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).

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