scholarly journals The spherical building and regular semisimple elements

1983 ◽  
Vol 27 (3) ◽  
pp. 361-379 ◽  
Author(s):  
G.I. Lehrer
Keyword(s):  
Finite Group ◽  
Finite Field ◽  
Algebraic Group ◽  
Weyl Group ◽  
Rational Points ◽  
Semisimple Element ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Equivariant Sheaves

Let G be a connected reductive algebraic group defined over a finite field k. The finite group G(k) of k-rational points of G acts on the spherical building B(G), a polyhedron which is functorially associated with G. We identify the subspace of points of B(G) fixed by a regular semisimple element s of G(k) topologically as a subspace of a sphere (apartment) in B(G) which depends on an element of the Weyl group which is determined by s. Applications include the derivation of the values of certain characters of G(k) at s by means of Lefschetz theory. The characters considered arise from the action of G(k) on the cohomology of equivariant sheaves over B(G).

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

scholarly journals Character sheaves and generalized Springer correspondence

2003 ◽  
Vol 170 ◽  
pp. 47-72 ◽  
Author(s):  
Anne-Marie Aubert
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Closure ◽  
Reductive Algebraic Group ◽  
Good For ◽  
Character Sheaves ◽  
Springer Correspondence

AbstractLetGbe a connected reductive algebraic group over an algebraic closure of a finite field of characteristicp. Under the assumption thatpis good forG, we prove that for each character sheafAonGwhich has nonzero restriction to the unipotent variety ofG, there exists a unipotent classCAcanonically attached toA, such thatAhas non-zero restriction onCA, and any unipotent classCinGon whichAhas non-zero restriction has dimension strictly smaller than that ofCA.

scholarly journals Permutation characters of finite groups of Lie type

1979 ◽  
Vol 27 (3) ◽  
pp. 378-384 ◽  
Author(s):  
David B. Surowski
Keyword(s):  
Fixed Points ◽  
Algebraic Group ◽  
Weyl Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Semisimple Element ◽  
Frobenius Endomorphism ◽  
Groups Of Lie Type

AbstractLet g be a connected reductive linear algebraic group, and let G = gσ be the finite subgroup of fixed points, where σ is the generalized Frobenius endomorphism of g. Let x be a regular semisimple element of G and let w be a corresponding element of the Weyl group W. In this paper we give a formula for the number of right cosets of a parabolic subgroup of G left fixed by x, in terms of the corresponding action of w in W. In case G is untwisted, it turns out thta x fixes exactly as many cosets as does W in the corresponding permutation representation.

scholarly journals Kazhdan-Lusztig Basis and a Geometric Filtration of an Affine Hecke Algebra

2006 ◽  
Vol 182 ◽  
pp. 285-311 ◽  
Author(s):  
Toshiyuki Tanisaki ◽  
Nanhua Xi
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Hecke Algebra ◽  
Reductive Algebraic Group ◽  
Affine Hecke Algebra ◽  
Nilpotent Variety ◽  
Affine Weyl Group

AbstractAccording to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

scholarly journals The real Chevalley involution

2014 ◽  
Vol 150 (12) ◽  
pp. 2127-2142 ◽  
Author(s):  
Jeffrey Adams
Keyword(s):  
Irreducible Representation ◽  
Algebraic Group ◽  
Semisimple Element ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
The Real ◽  
Chevalley Involution

AbstractThe Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$, takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

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 Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

scholarly journals Springer’s Weyl Group Representation via Localization

2017 ◽  
Vol 60 (3) ◽  
pp. 478-483 ◽  
Author(s):  
Jim Carrell ◽  
Kiumars Kaveh
Keyword(s):  
Weyl Group ◽  
Equivariant Cohomology ◽  
Group Representation ◽  
General Theorem ◽  
Torus Action ◽  
Cohomology Algebra ◽  
Reductive Algebraic Group ◽  
Point Set ◽  
Closed Subvariety ◽  
Springer Variety

AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.

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