Appearance of the Kashiwara-Saito singularity in the representation theory of $p$-adic $\mathop{GL}_{16}$

TL;DR

This paper demonstrates the appearance of the Kashiwara-Saito singularity in the representation theory of p-adic GL(16), revealing differences between Arthur and ABV-packets and challenging existing basis conjectures.

Contribution

It shows that the two bases of stable distributions in p-adic GL(16) are not equal by constructing a specific irreducible admissible representation involving the Kashiwara-Saito singularity.

Findings

Existence of a non-Arthur type irreducible representation with a singleton ABV-packet.

Identification of a representation whose ABV-packet contains exactly one other representation.

Strengthening of the main result concerning the Kashiwara-Saito singularity.

Abstract

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packet, as defined in earlier work, contains exactly one other representation; remarkably, this other admissible representation is of Arthur type. In the course of this study we strengthen the main result concerning the Kashiwara-Saito singularity. The irreducible admissible representations in this paper illustrate a fact we found surprising: for general linear groups, while all A-packets are singletons, some ABV-packets are not.