  For a family of weight functions $h_\kappa$ that are invariant under a
reflection group, the uncertainty principle on the unit sphere in the form of
$$
  \min_{1 \le i \le d} \int_{\mathbb{S}^{d-1}} (1- x_i) |f(x)|^2 h_\kappa^2(x)
d\sigma \int_{\mathbb{S}^{d-1}}\left |\nabla_0 f(x)\right |^2 h_\kappa^2(x)
d\sigma \ge c $$ is established for invariant functions $f$ that have unit norm
and zero mean, where $\nabla_0$ is the spherical gradient. In the same spirit,
uncertainty principles for weighted spaces on the unit ball and on the standard
simplex are established, some of them hold for all admissible functions instead
of invariant functions.
