Lusztig correspondence and Howe correspondence for finite reductive dual pairs

Let \((\textbf{G},\textbf{G}')\) $(\mathbf{G},{\mathbf{G}}^{\prime })$ be a reductive dual pair of a symplectic group and an orthogonal group defined over a finite field of odd characteristic with corresponding Frobenius maps F. The Howe correspondence establishes a correspondence between a subset of irreducible characters of \(\textbf{G}^F\) ${\mathbf{G}}^F$ and a subset of irreducible characters of \(\textbf{G}'^F\) ${\mathbf{G}}^{\prime F}$ . The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a rational semisimple element s in the connected component \((\textbf{G}^*)^0\) ${({\mathbf{G}}^{\ast })}^0$ of the dual group \(\textbf{G}^*\) ${\mathbf{G}}^{\ast }$ of \(\textbf{G}\) $\mathbf{G}$ and the set of unipotent characters of the centralizer \(C_{\textbf{G}^*}(s)^F\) $C_{{\mathbf{G}}^{\ast }}{(s)}^F$ . In this paper, we prove the commutativity (up to a twist of the sign character) between these two correspondences. As a consequence, the Howe correspondence can be explicitly described in terms of Lusztig’s parametrizations for classical groups.