We give an informal introduction to the theory of Deligne-Lusztig which gives all the irreducible representations of reductive groups over finite fields [DL], with an emphasis on the geometry of the Deligne-Lusztig variety. Disclaimer: by no means complete! Comments welcome / use at your own risk!

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic ...

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig’s results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theor...

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. prove that $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite construction is isomorphic to an variety at infinite level. their homology give geometric realizations Langlands Jacquet–Langlands correspondences in setting Weil paramete...