Geometry of Deligne-Lusztig varieties

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Representation theory, in other words the study of symmetries, is an important area of mathematics. For a certain class of “representations” Pierre Deligne and George Lusztig exhibited a close connection between the algebraic notion of representation, and certain geometric objects, now called Deligne-Lusztig varieties. Since then, Deligne-Lusztig varieties have been thoroughly investigated from a representation theoretic point of view. Much less is known about their geometric properties, though. The goal of the proposed project is to improve our understanding of the geometry of Deligne-Lusztig varieties using explicit and algorithmic methods in order to study concrete examples, and hopefully eventually to reach new general results. Deligne-Lusztig varieties are smooth quasi-projective varieties over finite fields. An important specific question is whether there exist Deligne-Lusztig varieties which are not affine varieties.