding hypoelliptic heat kernel using an integral formula over a
Euclidean space. As an application, we describe a short-time behavior of the
hypoelliptic heat kernel in our case.
