Dice Question Streamline Icon: https://streamlinehq.com

Vogan’s conjecture equating ABV-packets and Arthur packets

Prove that for any connected reductive p-adic group G and any Arthur parameter ψ, the ABV-packet Π_{φ_ψ}^∘(G) coincides with the Arthur packet Π_ψ^∘(G), where φ_ψ denotes the Langlands parameter attached to ψ.

Information Square Streamline Icon: https://streamlinehq.com

Background

Vogan’s conjecture proposes an equality between ABV-packets (defined via microlocal vanishing cycles and perverse sheaf machinery on varieties of Langlands parameters) and Arthur packets (defined by Arthur’s endoscopic classification). It is reformulated via the functorial framework developed in the ABV/CFMMX literature.

The authors rely on this conjecture to deduce genericity results for A-packets from their ABV-packet analogues, but they do not claim it in full generality beyond known cases (e.g., GL_n).

References

The following conjecture was originally posed by Vogan using the language of micolocal geometry, and reformulated in terms of the $$ functor in *{Conjecture 1. a), Section 8.3}.

For a connected reductive $p$-adic group $G$, and an Arthur parameter $\psi$, % $$\Pi_\psiG= \Pi_{\phi_\psi}G.

— Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters (2504.04163 - Balodis et al., 5 Apr 2025) in Section 1.3 (ABV-packets)