  We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of
EL-type. These spaces were used in Harris and Taylor's proof of the local
Langlands correspondence for $\mathrm{GL_n}$ and to show local-global
compatibilities of the Langlands correspondence. In this paper we consider
certain morphisms, $\mathrm{Mant}_{b, \mu}$, of Grothendieck groups of
representations constructed from the cohomology of the above spaces, as studied
by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work
of Fargues and Shin we have a description of $\mathrm{Mant}_{b, \mu}(\rho)$ for
$\rho$ a supercuspidal representation. In this paper, we give a conjectural
formula for $\mathrm{Mant}_{b, \mu}(\rho)$ for all $\rho$ and prove it when
$\rho$ is essentially square integrable. Our proof works for general $\rho$
conditionally on a conjecture appearing in Shin's work. We show that our
description agrees with a conjecture of Harris in the case of parabolic
inductions of supercuspidal representations of a Levi subgroup.
