JK Rowling directs armchair sleuths down relevant but invisible path. Harry Potter author encourages fans to ponder why Dumbledore was in possession of James Potter's cloak.
JK Rowling recently updated her web site (jkrowling.com) with a question she says she has never been asked by Harry Potter fans: "Why did Dumbledore have James' invisibility cloak at the time of James' death, given that Dumbledore could make himself invisible without a cloak? . . . There IS a significant - even crucial - answer."
Realizing You're Asking the Right Question
It's a question fans have been asking each other, at least, since the first book, Harry Potter and the Philosopher's Stone (Sorcerer's Stone to Yanks), was published. Speculation on fan forums has run the gamut from Dumbledore is James Potter's real father (a theory shot down by Rowling in the past) to James must have left it with Dumbledore intending to retrieve it later. It seems unlikely however that anyone thought it was a point 'crucial' to the series.
Connection Between Dumbledore and Invisibility Cloaks
There has been speculation in the past that Dumbledore is part Demiguise and this is what allows him to become invisible at will. This theory is based on the additional similarity that Demiguises (described in Fantastic Beasts & Where To Find Them (page 9), though never mentioned in the novels themselves) have long, silvery hair as does everyone's favorite elderly wizard. It is Demiguise hair that is spun into invisibility cloaks.
Lines Of Thought To Pursue
If there's any merit to this line of thinking about Dumbledore and invisibility cloaks, suppose Dumbledore had some skill in making and repairing invisibility cloaks. James' cloak may have been torn or tattered in any number of escapades. He was a Marauder and a member of the original Order of the Phoenix, after all. We do not know what, if anything, he did for a living but there's a good chance he was an auror as well. What if Dumbledore was repairing the invisibility cloak for James with the intention of returning it to him in the near future? Once James died, Dumbledore just held onto it until giving it to Harry in his first year at Hogwarts. How would this be significant though or 'crucial' to the series though?
Is it possible that Dumbledore deliberately deprived James of the invisibility cloak, with the purpose of leaving him more vulnerable to Voldemort? No. Rowling has confirmed in various interviews that Dumbledore's character often speaks for her and is the embodiment of all that is good within the world of her novels. So there must have been a reason for his possession of the cloak which is not only innocuous but actively intended for the good.
Another interesting fact from the description of the Demiguise is that this creature can be seen - even when in invisible mode - by certain wizards who hunt it for its hair. We also know that Mad-Eye Moody's mad eye can see through invisibility cloaks. In the fifth novel, Harry Potter and the Order of the Phoenix, Mr. Weasley is attacked while wearing an invisibility cloak - by Voldemort, who was possessing a snake at the time.
If Voldemort and/or certain others can see through invisibility cloaks, perhaps Dumbledore deliberately kept the item from James to keep him from doing something rash out of a false sense of security. Perhaps taking the invisibility cloak away from James ensured that he would stay in his hidden location with his wife and child, instead of running out to try to get to Voldemort before Voldemort got to the Potters. This thoughtful plan of course would have backfired when the Fidelius Charm was broken and the Secret Keeper, Peter Pettigrew, told Voldemort where the Potters were hiding. Then there was nowhere to hide, not even under a cloak. It would be an ironic twist if Voldemort could not in fact see through invisibilty cloaks. The snake sensed Mr. Weasley's presence and the invisibilty cloak had slipped off Mr. Weasley entirely just before he was bitten.
Alternatively, what if Dumbledore had borrowed the invisibility cloak from James to give to someone else in the Order to use - someone who was also at Godric's Hollow the night the Potters were killed. Snape or Pettigrew? Or were both of them there? Pettigrew with Voldemort and Snape under the cloak spying for the Order? How Dumbledore would have known something was going to happen that night, we do not know. Nor do we know how he or Sirius even knew where the Potters were hiding since neither was chosen to be the Secret Keeper. We do know that invisibility cloaks are rare and they are handed off amongst Order members as needed for dangerous assignments.
This last line of thinking in particular may lead the sharper theorists amongst Harry Potter fans to figuring out what the crucial mystery is surrounding Dumbledore's possession of the cloak at the time of James Potter's death. It's certainly a tantalizing clue to wrestle with between now and the publication of the final volume in the series, coming straight from the Mistress of Mystery herself. The last time Rowling directed fans towards a specific line of inquiry, she encouraged consideration of why Voldemort survived the night he attempted to kill baby Harry and why Dumbledore did not attempt to kill Voldemort when they met in the Department of Mysteries. The answer to those questions was at the heart of the sixth novel, Harry Potter and the Half-Blood Prince, as well as earlier volumes - unbeknownst to readers at the time. It is expected to anchor the plot of the seventh and final novel as well.