Appearance of the Kashiwara–Saito singularity in the representation theory of $p$-adic $\mathrm{GL}(16)$

Clifton Cunningham, Andrew Fiori and Nicole Kitt

Vol. 321 (2022), No. 2, 239–282

DOI: 10.2140/pjm.2022.321.239

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 ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_{16}$ whose ABV-packet, as defined by Cunningham et al. (2022b), contains exactly one other representation. Consequently, for general linear groups, while all A-packets are singletons, some ABV-packets are not. In the course of the proof of this result, we strengthen the main result concerning the Kashiwara–Saito singularity by Kashiwara and Saito (1997).

Keywords

admissible representations, Arthur parameters, L-packets, Langlands correspondence, Kashiwara–Saito singularity, perverse sheaves, vanishing cycles

Mathematical Subject Classification

Primary: 11F70, 22E50, 32S30

Secondary: 32S60

Milestones

Received: 16 December 2021

Revised: 13 August 2022

Accepted: 25 October 2022

Published: 21 March 2023