The Jacobson–Morozov Morphism for Langlands Parameters in the Relative Setting

Abstract

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.