I'm talking about the scene in season 2 episode 6, when Patyr Baelish visits Tywin to talk about Renly's death and a possible alliance between the crown and the Tyrells.
During this time, Arya is Tywin's cupbearer and is the only person, who's allowed to be in the room, as a servant.
There is a huge hint that Petyr actually recognized Arya. It's at the time when Arya accidentally spills wine next to Littlefinger.
'Pardon, my lord'
'It's only wine' Petyr replied and gave her this look:
Now, it seems highly likely that he recognized her at the moment, because he automatically lost the thread in his conversation with Tywin.
It took him more than a second to answer Tywin's next question. It's almost like for a second there he was calculating whether or not it would be profitable for him to tell Tywin about his cupbearer.
Apparently, he decides not to, and that's why I have this question. Why did he decide not to reveal Arya's identity? How could he benefit from that decision?