Complex representations of finite group of Lie type - inductive methods
Abstract
Finite groups of Lie type arise as the rational point over a finite field of a reductive linear algebraic group.
A standard technique to gain knowledge about representations of these groups and to classify them consist in detecting a suitable family of subgroups and building representations of the group by induction starting from the ones of the subgroups. The "classical" instance of this general idea Is the so called "Harish-Chandra theory", that is the study of representations by exploiting parabolic induction from Levi subgroups. Toward the end of last century, Deligne and Lusztig developed an enhancement of this theory, constructing a new induction that allows to keep track of "twisted" object.
My aim is to give an overview of some of the constructions involved and of the main results in these theories.
Mod p Langlands for GL2
Abstract
The mod p Langlands program is an attempt to relate mod p Galois representations of a local field to mod p representations of the p-adic points of a reductive group. This is inspired by the classical local Langlands (l-adic coefficients) and it is partially a stepping stone towards the p-adic Langlands (p-adic coefficients). I will explain this for GL2/Qp, where one can explicitly describe both sides, and I will relate it to congruences between modular forms.
The first Hochschild cohomology of twisted group algebras
Abstract
Given a group G and a field k, we can "twist" the multiplication of the group algebra kG by a 2-cocycle, and the result is a twisted group algebra. Twisted group algebras arise as direct sums of blocks of group algebras, and so are of interest in representation and block theory. In this talk we will discuss some recently obtained results on the first Hochschild cohomology of twisted group algebras, in particular that these cohomology groups are nontrivial whenever G is a finite simple group.