Canon:
Throughout the series, we see Vader showing some degree of compassion for Luke - relative to Vader's default setting of murderous ruthlessness and mercilessness, at least - whenever they interact after Vader discovers Luke's true identity.
The Empire Strikes Back:
At the end of their first face-to-face meeting, Vader refuses to kill Luke, and instead says "Let's be bros, kill the Emperor, and rule the Galaxy together. It'll be an awesome bonding experience!":
There is no escape. Don't make me destroy you. You do not yet realize your importance. You have only begun to discover your power. Join me and I will complete your training. With our combined strength, we can end this destructive conflict and bring order to the galaxy...
You can destroy the Emperor. He has foreseen this. It is your destiny. Join me, and together we can rule the galaxy as father and son. Come with me. It is the only way.
- Star Wars - Episode V: The Empire Strikes Back
The first thing we should note is that Vader was clearly going easy on Luke throughout their fight. Luke had spent very little time practicing with his lightsaber before this point - the only time we see him doing any lightsaber training is aboard the Millennium Falcon in Episode IV, with Obi-Wan and the remote droid. Vader has been wielding a lightsaber for decades, and was one of the most skilled lightsaber combatants in the Jedi Order before he fell to the Dark Side. There is little reason to believe that Luke would have been able to resist a Force-choke from Vader, yet Vader never bothers to attempt the maneuver. Even without using his superior knowledge of Force powers, Vader would have been able to chop Luke into a dozen pieces with his saber, without seriously exerting himself, if he wished to do so. He never even tried. The only serious wound he inflicts (the amputated hand) is one he knows Luke will survive, as Vader himself had, more than once. Vader's tactics throughout the duel were obviously not designed to kill Luke - rather, they were intended to test Luke's powers, to wear him down both mentally and physically, and to demoralize him so thoroughly that he would accept Vader's offer to join forces and turn against the Emperor together. In other words, Vader is showing a remarkable degree of mercy towards Luke - in fact, this is the only time we see him show mercy to anyone between his fall to the Dark Side and his attack on the Emperor.
Compare this to Vader's slaughter of the younglings in the Temple - he massacred them, despite the fact that they were harmless, and posed no threat to him. Compare it to his first fight with Obi-Wan - his best friend, master, brother, and only lifelong ally - where Vader clearly fought with the explicit intention of killing Kenobi; or his second fight with with Obi-Wan, where he actually did kill him. Luke is younger than Obi-Wan and Vader, but his superior agility and speed is far outweighed by his lack of training, experience, and skill - in regards to his abilities with both his lightsaber and the Force itself. Vader could have taken him apart in seconds, but deliberately refrained from doing so.
At the end of the fight, Vader sees that Luke is physically exhausted, completely cornered, defenseless, and disarmed (both figuratively and, more or less, literally). He also sees that despite being physically depleted, Luke's resolve is still rock solid - he will absolutely not turn to the Dark Side. Since Vader is blocking the only escape route, and Luke has made it clear that he won't join his father, Luke is left with only one option: leap to his death. Vader must realize what Luke is contemplating, and when he says "Come with me - it is the only way", we can almost hear him begging "Please, son - don't do this. I didn't kill you because you matter to me. I don't want to see you die."
Again, this is a level of concern we have never seen from Vader before. He didn't show such compassion for the Sand People, Mace Windu, Obi-Wan, the younglings, or even Padmé (and the offspring she was carrying when he Force-choked her). This only time we see Vader hesitate to kill. Not quite love, perhaps, but certainly an uncharacteristic level of humanity and restraint.
Return of the Jedi:
The next time they meet, Vader appears to be truly remorseful that he can't accommodate Luke's plea for the two of them to flee the Emperor together:
From the script - I mark the lines where Vader is clearly unhappy with himself, and what he is being forced to do, in bold:
VADER
The Emperor has been expecting you.
LUKE
I know, father.
VADER
So, you have accepted the truth.
LUKE
I've accepted the truth that you were once Anakin Skywalker, my father.
VADER (turning to face him)
That name no longer has any meaning for me.
LUKE
It is the name of your true self. You've only forgotten. I know there
is good in you. The Emperor hasn't driven it from you fully. That is
why you couldn't destroy me. That's why you won't bring me to your
Emperor now.
