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

Let $F$ be a $p$-adic field. In this article, we consider representations of split special orthogonal groups ${\mathrm{S}\mathrm{O}}_{2n+1}(F)$ and symplectic groups ${\mathrm{S}\mathrm{p}}_{2n}(F)$ of rank $n$. We denote by ${\pi }_1\times \dots \times {\pi }_r⋊\pi$ 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 $u_1\times \dots \times u_r⋊\pi$ is irreducible if and only if $u_i\times u_j$, $u_i\times u_j^{\vee }$ and $u_i⋊\pi$ are irreducible for all choices of $i\mathrm{\ne }j$. 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.