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 (${\mathbf{Sp}}_{2m}({\mathbb{F}}_q),{\mathbf{O}}_{2n}^{\pm }({\mathbb{F}}_q)$), 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.

22 pages