Vader looks down from Luke to the lightsaber in his own black-gloved
hand. He seems to ponder Luke's words.
VADER (indicating lightsaber)
I see you have constructed a new lightsaber.
Vader ignites the lightsaber and holds it to examine its humming,
brilliant blade.
VADER
Your skills are complete. Indeed, you are powerful, as the Emperor has
foreseen.
They stand for a moment, the Vader extinguishes the lightsaber.
LUKE
Come with me.
VADER
Obi-Wan once thought as you do.
Luke steps close to Vader, then stops. Vader is still.
VADER
You don't know the power of the dark side. I must obey my master.
LUKE
I will not turn... and you'll be forced to kill me.
VADER
If that is your destiny.
LUKE
Search your feelings, father. You can't do this. I feel the conflict
within you. Let go of your hate.
VADER
It is too late for me, son.
I think it is pretty clear that Vader is trying to convince himself of what he is saying far more than he is trying to convince Luke. Some of what he says is clearly untrue: "I must obey my master" is a strange thing to say to Luke, considering the fact that the only time they spoke previously, minutes after their first meeting, Vader was ready to throw his master under the bus and team up with Luke.
On a more basic level, Vader isn't handing Luke over to the Emperor because he wants to - he's doing it because he can't muster the willpower to not do it:
"I must obey my master": Not "I want to obey my master" or "I like obeying my master" or "I choose to obey my master" or "I want the exact same thing my master does - I'm totally on-board with his plans here". Vader is saying he has no choice, and he is powerless to go against Palpatine's will. He eventually learns that this isn't the case, and he throws his master to his death, but at the time Vader says this, he still believes it for the most part, and is trying to squash any lingering internal doubts/conflict about it.
"It is too late for me, son": This is how Vader responds when Luke says "You can't do this. I feel the conflict within you. Let go of your hate." Vader doesn't say "I can do this without reservation", or "There is no conflict", or "I like hate". He says "It's too late". In other words, he is saying "Doing this is killing me. There is conflict, and it is tearing me apart. I wish I could let go of my hate. But I can't. I've done too many terrible things, I have fallen too far, I let the Emperor sink his claws into me much too deep, and I am too worthless and awful to resist now. No matter how much I want to do what I know is right, I can't. I'm a horrid person, and I don't deserve to be saved."
"Obi-Wan once thought as you do": I think this is the only instance in the original trilogy in which Vader mentions Obi-Wan to Luke without any derision, mockery, or contempt. The last time he met Luke, he said Obi-Wan knew Luke would end up with Vader, and urges Luke to avoid Obi-Wan's fate; shortly after the exchange we're discussing here, Vader realizes Leia is Luke's sister and gloats that Obi-Wan's failure is now complete. But here, he seems nostalgic and melancholy when he tells Luke that Obi-Wan used to hold out hope of redemption for Vader, but eventually gave up on him.
'Vader looks down from Luke to the lightsaber in his own black-gloved hand. He seems to ponder Luke's words.': In the film as in the script, when Luke says that Anakin is Vader's true identity, there is still good in him, and this good is the reason he let Luke survive their first encounter, Vader truly does take a moment to consider what Luke is saying. Instead of confirming or denying this, he changes the subject and talks about Luke's new lightsaber. I believe he's feigning a lack of interest because he knows Luke is right, and this is extremely uncomfortable for him - it shakes his confidence more than he thought was possible.
Finally, Luke defeats Vader, prepares to kill him outright, then regains his composure, spares Vader's life, and tosses away his weapon. This is the ultimate act of mercy - he is leaving himself vulnerable to whatever the Emperor can throw at him (and appears to be completely confident that, even if Vader recovers quickly, he will not resume his attack on Luke). This sways Vader, who steps in to save Luke and kill Palpatine, thereby redeeming himself.
In the midst of this uproar, Luke is trying to carry the enormous
dead weight of his father's weakening body toward an Imperial shuttle.
Finally, Luke collapses from the strain. The explosions grow louder as
Vader draws him closer.
VADER (a whisper)
Luke, help me take this mask off.
LUKE
But you'll die.
VADER
Nothing can stop that now. Just for once... let me look on you with my
own eyes.
