The two major powers in Kurdish politics are the Patriotic Union of Kurdistan (PUK) of Jalal Talabani and the Kurdish Democratic Party (KDP) of Massoud Barzani. Within Kurdistan they jockey for power (and fought a blood civil war in the 1990s) but on the national stage they present a relatively united front, focused largely on securing Kurdish autonomy, lobbying for an extension of their territory, and ensuring that a maximum amount of oil revenue stays at home.
In the battle for control of Baghdad, they'll probably seek to play a balancing role, not interested in seeing a completely dominant sect emerge. In Kurdistan, they've been dealing with democracy challenges of their own, with a rise in the arrests of journalists and political activists by Kurdish security forces this year.