Given a compact Lie group $K$, and a maximal torus $T\leq K$, and choose a positive Weyl chamber $\mathfrak t^*_+\subset \mathfrak k^*$. Then $\mathfrak t^*_+$ can be decomposed into disjoint open faces, $\mathfrak t^*_+=\sqcup_{\sigma\in\Sigma}\sigma$. The question is to prove $K_\sigma$ is well-defined as the centralizer of any $\xi\in\sigma$, i.e. for any $\xi,\xi'\in\sigma$, $K_\xi=K_{\xi'}$.
This is what I read from Symplectic Implosion, Guillemin, Jeffrey, Sjamaar 's paper (this can be found after definition 2.1).