 Deligne-Lusztig variety

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 ${\mathcal{L}}_F^{-1}(B\dot{w}B)/B$, using the twisted right action identity of the Lang map. Its dimension is the length of $w$.