On supercuspidal part of the l-adic cohomology of the Rapoport-Zink space for GSp(4)


The Rapoport-Zink space for $\mathrm{GSp}(4)$ is a local counterpart of the Siegel threefold. Its $\ell$-adic etale cohomology is equipped with an action of the product of three groups: $\mathrm{GSp}_4(\mathbb{Q}_p)$, its non-trivial inner form $J(\mathbb{Q}_p)$, and the Weil group of $\mathbb{Q}_p$. This action is expected to be strongly related with the local Langlands correspondence for $\mathrm{GSp}_4$ and $J$. In this talk, I will explain how the $\mathrm{GSp}_4(\mathbb{Q}_p)$-supercuspidal part of the cohomology is described by the local Langlands correspondence. If time permits, I will also give some observations on the $\mathrm{GSp}(6)$ case.