Question: During the big battle between Kylo Ren and Luke Skywalker (as a force projection), Luke appears with his blue lightsaber. While I assume that a force projection probably appear any way he wants, and he did choose to look younger, Luke had lost his blue lightsaber decades ago during his battle with Vader on Bespin Cloud City. He had already built a new green one by the time Return of the Jedi started, so why would he appear with the blue one? It should have immediately given away to Kylo that he was a force projection and not the real Luke.
Answer: He knew that his blue lightsaber still existed because Rey brought it to him. He chose to show the blue lightsaber because it belonged to Anakin and Kylo Ren was obsessed with it. He knew that Kylo Ren would be angry at Luke wielding a lightsaber that looked like Anakin's and that rage would help keep him distracted. Here is a quote from Rian Johnson regarding this: "He knows that Kylo's Achilles heel is his rage, and so that's why he kind of makes himself look younger, the way Kylo would've last seen him in their confrontation at the temple, and that's why he decided to bring Kylo's grandfather's lightsaber down there - the lightsaber that Kylo screamed at Rey, 'that's mine, that belongs to me.'"