Related papers: Tilting representations of finite groups of Lie ty…

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there…

In this paper we study higher Deligne--Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations coincide with certain induced…

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations…

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ of characteristic $p$, with Deligne--Lusztig dual $G^\ast$. We show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of all bad primes for…

Let $p$ be a prime, $k$ a finite extension of $\mathbf{F}_p$ of cardinal $q$, $l$ a finite extension of $k$ of group $\Sigma=\mathrm{Gal}(l|k)$, and $T$ a subgroup of $l^\times$. Using the method of "little groups", we classify irreducible…

In this paper we introduce a family of Deligne--Lusztig type varieties attached to connected reductive groups over quotients of discrete valuation rings, naturally generalising the higher Deligne--Lusztig varieties and some constructions…

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel [Ha] proved that generalized tilting…

This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic…

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to…

In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a…

We give parameterizations of the irreducible representations of finite groups of Lie type in their defining characteristic.

A new approach to Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups is presented. Connections with the theory of cuspidal Deligne-Lusztig representations of finite groups of Lie type are also…

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by…

Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations…

In 1979 Lusztig proposed a conjectural construction of supercuspidal representations of reductive p-adic groups, which is similar to the well known construction of Deligne and Lusztig in the setting of finite reductive groups. We present a…

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…

In this paper we study higher level Deligne--Lusztig representations of reductive groups over discrete valuation rings, with finite residue field $\mathbb{F}_q$. In previous work we proved that, at even levels, these geometrically…

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…

Higher Deligne-Lusztig representations are virtual smooth representations of parahoric subgroups in a $p$-adic group. They are natural analogs of classical Deligne-Lusztig representations of reductive groups over finite fields. The most…

Let k be an algebraically closed field of characteristic p>0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions…