Nilpotent Singular Support in Geometric Langlands

• Nilpotent singular support is a microlocal condition that confines objects to the global nilpotent cone, ensuring structural integrity in geometric representation theory.
• It guarantees Hecke stability and spectral decomposition aligned with Langlands parameters, facilitating a bridge between the automorphic and spectral sides.
• The framework underpins categorical invariants and practical applications, from parabolic induction to factorization in both algebraic geometry and quantum field theory.

Nilpotent singular support characterizes a fundamental microlocal constraint on categories and sheaves in geometric representation theory, especially in the context of the Geometric Langlands program. It requires objects (such as sheaves or categories with group action) to be microsupported within the global nilpotent cone in an appropriate cotangent bundle or stack. This condition imposes deep structure on the categories in question, ensuring compatibility with key functorial operations (e.g., Hecke functors), controlling the spectral decomposition of objects, and relating automorphic and spectral sides of geometric Langlands duality.

1. Microlocal Foundations and Definition

Let $y$ be a smooth algebraic stack (often $Bun_G$ for $G$ a reductive group), or more generally an algebraic stack with an action of a group $G$. Given a sheaf-theoretic context (constructible $\ell$-adic, Betti, or D-modules), the singular support $SS(\mathcal{F})$ of an object $\mathcal{F}$ in $Shv(y)$ is a conic, closed subset of the classical cotangent stack $T^*y$ or, in the categorical setting, a conic, closed, $\mathrm{Ad}$-stable subset $Bun_G$0 associated to a $Bun_G$1-category $Bun_G$2 via the singular support of its sheaves of matrix coefficients (Arinkin et al., 2020, Dhillon et al., 2024).

Global nilpotent cone. For $Bun_G$3, each point in $Bun_G$4 is a pair $Bun_G$5 where $Bun_G$6 is a $Bun_G$7-bundle and $Bun_G$8. The global nilpotent cone $Bun_G$9 is defined by requiring that $G$0 is nilpotent in $G$1 for all $G$2 (Arinkin et al., 2020, Arinkin et al., 2020).

Nilpotent singular support condition. An object $G$3 (or a $G$4-category $G$5) is said to have nilpotent singular support if $G$6 (or $G$7), where $G$8 is the nilpotent cone (Arinkin et al., 2020, Dhillon et al., 2024).

Universal setting and families. For a family of curves $G$9 and the associated $G$0 over $G$1, the universal nilpotent cone $G$2 is constructed via the zero-fiber of the relative Hitchin map, or as a Lagrangian image of the zero-section under Lagrangian correspondences arising from Eisenstein series induction. This defines a horizontal condition uniform in families (Nadler et al., 1 Mar 2026).

2. Structural Theorems and Categorical Properties

The nilpotent singular support condition isolates a robust subcategory within automorphic or $G$3-equivariant sheaf categories. Several rigorous results characterize its behavior:

• Hecke stability: For any Hecke functor $G$4 associated to a representation $G$5 of $G$6, $G$7 preserves the nilpotent singular support subcategory; i.e., $G$8. Conversely, objects whose Hecke images have trivial support along $G$9 must already be nilpotent-supported (in the singular support sense) (Arinkin et al., 2020, Arinkin et al., 2020, Nadler et al., 1 Mar 2026).
• Spectral decomposition: $\ell$0 decomposes as a (possibly completed) direct sum indexed by semi-simple Langlands parameters, reflecting the spectral side of the Geometric Langlands program (Arinkin et al., 2020).
• Self-duality: The category $\ell$1 is canonically equivalent to its dual as a DG-category, with the key pairing implemented via integration over $\ell$2 of pairwise tensor products. This duality functor coincides with the restriction of miraculous duality on the whole $\ell$3 category, intertwining with the Serre functor (up to a technical "co" issue on non-quasi-compactness) (Arinkin et al., 2020).
• Categorical support for $\ell$4-categories: For dualizable DG-categories with a strong $\ell$5-action, nilpotent singular support (i.e., support lying in $\ell$6) can be characterized by the matrix-coefficient functor and checked against the vanishing of generalized Whittaker models attached to nilpotent orbits (Dhillon et al., 2024).

3. Constructions, Examples, and Classification

The nilpotent singular support condition admits explicit geometric and algebraic constructions in a wide range of settings:

• Character sheaves: On a reductive group $\ell$7, all character sheaves have singular support contained in $\ell$8, where $\ell$9 is the nilpotent cone in $SS(\mathcal{F})$0 (Psaromiligkos, 2022). The proof involves the use of tame perverse sheaves, conormality properties via stratifications such as Bruhat and Bott–Samelson resolutions, and explicit computation on Bruhat cells.
• Moduli of coherent sheaves: On stacks such as $SS(\mathcal{F})$1 (the moduli of coherent sheaves on an elliptic curve $SS(\mathcal{F})$2), nilpotent singular support is defined via the nilpotency of the Higgs field—$SS(\mathcal{F})$3 for some $SS(\mathcal{F})$4. The irreducible components of the global nilpotent cone are combinatorially classified using Harder–Narasimhan filtrations and partitions. Perverse sheaves with nilpotent singular support coincide with Eisenstein-type sheaves, and their classes biject with the components of the nilpotent cone via the characteristic cycle map (Hennecart, 2021).
• Families and universal cones: For families of curves, the universal nilpotent cone is a conic Lagrangian submanifold of the total cotangent bundle, constructed both as the zero fiber of the relative Hitchin map and as the Eisenstein image of the zero section via Lagrangian correspondences. This induces a singular support condition uniform over the entire family (Nadler et al., 1 Mar 2026).
• Categorical invariants: In $SS(\mathcal{F})$5-equivariant categorical contexts, nilpotent support can be detected via the vanishing of generalized Whittaker categories $SS(\mathcal{F})$6 for nilpotent $SS(\mathcal{F})$7, or via the behavior of matrix-coefficient sheaves (Dhillon et al., 2024).

4. Physical and Factorization-Theoretic Perspectives

Nilpotent singular support arises from and interacts naturally with topological quantum field theory and factorization algebra structures:

• In 4d $SS(\mathcal{F})$8 gauge theory, the nilpotent singular support condition corresponds to restricting to the zero vacuum in the moduli space of vacua $SS(\mathcal{F})$9. This restriction selects the category of boundary conditions compatible with the nilpotent vacuum, leading to the subcategory $\mathcal{F}$0 on the spectral (Higgs bundle) side (Elliott et al., 2017).
• Factorization algebra technology relates the structure of the singular support condition to global and microlocal symmetries, with factorization structures induced from the configuration spaces of eigenvalues; in particular, for $\mathcal{F}$1 the hidden factorization structure is expressed via the stratification of $\mathcal{F}$2 (Elliott et al., 2017).
• The physics of branes (e.g., $\mathcal{F}$3-branes in type IIB string theory) provides further justification for the factorization properties and the role of nilpotent conditions in symmetry breaking to Levi subgroups.

5. Functoriality, Induction, Whittaker Models, and Applications

Nilpotent singular support is stable and behaves predictably under several functorial constructions, leading to powerful applications:

• Parabolic induction and restriction: The singular support of parabolic induction and restriction functors at the categorical level is explicitly computable via functorial maps on conic subsets: $\mathcal{F}$4, $\mathcal{F}$5 (Dhillon et al., 2024).
• Generalized Whittaker categories: The vanishing and nonvanishing of generalized Whittaker models $\mathcal{F}$6 for a nilpotent $\mathcal{F}$7-category $\mathcal{F}$8 precisely reflects the presence of nilpotent orbits in $\mathcal{F}$9—maximal orbits inside the support yield nonzero Whittaker categories and carry information about the microstalks of character sheaves (Dhillon et al., 2024).
• Springer theory and $Shv(y)$0-algebras: The classification of finite-dimensional $Shv(y)$1-algebra modules with a fixed regular central character is realized as a subrepresentation of the Springer representation, with explicit connection to the rationalized Grothendieck group and categorical traces of wall-crossing functors (Dhillon et al., 2024).
• Categorical trace and automorphic functions: The self-duality and perfect pairing on $Shv(y)$2 enable comparison of categorical traces (e.g., of Frobenius endomorphisms) to spaces of classical automorphic functions, enacting the trace formula conjectures in the categorical setting (Arinkin et al., 2020).

6. Implications for Geometric Langlands and Further Directions

Imposing nilpotent singular support is a defining feature of modern categorical and spectral approaches to the Geometric Langlands program:

• Spectral–automorphic duality: The nilpotent singular support condition singles out the subcategory of $Shv(y)$3 corresponding to the "Betti automorphic" sheaves relevant for the geometric Langlands correspondence, and matches it to the subcategory $Shv(y)$4 on the spectral side (Arinkin et al., 2020).
• Hecke module structure and gluing: Hecke-stable categories of objects with nilpotent singular support enable the construction of gluing functors and module structures over local Hecke algebras, critical for the full geometric and categorical formulation of Langlands duality, including in families (Nadler et al., 1 Mar 2026).
• Local constancy and monodromy: In families of curves, local constancy of the nilpotent singular support category over contractible bases implies mapping class group actions and monodromy invariants, with recent advances extending proof of equivalence statements in rank one and beyond (Nadler et al., 1 Mar 2026).
• Extensions: The framework extends to deeper level structures, higher-dimensional base geometries, and expected refinement to derived and quantum settings, indicating a vast field of potential applications and generalizations (Nadler et al., 1 Mar 2026).

Nilpotent singular support thus acts as a technical and conceptual bridge, uniting microlocal sheaf theory, representation theory, algebraic geometry, quantum field theory, and the spectral–automorphic duality at the heart of the Geometric Langlands program.