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)
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.