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The irreducible unipotent characters of classical finite groups of Lie type have been classified by Lusztig using the combinatorical notion of "symbols", see "Irreducible representations of finite classical groups", Inv. Math., 1977. Since the construction is a little bit involved, I won't try to recall it here. Nonetheless here is a question regarding the dual operation on these unipotent characters.

Let $G$ be such a classical finite group, let $S$ be a symbol as defined by Lusztig, and let $\rho_S$ be the associated unipotent character of $G$. Is there a combinatorical way to express the symbol $S^{\vee}$ such that the dual character $(\rho_S)^{\vee}$ is equivalent to $\rho_{S^{\vee}}$ ? In particular, can we easily determine which unipotent characters are auto-dual ?

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