Picard Groups of Deligne-lusztig Varieties — with a View toward Higher Codimensions

Let (G;F) be a connected reductive algebraic group over an algebraically closed field k of positive characteristic p, equipped with an Fq -structure coming from a Frobenius morphism F : G ! G. Let L : G ! G be the corresponding Lang map taking an element g 2 G to g 1F(g). By the Lang-Steinberg Theorem [Bor92, Theorem 16.3] this morphism of varieties is surjective with finite fibres. From this result it follows that, by conjugacy of Borel subgroups, there exists an F-stable Borel subgroup B. Let π : G!G=B := X denote the quotient. There are then (with a slight abuse of notation) natural endomorphisms F :W !W and F : X !X of the Weyl group of G and the variety X of Borel subgroups of G. Let W be generated by the simple reflections s1; : : : ;sn and let l( ) be the length function with respect to these generators.