Covering Barbasch–Vogan Duality

• Covering Barbasch–Vogan duality is a refined extension that associates nilpotent orbits in dual groups with finite equivariant covers, enriching classical duality through additional geometric invariants.
• It extends the framework to nonlinear groups such as metaplectic and Brylinski–Deligne covers, thereby enhancing the classification of unipotent representations and primitive ideals.
• The duality underpins advancements in the geometric and categorical Langlands programs, establishing precise multiplicity formulas and transfer identities in Arthur packets.

Covering Barbasch–Vogan duality is an extension of the classical Barbasch–Vogan duality, enhancing the correspondence between nilpotent orbits, primitive ideals, and unipotent representations of complex, real, and $p$-adic groups to the setting of nonlinear (covering) groups and their associated covering duals, as well as accounting for finite equivariant covers of nilpotent orbits. This refinement is central for the representation theory of nonlinear groups including metaplectic covers, Brylinski–Deligne covering groups, and for the structure of Arthur packets via equivariant parametrization. The covering duality connects geometric and categorical invariants, symplectic duality, and multiplicity formulas, and it plays an essential role in ongoing developments around the geometric and categorical Langlands programs.

1. Classical Barbasch–Vogan Duality and Its Extensions

The original Barbasch–Vogan (BV) duality constructs a surjective, order-reversing map between nilpotent orbits in the Langlands dual Lie algebra $\mathfrak{g}^\vee$ and the special nilpotent orbits in $\mathfrak{g}$: $$D: \text{Nil}(\mathfrak{g}^\vee) \longrightarrow \text{Special}(\mathfrak{g})$$ This map transposes partitions in type $A$ and uses combinatorially defined induction/collapse rules in types $B$, $C$, and $D$, yielding a bijection between orbits with applications to the classification of special unipotent representations and primitive ideals, and governing the structure of Arthur packets. The map is characterized by the combinatorial transformation of partitions (transpose in type $A$, Lusztig–Spaltenstein rules in other types), order-reversal with respect to orbit closure, and compatibility with Richardson induction in Levi subalgebras (Hoang, 2024).

Covering Barbasch–Vogan duality generalizes this correspondence in several directions:

• It replaces simple orbits with finite $G$-equivariant covers of nilpotent orbits, capturing additional symplectic and representation-theoretic information.
• It incorporates nonlinear covering groups such as double covers of classical groups (e.g., metaplectic groups), $n$-fold Brylinski–Deligne covers, and more generally Hopf–Hecke and Barbasch–Sahi algebras.
• It realizes duality at the level of primitive ideals, associated varieties, and Dirac cohomology, and connects with symplectic duality and categorical representation theory (Losev et al., 2021, Mason-Brown et al., 2023, Gao et al., 18 Nov 2025, Flake, 2016).

2. Formalism for Covering Groups and Nilpotent Covers

Let $G$ be a complex semisimple group with Lie algebra $\mathfrak{g}$, possibly replaced by a nonlinear cover $\widetilde{G}$. The refinement of BV duality—sometimes termed "covering Barbasch–Vogan duality," "refined BVLS duality," or "extended covering duality"—takes the form: $$\widetilde{D} : \text{Nil}(\mathfrak{g}^\vee) \to \{\text{isomorphism classes of } G\text{-equivariant covers of nilpotent orbits in } \mathfrak{g}\}$$ In the framework of Losev–Mason-Brown–Matvieievskyi, given a nilpotent orbit $\mathcal{O}_{e^\vee} \subset \mathfrak{g}^\vee$, the image under $\widetilde{D}$ is constructed as a $G$-equivariant cover $\widetilde{\mathcal{O}_e} \to \mathcal{O}_e$ (where $\mathcal{O}_e = D(\mathcal{O}_{e^\vee})$) arising geometrically from the fiber of a moment map on $T^*(G/P)$, with monodromy governed by the component group $A(\mathcal{O}_e)$ (Hoang, 2024, Mason-Brown et al., 2023).

This refined duality is essential in settings where the naive duality at the level of orbits is insufficient, for example:

• In type $D$, very-even orbits require identifying double covers to distinguish the two types of orbits.
• In quantization and symplectic geometry, covers record crucial information for the proper classification of unipotent ideals and the construction of Harish–Chandra bimodules.
• For $p$-adic and metaplectic coverings, the covering duality selects the correct recipient for wavefront set formulations and multiplicity formulas (Gao et al., 18 Nov 2025, Barbasch et al., 2020).

3. Representation-Theoretic and Geometric Implications

Primitive Ideals and Harish–Chandra Bimodules

A fundamental property is that for each cover $\widetilde{\mathcal{O}} \to \mathcal{O}$, the canonical quantization yields a filtered algebra $\mathcal{A}_0$ equipped with a quantum co-moment map $\Phi_0 : U(\mathfrak{g}) \to \mathcal{A}_0$, whose kernel $I_0(\widetilde{\mathcal{O}})$ is a completely prime primitive ideal with associated variety $\overline{\mathcal{O}}$ (Losev et al., 2021).

