Deligne--Lusztig varieties, toric orbifolds, and the $q$-Klyachko algebra

Abstract

We investigate the geometry behind the $q$-Klyachko algebra, introduced by Nadeau--Tewari. When $q$ is a prime power, we show that the $q$-Klyachko algebra is the image of the pullback map on Chow rings $\mathrm{CH}({\mathrm{Fl}}_{n+1})\to \mathrm{CH}({\mathrm{DL}}_n)$, where ${\mathrm{DL}}_n\subseteq {\mathrm{Fl}}_n$ is a compactified Deligne--Lusztig variety inside the complete flag variety ${\mathrm{Fl}}_{n+1}$. When $q$ is a positive rational number, we establish a Kähler package for the $q$-Klyachko algebra through inputs from toric geometry.

10 pp. Accepted for FPSAC 2026