### Background

I am studying the paper "On the Green polynomials of classical groups" by Lusztig, in which he computes the values of the Deligne-Lusztig representation, corresponding to a Coxeter element of minimal length in a classical group, on unipotent elements. I am interested in computing the values of representations not corresponding to a Coxeter element of minimal length. (Note that this is done in a vast generalisation in later work by Lusztig, and in the work of Shoji. But I am not in a place to be able to use their methods)

### Question

Let $G$ is a classical group defined over a finite field with frobenius morphism $F$, $w$ an element of the Weyl group, and let $X(w)$ be the Deligne-Lusztig variety - all Borel subgroups $B$ of $G$ such that $B$ and $F(B)$ are in relative position $w$.

My specific question is: how do I translate this definition into the language of flags? I.e. I would like an alternative definition for $X(w)$ as the variety of flags satisfying some conditions involving to $F$.

In the original Deligne-Lusztig paper, my question is answered, for the case $w$ a Coxeter element of minimal length, in a short section.

`$w$`

have the same length (equal to the rank of the given group) and are conjugate. I'll have to look at the original papers to sort out better what is going on here, but the terminology confuses me at first. $\endgroup$Proc. LMS33 (1976) at a very early stage of what turned out to be a vast project to find all character values of finite groups of Lie type. This paper focuses on a "Coxeter torus". $\endgroup$