Given a reductive group $G$, the classical Jacobson--Morozov theorem states that, up to conjugacy, the nilpotent elements of $\mathrm{Lie}(G)$ correspond bijectively to the morphisms of algebraic groups $\mathrm{SL}_2\to G$. This result has an analogue in the theory of complex Langlands parameters which states that, up to equivalence, there is a bijection between the two common forms of Langlands parameters: Frobenius semi-simple Weil--Deligne parameters $(\varphi,N)$, and Frobenius semi-simple $\mathrm{SL}_2$-paramers $\psi$. Recent work of Fargues--Scholze, Dat--Helm--Kurinczuk--Moss, and Zhu has established that the moduli space $\mathsf{WDP}_G$ of Weil--Deligne parameters, and in particular its fine geometric structure, plays an important role in the study of the Langlands program. In this talk I discuss the construction of a moduli space $\mathsf{LP}_G$ of $\mathrm{SL}_2$-parameters and a *Jacobson--Morozov morphism* $\mathsf{JM}\colon \mathsf{LP}_G\to\mathsf{WDP}_G$ geometrizing the map of sets over $\mathbb{C}$. The Jacobson--Morozov morphism has several good properties, chief amongst them being weak birationality (i.e. it is an isomorphism over a dense open subset of $\mathsf{WDP}_G$). In addition to this result clarifying the classical picture, it is also significant for its potential use in the detailed study of the generic geometric structure of $\mathsf{WDP}_G$. This is particularly useful as $\mathsf{LP}_G$ enjoys many favorable geometric properties that $\mathsf{WDP}_G$ does not (e.g. it is smooth over $\mathbb{Q}$ and has explicitly parameterized affine geometric connected components).

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