## Points on Deligne-Lusztig varieties: Interpreting Borels in relative position as flags with conditions

### 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.

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A small question about the formulation: What do you mean by "a Coxeter element of minimal length"?    In a finite Coxeter group such as a Weyl group, all Coxeter elements `$w$` 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. – Jim Humphreys Jul 9 at 22:56
I am aware of this fact, and it confuses me as well. The words "a Coxeter element of minimal length" are in the third paragraph of the article I cite. – Dror Speiser Jul 9 at 23:00
@Dror: To clarify further, Lusztig is working with a special set-up just for certain classical linear groups.  One Weyl group might be a subgroup of another (fixed by a twisted Frobenius map); thus a Coxeter element (in Steinberg's twisted sense) would be defined relative to the larger Weyl group, etc.   The paper has lots of ad hoc notation and may or may not be a useful place to start.   It was published in Proc. LMS 33 (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". – Jim Humphreys Jul 10 at 14:14

## 1 Answer

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 $w$.

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Indeed, I was having trouble with what relative position means in terms of flags, but your last sentence solved that. Thank you very much. – Dror Speiser Jul 10 at 7:34
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