I could come next week, or the week after.

His people loved and remembered him for centuries after.

【答】英语中，有的“小品词”既是介词又是副词，身兼两职。区分的办法很简单：有宾语时是介词，无宾语时是副词。

你问的两个句子中的after，都是副词。当然，为了更好的理解句子意思，可以看作其后省略了已知的有关宾语。如：

I could come next week, or the week after.

= I could come next week, or the week after next week.

His people loved and remembered him for centuries after.

=His people loved and remembered him for centuries after his death.

既是介词又是副词的“小品词”一览：