Related papers: On the definition of unipotent representations

Let \hat G be the semidirect product of a connected reductive group G over F_q with a finite cyclic group generated by a quasisemisimple automorphism of G defined over F_q. In this paper we prove a conjecture of G. Malle concerning the…

We discuss a cohomological construction of representations of a reductive group over the ring of power series over a finite field modulo the r-th power of the maximal ideal. The case r=1 goes back to Deligne and the author. The case where r…

Let G be a connected reductive group defined over a finite field F_q. We give a parametrization of the irreducible representations of G(F_q) in terms of (twisted) categorical centres of various monoidal categories associated to G. (Results…

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize…

Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This…

Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what…

Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q).

In a recent paper I defined a new basis for the Grothendieck group of unipotent representations of an almost simple Chevalley group over a finite field. The definition for classical types was different from that for exceptional types. In…

Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of…

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also…

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and…

The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of…

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on…

Let $G$ be a connected reductive algebraic group defined over a finite field with $q$ elements. In the 1980's, Kawanaka introduced generalised Gelfand-Graev representations of the finite group $G(F_q)$, assuming that $q$ is a power of a…

Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent…

We consider quantum group representations for a semisimple algebraic group G at a complex root of unity q. Here q is allowed to be of any order. We revisit some fundamental results of Parshall-Wang and Andersen-Polo-Wen from the 90's. In…

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…

The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and…

We recall the cohomological interpretation of the unipotent quotients of the fundamental groupoid of an algebraic complex variety (Beilinson, Deligne-Goncharov). We then give a construction of the resutting transition morphisms in terms of…

Consider a Chevalley group over a finite field $F_q$ such that the longest element of the Weyl group is central. We construct an involution $\xi\mapsto\xi^!$ of the set of unipotent representations of this group such that the degree…