Quoting Quirrell from Harry Potter and the Philosopher's Stone, chapter 17, "The Man with Two Faces":
"No, no, no. I tried to kill you. Your friend Miss Granger accidentally knocked me over as she rushed to set fire to Snape at that Quidditch match. She broke my eye contact with you. Another few seconds and I'd have got you off that broom. I'd have managed it before then if Snape hadn't been muttering a countercurse, trying to save you."
Why was there no further investigation after this incident?
Snape merely decided to referee the next match to protect Harry. But why wasn't Quirrell interrogated/sacked immediately for attempted murder of a student? They were already suspecting Quirrell and some basic investigations must have revealed that it was him cursing Harry's broomstick.
Unfortunately, while everyone else was running around looking for it, Snape, who already suspected me, went straight to the third floor to head me off -- and not only did my troll fail to beat you to death, that three-headed dog didn't even manage to bite Snape's leg off properly.
"Yes," said Quirrell idly, walking around the mirror to look at the back. "He was on to me by that time, trying to find out how far I'd got. He suspected me all along. Tried to frighten me -- as though he could, when I had Lord Voldemort on my side..."
Harry leaned forward. Quirrell was mumbling something. Snape interrupted him.
"Have you found out how to get past that beast of Hagrid's yet?"
"B-b-but Severus, I--"
"You don't want me as your enemy, Quirrell," said Snape, taking a step toward him.
"I-I don't know what you--"
"You know perfectly well what I mean."
Harry Potter and the Philosopher's Stone, Chapter 13, Nicholas Flamel
Dumbledore turned a page, and said, without looking up, “Keep an eye on Quirrell, won’t you?”
Harry Potter and the Deathly Hallows, chapter 33, "The Prince's Tale"
Why wasn't Quirrell sacked, and arrested/questioned further about Voldemort if they knew he was trying to steal the Philosopher's stone for Voldemort, and was in contact with him?
Why didn't Dumbledore act more seriously? How could he have let Quirrell continue as a teacher after all this?