The geometry of the unipotent component of the moduli space of Weil-Deligne representations

TL;DR

This work studies the geometry of the moduli space of unipotent Weil–Deligne representations valued in a split reductive group $G$, focusing on the irreducible components and their smoothness. The authors develop a framework using $q$-considerateness and integral deformations to relate the unipotent moduli $S_{G,\mathcal{O}}$ to the tame L-parameter space, decompose components via nilpotent orbits, and prove a sharp smoothness criterion: components corresponding to the zero orbit or to distinguished nilpotent orbits are smooth, while components from non-distinguished orbits are singular. They apply these geometric insights to automorphic forms on unitary groups, constructing big ordinary Hecke algebras and employing Taylor–Wiles–Kisin patching to show that the ordinary automorphic forms space is locally freely generated over the global deformation ring, yielding $R^{\mathrm{univ}}_{\mathcal{S}}[1/l]\cong \mathbb{T}[1/l]$ and constant multiplicities along connected components. The results bridge the geometry of unipotent L-parameter spaces with automorphic and Galois deformation theories, providing precise smoothness criteria and powerful patching arguments that control multiplicities in Hida families for unitary groups.

Abstract

In this paper, we study the moduli space of unipotent Weil-Deligne representations valued in a split reductive group $G$ and characterise which irreducible components are smooth. We apply the smoothness results proved to show that a certain space of ordinary automorphic forms is a locally generically free module over the corresponding global deformation ring.

Figures (5)

• Figure 1: The Distinguished weighted Dynkin diagrams of Levi subgroups of $F_4$
• Figure 2: An example of $\alpha$ in the case where $G$ is of type $E_8$ and $L$ of type $D_4\times A_2$.
• Figure 3: The weighted Dynkin diagrams of distinguished orbits of type $D_n$
• Figure 4: The weighted Dynkin diagrams of distinguished orbits of type $E_6$
• Figure 5: The weighted Dynkin diagrams of distinguished orbits of type $E_7$

Theorems & Definitions (67)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['theorem locfree']}
• Definition 2.1
• Definition 2.2
• Remark
• Proposition 2.3
• proof
• Remark
• Definition 2.4