While Darth Vader's obsession with Luke Skywalker in The Empire Strikes Back is seemingly due to the latter's destruction of the Death Star battle station, his revelation to Luke and brief dialogue with Emperor Palpatine recontextualizes his pursuit.
It was the fact that Luke was his son that caused turmoil within Vader. It was his love for his son that caused his loyalty to Palpatine and the Dark Side to falter. If Vader didn't know who Luke was he would have been just another force sensitive person to be trained to serve Palpatine like an Inquisitor or be killed.
Vader wants Luke to help him overthrow the Emperor, then they can “rule the galaxy as father and son.” Palpatine wants Luke to kill his father and take Vader's place as his apprentice.
The implication is clear - that this is Luke Skywalker at his peak, equal in power to Darth Vader but a champion of the light nonetheless - but Star Wars is yet to confirm that this was indeed the case.
The First Time Darth Vader SHOWED His LOVE For Luke!!(CANON) - Star Wars Comics Explained
Why did Vader stop Luke from killing Palpatine?
Vader's redemption in Return of the Jedi was not just about saving his son; it was also a defiance of the dark side's corruption and his own past as a Sith Lord. Vader's decision to block Luke from killing Palpatine was motivated by his knowledge that it would corrupt Luke and lead him down a similar path of suffering.
Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
While Darth Vader's obsession with Luke Skywalker in The Empire Strikes Back is seemingly due to the latter's destruction of the Death Star battle station, his revelation to Luke and brief dialogue with Emperor Palpatine recontextualizes his pursuit.
Darth Vader didn't learn he had a daughter until Return of the Jedi, when he stood before Emperor Palpatine. The Emperor's power probed Luke's mind, attempting to goad him to fall to the dark side, and he discovered Leia's existence.
Yoda. Yoda's knowledge of Luke Skywalker's true parentage was also confirmed in Return of the Jedi. As with Obi-Wan, Yoda believed that Anakin's fall to the dark side made him unsalvageable, with the Darth Vader persona being his only identity following his ascension to Sith Lordship.
During the battle, Anakin, who had been known as the Sith Lord Darth Vader, was redeemed by Luke and brought balance to the Force. However, the redemption cost Anakin his life, having been mortally wounded by the Emperor, Darth Sidious, while killing his former Master. After his death, Anakin became one with the Force.
She finds it easy to accept Luke as her brother, but she is horrified by the idea Darth Vader was her father. Leia couldn't even understand why Luke accepted it so easily, and somehow believed there was good in their father.
R2-D2 was present when Padmé gave birth to Leia and Luke, so he knew Luke was Anakin's son – but what he didn't know was that Anakin became Darth Vader. In addition to that, Anakin's fall was surely a painful event in R2-D2's life, one that he definitely wouldn't like to remember or even talk about.
secondly, in Legends, R2 was given programming that actually restricted him. from sharing any information with Luke or Leia. about their parents or relation to one another. this was all part of the plan. to conceal this information from the twins.
Since Anakin's defeat on Mustafar, he needed frequent visits to his bacta tank to survive, and even the Vader suit itself was crude enough for Palpatine to control him. So with an aged body, a mind out of balance and the surprise of Luke's attack, Vader was caught at his weakest point.
The novel explained that, had Vader killed Obi-Wan on Mustafar, he would have tried to kill Palpatine too, but given how badly wounded he was after the duel, his power wasn't the same and he was weak, both physically and mentally.
Vader's only goal in the fight was to make Luke lose control of his emotions and bring him closer to the dark side -- and that's exactly what happened.
While Vader may be stronger, Luke had several advantages of his own. He was both swifter and nimbler than Vader. But the biggest edge was that Luke didn't give in to his hatred and anger.
