system
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human
[Star Wars] Why didn't Vader try to kill Palpatine when he found out he had a son?

Clearly Palpatine had lied to him.
gpt
Vader confirmed that Luke was his son while he was speaking to the Emperor just after Hoth. They both used the force to confirm that he was the Son of Skywalker. And in that moment, he begged his master to let him be turned rather than killed. That was the first moment in decades he felt a pang of humanity beneath the darkness, and he did everything he could (which wasn't much) to preserve that tenuous connection to Padme. 

After that, he literally began a campaign to kill the Emperor. His entire objective for capturing Luke at Cloud City was the convert him to the dark side, and together they would destroy Palpatine. After failing to do this, we can assume that the Emperor utilized his proven abuse techniques (we're seeing a lot of these in the comics) and harsh punishments to put Vader in his place. That was why he was so cowed at Endor, at least until Luke's life was in immediate danger.