Related papers: Reductive groups and Steinberg maps
This paper is an introduction, in a simplified setting, to Lusztig's theory of character sheaves. It develops a notion of character sheaves on reductive Lie algebras which is more general then such notion of Lusztig, and closer to Lusztig's…
In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of…
We give parameterizations of the irreducible representations of finite groups of Lie type in their defining characteristic.
We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…
This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but…
This survey article is an introduction to some of Lusztig's work on the character theory of a finite group of Lie type $G(F_q)$, where $q$ is a power of a prime~$p$. It is partly based on two series of lectures given at the Centre Bernoulli…
These notes are our contribution to the Proceedings of the ICM 2026. We discuss some results we have obtained (in part jointly with coauthors) regarding the representation theory of reductive algebraic groups over algebraically closed…
We present various results on disconnected reductive groups, in particular about the characteristic 0 representation theory of such groups over finite fields.
This paper contains an exposition of the theory of character sheaves for reductive groups and some attempts to extend it to other cases: unipotent groups, reductive groups modulo the unipotent radical of a parabolic.
The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between…
We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.
We determine the irreducible constituents of the Steinberg character of an orthogonal group over a finite field restricted to the orthogonal group of one less dimension
We give a purely algebraic treatment of reduction theory for connections over the formal punctured disc. Our proofs apply to arbitrary connected linear algebraic groups over an algebraically closed field of characteristic 0. We also state…
Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our…
Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over…
This book is an introduction to a fast developing branch of mathematics - the theory of representations of groups. It presents classical results of this theory concerning finite groups.
In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite…
These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the…
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…