The only authority that dwarfs on Discworld truly accept
The Low King resides in the vast mines under the fiefdom of Uberwald, probably somewhere below its capital, Bonk. The current (at the time of Discworld Novel 27) is the rather moderate and modern(ish) Rhys Rhysson, a dwarf from the mines of Llamedos.
There was plenty of pulling of beards, chopping of Kneecaps and axe throwing involved in his inauguration, as the dwarfs in the largest dwarf city outside Uberwald, Ankh-Morpork, were against the more traditional candidate, Albrecht Albrechtson, who apparently never goes above ground as he regards this as un-dwarfish.
Making matters worse, a conspiracy of orthodox dwarfs, werewolfs and vampires tried to take advantage of the unstable political situation in the run-up to the inauguration and try to turn dwarf against dwarf, resulting in a subterranean civil war in which plenty of mines would have mysteriously collapsed.
This was fortunately avoided by the Duke of Ankh-Morpork, with the help of the representatives of Ankh's biggest ethnic minorities, Cheery Littlebottom and Detritus, by prodding some serious buttock.
The Low King serves an important role in the Dwarf community: it is him who rules over mining disputes, sitting on the Scone of Stone, just like B'hrian Bloodaxe 1500 years before , interpreting the kruk or dwarf mining law.
The new Low King has his work cut out, with the tensions between the Copperhead, Uberwald and Ankh-Morpork Dwarfs.
It will probably help that he's actually a woman - as dwarfs don't really care what's underneath all of that chain mail - and will possess superior diplomatic skills.