That's what I put because I would think "lets" suggests permission whereas in English as we speak it in Ireland anyway "leaves" means that he found it open and left it that way, or opened it himself and didn't close it again
This phrasing was used in the past I think, which is why you don't here it now so much. I certainly heard this in school "Can you let window open?" and it meant to open it. That said I don't know if its from Hiberno-English...
I am well aware of the different meanings of prepositions used with the verbal noun. What I am unclear about is what this sentence is getting at by "lig" with a phrase describing a state rather than a verb.
This is not hiberno-english or anything colloquial. The only way this sentence makes sense to me is in the sense that "he does not obstruct the window from opening", i.e. the window is opening under the agency of someone/something else and he does not stop it from doing so.
This is a strange sentence alright. I could understand if he was allowing the window to open but that would be 'a oscailt'. Would it make more sense to say something like 'Ligeann sé an fhuinneog fanacht ar oscailt.' - He allows the window to remain opened?