In Harry Potter and the Order of the Phoenix, there is an important moment in which ...
... Sirius Black is killed.
Film: Bellatrix uses killing curse, definitively killing him, then he falls back into the mysterious veil and disappears.
Book: Sirius dodges Bellatrix's red jet of light (not killing curse, which is green), he laughs and taunts her, consequently gets hit squarely in the chest by a "second jet of light" (killing curse?), he falls back into the veil and disappears.
The main difference being that, in the film, we are certain Sirius died before falling into the veil and disappearing, whereas in the book we aren't certain whether the cause of death was Bellatrix's "second jet of light" or the veil.
The relevant scene in the film can be watched by clicking here.
The relevant scene in the books is described in the following excerpt from Harry Potter and the Order of the Phoenix, Chapter 35, Beyond the veil.
Harry saw Sirius duck Bellatrix’s jet of red light: He was laughing at her. “Come on, you can do better than that!” he yelled, his voice echoing around the cavernous room.
The second jet of light hit him squarely on the chest.
The laughter had not quite died from his face, but his eyes widened in shock.
Harry released Neville, though he was unaware of doing so. He was jumping down the steps again, pulling out his wand, as Dumbledore turned to the dais too.
It seemed to take Sirius an age to fall. His body curved in a graceful arc as he sank backward through the ragged veil hanging from the arch. . . .
And Harry saw the look of mingled fear and surprise on his godfather’s wasted, once-handsome face as he fell through the ancient doorway and disappeared behind the veil, which fluttered for a moment as though in a high wind and then fell back into place.
Harry heard Bellatrix Lestrange’s triumphant scream, but knew it meant nothing — Sirius had only just fallen through the archway, he would reappear from the other side any second. . . .
But Sirius did not reappear.
- Harry Potter and the Order of the Phoenix, Chapter 35
Harry believes for quite some time that Sirius will reappear from the veil, he is sure of it, until he realizes otherwise, and he goes toward the veil with an idea of rescue or retrieval in mind, but Lupin is there to stop him.
Similarly to the film adaptation, Lupin grabs hold of Harry and tries to console him and convince him that ...
Sirius really is dead, and there is nothing that can be done
Eventually Harry comes to terms with reality and agrees, then goes ...
... full revenge mode on Bellatrix.
So, my question is:
Why did nobody care to investigate further?
Within that question are a few lead-up questions like
How did Lupin know that a) Sirius was definitely dead, and b) enough about the veil to know that "there's nothing you can do, Harry"/"It's too late"?
Fair enough that Harry accepted the truth with Lupin there to sternly tell him so, but how did Lupin know? Why was nobody interested in investigating the possibility of rescue? We as an audience may know a lot of meta-things, but how did the relevant characters know enough to decide not to investigate the issue?