Showing papers on "Reductive group published in 1975"

Linear Algebraic Groups

Abstract: Algebraic geometry affine algebraic groups lie algebras homogeneous spaces chracteristic 0 theory semisimple and unipoten elements solvable groups Borel subgroups centralizers of Tori structure of reductive groups representations and classification of semisimple groups survey of rationality properties.

Extension theorems for reductive group actions on compact Kaehler manifolds

Abstract: It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

On coverings and hyperalgebras of affine algebraic groups

Abstract: We will continue to study relationships between etale group coverings and the hyperalgebra of a connected affine algebraic group. Let k be a field, and let G be a connected affine algebraic k-group. The Hopf algebra of representative functions on the hyperalgebra hy(G) will determine a pro-affine algebraic k-group G and a covering 7r: G -G. In the previous paper [2] we studied basic properties of the covering (G, ir) and explained how it plays an essential role in the theory of etale group coverings. Sullivan [5] gave another description of G in positive characteristic, and he provided a transparent proof of the fact that ir is a central extension in [7]. In this article, we will give more properties of (G, ir) and improve our previous results to characterize those affine algebraic k-groups which have universal group coverings. The properties of (G, ir) we want to prove are the following two:

Extension Theorems for Reductive Group Actions on Compact Kaehler Manifolds

Abstract: It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Classification of three-dimensional affine algebraic varieties that are quasi-homogeneous with respect to an algebraic group

Abstract: In this article, we find all irreducible three-dimensional affine algebraic varieties that admit a quasi-transitive algebraic group of biregular automorphisms (that is, there is an orbit under the group action whose complement has dimension at most zero). The ground field is algebraically closed and has characteristic zero.Bibliography: 29 items.

$k$-regular elements in semisimple algebraic groups

Abstract: In this paper, Steinberg's concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a k-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent k-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all k-regular unipotent elements is given. The number of minimal parabolic subgroups containing a kregular element is given. The number of conjugacy classes of R-regular unipotent elements is given, where R is the real field. The number of conjugacy classes of Qp-regular unipotent elements is shown to be finite, where Q is the fileld of p-adic numbers.

Singularities in the nilpotent scheme of a classical group

Abstract: If (X, x) is a pointed scheme over a ring k, we introduce a (generalized) partition ord(x, X/k). If G is a reductive group scheme over k, the existence of a nilpotent subscheme N(G) of Lie(G) is discussed. We prove that ord(x, N(G)/k) characterizes the orbits in N(G), their codimension and their adjacency structure, provided that G is Gln, or SPn and 1/2 E k. For SOn only partial results are obtained. We give presentations of some singularities of N(G). Tables for its orbit structure are added. Introduction. Let G be a reductive algebraic group over a field of characteristic p. Let g be its Lie-algebra and N(G) the closed subset of the nilpotent elements of g, cf. [19]. The G-orbits in N(G) are characterized by weighted Dynkin diagrams,cf. [20, III]. Consider the following question. Is it possible to classify the orbits in N(G) using only the local structure of the variety N(G)? We prove in (4.3) that the answer is positive if G is Gln or if G is SPn and p * 2. To this end we introduce a local invariant "ord" for any pointed scheme in ? 1. We develop the theory of N(G) over an arbitrary ground ring k in ?2. In ?3 we restrict our attention to the classical group schemes. Using a cross section we obtain information about the orbit structure of N(G). Our main theorem (4.2) relates ord(x, N(G)/k) to the Jordan normal form of the nilpotent endomorphism induced by x in the classical representation. This paper is a condensed version of [13]. The author wishes to express his gratitude to his thesis adviser, Professor T. A. Springer. Conventions and notations. The cardinality of a set V is denoted by # V Any infinite cardinal is represented by oo. If x is a real number then [x] is thc greatest integer in x. All rings are commutative with 1. Let M be a module over a ring A. If M is free the rank of M is denoted by rgAM. An element r E A is called M-regular if a: M M is injective. Let a = (a,, .. . , ar) be a Received by the editors March 18, 1975. AMS (MOS) subject classifications (1970). Primary 14B05, 14L15; Secondary 05A17, 10C30, 13H15, 20G35.

Prehomogeneous vector space defined by a semisimple algebraic group

Abstract: The purpose of this department is to provide early announcement of outstanding new results, with some indication of proof. Research announcements are limited to 100 typed lines of 65 spaces each. A limited number of research announcements may be communicated by each member of the Council who is also a member of a Society editorial committee. Manuscripts for research announcements should be sent directly to those members of the Publications Committees listed on the inside back cover.

