On the values of unipotent characters of finite Chevalley groups of type $E_7$ in characteristic 2
Jonas Hetz
备注：
19 pages. Correction in the proof of Proposition 5.2 (whose statement remains valid); some minor improvements. We thank an unknown referee for a careful reading of the manuscript and useful comments. To appear in Osaka J. Math
Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to considering cuspidal character sheaves. We will deal with the latter for the specific case where $G=E_7(q)$, and $q$ is a power of the bad prime $p=2$ for $E_7$.
We show that the two cuspidal unipotent characters of a finite Chevalley group $E_7(q)$ have Schur index~2, provided that $q$ is an even power of a (sufficiently large) prime number $p$ such that $p\equiv 1 \bmod 4$. The proof uses a refinement of Kawanaka's generalized Gelfand--Graev representations and some explicit computations with the {\sf CHEVIE} computer algebra system.
Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power o…
Let $q$ be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with $q$ elements. Marcelo--Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that $q$ is a power of~$2$, we explain how to construct the complete character table of~$F_4(q)$.
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 this paper we reconcile the two definitions.
We discuss various computational issues around the problem of determining the character values of finite Chevalley groups, in the framework provided by Lusztig's theory of character sheaves. Some of the remaining open questions (concerning certain roots of unity) for the cuspidal unipotent character sheaves of groups of exceptional type are resolved.
Let $G$ be the symplectic group, $Φ=C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $Φ$. To each involution $σ\in W$ one can assign the $B$-orbit $Ω_σ$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that $Ω_σ$ is contained in the Zariski closure of $Ω_τ$ if and only of $σ\leqτ$ with respect to the Bruhat--Chevalley order. We also prove that $\dimΩ_σ$ is equal to $l(σ)$, the length of $σ$ in $W$.
Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hit…
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These are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The slides can be used as a short introduction to the theory of characters and character sheaves for unipotent groups in positive characteristic, developed by the authors in a series of articles written between 2006 and 2011. We give an overview of the main results of this theory along with a bit of motivation.
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It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this paper we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups $U_n$), and under a certain hypothesis relating the characteristic $p$ to both $n$ and the dimension $d$ of a representation (specifically, $p \geq \text{max}(n,2d)$), Lie theory is completely sufficient to determine the …
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Soit G un groupe algebrique reductif sur la cloture algebrique d'un corps fini F_q et defini sur ce dernier. L'existence du support unipotent d'un caractere irreductible du groupe fini G(F_q), ou d'un faisceau caractere de G, a ete etablie dans differents cas par Lusztig, Geck et Malle, et le second auteur. Dans cet article, nous demontrons que toute classe unipotente sur laquelle la restriction du faisceau caractere ou du caractere donne est non nulle est contenue dans l'adherence de Zariski de son support unipotent. Pour etablir ce resultat, nous etudions certaines representations des groupe…
This paper provides a construction of the unipotent representations for classical complex groups in terms of the Theta correspondence as introduced and studied by R. Howe. The K-type structure of unipotent representations is obtained as a consequence of the character formulas for unipotent representations of D. Vogan and the author. This provides a tight link between unipotent representations and the orbit philosophy. A parametrization of unipotent representations for the Spin groups is obtained.
In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups.
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The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are $q$ and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this …
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In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In this paper, we consider a similar problem for the case of classical groups Sp_{2n}, SO_N with arbitrary characteristic. Our result enables us, in particular, to compute Green functions in bad characteristic case, which was not known before.
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Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give …
