Dong Gyu Lim

Tuesday, January 31, 2023 - 1:45pm

PDL C-38

**Pretalk title**: What is a Deligne-Lusztig variety?

**Pretalk abstract**:Representation theory of finite groups of Lie type is often called Deligne-Lusztig theory. In their seminal paper, Deligne and Lusztig constructed geometric objects whose cohomology groups contain all irreducible representations in a suitable sense. I will briefly talk about the SL(2) case which is due to Drinfeld and then discuss the relation between the classical one and the affine version of that.

**Main talk title**: Some basic questions on affine Deligne-Lusztig varieties

**Main talk abstract**:Affine Deligne-Lusztig varieties can be understood as a

*p*-adic generalization of a

*classical*Deligne-Lusztig variety. One of the most basic questions is 'when it is nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or the set B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest my own conjecture in the

*basic*case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case for which no conjectural formula existed so far.