Deligne-Lusztig variety
= Deligne-Lusztig variety
{c}
{title2=$X(w)=\{B':(B',F(B'))\in O(w)\}$}
For a connected <reductive algebraic group> and a <Weyl group> element $w$, this locally closed subset of the <flag variety> consists of flags in relative position $w$ to their Frobenius image. For rational $B$ it is $\mathcal L_F^{-1}(B\dot wB)/B$, using the twisted right action identity of the <Lang map>. Its dimension is the length of $w$.