Tag - Deligne-Lusztig theory

Charlotte Chan: Generic character sheaves on parahoric subgroups

27th May, 2024

Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the 'generic' part of this desired theory. In the simplest non-trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne-Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands programme.

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Lucas Mason-Brown: What is a Unipotent Representation?

22nd February, 2021

The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(𝔽q) be the group of 𝔽q-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(𝔽q). He showed:

   1.  All irreducible representations of G(𝔽q) can be constructed from a finite set of building blocks -- called 'unipotent representations.'
   2.  Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(𝔽q).

Now, replace 𝔽q by ℂ, the field of complex numbers, and replace G(𝔽q) with G(ℂ). There is a striking analogy between the finite-dimensional representation theory of G(𝔽q) and the unitary representation theory of G(ℂ). This analogy suggests that all unitary representations of G(ℂ) can be constructed from a finite set of building blocks - called 'unipotent representations' - and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of 'special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Rong Zhou: Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

25th October, 2018

Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to use techniques from local harmonic analysis to compute the asymptotics of a certain twisted orbital integral which counts the number of 𝔽q-points on the ADLV as q goes to infinity. This is joint work with Yihang Zhu.

Charlotte Chan: Towards a p-adic Deligne-Lusztig theory

27th September, 2018

The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In this talk, we will discuss recent progress on studying analogues of Deligne-Lusztig varieties attached to p-adic groups.