He did know what Luke was his son, once he knew that Luke's last name was Skywalker. After Vader sensed the strength of the Force in Luke, he spent time trying to find the identity of the young Force-wielder who had destroyed the Death Star.
Vader's Quest takes place before Star Wars #35 and is specifically set two months after A New Hope, giving it the most specific answer to when Darth Vader learned that Luke Skywalker is his son in the Star Wars franchise.
Vader was hunting down the pilot who blew up the Death Star, and hired Boba Fett to find them. Boba was unable to, but he did give Vader a name: “Skywalker.” This, combined with his knowing Luke was Force-sensitive and had been trained by Obi-Wan, was what made him realize the truth.
With Leia, Luke, and Darth Vader all being major characters in "Obi-Wan Kenboi," the show is having to tread lightly as the "Star Wars" canon is pretty explicit. We know exactly when Vader discovered that Luke was his son, and we know when he figured out that Leia was connected to him as well.
However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover she was his daughter. That serves to justify Vader's lack of awareness of his connection to Leia, although, perhaps, not as satisfyingly as if Lucas had planned for the relationship from the beginning.
Does Darth Vader know Leia is his daughter at the beginning of A New Hope? No. Vader is not aware that his child(ren) survived Padme's death until he learns Luke's name after the Battle of Yavin. He doesn't know about Leia until he reads Luke's mind during their fight in Return of the Jedi.
George Lucas didn't plan on making Darth Vader the secret father of Luke Skywalker. As much as fans have been told that Lucas had a massive plan for the Star Wars saga prior to 1977, the sources closest to that process say the opposite.
Anakin, who has since become the Sith Lord Darth Vader, is unaware of his children's birth. The twins soon became part of the Rebel Alliance following the deaths of their adoptive families at the hands of the Empire.
It was only then that Darth Vader realized his son was alive, and he began his quest to find the boy. Darth Vader didn't learn he had a daughter until Return of the Jedi, when he stood before Emperor Palpatine.
Leia Organa Solo (born Leia Amidala Skywalker), a Force-sensitive Human female was at various stages of her life, a politician, revolutionary, and Jedi Knight of the New Jedi Order.
While their affection had to remain hidden, the two kept in frequent contact, even throughout the Clone Wars. But despite this, Anakin found out that Padme was pregnant only a couple of weeks before the twins were born.
First, Vader didn't even know he had a son, so there was no reason for him to even look. Second, Anakin couldn't wait to get off Tatooine, and there was no reason for him to return.
Eventually, Darth Vader was able to redeem himself back as Anakin Skywalker, a personality last seen in Revenge of the Sith, as he sacrificed himself and threw Palpatine down the Death Star's shaft, saving Luke's life.
It's well known that Star Wars lead characters Luke Skywalker and Princess (now General) Leia are brother and sister. The estranged siblings didn't know about their familial relationship until "Return of the Jedi." They even shared a passionate kiss in the previous movie, "The Empire Strikes Back."
Does Darth Vader remember being Anakin? Yes. Yes, he remembers his name but feels it has no meaning to him anymore.
Star Wars is finally exploring Leia's emotional reaction to the revelation Darth Vader was really her father - as her identity is transformed. Star Wars has finally revealed Princess Leia's reaction to learning Darth Vader was her father.
Reva knows that Anakin Skywalker is Darth Vader because she was one of his targets—a child—during the Order 66 killings that took place during the events of Star Wars Episode III: Revenge of the Sith.
She does not have a baby bump, her hand never goes to her stomach to indicate she's carrying a child, and there wasn't exactly any time for her to have conceived a child with anyone in the course of the film, including Kylo Ren/Ben Solo.
Because, as Obi-Wan explains, the man he knew as Luke's father was dead to him, after being transformed into Darth Vader. Additionally, if Luke had known that Darth Vader was his father, he would have rushed into a confrontation too early, which would have undoubtedly resulted in him turning to the Dark Side.
Because R2-D2 did not know. The last time R2 saw Anakin was when he accompanied Padme and Obi-Wan to Mustafar; while Anakin and Obi-Wan fought, R2 and C-3PO loaded Padme onto the ship, then saw Obi-Wan return alone, upset, and holding Anakin's lightsaber.
Yoda's connection to Luke is never mentioned by either Vader or the Emperor in the films.
She also knows Obi-Wan, and she knows him well. Her holographic plea for help, already iconic, will never be the same. She knows Obi-Wan's real name as well as his alias of “Ben.” When Luke comes barging into her cell and says, “I'm here with Ben Kenobi,” Leia jumps off the bunk in an instant.
Star Wars Just Confirmed Darth Vader's Undying Love for Padme Amidala. Star Wars: Darth Vader #23 revealed just how much Anakin Skywalker loved Padme Amidala even after his transformation into the Sith Lord.
In Legends, Luke and Leia do eventually discover their mother's identity during the Swarm War Trilogy.