The police are re-investigating the old case due to new evidence that has emerged.

re-investigating 是动词 re-investigate 的现在进行时形式。re-investigate 由前缀 re-（表示"再次"）和动词 investigate（调查）组成，字面意思是"再次调查"。 investigate 本身由词根 vestige（痕迹）和动词后缀 -ate 组成，原意是"追踪痕迹"，引申为"调查"。 re-investigating 表示"重新调查"的动作正在进行中。