Koszul duality for generalised Steinberg representations of $p$-adic groups

Let $G$ be a semisimple group, split over a non-Archimedean field $F$. We prove that the category of modules over the extension algebra of generalised Steinberg representations of $G(F)$ is equivalent to a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations. Specifically, we establish an equivalence $$\textbf{Mod}(\text{Ext}_G^\bullet(\Sigma_\lambda, \Sigma_\lambda)) \simeq \textbf{Per}_{\widehat{G}}^\circ(X_\lambda),$$ where $\Sigma_\lambda$ is the direct sum of generalised Steinberg representations and $\textbf{Per}_{\widehat{G}}^\circ(X_\lambda)$ is the subcategory of perverse sheaves on the variety of Langlands parameters $X_\lambda$ corresponding to these representations under Vogan's geometrisation of the Langlands correspondence. Furthermore, we demonstrate that this equivalence is a true Koszul duality by showing that the extension algebra of generalised Steinberg representations is Koszul dual to the endomorphism algebra of the direct sum of corresponding equivariant perverse sheaves, taken in the equivariant derived category $D_{\widehat{G}}^b(X_\lambda)$.