### 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.



Points on Deligne-Lusztig varieties: Interpreting Borels in relative position as flags with conditions
It might help if you specified what part of the translation your having trouble with (for example, it's unclear from what you wrote if you're comfortable with relative position).


For $SL\_n$, a Borel corresponds to a flag in affine $n$-space over an extension of your field (essentially, the one you need to diagonalize the elements of a torus in the Borel). Applying the Frobenius means doing it to the coordinates on $n$ space, and looking at the resulting flag. Relative position w just means that the dimensions of the intersections of the pieces of the flag are the same as those between the standard flag, and that gotten by per muting the basis by $w$.


