NCMW - Finite groups of Lie type (2021)

Speakers and Syllabus

Our plan for the lectures and tutorials is as follows: We begin the programme on a Saturday and have two lectures on the mornings of the first weekend, one per day. The video recordings of the lectures will be made available to students by the same day and the lecture notes will also be made available by the next day. The tutorial problems will also be provided with the notes. The tutorials will be conducted on the Thursday and Friday of the following week.
This pattern of lectures and tutorials will continue for all the weeks. In the last week, we plan to have two lectures devoted to current research in the area.

Topics for the workshop:
We intend to run 2 short courses of 9 lectures each in this meeting mainly following two themes (details below). Each course will consist of 13.5 hours of lectures and 8 hours of informal discussions (tutorial sessions). The details of the courses are as follows:

Course 1
Finite Groups of Lie type: This topic would be a natural follow up of the 2019 AIS on Algebraic groups where the theory over algebraically closed fields was considered. This topic intends to expose the participants to the structure of reductive groups defined over a finite field and their representations with emphasis on Deligne-Lusztig theory.

Course 2
Word maps and linear groups: It was a conjecture of Oystein Ore, from 1951,that every element of a non-abelian finite simple group G is a commutator. After almost 60 years, this conjecture was settled in affirmative in the celebrated paper of Liebeck, O’Brien,Shalev, and Tiep, well-known in the community as LOST. Ore’s conjecture asserts that the function, not a group homomorphism, from G x G to G which sends a pair (g, h) to the element ghg^{-1}h^{-1} is surjective. It was then natural to consider various such maps, called word maps, and investigate their properties.

The theory of word maps on groups and their applications has developed significantly in the past two decades, with emphasis on finite simple groups, infinite linear groups, profinite groups and related objects. Word width was studied, leading to solutions to a question of Serre (by Nikolov and Segal) and the conjecture of Ore, stated above. Probabilistic aspects of word maps were also explored, leading to a probabilistic Tits alternative for linear groups,characterizations of probabilistically nilpotent groups, as well as recent solutions to probabilistic Waring problems. This series of lectures will describe these developments, related results, and
tools used in the proofs.

Tutors:
1.(AB) Anirban Bose, IISER Mohali
2.(SB) Sushil Bhunia, IISER Mohali
3.(PC) Pratyusha Chattopadhyay, BITS Pilani, Hyderabad
4.(SPP) Shiv Prakash Patel, IIT Delhi

SPEAKERS ALONGWITH TITLE AND ABSTRACTS

1. Finite Groups of Lie type

1.1. DP - 1 lecture. Dipendra Prasad, IIT Bombay, India (Representations of finite groups of Lie type)This lecture kick-starts the workshop with a brief overview of the celebrated Deligne-Lusztig theory of representations of finite groups of Lie type.

1.2. MM - 3 lectures. Manish Mishra, IISER Pune, India (Supercuspidal representations for finite and p-adic groups) Let G be a connected reductive group defined over a finite or a non-archimedean local field F . We show that G(F ) admits cuspidal representations when F is finite and supercuspidal representations when F is non-archimedean local. We also determine precisely when G(F ) admits self-dual representations. For the results on self-duality, we assume some mild hypothesis on G. These hypothesis disallow G to have certain small rank factors when the field (in case F is finite) or the residue field (in case F is non-archimedean local) is of cardinality ≤ 5.

• Lecture 1: A brief review of representations of finite groups of Lie type, with emphasis on Deligne Lusztig theory.
• Lecture 2: Proofs of the main results in finite reductive group case.
• Lecture 3: Proofs in the case of p-adic groups.

1.3. SMG - 3 lectures. Shripad Garge, IIT Bombay, Mumbai, India (Structure of finite groups of Lie type) We aim to prepare the audience for the following two lecture courses. We give a brief review of representations of finite groups of Lie type in the first lecture. In the remaining lectures, we will study the structure of centralisers of elements in finite groups of Lie type.

• Lecture 1: The structure of finite groups of Lie type
• Lecture 2: Centralisers of elements in finite groups of Lie type-I
• Lecture 3: Centralisers of elements in finite groups of Lie type-II

1.4. CRV - 2 lectures. C. R. Vinroot, College of William and Mary, VA, USA (Real representations of finite groups of Lie type)The main purpose of these lectures is to give both an overview and some details on the current state of results on the classification of real representations of finite groups of Lie type.

