  In this notes, we will give an exposition of some results on the method of
partial conjugation action. We first discuss the partial conjugation action of
a parabolic subgroup of a Coxeter group. We then discuss some applications to
Lusztig's $G$-stable pieces and its affine generalization. We also discuss some
recent work on the $\s$-conjugacy classes of loop groups and affine
Deligne-Lusztig varieties.
