scholarly journals Loewy series of parabolically induced -Verma modules

2014 ◽  
Vol 14 (1) ◽  
pp. 185-220 ◽  
Author(s):  
Abe Noriyuki ◽  
Kaneda Masaharu
Keyword(s):  
Algebraic Group ◽  
Positive Characteristic ◽  
Algebraic Groups ◽  
Minimal Length ◽  
Closed Field ◽  
Verma Modules ◽  
Parabolic Subgroups ◽  
Reductive Algebraic Group ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

AbstractWe show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

scholarly journals UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF

2019 ◽  
Vol 71 (1) ◽  
pp. 197-206
Author(s):  
Michel Gros ◽  
Kaneda Masaharu
Keyword(s):  
Nous Avons ◽  
Algebraic Group ◽  
Positive Characteristic ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Frobenius Endomorphism ◽  
Simply Connected ◽  
Simplement Connexe ◽  
Field Of Positive Characteristic

Abstract Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants. For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Falk Bannuscher ◽  
Alastair Litterick ◽  
Tomohiro Uchiyama
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Completely Reducible ◽  
Rationality Problems ◽  
Reductive Algebraic Groups

Abstract Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

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).

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

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.

DIFFERENTIAL OPERATORS ON AND THE AFFINE GRASSMANNIAN

2014 ◽  
Vol 14 (3) ◽  
pp. 493-575 ◽  
Author(s):  
Victor Ginzburg ◽  
Simon Riche
Keyword(s):  
Algebraic Group ◽  
Affine Space ◽  
Equivariant Cohomology ◽  
Differential Operators ◽  
Dual Group ◽  
Intertwining Operators ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Affine Grassmannian ◽  
Basic Affine

We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of $\mathscr{D}$-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parameterized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman and Finkelberg.

Support varieties of baby Verma modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850211
Author(s):  
Yiyang Li ◽  
Bin Shu ◽  
Yufeng Yao
Keyword(s):  
Verma Module ◽  
Nilpotent Orbit ◽  
Closed Field ◽  
Verma Modules ◽  
Prime Characteristic ◽  
Reductive Algebraic Group ◽  
Type Formula ◽  
Coxeter Number ◽  
Support Variety ◽  
Support Varieties

Let [Formula: see text] be a connected reductive algebraic group over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text]. For a given nilpotent [Formula: see text]-character [Formula: see text], let [Formula: see text] be a baby Verma module associated with a restricted weight [Formula: see text]. A conjecture describing the support variety of [Formula: see text] via that of its restricted counterpart is given: [Formula: see text]. Under the assumption of [Formula: see text](the Coxeter number) and [Formula: see text] [Formula: see text]-regular, this conjecture is proved when [Formula: see text] falls in the regular nilpotent orbit for any [Formula: see text] and the subregular nilpotent orbit for [Formula: see text] being of type [Formula: see text]. We also verify this conjecture whenever [Formula: see text] is of type [Formula: see text] and [Formula: see text] falls in the minimal nilpotent orbit.

scholarly journals The construction problem for Hodge numbers modulo an integer in positive characteristic

2020 ◽  
Vol 8 ◽  
Author(s):  
Remy van Dobben de Bruyn ◽  
Matthias Paulsen
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Construction Problem ◽  
Algebraically Closed Field ◽  
Serre Duality ◽  
Hodge Numbers ◽  
Field Of Positive Characteristic

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

Sign in / Sign up

Export Citation Format

Share Document