Combinatorial Approach to ABV-packets for $\mathbf{GL_n}$

Abstract

There exists a significant conjecture in the local Langlands correspondence that A-packets are ABV-packets. For the case $G=GL_n$, the conjecture reduces to ABV-packets for orbits of Arthur type in $GL_n$ being singletons, which is a specialisation of the wider conjecture known as the Open-Orbit conjecture. In this paper, we will prove the reduced conjecture since there exists a nice combinatorial description. The result first appeared in the associated Master's thesis, however we aim to use a slightly more simplified and succinct approach in this paper using results of Knight and Zelevinskii. We will also prove the partial ordering relation associated to the conjecture for multisegments of ladder type.