On the Unicity and the Ambiguity of Lusztig Parametrizations for Finite Classical Groups

TL;DR

This paper investigates the uniqueness and ambiguity of Lusztig parametrizations for finite classical groups, proposing conditions under which the Lusztig correspondence can be made unique.

Contribution

It demonstrates that Lusztig correspondence for classical groups can be uniquely determined when compatible with parabolic induction and finite theta correspondence.

Findings

Lusztig correspondence can be made unique under certain compatibility conditions

Ambiguity in Lusztig parametrizations is analyzed and characterized

Provides criteria for the unicity of Lusztig parametrizations in classical groups

Abstract

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.