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

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 $\mathrm{C}\mathrm{H}({\mathrm{F}\mathrm{l}}_{n+1})\to \mathrm{C}\mathrm{H}({\mathrm{D}\mathrm{L}}_n)$, where ${\mathrm{D}\mathrm{L}}_n\subseteq {\mathrm{F}\mathrm{l}}_n$ is a compactified Deligne–Lusztig variety inside the complete flag variety ${\mathrm{F}\mathrm{l}}_{n+1}$. When $q$ is a positive rational number, we establish a Kähler package for the $q$-Klyachko algebra through inputs from toric geometry.