We give a geometric proof that any minimal length element in a (twisted) conjugacy class of a finite Coxeter group has remarkable properties with respect to conjugation, taking powers in the associated braid monoid and taking the centralizer in .
Xuhua He. Sian Nie. "Minimal length elements of finite Coxeter groups." Duke Math. J. 161 (15) 2945 - 2967, 1 December 2012. https://doi.org/10.1215/00127094-1902382