Showing papers on "Reductive group published in 1990"
TL;DR: In this article, the authors give a combinatorial description of the inclusion relations between the Zariski-closures of double cosets, which can be viewed as a generalization of Chevalley's description of inclusion relations.
Abstract: Let G be a connected reductive linear algebraic group over an algebraically closed field of characteristic not 2. Let θ be an automorphism of order 2 of the algebraic group G. Denote by K the fixed point group of θ and by B a Borel group of G. It is known that the number of double cosets BgK is finite. This paper gives a combinatorial description of the inclusion relations between the Zariski-closures of such double cosets. The description can be viewed as a generalization of Chevalley's description of the inclusion relations between the closures of double cosets BgB, which uses the Bruhat order of the corresponding Weyl group.
270 citations
141 citations
TL;DR: In this paper, the moment map on the cotangent bundle is used to define a finite cristallographic reflection group Wx, which generalizes the little Weyl group of a symmetric space.
Abstract: Summary. Let G be a connected, reductive group defined over an algebraically closed field of characteristic zero. We assign to any G-variety X a finite cristallographic reflection group Wx by means of the moment map on the cotangent bundle. This generalizes the "little Weyl group" of a symmetric space. The Weyl group Wx is related to the equivariant compactification theory of X. We determine the closure of the image of the moment map and the generic isotropy group of the action of G on the cotangent bundle. As a byproduct we determine the ideal of elements of 11(g) which act trivially on X as a differential operator.
132 citations
TL;DR: A series of algebraic geometric codes is constructed using a class of curves which have many rational points and has the maximal number of GFq-rational points, which is effectively smaller than the Hasse—Weil bound.
Abstract: We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq
2 over
$$\mathbb{F}$$
q
, whereq = 2q
0
2
andq
0 = 2
n
, such that dimension + minimal distance ≧q
2
+ 1 − q
0
(q − 1). The codes are ideals in the group algebra
$$\mathbb{F}$$
q
[S], whereS is a Sylow-2-subgroup of orderq
2 of the Suzuki-group of orderq
2
(q
2
+ 1)(q − 1). The curves used for construction have in relation to their genera the maximal number of
$$\mathbb{F}$$
GF
q
-rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.
102 citations
TL;DR: In this paper, the relation between the notions of complexity and rank of an algebraic variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup is elucidated.
Abstract: We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.
62 citations
TL;DR: In this article, the authors describe the action of the inertia group I p on the quotient groups of π 1( X )l by the higher commutator subgroups of a smooth proper connected algebraic curve defined over an algebraic number field K.
50 citations
01 Jan 1990
TL;DR: In this paper, the identity component of a connected reductive group defined over a finite field Fq (q odd) with a given involution θ is defined over Fq and the pair (G, θ) is called a symmetric space.
Abstract: Let G be a (connected) reductive group defined over a finite field Fq (q odd) with a given involution θ:G → G defined over Fq. The pair (G, θ) will be called a symmetric space (over Fq), we shall fix a closed subgroup K of the fixed point set Gθ such that K is defined over Fq and K contains the identity component (Gθ)0 of Gθ.
48 citations
TL;DR: In this paper, extending morphisms of finite groups of Lie type has been studied in the context of algebraic groups and maximal subgroups. Reference CTG-ARTICLE-1990-001
30 citations
TL;DR: The observable subgroups of linear algebraic groups over an algebraically closed field of characteristic zero are described in this article, where the observable subgroup is defined as a subgroup of a linear algebra.
Abstract: The observable subgroups of linear algebraic groups over an algebraically closed field of characteristic zero are described. Bibliography: 8 titles.
25 citations
TL;DR: Sato and Kimura as mentioned in this paper constructed an irreducible relative invariant of a prehomogeneous vector space from a castling transformation of a connected reductive group defined over the complex number field.
Abstract: Let G be a connected reductive group defined over the complex number fieldC, V a finitedimensional vector space and p: G->GL(V) a rationalrepresentation of G. Such a triplet(G, p, 7) is calleda prehomogeneous vectorspace if 7 has an open G-orbit,and calledirreducibleifpisan irreduciblerepresentation. A complete listofirreducible prehomogeneous vector spaces is given by M. Sato and T. Kimura [12]. The purpose of thispaper is to construct explicitlyan irreduciblerelativeinvariant for every irreducible prehomogeneous vector space. If(G, p, V) and (G', p',V) are in the same castling class,then an irreduciblerelativeinvariant of(G, p, V) can be constructed from that of (G', p',V). (See proposition 18in [12,section4].) Hence itis enough to consider irreducible reduced prehomogeneous vector spaces. (See [12, section 2] for the generalitiesconcerning the castling transformations.) In the tables I and II of [12, section7],irreduciblerelativeinvariants are given except forthe following six cases;
TL;DR: It is shown that all Borel probability measures invariant under unipotent translations on homogeneous spaces of connected Lie groups are algebraic.
Abstract: It is shown that all Borel probability measures invariant under unipotent translations on homogeneous spaces of connected Lie groups are algebraic.
