Geometry of Deligne-Lusztig varieties

• Applicant: Professor Dr. Ulrich Görtz
• Subject Area: Mathematics
• Term: from 2010 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 171349465

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory