Ajax's shade is the hero who refuses to speak to Odysseus in the Underworld. This is Ajax the Great, not Ajax the Lesser, the other Greek warrior with the same name.
Odysseus encounters Ajax during his visit to the realm of the dead in Book 11 of the Odyssey. Odysseus tries to reconcile with him, but Ajax remains silent and turns away. Their hostility began after Achilles' death, when Odysseus and Ajax competed for Achilles' armor. Odysseus won the judgment, while Ajax's wounded honor drove him to madness and suicide in later versions of the myth.
The scene is striking because other dead heroes, including Elpenor, Tiresias, Agamemnon, and Achilles, communicate with Odysseus. Ajax's silence therefore becomes a dramatic symbol of his lasting resentment. A common mix-up is choosing Achilles: Odysseus does speak with Achilles, who famously says that he would rather be alive as a poor laborer than rule over all the dead.