It is a widely accepted belief in the fandom that Sesshoumaru is the Lord of the Western Lands and/or governs them and it's much used in fics.
But again, there is not really anything to support it in the actual canon of the series.
The whole Lord of Western Lands thing comes from a reference to Inu no Taishou early on in the series. And I think part of the confusion could've been because of translation but I'm not 100% sure on that.
Here's a bit of Trivia written on the
Inuyasha Wikia's page on Sesshoumaru's father (not the most reliable source, perhaps, I grant you that)
:
Many fans refer to him as Lord of the Western Lands; however, such a designation appears nowhere in any incarnation of the series. He was never stated to have any land holdings or official political power other than holding the title of "General" and being "based in the western provinces". After Sesshoumaru discovered that his father had intended for Tenseiga to be re-absorbed by Tessaiga, he lamented that this meant that his father intended to "leave him with nothing," indicating that Tenseiga was his sole inheritance from his father, and he did not inherit any land, title, or followers.