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^*) ^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.