[Jack:] "Governor Stark wants you to be director of the new hospital and medical center." (6.71)
Though this quickly escalates to a political situation, initially it is not. Willie wants Adam to run the hospital because Adam is the best. But Adam and others think Willie is using him as a political tool.
[Willie:] "I told you to dig on Irwin. What did you get?" (8.222)
When the Boss has nowhere left to turn, with regard to the Sibyl Frey situation, he turns to Judge Irwin. This shows how politics can become inextricably linked to the personal lives of Jack, Willie, and the Judge.
[Willie:] "And I – I had to buy, the sons of bitches made me buy!" (9.69)
It's a point of pride for Willie to never have to bribe anybody. He certainly prefers blackmail. With the Judge dead, Willie doesn't have any competing dirt to counteract the dirt on Tom. So he has to resort to filthy lucre. This is the beginning of the end of the politics in this novel.