Square integrable representations of classical $p$-adic groups corresponding to segments

Let $S_n$ be either the group $Sp(n)$ or $SO(2n+1)$ over a $p$-adic field $F$. Then Levi factors of maximal parabolic subgroups are (isomorphic to) direct products of $GL(k)$ and $S_{n-k}$, with $1\leq k\leq n$. The square integrable representations which we define and study in this paper (and prove their square integrability), are subquotients of reducible representations Ind$_P^{S_n}(\delta\otimes\sigma),$ where $\delta$ is an essentially square integrable representation of $GL(k)$, and $\sigma$ is a cuspidal representation of $S_{n-k}$. These square integrable representations play an important role in a construction of more general square integrable representations.