lionhead

Question: What happened to Luke's hand after Vader cut it off?

Answer: It fell into the reactor shaft alongside his lightsaber, unrecoverable. It probably fell outside like Luke did and dropped into the clouds of the gas giant Bespin.

lionhead

Well, what about the thing that fell after Luke landed on the satellite dish outside? Was that Luke's hand?

For what it's worth, the descriptive audio identifies the falling object as a piece of the antennae that Luke is hanging from.

TonyPH

No, nothing as macabre as that. Probably a piece of cloth or something that fell out of his pocket. Or, possibly, his lightsaber. But I doubt they would have him watch his own hand fall.

lionhead

I always assumed that was his blaster, falling out of the holster.

In the Legends continuity someone found Luke's severed hand and used it to create Luke's evil clone, known as Luuke.

Answer: In the books before Disney, the hand was recovered by an Ugnaught (the pig-like people in Cloud City) and was taken with his lightsaber to the Emperor's secret storehouse on the planet Wayland. It was later used (in the Thrawn trilogy) to create a clone of Luke.

LorgSkyegon