For any finite reflection group W on RN and any irreducible W-module V there is a space of polynomials on RN
Nearby in the stack
with values in
V
. There are Dunkl operators parametrized by a multiplicity function, that is, parameters associated with each conjugacy class of reflections. For certain parameter values, called singular, there are nonconstant polynomials annihilated by each Dunkl operator. There is a Gaussian bilinear form on the polynomials which is positive for an open set of parameter values containing the origin. When
W
has just one class of reflections and
dimV>1
this set is an interval bounded by the positive and negative singular values of respective smallest absolute value. This interval is always symmetric around
0
for the symmetric groups. This property does not hold in general, and the icosahedral group
H3
provides a counterexample. The interval for positivity of the Gaussian form is determined for each of the ten irreducible representations of