Covering Barbasch-Vogan duality and wavefront sets of genuine representations

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of $p$-adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.

💡 Research Summary

The paper introduces a “covering Barbasch‑Vogan duality” for Brylinski–Deligne central covers of split reductive p‑adic groups and uses it to formulate and partially prove an upper‑bound conjecture for wave‑front sets of genuine (i.e., µₙ‑genuine) representations.

Main constructions.
Let G be a split connected reductive group over a non‑archimedean field F and let G^{(n)} be an n‑fold Brylinski–Deligne central cover (assume µₙ⊂F×). The complex dual group G^∨ is defined using the modified root datum (Y_{Q,n},Φ^∨{Q,n}) that incorporates the quadratic form Q of the cover. For a nilpotent orbit O⊂𝔤^∨, the usual Barbasch‑Vogan construction starts from the dominant semisimple element h_O∈Lie(T^∨) and takes h_O/2 as a weight for the complex group G_ℂ. In the covering setting the authors apply the linear map s:Lie(T^∨)→Lie(T^∨) induced by the inclusion X→X{Q,n} to obtain a new weight
 h^{(n)}O/2 = s(h_O/2).
They then consider the Verma module I(λ) with λ = h^{(n)}O/2, take its unique irreducible quotient L(λ), and define the covering duality by
 d^{(n)}{BV,G}(O) = AV{𝔤_ℂ}(L(λ)),
the associated nilpotent orbit in 𝔤. When n=1 this recovers the classical Barbasch‑Vogan duality; for the metaplectic double cover of Sp_{2r} it coincides with the metaplectic duality of Barbasch‑Ma‑Sun‑Zhu. The paper supplies explicit combinatorial formulas for h^{(n)}O/2 and for d^{(n)}{BV,G} in all classical types (A, B, C, D) using the work of Bai‑Ma‑Wang, and for exceptional types using results of Bai‑Gao‑Xie‑Wang. The authors prove that d^{(n)}{BV,G} is order‑reversing and compatible with Levi induction:  d^{(n)}{BV,G}∘Sat_{G^∨→M^∨} = Ind_{M→G}∘d^{(n)}_{BV,M}.

Wave‑front set conjecture.
For a genuine irreducible representation π of G^{(n)} let φ_π:WD_F→^LG be its (conjectural) L‑parameter and O(φ_π) the corresponding nilpotent orbit in G^∨. The Aubert–Zelevinsky involution AZ is denoted by AZ(π). The conjecture (Conjecture 1.2) states:

1. For every genuine π,  WF_{geo}(AZ(π)) ≤ d^{(n)}_{BV,G}(O(φ_π)).
2. Equality is attained for at least one member of each tempered L‑packet.

The inequality is equivalent to WF_{geo}(π) ≤ d^{(n)}{BV,G}(O(φ{AZ(π)})). The authors show that proving the inequality for all genuine π reduces to proving it for discrete series of all Levi subgroups (Theorem 6.5), mirroring the reduction in the linear case proved by Hazeltine‑Liu‑Lo‑Shahidi.

Evidence and proofs.

• For unramified theta representations (the covering Steinberg representation) the equality is already known from Karasiewicz‑Okada‑Wang, providing evidence for part (ii).
• The paper proves the full conjecture for Kazhdan‑Patterson covers of GL_r. Under a “working hypothesis” concerning the behavior of certain intertwining operators (Hypothesis 5.2) and assuming (p,n)=1, the authors consider genuine Speh representations Z(ρ,