On the unicity and the ambiguity of Lusztig parametrizations for finite classical groups

The Lusztig correspondence is a bijective mapping between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^{\ast }{)}^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^{\ast }$. In this article we discuss the unicity and ambiguity of such a bijective correspondence. In particular, we show that the Lusztig correspondence for a classical group can be made to be unique if we require it to be compatible with the parabolic induction and the finite theta correspondence.