Vader's spy in Jabba's palace sends regular reports, though there has been nothing of interest in them in the months since Fett delivered Solo to the Hutt. When one finally comes through flagged with an urgent code, Vader wills himself to remain calm. It's been months since he's had more than the briefest intel on Luke's--on his son's--whereabouts; the more general reports on the Rebel Alliance mention him only fleetingly, though Vader knows that even reams of information wouldn't be enough to sate his burning hunger for knowledge about the boy.
There is pride mingled with annoyance when Luke claims to be a Jedi, and a brief flare of ever-present anger at the evidence of Obi-Wan's influence on him. Of course, Luke's plan doesn't work--he feels a brief burst of smug satisfaction at that, along with a burst of rage that Jabba dares to threaten his son. Even knowing the events in the holovid are hours, if not days, old isn't enough to assuage that anger. If Luke dies by Jabba's hand, not even the Emperor will save the Hutt from Vader's wrath.
The scene cuts to a sail barge and Vader wonders queasily if he's going to watch his son die being fed to the sarlacc (and if the agent who's sent him this report knows his life will be forfeit for bringing him such bad news), ending both his and the Emperor's plans for the boy. Then Artoo supplies Luke with his lightsaber and the tables turn. Vader can't help it if the corner of his mouth twitches into a tiny half-smile in seeing how much Luke's abilities have grown over the past few months. He is becoming a formidable warrior. Vader allows himself a moment of fatherly pride.
And then his attention is drawn back to Jabba's barge, where a woman he suddenly recognizes as the Princess Leia Organa--rebel terrorist and persistent thorn in his side--is wrapping a slave chain around Jabba's slimy bulk and bracing herself against it. He leans forward, as if it will help him see more clearly, as she strains to strangle Jabba. He finds himself in the unexpected and novel position of rooting for her, this small young woman who has never seemed cowed by his presence, who has gone toe-to-toe with him repeatedly and emerged intact, if not unscathed. She's had his grudging respect since her refusal to break under torture the first time he captured her, and if pressed, he'd admit to a certain enjoyment in finding an enemy worthy of his attention.
Now, he watches as she kills the most notorious crime boss on the Outer Rim with little more than leverage, brute strength, and sheer will. That piques his interest; Bail Organa had had no particular affinity for the Force, but Bail was not her biological father.
Vader wonders if Luke has been working with her. It might be worth investigating. He's not deluded enough to believe he could break her and remake her into an Inquisitor, but it might be worth the effort to try. Certainly, she'd be a formidable weapon against his enemies (and not just the one they have in common), and perhaps a useful goad for his son.
He recalls her sharp, angry presence on the Death Star and her refusal to give up her rebel allies; how he'd had to hold her back from launching herself at Tarkin after the destruction of Alderaan; the way she'd confronted him fearlessly on Vrogas Vas; the sickening display of affection when Solo was lowered into the carbon freezing unit.
She reminds him of Padmé, he thinks before he can stop himself, with her belief in justice and democracy and doing the right thing, and like Padmé, she appears to have no idea of the true cost of order, of peace. The price only he is willing to pay. For a brief moment, he connects the two women in his mind--small and fierce and willful--and the Force thrums triumphantly at the link.
He basks in the wonder of it for a few moments, all his previous ideas to torture or dispose of her wiped away by this potential truth, by the idea of a daughter in addition to a son. It's been many years since Darth Vader laughed, many years since he had the ability, but at this moment, he feels the desire, stronger even than his anger that Obi-Wan kept another child from him. A child even the Emperor is unaware of.
He turns off the holoprojector and pushes away from his desk. There should be a sample of Leia Organa's DNA in the Imperial databank. It will be easy enough to have his personal medical droid run the test and wipe the results before anyone else is aware it's even been done. He can't protect Luke from the Emperor's vision (or his own plans), but if this sudden revelation proves true, it will be a secret he'll try to take to his grave. It's the only way to keep Padmé's daughter safe.