What do the four tempters represent in Murder in the Cathedral?

The tempters represent

• pleasure,
• secular power,
• the desire to fight tyranny, and
• the desire to seek martydom and sainthood for personal, spiritual glory.

Here are the details.

The First Tempter represents the temptation to indulge the senses in parties, entertainment, and light pursuits.

He offers Thomas the prospect of living the social life he had enjoyed in his youth with the king. Laughter, "singing at nightfall" and enjoying "wit, wine, and wisdom." Giving into this wouldn't just be pleasurable, it would be safe. Being "easy" -- offering easy companionship to other pleasure-seekers -- would prevent Thomas from coming into conflict with the king. If Thomas refuses to take this path

"…[Y]our goose may be cooked and eaten to the bone."

The Second Tempter represents the temptation to seek worldly, secular power and the "life lasting" glory that may grow from it:

"...You, master of policy
Whom all acknowledged, should guide the state again."

This temptation isn't simply about obtaining a glorious reputation. The tempter appeals to Thomas's desire to help people in their everyday lives:

"Disarm the ruffian, strengthen the laws
Rule for the good of the better cause…"

This is a more difficult temptation to overcome, but Thomas rejects it. He already possesses the more important power -- spiritual authority. He won't "descend to desire a punier power."

The third tempter is a "rough, straightforward Englishman" representing the barons and the forces that would weaken the king's concentrated power. By succumbing to this tempter, Thomas could join the "fight for liberty" and end the "tyrannous jurisdiction" of the king over the bishops and barons.

Thomas rejects this temptation as well, noting that men who plot against the king are not trustworthy allies, and resolving that "No one shall say that I betrayed the king."

The fourth tempter represents Thomas's desire to become a martyr and saint -- to seek out this fate because it appeals to his pride and selfish wishes for spiritual glory that will outlast his death. This is the most difficult temptation to overcome. Thomas says it's also the "greatest treason / To do the right deed for the wrong reason." The servant of God has the capacity to commit a worse sin than the servant of the king. He can end up using the pretext of doing God's work to further his own personal ends.

The first three tempters offer Becket reward in this life in return for his taking back his judgments against the King.  They are, in some ways, reminiscent of the three temptations of Jesus at the start of his ministry, offering him rewards in this world.   As with Jesus, Becket rejects these three.  The fourth tempter is a bit different because he appeals not to his relationship with the king, but to his own desire to become a saint and a martyr for his work.  He is told that without sainthood, he will soon be forgotten. Appealing as this may be, it leads him to the most important decision in the play which he expresses in this way:  "The last temptation is the greatest treason:/To do the right deed for the wrong reason."  He rejects the temptation to act out of pride and conceit (the wrong reason), and to stay and fight (the right deed.)

What role do the four tempters play in T.S. Eliot's Murder in the Cathedral?

The four temptations faced by Thomas Becket, the Archbishop of Canterbury, are variations on the three temptations Jesus faced when Satan tempted him in the desert, plus a fourth. As the play opens, Becket knows he has to follow his conscience in facing down the demands of Henry II and knows he will probably be killed for opposing the king.

It would be easy and completely ordinary for Becket to rationalize himself out of the situation.  The first tempter offers Becket a variation of "turn these stones into bread." Becket has the power simply to retreat into a materially secure situation and take care of  himself and his own needs.

The second tempter offers a variation on Satan's offer to give Jesus all the kingdoms of the world if he will bow down and worship him. In this case, Becket is tempted to work with the king, rationalizing it as the best way to create a stable state that will help the poor and bring justice to the land. He is tempted to try to use the power of the "devil," in this case Henry II, to do good.

The third tempter offers a variation on Satan's challenge that Jesus fling himself out of a tower to prove he is God's chosen one—God's angels, Satan says, will not allow him to hit the ground below. In Becket's case, he's tempted to join with others to overthrow the king, a risky venture.

The fourth temptation is the hardest because it tempts Becket most strongly: to invite martyrdom for the wrong reasons. Becket has to wrestle with whether he is acting out of his own personal desire for glory, his wish to become a famous saint in the church, or because opposing Henry is what God is calling him to do:

The last temptation is the greatest treason: To do the right deed for the wrong reason

It helps in understanding the play to know that the Archbishop of Chichester commissioned it in the 1930s as a response to Hitler's tyrannies in Germany, in order to encourage people to explore their consciences to offer a principled response to events in that country.