Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field

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 $\pi_1 \times \ldots \times \pi_r \rtimes \pi$ the normalized parabolically induced representation of either. Now let $u_i$ be essentially Speh representations and $\pi$ a representation of Arthur type. We prove that the parabolic induction $u_1 \times \ldots \times u_r \rtimes \pi$ is irreducible if and only if $u_i \times u_j$, $u_i \times u_j^\vee$ and $u_i \rtimes \pi$ 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.