A short note on Arthur parameters with singular orbit closures

Abstract

This short article resolves a question about Arthur parameters highlighted in previous work of Balodis, Cunningham, Fiori and Zhang, in the case of ${\mathrm{GL}}_n/F$ where $F/{\mathbb{Q}}_p$ is a $p$-adic field. Namely, we prove that every Arthur parameter $\psi$ of ${\mathrm{GL}}_n/F$, can be factored into Arthur parameters $\psi \cong {\psi }_1\oplus \cdots \oplus {\psi }_r$ such that the orbit $C_{\psi }=C_{{\psi }_1}\times \cdots \times C_{{\psi }_r}\subseteq V_{{\lambda }_{{\psi }_1}}\times \cdots \times V_{{\lambda }_{{\psi }_r}}$ in the associated Vogan variety has smooth closure if and only if each ${\bar{C}}_{{\psi }_i}$ is smooth for $i=1,\dots ,r$, and that $C_{{\psi }_i}$ has smooth closure if and only if it is open or closed.

5 pages