  Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal
group over a finite field of odd characteristic. The Howe correspondence
establishes a correspondence between a subset of irreducible characters of $G$
and a subset of irreducible characters of $G'$. The Lusztig correspondence is a
bijection 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 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
parametrization for classical groups.
