It is known from Dyson, Macdonald $et$ $al$ that the constant term asscoiated to the expansion of the following expression
\begin{equation} \prod_{\alpha \in \Delta} (1-e^{\alpha})^k \end{equation}
is given by
\begin{equation} \prod_{i} \left( \begin{array}{ccc} k d_i\\ k \end{array} \right) \end{equation}
where $d_i$ are the fundamental invariants of the Weyl group. I wonder if there is an intuitive way to understand this expression; it only depends on the group and on the value of external parameter $k$. What kind of group invariants it tries to compute, given a sequence of $k$? For example, if $k=1$ then one is counting the product of degree of invariant polynomials; but if $k=2$, what quantity of the group is reflected in the constant term?