Defining normal subgroups of unipotent algebraic groups

Abstract: Let G be a connected unipotent algebraic group defined over the perfect field k. We show that polynomial generators x1, , x for the ring k[G] can be chosen so that if N is any connected normal kclosed subgroup of G, then I(N) can be generated by codim N p-polynomials in x1 X * xn where p = char k. Moreover k[G/N] cani also be generated as a polynomial algebra over k by p-polynomials. Introduction. These results are essentially an extension of a theorem of Rosenlicht [4, Theorem 1]. We use the notation and conventions of [1] throughout this paper. Recall that a p-polynomial in k[T] is a linear form if p = 0 and a polynomial all of whose exponents are powers of p if p > 0. A p-polynomial in k[xl, *. , xn] is a sum of p-polynomials in each of the single variables xl, x . A function f E k[G] will be called additive if /(ab) = /(a) + /(b) for all closed points a, b in G. 1. Frattini coordinates. Let G be a unipotent algebraic group. The Frattini subgroup of G is the intersection of all closed subgroups of codimension one. We shall denote this group by Fr(G). Proposition 1. If G is a unipotent algebraic group then Fr(G) is a closed characteristic subgroup of G. If G is connected and defined over the perfect field k, then Fr(G) is connected and defined over k. Moreover in the connected case G/Fr(G) has the structure of a vector group (over k if G is defined over k) and is the maximal such quotient. Proof. The first assertion is immediate. Let H C G be a closed subgroup of G of codimension one. Since G/H Ga7 H contains the commutator subgroup of G and the group generated by the pth powers of the elements of G. It follows that Fr(G) also contains these subgroups. Thus G/Fr(G) is connected, commutative and of exponent p hence by Received by the editors December 28, 1973. AMS(MOS) subject classifications (1970). Primary 14L10, 20G15.

On cuspidal representations of $p$-adic reductive groups

Abstract: Let k be a p-adic field, and G a reductive connected algebraic group over k. Fix a maximal torus T of G which splits in an unramified extension of k, and which has the same split rank as the center of G. For each character 6 of T(k), satisfying some conditions, there is a cuspidal representation y$ of G{k) which is a sum of a finite number of irreducible representations; the correspondence 0 |-*JQ is one-to-one on the orbits of such characters by the little Weyl group of T; furthermore, the formulas for the formal degree of JQ and its character for sufficiently regular elements of T(k) are given: they are formally the same as is the discrete series for real reductive groups. 1. Unramified maximal tori. Let k be a p-adic field, that is a finite extension of Q or a field of formal series over a finite extension of F p . We denote by k the residue field of order q. Let G be a reductive connected algebraic group defined over k, the derived group Gde r of which is simply connected. A maximal torus of G defined over k is called minisotropic if it normalizes no (proper) horocyclic subgroup of G defined over k. LEMMA. Suppose there exists a minisotropic maximal torus T of G which splits in a finite unramified extension L of G. Then the Galois group r of L over k has a unique fixed point v in the apartment of T in the building ofGder(L) [2] ; moreover, the face ofv is minimal amongst the faces in this apartment which are invariant by T. 2. Characters. We conserve notations and hypotheses of §1 and the Lemma. Let 0 be a continuous character of 7\k). For each X £ ^(T), the lattice of rational one-parameter subgroups of T, we define a character 6X of Lby AMS (MOS) subject classifications (1970). Primary 22E50, 20G25; Secondary 20C15.

Finite Fields and Coding Theory

Abstract: Many introductory books on algebra contain a section on finite fields and prove some of their basic properties. Often their interest is to discuss Wedderburn's theorem on finite division rings or to consider the structure of groups of transformations of vector spaces over fields. For these ends, only the elementary properties of finite fields are needed. This chapter presents a more detailed account of the theory of finite fields, including material on polynomials over finite fields and linear transformations of vector spaces over finite fields. An introductory course of algebra allows an efficient development of the properties of finite fields and places them in their proper algebraic perspective as particular and interesting examples of a more general theory. The chapter reviews a few of the theories of fields, extensions of fields, and polynomials over fields. The theory of finite field is used in the construction and analysis of codes.

Submersive and unipotent group quotients among schemes of a countable type over a field $k$

Abstract: An algebraic group G is called submersive if every quotient in affine schemes cG: Spec A -4 Spec AG which is surjec. tive is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if cG(Spec A) is open in Spec AG or if some restrictions on the action of G on A are made, cG is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided. 0. Introduction. Let X be an affine scheme of finite type over a field k and suppose that G is an irreducible algebraic group over k which has a closed action on X via a k morphism a: G x X -4 X. If X = Spec A, AG consists of the functions in A invariant under G and XG = Spec AG, we call the map cG: X -4 XG induced by the inclusion AG .A the algebraic group quotient of X by G. It is easily seen that if G is affine, the mapping cG is the coequalizer of a and P2, the projection of G x X onto X, in the category of affine schemes of a countable type over k (see [2]). Definition 1. Let X be a free variable. If cG: X -__ XG is surjective implies that c G: X -+ XG is submersive, we say that G is submersive. Every reductive group is submersive. See Mumford [6, p. 27, Theorem 1.1]. In ?4, we show that every unipotent group is submersive. The basic notation to be used here is that of Mumford [6]. However, we present now some of the conceptions to be found in Mumford [6] in a more compact form using some category theory. Definition 2. Let D1 be a subcategory of D2 and D2 be a subcategory of D3 (thus, D1, D2 and D3 are categories). Suppose that f, g: Y1 2 Received by the editors May 10, 1974. AMS (MOS) subject classifications (1970). Primary 14A15, 14D20, 14CGb.

Structure of Reductive Groups

Abstract: By studying the actions of tori and their centralizers on G/B, we showed in Chapter IX that a reductive group is generated by the centralizers of singular tori (the latter being precisely the connected kernels of roots). Moreover, we showed that the quotient of such a centralizer by its center is essentially PGL(2, K). The goal of this chapter is a more detailed description of G: properties of the root system, structure of normal subgroups of G, “normal form” for elements of G, structure of parabolic subgroups.