The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

Let $(\mathrm{G},b,\mu)$ be a PEL-type Rapoport–Zink datum, with $\mathrm{G}$ a connected reductive group over $\mathbb{Q}_p$, $\mu$ a minuscule cocharacter, and $b\in\mathbf{B}(\mathbb{Q}_p,\mathrm{G},-\mu)$. Let $\mathrm{J}_b$ be the associated inner form of a Levi subgroup of $\mathrm{G}$, and suppose that $b$ is basic. For an irreducible admissible representation $\rho$ of $\mathrm{J}_b(\mathbb{Q}_p)$ with supercuspidal $L$-parameter $\psi_\rho$, let $\mathrm{Mant}_{\mathrm{G},b,\mu}(\rho)$ denote the associated element of $\mathrm{Groth}(\mathrm{G}(\mathbb{Q}_p)\times W_{E_\mu})$. Kottwitz's conjecture. For every such $\rho$,

$$\mathrm{Mant}_{\mathrm{G}, b, \mu}( \rho )= \sum\limits_{\pi \in \Pi_{\psi_{\rho}}(\mathrm{G})} [\pi][ \operatorname{Hom}_{\overline{\mathcal{S}}_{\psi_{\rho}}}(\iota_{\mathfrak{w}}(\rho) \otimes \iota_{\mathfrak{w}}(\pi)^{\vee}, r_{-\mu} \circ \psi_{\rho}) \otimes | \cdot |^{-\langle \rho_{\mathrm{G}}, \mu \rangle}],$$

where $\Pi_{\psi_\rho}(\mathrm{G})$ is the $L$-packet attached to $\psi_\rho$. This is the predicted description of the alternating compactly supported cohomology of the Rapoport–Zink tower in terms of local Langlands data. The paper proves the conjecture for PEL-type Rapoport–Zink spaces associated with unramified unitary similitude groups over $\mathbb{Q}_p$ in an odd number of variables; the general statement remains open.