Let $\mathbb{F}_q$ be a finite field with $q$ elements and odd characteristic. A pair $(G,G')$ of mutually centralized reductive subgroups of $Sp_{2n}(\mathbb{F}_q)$ is called a reductive dual pair. By means of the Weil representation of $Sp_{2n}(\mathbb{F}_q)$, Roger Howe introduced a correspondence $\Theta:\mathscr{R}(G)\rightarrow\mathscr{R}(G')$ between the category of complex representations of these subgroups. Here we discuss how this correspondence relates the Harish-Chandra series of $G$ to those of $G'$. If time allows, we will discuss how this correspondance can be expressed as a correspondance between unipotent Harish-Chandra series.