Kristaps John Balodis

arXiv2026

A short note on Arthur parameters with singular orbit closures

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 $ψ$ of $\mathrm{GL}_n/F$, can be factored into Arthur parameters $ψ\cong ψ_1\oplus\cdots \oplusψ_r$ such that the orbit $C_ψ=C_{ψ_1}\times\cdots\times C_{ψ_r}\subseteq V_{λ_{ψ_1}}\times \cdots \times V_{λ_{ψ_r}}$ in the associated Vogan variety has smooth closure if and only if each $\bar{C}_{ψ_i}$ is smooth for $i=1, \ldots, r$, and that $C_{ψ_i}$ has smooth closure if and only if it is open or closed.

math.RT↗

arXiv2026

Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$

In this article, we prove the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}_n(F)$. While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes and clarifies the approach via affine Hecke algebras of Chriss and Ginzburg. In particular, this article serves as an opportunity to articulate several results which are undoubtedly known to experts, but have not been formally recorded in the literature.

math.RT↗