The investigator is another of the terms used for the character of the lady Chief Inspector. She is at the museum when you arrive, and arrests you that night. Fortunately, she needs your help.
The key to the investigator's chest on Counterfeit Island is hidden in the bushes to the right of the cottage. You need to use a stick to reach it and then unlock the investigator's chest to get the next clue.
night right after you get caught the thief is the investigator
it's the little cabin at the very end of the island, if you need help check your map :)
The investigator's house is all the way right (as far as you can go).
You don't defeat the investigator -- Black Widow, an art thief, is posing as the investigator to steal The Scream. When you find the painting in the investigator's cottage, she captures you and takes you to her underground lair. (see the related question)
Go inside the house and peel the painting off the one on the top floor
You will not see the investigator anywhere after you finish the security tapes in the museum. However, now that you have the key to her house (in the Country Side), you should find something there.
Jump on the light hanging above the Scream to catch the theif and wait for the "investigator" with the police.
The lady Chief Inspector is nowhere to be found once you leave the Security Room with the photo of the art thief.
Once you have found and chased the thief, the investigator (Chief Inspector) is nowhere to be found. Once you have the key from the curator, you can open the Inspector's house in the Countryside to locate the painting.
The investigator (aka Chief Inspector) can be found outside the museum, or in her lair after she is revealed as the Black Widow. You get into her lair by using the curator's key to get into her house, then clicking on the peeling corner of the painting.
The investigator (who has been renamed the Chief Inspector) is nowhere to be found. The curator was strangely vague about who might be secretly in cahoots with the art thieves. But she gave you the key to keep you looking for clues.