In this paper, we survey some mathematical developments that followed from the discovery of simple supercuspidal representations p-adic groups.

For a connected reductive group $G$ defined over $\mathbb{F}_q$ and equipped with the induced Frobenius endomorphism $F$, we study relation among following three $\mathbb{Z}$-algebras: (i) $\mathbb{Z}$-model $\mathsf{E}_G$ of algebras Gelfand-Graev representations $G^F$; (ii) Grothendieck $\mathsf{K}_{G^\ast}$ category $G^{\ast F^\ast}$ $\overline{\mathbb{F}_q}$ (Deligne-Lusztig dual side); (ii...

Introduction The irreducible admissible representations of a real reductive group such as GL(n,R) have been classified by work of Langlands, Knapp, Zuckerman and Vogan. This classification is somewhat involved and requires a substantial number of prerequisites. See [13] for a reasonably accessible treatment. It is fair to say that it is difficult for a non-expert to understand any non-trivial c...

The first part of this report describes the class of representations of Galois groups of number fields that have been attached to automorphic representations. The construction is based on the program for analyzing cohomology of Shimura varieties developed by Langlands and Kottwitz. Using p-adic methods, the class of Galois representations obtainable in this way can be expanded slightly; the lin...

We give a new characterization of Lusztig’s canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assi...