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 these 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 $\rho $ an admissible representation 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.

我们研究了EL 型未分支 Rapoport-Zink 空间的l -adic 上同调。这些空间用于 Harris 和 Taylor 对 $\mathrm {GL_n}$ 的局部朗兰兹对应的证明，并显示了朗兰兹对应的局部-全局兼容性。在本文中，我们考虑 由这些空间的上同调构造的 Grothendieck 表示群的某些态射 $\mathrm {Mant}_{b, \mu }$ ，正如 Harris 和 Taylor、Mantovan、Fargues、Shin 等人所研究的那样。由于 Fargues 和 Shin 的早期工作，我们对 $\mathrm {Mant}_{b, \mu }(\rho )$ 进行 了描述，因为 $\rho $ 是一个超尖表示。在本文中，我们给出了一个猜想公式 $\mathrm {Mant}_{b, \mu }(\rho )$ 为 $\rho $ 一个可接受的表示，并在 $\rho $ 本质上是平方可积时证明它 我们的证明适用于一般 $\rho $ 有条件地基于 Shin 工作中出现的猜想。我们表明，我们的描述与哈里斯的猜想一致，在 Levi 亚群的超尖顶表征的抛物线归纳的情况下。