In this map, the heroes can only place in an adjacent empty space the shadow key, but they can not pick up the shadow key?

Thanks for your help

Correct. A master monster adjacent to the key can still pick up the key, as a hero still adjacent to a search can pick up a search.

The rules don't say anything about heroes placing the key. They say "A figure may place the Shadow Key..." Since heroes don't have the ability to pick up the key, they won't ever have the opportunity to place it anywhere. This rule basically exists for the Overlord to be able to hand off the key from one master monster (or Baron Zachareth) to another if so desired.