Slowly, hesitantly, Luke removes the mask from his father's face. There
beneath the scars is an elderly man. His eyes do not focus. But the
dying man smiles at the sight before him.
ANAKIN (very weak)
Now... go, my son. Leave me.
LUKE
No. You're coming with me. I can't leave you here. I've got to save
you.
ANAKIN
You already have, Luke. You were right about me. Tell your sister... you
were right.
- Star Wars: Episode VI: Return of the Jedi
What was Luke right about? Every hopeful, positive, understanding, forgiving thing he had said about Vader. These have all been addressed above, but they are best summarized in the Luke's explanation to Leia of why he had to confront Vader on Endor:
Because... there is good in him. I've felt it. He won't turn me over to
the Emperor. I can save him. I can turn him back to the good side. I
have to try.
- Star Wars: Episode VI: Return of the Jedi
Luke loved Vader because he believed that Anakin was still hidden somewhere under the black armor. Luke was the only person who never gave up the hope that Vader could be saved; Luke - and only Luke - loved Vader, despite all the monstrous reasons he had to hate him; and Luke did something Vader had forgotten was possible - he spared Vader's life when he had every reason not to. Because of this, Vader discovered (or admitted, or acknowledged) that he loved Luke as well, and sacrificed his life to save his son. In his last words, he made it clear that his love now extended to Leia as well.
And finally, the expression on the face of Vader's Force-ghost is quite plainly one of love and pride.
This isn't entirely unforeseen, either - Palpatine was a bit suspicious of Vader's feelings towards Luke earlier in the film:
EMPEROR
I told you to remain on the command ship.
VADER
A small Rebel force has penetrated the shield and landed on Endor.
EMPEROR (no surprise)
Yes, I know.
VADER (after a beat)
My son is with them.
EMPEROR (very cool)
Are you sure?
VADER
I have felt him, my Master.
EMPEROR
Strange, that I have not. I wonder if your feelings on this matter are
clear, Lord Vader.
Vader knows what is being asked.
VADER
They are clear, my Master.
In fact, the Emperor even knew that Luke loved Vader - he was only mistaken about who would pay the price for this love.
EMPEROR
Then you must go to the Sanctuary Moon and wait for them.
VADER (skeptical)
He will come to me?
EMPEROR
I have foreseen it. His compassion for you will be his undoing. He will
come to you and then you will bring him before me.
VADER (bows)
As you wish.
Novels:
In the Disney canon novel Bloodline, this is made explicit:
Whenever she’d talked with Luke about their birth father, this was the part where he’d refuse to use the name Darth Vader. He was Anakin Skywalker when he fell in love with our mother, Luke would say, taking her hand gently in his. And he became Anakin Skywalker again in the last hour of his life. He came back from the dark side, Leia. They said it could never be done, but our father did it. He made that journey because of his love for us.
- Star Wars: Journey to The Force Awakens: Bloodline*
The novelization of Return of the Jedi agrees:
Together they lifted the heavy helmet from Vader’s head— inside the mask portion, a complicated breathing apparatus had to be disentangled, a speaking modulator and view-screen detached from the power unit in back. But when the mask was finally off and set aside, Luke gazed on his father’s face.
It was the sad, benign face of an old man. Bald, beardless, with a mighty scar running from the top of his head to the back of the scalp, he had unfocused, deepset, dark eyes, and his skin was pasty white, for it had not seen the sun in two decades. The old man smiled weakly; tears glazed his eyes, now. For a moment, he looked not too unlike Ben.
It was a face full of meanings, that Luke would forever recall. Regret, he saw most plainly. And shame. Memories could be seen flashing across it... memories of rich times. And horrors. And love, too...
This boy had pulled him from that pit— here, now, with this act. This boy was good.
The boy was good, and the boy had come from him — so there must have been good in him, too. He smiled up again at his son, and for the first time, loved him. And for the first time in many long years, loved himself again, as well...
Luke stood in a forest clearing before a great pile of logs and branches. Lying, still and robed, atop the mound, was the lifeless body of Darth Vader. Luke set a torch to the kindling.
As the flames enveloped the corpse, smoke rose from the vents in the mask, almost like a black spirit, finally freed. Luke stared with a fierce sorrow at the conflagration. Silently, he said his last goodbye. He, alone, had believed in the small speck of humanity remaining in his father. That redemption rose, now, with these flames, into the night.
