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

Technical Summary: The Geometry of the Unipotent Component of the Moduli Space of Weil-Deligne Representations

Problem Statement
The paper investigates the geometry of the moduli space of unipotent Weil-Deligne representations valued in a split reductive group $G$ over a local field. Specifically, it focuses on the scheme $S_{G,O}$, which parametrizes pairs $(\Phi, N)$ where $\Phi$ is a semisimple element and $N$ is a nilpotent element satisfying the relation $\text{Ad}(\Phi)N = qN$. This scheme serves as a framed moduli space for unipotent Weil-Deligne representations and is closely related to local Galois deformation rings, which are central to the Taylor-Wiles-Kisin patching method used in the proof of modularity lifting theorems.

The primary geometric problem addressed is the characterization of the smoothness of the irreducible components of $S_{G,O}$. While the scheme is known to be a union of irreducible components indexed by nilpotent orbits, the conditions under which these components are smooth (regular) versus singular were not fully established for general reductive groups, particularly in the context of the "considerate" condition required for deformation theory.

Methodology
The author employs a combination of algebraic geometry, representation theory, and commutative algebra to analyze the local structure of $S_{G,O}$.

1. Considerateness and Banality: The paper introduces and utilizes the notion of "$q$-considerateness," a condition on the prime power $q$ relative to the Coxeter number $h_G$ of the group $G$. This ensures that $q$ behaves well with respect to the group's structure (specifically, that $1, q, \dots, q^{h_G}$ are distinct in the residue field). This is linked to the concept of "$G$-banality" and "geometrically-$G$-banality" from the work of Dat, Helm, Kurinczuk, and Moss.
2. Bala-Carter Theory and Cocharacters: The analysis relies heavily on Bala-Carter theory to classify nilpotent orbits. For a nilpotent element $e$, the author associates a cocharacter $\lambda: \mathbb{G}_m \to G$ and decomposes the Lie algebra $\mathfrak{g}$ into eigenspaces of $\text{Ad}(\lambda(t))$. The distinction between "distinguished" nilpotent orbits (where the centralizer has a specific structure) and non-distinguished orbits is crucial.
3. Smoothness Proofs:
   • Smoothness: For components corresponding to distinguished nilpotent orbits (or the zero orbit), the author constructs a smooth surjective morphism from a product of the group $G$ and a specific smooth subscheme $Y$ (defined via the Levi decomposition of the centralizer of the nilpotent element). By showing this map is a pullback of a smooth morphism, smoothness of the component is established.
   • Singularities: For non-distinguished orbits, the author constructs a specific point in the component and demonstrates that the dimension of the tangent space at this point strictly exceeds the dimension of the component. This is achieved by analyzing the interaction between the tangent spaces of the orbit, the torus, the nilpotent cone, and unipotent subgroups, utilizing the properties of "exposed" roots in the Dynkin diagram.
4. Patching and Automorphic Forms: The geometric results are applied to the arithmetic setting using the Taylor-Wiles-Kisin patching method. The author considers Hida families of ordinary automorphic forms on unitary groups and constructs a global deformation ring $R^{\text{univ}}_S$.

Key Contributions and Results

1. Characterization of Smooth Components (Theorem 1.2):
   The paper proves that under the assumption that $q$ is considerate towards $G$:

   • The irreducible component $X_C$ of $S_{G,O}$ corresponding to a distinguished nilpotent orbit $C$ (or the zero orbit) is smooth over the base ring.
   • Conversely, if $C$ is a non-trivial, non-distinguished nilpotent orbit, the component $X_C$ is singular.
     This result generalizes previous work by Bellovin and Helm, providing a complete geometric classification of the smoothness of these moduli spaces.

2. Regularity of Deformation Rings:
   By establishing the smoothness of the local deformation rings associated with distinguished orbits (specifically Steinberg and discrete series representations), the paper confirms that the generic fibers of these local deformation rings are regular local rings. This is a prerequisite for the patching argument to succeed.

3. Freeness of Ordinary Automorphic Forms (Theorem 1.3):
   Applying the geometric results to the global setting, the author proves that the module of ordinary automorphic forms $S^{\text{ord}}(U(p^\infty), L/\mathcal{O})^\vee$ is a finite locally free module over the global deformation ring $R^{\text{univ}}_S[1/l]$.

   • This implies that the Hecke algebra acting on these forms is isomorphic to the global deformation ring (after inverting $l$).
   • Crucially, it establishes that the multiplicity of automorphic forms with a given characteristic zero Galois representation is constant along the connected components of the deformation ring.

Significance and Claims
The paper claims to provide a rigorous geometric foundation for the smoothness of local deformation rings used in modern modularity lifting theorems. By explicitly identifying which components of the Weil-Deligne moduli space are smooth, the work justifies the regularity assumptions often made in patching arguments.

The application to Hida families is significant because it extends multiplicity results from classical automorphic forms to non-classical Hida families. The author notes that if the multiplicity is 1 for classical forms on a specific connected component, the geometric freeness result implies the multiplicity is 1 for all forms (classical and non-classical) in that component. This bridges the gap between classical number theory and the geometry of $p$-adic families, ensuring that the "multiplicity one" property is preserved across the family.

The work does not propose new experimental applications or future conjectures beyond the immediate scope of establishing the freeness of the module and the isomorphism between the Hecke and deformation rings in the ordinary setting for unitary groups. It relies on and synthesizes existing machinery (Taylor-Wiles-Kisin, Hida theory) with new geometric insights regarding the singularities of nilpotent orbit closures in the context of Weil-Deligne parameters.