• Lecture 1: First, we will state the main problem, along with the broader problem of Brauer to describe the number of irreducible complex representations of a finite group which are real. The main focus of this lecture will be to give the main idea of several proofs of the fact that all irreducible real-valued characters of the finite general linear group are afforded by real representations. We will explain how some of these methods can be extended to other groups, and why some cannot. One focus will be the use of generating functions.
• Lecture 2: In the second lecture, the goal will be to show how the generating function method from Lecture 1 can be adapted with Deligne-Lusztig theory to classify the real representations of finite classical groups in characteristic 2. We will also discuss computational methods used to classify the real representations of finite exceptional groups, including E8 (q). Finally, some general conjectures will be discussed, along with their current status.

2. Word maps on linear groups
2.1. AKS - 1 lecture. Anupam Singh, IISER Pune, India (Revision of algebraic groups)This lecture will briefly recall the prerequisite for this series of lectures, namely, algebraic geometry, algebraic groups, word maps etc.

2.2. BK - 4 lectures. Boris Kunyavskiĭ, Bar-Ilan University, Israel (Word maps on linear algebraic groups)Given a group word w = w(x1, . . . , xd ) on d letters and a group G, one can consider the evaluation map w : Gd → G induced by w. In this minicourse, we shall consider the case where G = G(K) is the group of K-points of a linear algebraic K-group G. Our focus is on the special ground fields (complex, real, p-adic, number, or close to such) and on simple K-groups (or close to such). The course is built as a survey of results and open problems concerning the image of a word map, which corresponds to the solvability of the equation w = w(x1 , . . . , xd ) = g for an arbitrary (or ‘generic’) g, as well as on the fibres of this map, which corresponds to distributio of solutions of such an equation. Various analogies with polynomial maps on associative and Lie algebras will also be discussed.The course is based on (but not limited by) the material contained in the survey papers listed
below.It is assumed that the participants possess necessary background in algebra, particularly in algebraic geometry (first chapters of Hartshorne or Shafarevich) and algebraic groups (Borel,Springer, or Humphreys).
References:

1. T. Bandman, S. Garion, B. Kunyavskiĭ, Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics, Central Eur. J. Math. 12 (2014), 175-211.
2. N. Gordeev, B. Kunyavskiĭ, E. Plotkin, Geometry of word equations in simple algebraic groups over special fields, Uspekhi Mat. Nauk 73 (2018), no. 5, 3–52; English transl. in Russian Math. Surveys 73 (2018), 753–796.
3. A. Kanel-Belov, B. Kunyavskiĭ, E. Plotkin, Word equations in simple groups and polynomial equations in simple algebras, Vestnik St. Petersburg Univ. Math. 46 (2013), no. 1, 3-13.
4. B. Kunyavskiĭ, Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry, in: “Arithmetic and Geometry”, L. V. Dieulefait et al. (Eds.), LMS Lecture Notes, vol. 420, Cambridge Univ. Press, 2015, pp. 264-282.

2.3. AK - 2 lectures. Amit Kulshrestha, IISER Mohali, India (Images of word maps and chi-rality) Let G be a finite simple group. In 2014, Lubotzky proved that if G is a finite simple group then every subset of G which contains identity and is invariant under automorphisms of G, occurs as image set for a suitable word. After a brief idea of the proof, we shall discuss word images in symmetric groups (work of Matthew Levy) and chirality of words (work of Cocke-Ho).

2.4. AKS - 2 lectures. Anupam Singh, IISER Pune, India (Power maps on finite groups of Lie type)One of the simplest example of word map is the power map. Let M ≥ 2 be an integer. Let G be a reductive group defined over a finite field Fq . We consider the word map x 7→ xM on G(Fq ).We study the following question. What is the size of the image under this map? The reference to these lectures will be the work done by the speaker in collaboration with Amit Kulshrestha and Rijubrata Kundu.

2.5. BS - 2 lectures. B. Sury, ISI Bengaluru (Combinatorial theory of arithmetic groups)In the 1980’s, some abstract and combinatorial group theoretic notions started to be studied in relation to lattices in Lie groups and arithmetic groups in particular. One of them - called Bounded generation - turned out to have unexpectedly strong implications to aspects like representation theory, super-rigidity and the congruence subgroup problem. In these two talks, we survey the developments right up to some very recent ones that address S-arithmetic groups in anisotropic algebraic groups over number fields. The various combinatorial facets of arithmetic groups use
interestingly diverse methods.

Time Table