- Star Wars: Episode VI: Return of the Jedi (novelization)
Legends:
In the Legends novel The Rise and Fall of Darth Vader, we learn that Vader decides to kill the Emperor because:
Luke is his son.
Luke is Padme's son.
Luke loves Vader.
It is hard to imagine that someone as evil as Vader would be swayed to kill his master (and himself) by Luke's love if that love didn't kindle his own love for Luke. And when Luke struggles to drag Vader to his ship, Vader decides to have Luke remove his mask - killing him - because he he sees that Luke will not escape the Death Star if he has to worry about keeping Vader alive. This kind of selflessness can only be understood as a sign of Vader's love.
Vader watched the Emperor extend his gnarled fingers and unleash blinding bolts of blue lightning from his fingertips. The lightning struck Luke, who tried to deflect the crackling bands of energy, but was so overwhelmed that his body crumpled to the floor.
No, Vader thought. No. Not like this.
As the Emperor continued to strike Luke with his barrage of Sith lightning, Vader struggled to his feet. One leg was broken, and the other wasn’t working right. Moving awkwardly, he shifted his bulk to stand beside his Master. On the floor, Luke writhed in agony, and was on the verge of death as he groaned, “Father, please. Help me.”
Vader watched Luke curl into a fetal position as the Emperor hurled an even more staggering wave of lightning at his victim. Vader had no doubt that Luke was about to die. His son screamed.
Not just my son...
The Emperor unleashed another round of lightning.
...or Padmé's son...
Luke screamed louder.
...but my son... who loves me.
Luke's clothes began to smoulder as his body involuntarily spasmed. Suddenly, Vader realised he was no longer concerned about his own personal future. Despite all the terrible, unspeakable things he'd done in his life, he knew he could not stand by and allow the Emperor to kill Luke. And in that moment of awareness, he was Darth Vader no more.
He was Anakin Skywalker.
It took all of his remaining strength to seize the Emperor from behind. The wretched Emperor continued to release lightning bolts, but they veered away from Luke and arced back to crash down upon him and his insurgent apprentice...
Despite his own injuries, Luke managed to haul his father to the hangar that contained Vader’s shuttle. The journey was made even more difficult by the fact that the Rebels had disabled the energy shield projector on the Sanctuary Moon, and the Death Star was now under heavy attack. Trying to keep his own legs steady as the battle station was wracked by explosions, Luke dragged his father to the shuttle’s landing ramp before he collapsed from the effort.
He’s not going to make it, Anakin thought. Not with me.
“Luke,” he gasped, “help me take this mask off.”
Luke knelt beside him and said, “But you’ll die.”
“Nothing can stop that now,” Anakin said. “Just for once... let me look on you... with my own eyes.”
- The Rise and Fall of Darth Vader
We can take this to mean either of two things:
But these really amount to the same thing: Luke loved Vader, and that love awoke Vader's own love for his children.
Word of god:
"It really has to do with learning," Lucas says, "Children teach you compassion. They teach you to love unconditionally... He doesn't right the wrongs, but he stops the horror. The end of the Saga is simply Anakin saying, I care about this person, regardless of what it means to me. I will throw away everything that I have... and throw away my life, to save this person. And I'm doing it because he has faith in me; he loves me despite all the horrible things I've done... he still cares about me, and I can't let that die. Anakin is very different in the end... He takes the one ounce of good still left in him and destroys the Emperor out of compassion for his son."
- George Lucas, The Making Of Revenge Of The Sith; page 221
...
"You learn that Darth Vader isn’t this monster. He’s a pathetic individual who made a pact with the Devil and lost. And he’s trapped. He’s a sad, pathetic character, not a big evil monster. I mean, he’s a monster in that he’s turned to the Dark Side and he’s serving a bad master and he’s into power and he’s lost a lot of his humanity. In that way, he’s a monster, but beneath that, as Luke says in Return of the Jedi, early on, “I know there’s still good in you, I can sense it.” Only through the love of his children and the compassion of his children, who believe in him, even though he’s a monster, does he redeem himself."
- George Lucas, quoted in J. Windolf, “Star Wars: The Last Battle,” Vanity Fair, 2005
