Showing papers on "Reductive group published in 1988"

On the Classification of Irreducible Representations of Real Algebraic Groups

Abstract: Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the representation ρ of G(A(F )) on the space of slowly increasing functions on the homogeneous space G(F )\G(A(F )). It is of particular interest to study the irreducible constituents of ρ. In a lecture [9], published some time ago, but unfortunately rendered difficult to read by a number of small errors and a general imprecision, reflections in part of a hastiness for which my excitement at the time may be to blame, I formulated some questions about these constituents which seemed to me then, as they do today, of some fascination. The questions have analogues when F is a local field; these concern the irreducible admissible representations of G(F ). As I remarked in the lecture, there are cases in which the answers to the questions are implicit in existing theories. If G is abelian they are consequences of class field theory, especially of the Tate-Nakayama duality. This is verified in [10]. If F is the real or complex field, they are consequences of the results obtained by Harish-Chandra for representations ∗Preprint, Institute for Advanced Study, 1973. Appeared in Math. Surveys and Monographs, No. 31, AMS (1988)

Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups

Abstract: Let G be a reductive algebraic group over an algebraically closed field F of characteristic zero and let L(G) be the Lie algebra of G. Then G acts on G by inner automorphisms and it acts on L(G) by the adjoint representation. By taking the diagonal actions, we get actions of G on the spaces of n-tuples G n and L(G)". In this paper we will prove a number of geometric properties of the orbits of G on these spaces of n-tuples.

Homology of the zero-set of a nilpotent vector field on a flag manifold

Abstract: 0.1. Let X be a linear transformation of a finite-dimensional vector space V. The configuration of flags in V which are fixed by X has rather remarkable properties when X is unipotent. Though this case is especially interesting, the proper generality in which to study such configurations is in the theory of reductive algebraic groups, where their definition can be reformulated in the language of Borel subalgebras as follows. Let G be a connected reductive group over C, with Lie algebra g, and let N E g be a nilpotent element. Let q be the variety of all Borel subalgebras of g and let

Sur les représentations non ramifiées des groupes réductifs $p$-adiques ; l’exemple de $GSp(4)$

Abstract: We correct two errors in the paper "Sur les representations non ramifiees des groupes reductifs p-adiques; l'exemple de GSp(4)": the first in the study of an involution on the irreducible unramified representations of a semi-simple group, the second in the description of representations of the group GSp(4).

Character, Character Cycle, Fixed Point Theorem and Group Representations

Abstract: Publisher Summary This chapter explores that among many methods to derive Weyl's character formula, there is an application of the fixed point theorem to a line bundle on the flag variety. Namely, any finite-dimensional irreducible representation of a reductive group G is obtained as the cohomology group of an equivariant line bundle on the flag variety. Therefore, the trace of the action of an element g of G is obtained as the sum of the contributions at each fixed point. On the other hand, Harish-Chandra defined the character of an infinite-dimensional representation of a real semisimple group G R as an invariant eigendistribution. The chapter presents a character formula in terms of the geometry of flag manifold as a conjecture and presents a proof of it for discrete series. The correspondence of Harish-Chandra modules and K -equivariant sheaves is completed by adding representations of G R and G R -equivariant sheaves. Thereafter, the character is calculated from G R -equivariant sheaves.

The algebraic nature of the elementary theory of PRC fields

Abstract: A field K is PAC if every nonempty absolutely irreducible variety V defined over K has a K rational point. Similarly K is PRC if each such V has a K-rational point provided it has a simple K-rational point for every real closure K of K. The elementary theory of algebraic PAC fields determines the elementary theory of all PAC fields. Thus a sentence θ is true in each PAC field of characteristic 0 if it is true in each PAC field which is algebraic over Q [FJ, Corollary 20.25]. The goal of the present note is to use the methods that lead to this result and the prove the analogous one for PRC fields and for maximal PRC fields. We also prove that the absolute Galois group of a maximal PRC field is a free product of groups of order 2 in the category of pro-2 groups. Conversely each such group is isomorphic to the absolute Galois group of a maximal PRC field.

Spherical functions on Cartan motion groups

Abstract: This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general K-type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of K in the universal enveloping algebra of the corresponding reductive group. Introduction. This paper gives a reasonably complete treatment of the harmonic analysis of Cartan motion groups. We recall the definition. Let G be a connected semisimple Lie group and let K C G be a maximal compact subgroup. Let g = t + s be the orthogonal (Cartan) decomposition of the Lie algebra of G. Then the Cartan motion group associated to G is the semidirect product H = K x s using the adjoint representation of K on s. In ?4 of this paper we obtain an explicit parameterization of the nonunitary dual of the Cartan motion group H (Lemma 22), and of the Fell topology on it (Theorem 24). (Lemma 25 is an interesting general result on Fell topologies.) Mackey's theory of unitary induction is ideally suited to Cartan motion groups, making the computation of the unitary dual easy. It turns out that the nonunitary dual is just about what you might expect if Mackey's machine worked for nonunitary representations (but of course it does not). In Theorem 28 we find an explicit expression for the Plancherel measure of H, including the normalizing constant. In ?5, Theorem 35, and its corollary, is the Paley-Wiener theorem for these groups. Lemma 30 and Corollary 33 give a result on the asymptotic growth of the matrix entries of these representations. They use a variation on the method of stationary phase, with complex parameters. (The matrix entries are essentially generalized Bessel functions.) I wish to thank the referee for, suggesting this greatly improved version of the result. Actually most of this paper is concerned with parameterizing these matrix entries, or spherical functions. In the first section, we gather the notation and definitions and some abstract nonsense used in the rest of the paper. These should be more or less familiar to anyone who knows semisimple theory, so no proofs are included. ??2 and 3 are devoted to the proof of Theorem 19, a sort of Chevalley restriction theorem. (This is related to Kostant's J and Q matrices.) Let V, r be a unitary Received by the editors August 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E30, 22E45, 22E47, 43A90, 15A72; Secondary 33A40, 33A75, 14L30. @1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page This content downloaded from 157.55.39.55 on Sun, 12 Jun 2016 04:42:11 UTC All use subject to http://about.jstor.org/terms

