The Kazhdan–Lusztig conjecture for $p$-adic representation pairings

Let $H$ be a $p$-adic reductive group, let $\lambda^{\mathbb{Q}_p}$ be an infinitesimal character, and let $\xi=(C,\mathcal{V})$ and $\xi'$ be parameters indexing the corresponding irreducible representations $\pi(\xi)$, standard representations $M(\xi)$, simple perverse objects $\mathcal{P}(\xi)$, and standard objects $\mu(\xi')$ in the equivariant derived category. Let

$$\langle-,-\rangle:K\operatorname{Rep}_{{\mathrm{m} \,\mathrm{pure}}}(\lambda^{\mathbb{Q}_p},H)\times K D^b(H_{\lambda^{\mathbb{Q}_p}}\backslash V_{\lambda^{\mathbb{Q}_p}})\longrightarrow\mathbb{C}$$

be the perfect pairing specified above. Kazhdan–Lusztig conjecture. Under this pairing,

$$\langle M(\xi),\mu(\xi')\rangle=(-1)^{\dim C}e(H_\xi)\delta_{\xi,\xi'}.$$

This is intended to express the expected compatibility between standard representations and standard objects under the representation-theoretic and geometric parametrizations; the source supplies no resolution status.