But Scruffy brings up a good point I never thought of before. I guess it is a good thing that Luke had never been able to apply to the Imperial Academy. Geez, did Owen and Beru ever think about that? But I really think that Palpatine would have been familiar with the name Skywalker, and since Anakin changed it to Darth Vader immediately after joining Palpatine, it's no stretch to think that the name wasn't too well known. And I doubt the Emperor does his own hiring, so Luke could become some petty officer and never be found out for being Darth Vader's son.
But Scruffy brings up a good point I never thought of before. I guess it is a good thing that Luke had never been able to apply to the Imperial Academy. Geez, did Owen and Beru ever think about that? But I really think that Palpatine would have been familiar with the name Skywalker, and since Anakin changed it to Darth Vader immediately after joining Palpatine, it's no stretch to think that the name wasn't too well known. And I doubt the Emperor does his own hiring, so Luke could become some petty officer and never be found out for being Darth Vader's son.