The Jacobson--Morozov morphism for Langlands parameters in the relative setting

We construct a moduli space $\mathsf{LP}_G$ of $\mathrm{SL}_2$-parameters over $\mathbb{Q}$, and show that it has good geometric properties (e.g. explicitly parametrized geometric connected components and smoothness). We construct a Jacobson--Morozov morphism $\mathsf{JM}\colon \mathsf{LP}_G\to\mathsf{WDP}_G$ (where $\mathsf{WDP}_G$ is the moduli space of Weil--Deligne parameters considered by several other authors). We show that $\mathsf{JM}$ is an isomorphism over a dense open of $\mathsf{WDP}_G$, that it induces an isomorphism between the discrete loci $\mathsf{LP}^{\mathrm{disc}}_G\to\mathsf{WDP}_G^{\mathrm{disc}}$, and that for any $\mathbb{Q}$-algebra $A$ it induces a bijection between Frobenius semi-simple equivalence classes in $\mathsf{LP}_G(A)$ and Frobenius semi-simple equivalence classes in $\mathsf{WDP}_G(A)$ with constant (up to conjugacy) monodromy operator.