Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field
Barbara Bošnjak, Alexander Stadler
TL;DR
The paper addresses the irreducibility of parabolic inductions of the form $u_1\times\cdots\times u_r\rtimes\pi$, where the $u_i$ are essentially Speh representations of GL factors and $\pi$ is an irreducible representation of Arthur type on a split classical group. It proves a precise criterion: $u_1\times\cdots\times u_r\rtimes\pi$ is irreducible if and only if every pairwise induction $u_i\times u_j$, $u_i\times u_j^{\vee}$, and $u_i\rtimes\pi$ is irreducible for $i\neq j$; when the $u_i$ are actually Speh representations, the paper also determines the full composition series and constructs new unitary representations. The approach integrates derivative theory for GL representations, Jacquet-module analysis, Atobe’s socle results, and the extended multi-segment framework for Arthur packets, to bridge Speh- and Arthur-type data under parabolic induction. This advances the understanding of the unitary dual for classical p-adic groups by providing a concrete irreducibility criterion and a pathway to unitary constituents arising from Arthur-type data.
Abstract
Let $F$ be a $p$-adic field. In this article, we consider representations of split special orthogonal groups $\mathrm{SO}_{2n+1}(F)$ and symplectic groups $\mathrm{Sp}_{2n}(F)$ of rank $n$. We denote by $π_1 \times \ldots \times π_r \rtimes π$ the normalized parabolically induced representation of either. Now let $u_i$ be essentially Speh representations and $π$ a representation of Arthur type. We prove that the parabolic induction $u_1 \times \ldots \times u_r \rtimes π$ is irreducible if and only if $u_i \times u_j$, $u_i \times u_j^\vee$ and $u_i \rtimes π$ are irreducible for all choices of $i\neq j$. If $u_i$ are Speh representations, we determine the composition series of the above parabolically induced representation. Through this, we are able to produce a new collection of unitary representations.