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[Star Wars] Vader corrupts Luke and they kill the emperor together, how do they set about ruling the Galaxy?

This isn't a typical Sith "you keep what you kill" deal. Palpatine had genuine political power, Vader did not; he was more akin to a special agent, from what we've seen; above the law, given some level of deference and respect, but definitely not in charge.



So Vader and Luke lay the smackdown on Palpatine. Then what? 


Best case scenario: The Moffs, Military officers et al assume control in absence of a direct leader. Some infighting occurs but mostly they limp on with the existing system, only without an ominous emperor looming over their heads. Vader is just seen as a bit of an odity/ the former emperor's pet, no one really knows what to do with him and he's just sort of kept around as a "well, it's always been the case, why break tradition?" thing?


Worst case scenario: As above, but Vader and Luke are vilified as assassins and hunted wherever they go.


In any case, it doesn't seem that Vader was a named successor, and I don't really a route from "kill Palpatine" to "rule the Galaxy" that would actually work.
gpt
Deep in the second Death Star, Luke, pushed beyond all reasonable limits, looks to his father's hand, outstretched in the Emperor's throne room and whispers his answer. "Yes." 

The Emperor, enraged, blows Vader across the room with a bolt of force lightning and then turns to the young Jedi. "You fool! You seek to ally with your father against *me*? He laughs at the idea that his crippled apprentice and his poorly trained son offer any threat and decides to torture them, he needs neither of them. 

Force lightning surges through Luke and the young boy man screams. Sideous is impressed with how much pain he can take and decides that he will keep the boy as his new apprentice and just dispose of Vader. Turning to see where the wreck of his Vader landed, he finds the back suit lumbering towards him at speed. 

He is surprised, Vader should be dead, but something has kept him alive and in that moment's surprise Vader strikes, flinging himself forward and smashing the Emperor into the deep pit to his death. It is an unlikely and unusual way for such a powerful Sith to die. 

Vader checks his suit, miraculously he is largely unharmed, but his boy has been badly hurt. Good, he needs to learn pain and fear if he is to be a worthy apprentice. He kneels down and touches his son's head, after all this time he has finally found his child. 

The fleet is in disarray and Vader immediately orders a retreat, leaving the second Death Star and transferring to a ship with a hyperdrive. He is intimately familiar with all ships, codes and the plans for the attack and quickly pulls everything back to the nearest base to reorganise. Everyone obeys, he is very high ranking and no one would dare disobey an order issued in the Emperors name. The Second Death Star, abandoned, is destroyed and the rebels rejoice. 

Vader acts quickly, he heads back to Coruscant and sends out word for all Moffs to return there for an emergency meeting. Most do, but a few, feeling strangely free, do not. They will be trouble, but later. 

90% of the Moffs return to Coruscant to find Vader and his son have seized power and are offered a choice, swear their allegiance, or die. They all swear and most, having meant it, are allowed to return with new "advisers" who will stay with them day and night to ensure they behave. A few, who are sensed to be disingenuous in their pledge, are separated and the new Sith Lord, kills them with his father's red lightsaber. 

Vader will spend the next few moths taking control of the Empire and concentrating on training Luke, giving him access to all the power of the dark side and sending him on his first missions to bring the errant Moffs in line. Vader knows every inch of the Empire and it is easy to kill those who would cause him trouble and replace them with loyal stooges. The rebellion is ruthlessly hunted, no quarter is given and they are hounded from one end of the Galaxy to the next. 

Finally, Luke is ready and his final mission is to track down and destroy the leaders of the rebellion. Under the guise of having escaped Vader, he finds them without much trouble and once he is inside their defences, he slaughters them all, except Leia. He returns to Vader with her and Vader begins the process of breaking his daughter. It takes months, but he succeeds and unlocks her terrifying potential. 

But trouble is brewing. Vader is an unpopular figure among those who know he had seized power and many do not with to follow him, he needs a figurehead, a young, attractive person who can be thrust into the light. It's handy that Leia is finally ready. 

Luke is his fist, immensely strong, vicious and amoral, he is sent to wipe out whole systems that pose a threat, but Leia offers a respectable face. She was a well known opponent of the former Chancellor and now many are confused, has she brought about the change in leadership she sought? She must have, but the Empire still exists? 

Leia, a trained diplomat, moves to create a respectability to the Empire that neither Vader nor Sideous understood. She reforms the puppet Senate, changes the Empire back to the Republic and declares that the people are now free. Of course, nothing changes, but everyone feels much better. 

Vader is now hidden in the background, he spends his time working with the remains of Palpatine's work, looking for a way to expend life. Luke restarts the Jedi order and soon force potentials are flooding in from all over the Empire, Luke is building an army. 

The three Skywalkers now control the Galaxy - Leia seemingly the ruler, Luke the head of the Jedi order and ridiculously powerful in the force and Vader, working behind the scenes to control it all and look for the answer he has sought for many years. 

At last he finds what he needs. Palpatine had found how to produce clone bodies and transfer his essence to a new form. Vader, sick of his life, transfers into a new body after years of work uncovering Palpatine's secrets and... is free of pain, free of anger, suddenly able to see more clearly than he has been in decades. 

In his new body, he finally has access to all the potential he should have had and his eyes open. His old mentor, Obi Wan, appears to him as a force ghost to give one last lesson, that he, Anakin, is the chosen one and no matter how badly he has fallen, he can redeem himself. Vader sees everything and agrees, he has done what no one else has done, he has returned to the light side without help from anyone else. 

Now Vader has a problem, he expected to be the puppet master behind this new evil Republic but be in a new, strong body and be able to take back some of the more direct power he gave to his children, but they have grown obscene with power and he does not know how to undo what he has done. If he directly challenges them he is not sure that he will be victorious against them both and they are close allies. 

Anakin Skywalker, the chosen one, is all alone against a galaxy that he himself has corrupted to the dark side. He must undo what he has done, defeat or redeem his children and finally bring balance to the force.