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

Abstract

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 $\mathsf{JM}:{\mathsf{LP}}_G\to {\mathsf{WDP}}_G$ (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 ${\mathsf{LP}}_G^{\mathrm{disc}}\to {\mathsf{WDP}}_G^{\mathrm{disc}}$, 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.

41 pages