On the Howe correspondence and Harish-Chandra series

Abstract: We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs ($\mathbf{U}m(\mathbb{F}_q),\mathbf{U}_n(\mathbb{F}_q$)) and ($\mathbf{Sp}{2m}(\mathbb{F}q),\mathbf{O}^\pm{2n}(\mathbb{F}_q)$), where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd). We use the Lusztig correspondence to reduce the study of the Howe correspondence between Harish-Chandra series to a correspondence between unipotent Harish-Chandra series. Finally, we provide two applications of these results.