[Aces McGee]

It seems unlikely that he'll fold a jack, and it seems unlikely he'll call with just overcards.

I think you should check the river and hope to induce a bluff.

It's also possible that he holds a jack, won't fold it, but would be scared to bet it, too. Checking saves you a bet in that case.

There's also a small chance that he's made his flush on the river. I know his play isn't totally consistent with a flush draw given that he didn't take the free card on the turn, but he could have recalled that other hand you posted (where he played postflop similarly).

So check-calling has the added benefit of saving a bet when he does have the hearts.

At least, that's my take on it.

-McGee