  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 $(G^*)^0$ of the dual group of $G$ and the set of irreducible
unipotent characters of the centralizer of $s$ in $G^*$. 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.