The covering BV duality asserts a bijection at the level of primitive ideals: $$I_{\max(\frac{1}{2}h^\vee)} = I_0\bigl(\widetilde{D}(\mathcal{O}^\vee)\bigr)$$ Crucially, unipotent representations attached to $\widetilde{\mathcal{O}}$ are parameterized by representations of the fundamental group $\pi_1^G(\widetilde{\mathcal{O}})$, generalizing the role of Lusztig's canonical quotient in the linear case (Losev et al., 2021, Mason-Brown et al., 2023).

Arthur Packets and Geometric Transfer

In the context of complex classical groups, Arthur packets $\Pi(\psi, G)$ and their internal parametrization by the dual group of the component group $A(\psi) = S_\psi / (S_\psi^0 Z(^LG))$ coincide exactly with Barbasch–Vogan's packets in the unipotent "good parity" case. Their construction by parabolic induction from lower-rank (especially unipotent) packets is irreducible, and multiplicity formulas and transfer identities match across both frameworks (Moeglin et al., 2016).

In $p$-adic settings, microlocal vanishing cycles and perverse sheaves on the parameter variety organize ABV-packets, with the geometric data of conormal bundles and local systems encoding the spectral transfer identities and recovering Arthur's packet multiplicities (Cunningham et al., 2017).

4. Combinatorial Description: Nilpotent Orbits and Partition Dualities

For classical types, covering BV duality admits a combinatorial characterization, generalizing classical partition rules:

• In type $A_n$, the duality remains given by the transpose of partitions.
• In types $B,C,D$, one employs dominance order, Lusztig–Spaltenstein's induction and collapse techniques, and partitions subject to parity or evenness constraints (Hoang, 2024, Gao et al., 18 Nov 2025, Barbasch et al., 2020).
• For covers, partitions may code for the isomorphism class of the covering, with marks or decorations specifying the relevant subgroup of the component group.

The formulae for the duality can be distilled into explicit recipes in terms of partition manipulation, collapse, and marking, ensuring that all structural properties (order-reversing, compatibility with induction, etc.) are preserved in the covering context (Gao et al., 18 Nov 2025, Barbasch et al., 2020).

5. Applications to Genuine Representations and Wavefront Sets of Covering Groups

The extension of BV duality to covering groups has direct representation-theoretic consequences:

• Wavefront set bounds: For $n$-fold Brylinski–Deligne covers or metaplectic groups, the conjectural correspondence assigns to a genuine irreducible representation an upper bound on its wavefront set via the covering BV dual of the nilpotent orbit attached to its $SL_2$-type Langlands parameter. For Kazhdan–Patterson covers and other covering types, this bound is explicit and proven in key families (Gao et al., 18 Nov 2025).
• Multiplicity and transfer conjectures: The refined parametrization by covers is crucial for multiplicity formulas in Arthur packets for both linear and nonlinear groups, as well as for transfer identities involving endoscopic and stable distributions (Moeglin et al., 2016, Barbasch et al., 2010).
• Metaplectic special unipotents: For real metaplectic groups, metaplectic Barbasch–Vogan duality defines metaplectic-special nilpotent orbits, generalizes the relationship between infinitesimal character and associated variety, and aligns with the theory of Weyl group double cells (Barbasch et al., 2020, Barbasch et al., 2017).
• Symplectic duality and Hikita conjecture: The upgraded duality aligns geometric and representation-theoretic dualities, substantiating versions of the Hikita conjecture relating cohomology of Springer fibers and algebras of functions on fixed points of covers (Hoang, 2024).

6. Algebraic Structures: Barbasch–Sahi Algebras and Dirac Cohomology

Within the theory of noncommutative algebras, covering BV duality is realized in the class of Hopf–Hecke and Barbasch–Sahi algebras. These generalize the universal enveloping and graded affine Hecke algebras, and support Dirac operators whose cohomology detects central characters and their images under the duality, thereby providing a categorical generalization of Vogan's conjecture (Flake, 2016). The existence of pin (or metaplectic) covers at the level of Hopf algebras enables the uniform treatment of all previously known Dirac cohomology and unipotent representation phenomena.

7. Interplay with Stable Combinations and Special Pieces

Barbasch–Vogan duality and its covering refinement explain the structure of the $\mathbb{Z}$-span of stable combinations of special unipotent representations, which are often, but not always, generated by Arthur packets alone. The canonical basis is indexed by rational forms of the special piece in the nilpotent cone of the dual algebra, and the full stable character theory is governed by the refined geometry of these pieces and their conormal cycle classes (Barbasch et al., 2010).

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References:

(Hoang, 2024, Moeglin et al., 2016, Gao et al., 18 Nov 2025, Losev et al., 2021, Mason-Brown et al., 2023, Barbasch et al., 2020, Barbasch et al., 2010, Cunningham et al., 2017, Barbasch et al., 2017, Flake, 2016, Crofts, 2009)