In the paper Constant terms in powers of a Laurent
we proved the conjecture of Mathieu for commutative K.
In response to a question of Springer we now observe that we may drop the
condition that K be connected. So the result still holds for a commutative
compact group K. The reason is very simple. For every connected component
one can build a generating function F as in the paper. The generating function
for the full group is simply the sum of the generating functions for the
components. For one component the generating function is either zero (when
the origin is outside the Newton polytope) or it is asymptotic to -1/t
by the arguments of the paper.
Here we have normalized the Haar measure so that a component has measure one.
The total generating
function is thus asymptotic to m times -1/t, where m is the number of
components for which the origin lies on the Newton polytope.
The reasoning ends as before. It is amusing to check what this proof
amounts to when K is finite. It also becomes clear that there are variations
in which K is replaced by a space with a group acting on it.
Wilberd van der Kallen