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In this article we propose a geometric description of Arthur
packets for padic groups using vanishing cycles of perverse
sheaves. Our approach is inspired by the 1992 book by Adams,
Barbasch and Vogan on the Langlands classification of admissible
representations of real groups and follows the direction indicated
by Vogan in his 1993 paper on the Langlands correspondence. Using
vanishing cycles, we introduce and study a functor from the
category of equivariant perverse sheaves on the moduli space of
certain Langlands parameters to local systems on the regular part
of the conormal bundle for this variety. In this article we
establish the main properties of this functor and show that it
plays the role of microlocalization in the work of Adams, Barbasch
and Vogan. We use this to define ABV-packets for pure rational
forms of p-adic groups and propose a geometric description of the
transfer coefficients that appear in Arthur's main local result in
the endoscopic classification of representations. This article
includes conjectures modelled on Vogan's work, including the
prediction that Arthur packets are ABV-packets for p-adic groups.
We gather evidence for these conjectures by verifying them in
numerous examples.
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