On the Schur indices of certain irreducible characters of reductive groups over finite fields

Abstract: Introduction. Let Fq be a finite field with q elements, of characteristic p. Let G be a connected, reductive linear algebraic group defined over Fqy with Frobenius endomorphism F, and let G denote the group of F-fixed points of G. In [13], we investigated, under the assumption that the centre Z of G is connected, the rationality-properties of the characters \\° of G induced by certain linear characters λ of a Sylow ^-subgroup of G and, using the results obtained there, proved some propositions concerning the Schur indices of the semisimple or regular irreducible characters of G. In this paper, we shall treat the general case, that is, the case that Z is not necessarily connected. The main results are stated and proved in § 2. In particular, we get the following (see Corollary 1 to Proposition 1, §2):

Algebraic monoids whose nonunits are products of idempotents

Abstract: Let M be a connected regular linear algebraic monoid with zero and group of units G. Suppose G is nearly simple, i.e. the center of G is one dimensional and the derived group G' is a simple algebraic group. Then it is shown that S = M\G is an idempotent generated semigroup. If M has a unique nonzero minimal ideal, the converse is also proved. It follows that if Go is any simple algebraic group defined over an algebraically closed field K and if 4T: Go -GL(n, K) is any representation of Go, then the nonunits of the monoid M(b) = KD(Go) form an idempotent generated semigroup. It has been shown by J. Erdos [3] (see also [2]) that any nonsingular matrix over a field is a product of idempotent matrices. Let K be an algebraically closed field. Our interest is in connected linear algebraic monoids with zero. This means by definition that the underlying set is an irreducible affine variety and that the product map is a morphism (i.e. a polynomial map). Let G denote the group of units of M, S = M\G. We are interested in knowing when S is idempotent generated. We will only consider the situation when G is a reductive group. This means [1] that the unipotent radical of G is trivial. Then by [8, 10], M is unit regular, i.e. M = E(M)G where E = E(M) = {e E Mle2 = e}. If X C M, let E(X) = X n E(M) and (X) the semigroup generated by X. Let >F,Wy denote the usual Green's relations on M (see [5]). If a, b E M, then a J b means MaM = MbM, a J b means aM = bM, a Y b means Ma = Mb, X f n S. The following result was proved by the author [9, Theorem 2.7]. THEOREM 1 [9]. Suppose S = M\G has a maximum f-class (i.e. S is an irreducible variety). Then S is an idempotent generated semigroup. Let R, G' denote the radical, derived group of G, respectively. Then [1, ?1.8], R is contained in the center of G, G' is semisimple and G = RG'. Suppose dim R > 1. Then there exists e E E(R), e =/ 0,1. There exists g E G such that eg :A e. If eg e (E(M)), then eg = e1 ek for some ei E E(M), i = 1,... , k. Then since e is a central idempotent, eg = e'l * e', where e' = eei E E(M) and e' < e. This is a contradiction since M is a matrix semigroup. Thus S = M\G is not idempotent generated. We will say that G is nearly simple if dim R = 1 and G' is a simple algebraic group, i.e. G' has no nontrivial closed connected normal subgroups. THEOREM 2. Suppose G is nearly simple. Then S = M\G is an idempotent generated semigroup. Received by the editors February 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20G99, 20M10. (D1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

On the fundamental group of the fixed points of an uhipotent action

Abstract: We show that for a meromorphic action of an unipotent linear algebraic group over on a compact connected reduced complex space, the fundamental group of the fixed point set of the action surjects onto the fundamental group of the space.

On the cyclicity and cocyclicity ofG-modules

Abstract: LetG be a linear algebraic group over an algebraically closed fieldK. We call a (rational)G-module cyclic if it is generated by one element, and call it cocyclic if its dual is cyclic. We callG a c.c. group if the cyclicity is equivalent to the cocyclicity for anyG-module. IfG is not a c.c. group, the critical number ofG is the greatest integerc(G) such that the cyclicity is equivalent to the cocyclicity for anyG-module of dimension ≤c(G).

Homogeneous affine bundles

Abstract: Let G be a semi-simple linear algebraic group defined over an algebraically closed field k. We describe the subgroups H of a parabolic subgroup P such tht P/H is the affine line.