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Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-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$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator. 

Scholze and Shin [J. Amer. Math. Soc. 26 (2013), pp. 261–294] gave a conjectural formula relating the traces on the automorphic and Galois sides of a local Langlands correspondence. Their work generalized an earlier formula of Scholze, which he used to give a new proof of the local Langlands conjecture for GL_n. Unlike the case for GL_n, the existence of non-singleton L-packets for more general reductive groups constitutes a serious representation-theoretic obstruction to proving that such a formula uniquely characterizes such a correspondence. We show how to overcome this problem, and demonstrate that the Scholze–Shin equation is enough, together with other standard desiderata, to uniquely characterize the local Langlands correspondence for discrete parameters.