Geometric analogues to local A-packets: Odd special orthogonal groups
In 1992, Adams, Barbasch and Vogan suggested a geometric approach to local Arthur packets for real groups, which Vogan extended to $p$-adic groups in 1993. In 2013, Arthur wrote the endoscopic classification of representations for orthogonal and symplectic groups, establishing what is now known as the theory of local Arthur packets. We call the proposed geometric analogues to local Arthur packets ABV-packets. We prove that ABV-packets generalize local Arthur packets for the split $p$-adic group $SO_{2n+1}$, by showing that every Arthur packet is an ABV-packet (but not vice-versa), as predicted by Vogan in 1993, and that the coefficients appearing in Arthur's main local result can be calculated using vanishing cycles of perverse sheaves and the Vogan-Langlands correspondence, as conjectured in Cunningham-Fiori-Moussaoui-Mracek-Xu in 2022.