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

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 ${\mathrm{GL}}_{16}$ 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.

To appear in the Pacific Journal of Mathematics

References in corpus (1)

• Arthur packets for $G_2$ and perverse sheaves on cubics

Cited by in corpus (1)

• Vogan's Conjecture on local Arthur packets of $p$-adic ${\mathrm{GL}}_n$ and a combinatorial Lemma