The geometric properties conjecture for Weil–Deligne parameter stacks

Let $F$ be a non-archimedean local field, let $\check G$ be the dual group of a split reductive group, and let $X_{\check G}^{\rm WD}$ be the scheme of $\check G$-valued Weil–Deligne representations. For a parabolic subgroup $\check P\subset\check G$ with Levi quotient $\check M$, let ${\bf X}_{\check P}^{\rm WD}$ be the associated derived scheme and let

$$\alpha_{\check P}^{\rm WD}:[{\bf X}_{\check P}^{\rm WD}/\check P]\longrightarrow[X_{\check M}^{\rm WD}/\check M],\qquad \beta_{\check P}^{\rm WD}:[{\bf X}_{\check P}^{\rm WD}/\check P]\longrightarrow[X_{\check G}^{\rm WD}/\check G]$$

be the induced morphisms. The geometric properties conjecture. The scheme $X_{\check G}^{\rm WD}$ is a local complete intersection; $\alpha_{\check P}^{\rm WD}$ has finite Tor-dimension; and there is a unique morphism

$$X_{\check M}^{\rm WD}/\!/\check M\longrightarrow X_{\check G}^{\rm WD}/\!/\check G$$

induced by these stack morphisms and making the stated quotient diagram commute. These properties are expected to provide the geometric foundation for the derived Langlands conjecture, but are not proved in the source.