Combinatorial approach to ABV-packets for GLn

Abstract

There exists a significant conjecture in the local Langlands correspondence that A-packets are ABV-packets. For the case G=GLn, the conjecture reduces to ABV-packets for orbits of Arthur type being singletons, which is a specialisation of the wider conjecture known as the Open-Orbit conjecture. We can reduce the complexity of this problem by considering the combinatorial geometry of these objects using multisegments, since there exists a natural relationship between this description and the structure of ABV-packets. The first part of this thesis investigates interpretations of the Zelevinskii Involution. We then use combinatorial approaches involving endoscopic decompositions and numerical invariants to study the partial ordering in the Open-Orbit conjecture, which will lead to the proof that ABV- packets for orbits of Arthur type in GLn are singletons. Finally, we use a numerical-based argument to conjecture families of ABV-packets for which the partial ordering relation is not satisfied for.