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

TL;DR

This paper explores the geometric structure of the $q$-Klyachko algebra, linking it to Deligne--Lusztig varieties and toric orbifolds, and establishes new algebraic and geometric properties for various values of $q$.

Contribution

It reveals the geometric realization of the $q$-Klyachko algebra via Chow rings of Deligne--Lusztig varieties and develops a K"ahler package for it using toric geometry.

Findings

For prime power $q$, the $q$-Klyachko algebra is the image of the Chow ring pullback from a Deligne--Lusztig variety.

For positive rational $q$, a K"ahler structure for the algebra is established.

Connections between algebraic structures and geometric objects like toric orbifolds are demonstrated.

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 K\"ahler package for the $q$-Klyachko algebra through inputs from toric geometry.