Extremal unipotent representations for the finite Howe correspondence (1708.02823v2)

Abstract: We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}{2m}(\mathbb{F}q),\text{O}^\epsilon{2n}(\mathbb{F}_q))$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd) and $\epsilon=\pm 1$. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of $\pi$ under the correpondence of a unipotent representation $\pi$ of $G$.