TL;DR: In this article, the authors analyse more closely the cases of embeddings of X(2) and PGL(2), where each embedding is represented by a diagram containing combinatorical information about the local rings of orbits.
TL;DR: In this paper, the cohomology of a higher Frobenius kernel of a Borel subgroup B of a reductive algebraic group G over an algebraically closed field K of a positive characteristic p is studied.
Abstract: In this paper we study the cohomology H'(Br, K) of a higher Frobenius kernel Br, r > 1, of a Borel subgroup B of a reductive algebraic group G over an algebraically closed field K of a positive characteristic p. For the first Frobenius kernel B 1 the cohomology has been completely determined by H.H. Andersen and J.C. Jantzen [3] (in case p > h the Coxeter number of G). In low degrees the cohomology is independent of r > 1, and their results allow us to compute the first different case H zp-I(Br, K) for S Lt+I (3.6). In higher degrees, however, we have to resort to a different method. Although H ' ( B 2, K) has already been computed for SL z also in [3], it is a considerably harder problem for SL 3. We apply algebraic Steenrod operations and the transgression theorem to the Lyndon-HochschildSerre or Cartan-Eilenberg spectral sequence arising from a central series of the unipotent radical U of B to find H~ K) for SL 3 (5.1). The method we employ here was originally developed by M. Tezuka and N. Yagita [21] in order to investigate H ' (U(Fq) , K), q a power ofp. In fact, the results of Sections 4 and 5 were first conceived from the corresponding results of H'(U(Fq), K). Aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the Bl-cohomology to G. Lusztig's conjectural character formula for the irreducible modules of the reductive group G, which is described in Section 2. Hoping that the representation theory may shed more light on some topological problems, we intend to take up the Steenrod algebra and the A-algebras in a subsequent paper.
TL;DR: It is pointed out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of aDifferentially algebraic group over K .
Abstract: We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K . This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem). I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B]. What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known. Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1 , d ). Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology . Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹ › ( where k ‹ › denotes the differential field generated by k and ). Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable ) .
TL;DR: In this article, the authors studied the structure of rational injectivc 7%,-modules from the point of view of the parabolic subgroup schemes of 7'G, and showed that the multiplicities of L,(p) in Tndyi;(/I) coincide with the multiplicity of L(p), in IndE(/i): L(m), where m denotes the rational irreducible G-module of high weight /i relative to B.
TL;DR: In this paper, it was shown that for groups of rational points, if G is a simply connected anisotropic absolutely simple algebraic group with rank n ≥ 2 defined over an algebraic number field and decomposable over a quadratic extension, then the group G(K) is projectively simple, i.e. the factor group modulo the center is simple.
Abstract: It is proved that, if G is a simply connected anisotropic absolutely simple algebraic group with rank n ≥ 2 defined over an algebraic number field and decomposable over a quadratic extension, then the group G(K) of rational points is projectively simple, i.e. the factor group modulo the center is simple. Projective simplicity of algebraic groups of type Bn, Cn, G2, F4, E7 is obtained as a corollary, and also the same for groups of type E8 whenever the Hasse principle holds. In addition the problem of projective simplicity for groups of type (1)Dn, (2)Dn (n ≥ 4) is reduced to the case of groups of type A3. Bibliography: 18 titles.
TL;DR: In this article, the authors studied the G -representations arising from G -actions on algebraic varieties X and G -sheaves F in terms of the geometry of the action.
TL;DR: In this article, a simple algebraic group G = M = M(G) generated by proj, PEA Let PEA, P # G, L a Levi factor of P Let H, denote the group of automorphisms of &I(L) induced by N,(L).
01 Jan 1990
01 Jan 1990
TL;DR: In this article, the sum of the identity component of a reductive group defined over a finite field F 9 ( q odd) with a given involution over F 9 is computed.
Abstract: Let G be a ( connected) reductive group defined over a finite field F9 ( q odd) with a given involution 9: G -+ G defined over F9 • The pair ( G, 9) will be called a symmetric space ( over F9 ); we shall fix a closed subgroup K of the fixed point set G' such that K is defined over F9 and K contains the identity component ( G')0 of G'. Let T be a maximal torus of G defined over F11 and Iet ..\: T( F9) -+ Q; be a character (.i is a fixed prime not dividing q); Iet .84be the virtual representation of G(F9) attached in [2] to (T,..\). This paper is concerned with the computation of the sum
01 Dec 1990
01 Jan 1990
TL;DR: In this paper, the authors consider the linearization problem of linear automorphisms, i.e., the automorphism is conjugated to a linear action with a fixed point.
Abstract: Let V be the affine space k n over an algebraically closed field k, G a linearly reductive group and A: G×V → V a group action with a fixed point, say the origin Then for all g ∈ G let me denote by A(g) the corresponding automorphism of V We have
$$A(g) = L(g) + D(g)$$
where L(g), D(g)∈ End V, L(g) linear and D(g) the sum of terms of higher degrees Let me recall the well known linearization problem: is the action A linearizable, ie conjugated to the linear action L: G ×V →V (see eg [B] and [K])? Recently counter-examples have been found, see [S] and [K + S], so it is reasonable to study additional assumptions on the action A One of them is considered in the present paper