scholarly journals Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics

1987 ◽  
Vol 108 (1) ◽  
pp. 195-201 ◽  
Author(s):  
Sharad V Keny
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Simple Algebraic Group ◽  
Nilpotent Elements

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

scholarly journals Maximal finite abelian subgroups of E8

2017 ◽  
Vol 147 (5) ◽  
pp. 993-1008 ◽  
Author(s):  
Cristina Draper ◽  
Alberto Elduque
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Homogeneous Component ◽  
Closed Field ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Irreducible Modules ◽  
Abelian Subgroups

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.

scholarly journals Derived subalgebras of centralisers and finite -algebras

2014 ◽  
Vol 150 (9) ◽  
pp. 1485-1548 ◽  
Author(s):  
Alexander Premet ◽  
Lewis Topley
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Dimension ◽  
Closed Field ◽  
Natural Action ◽  
Simple Algebraic Group ◽  
Finite Algebras ◽  
Nilpotent Elements ◽  
Primitive Ideals ◽  
Exceptional Lie Algebras

AbstractLet$\mathfrak{g}=\mbox{Lie}(G)$be the Lie algebra of a simple algebraic group$G$over an algebraically closed field of characteristic$0$. Let$e$be a nilpotent element of$\mathfrak{g}$and let$\mathfrak{g}_e=\mbox{Lie}(G_e)$where$G_e$stands for the stabiliser of$e$in$G$. For$\mathfrak{g}$classical, we give an explicit combinatorial formula for the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and use it to determine those$e\in \mathfrak{g}$for which the largest commutative quotient$U(\mathfrak{g},e)^{\mbox{ab}}$of the finite$W$-algebra$U(\mathfrak{g},e)$is isomorphic to a polynomial algebra. It turns out that this happens if and only if$e$lies in a unique sheet of$\mathfrak{g}$. The nilpotent elements with this property are callednon-singularin the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element$e\in \mathfrak{g}$is non-singular if and only if the maximal dimension of the geometric quotients$\mathcal{S}/G$, where$\mathcal{S}$is a sheet of$\mathfrak{g}$containing$e$, coincides with the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element$e$in a classical Lie algebra$\mathfrak{g}$the closed subset of Specm $U(\mathfrak{g},e)^{\mbox{ab}}$consisting of all points fixed by the natural action of the component group of$G_e$is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.

scholarly journals Varieties of a class of elementary subalgebras

2021 ◽  
Vol 7 (2) ◽  
pp. 2084-2101
Author(s):  
Yang Pan ◽  
◽  
Yanyong Hong ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Positive Characteristic ◽  
Maximal Dimension ◽  
Closed Field ◽  
Characteristic P ◽  
Simple Algebraic Group ◽  
Restricted Lie Algebra ◽  
Irreducible Components ◽  
Type C

<abstract><p>Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p &gt; 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.</p></abstract>

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

scholarly journals The Modality of a Borel Subgroup in a Simple Algebraic Group of Type E8

2018 ◽  
Vol 29 (3) ◽  
pp. 326-327
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Simple Algebraic Group

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

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