Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

Finite arithmetic subgroups ofGL n,III

LetG be an algebraic group inGL _n (C) defined over Q, andK an algebraic number field with the maximal orderO _k . If the groupG(O _k ) of rational points ofG inM _n (O _k ) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O _k ) is generated by roots of unity inK andG(Z). Dedicated to the memory of Professor K G Ramanathan     

On irreducible elements of semilattices

The concept of (join)-irreducible elements works well, especially for distributive lattices. Therefore our definition of elements of a given degree of irreducibility employs the notion of distributivity as much as possible, even if the irreducibility is defined for elements of a (meet)-semilattice. Via the lattice of hereditary subsets of the poset ofk-irreducible elements of a semilattice (wherek is a cardinal) we obtain a new construction of a D1_k-reflection (a sort of free distributive extension) of the semilattice, provided that there are sufficiently manyk-irreducible elements. The last property is satisfied, for example, if the original semilattice is the dual of an algebraic lattice [Dilworth and Crawley, 1960], but this condition is too restrictive for semilattices. It turns out that, under certain limitations, the D1_k-reflection of a semilattice both preserves and reflects the degree of irreducibility.Presented by R. Freese.     

Irreducibility Criterion for Quasiregular Representations of the Group of Finite Upper Triangular Matrices

An analog of the quasiregular representation is defined for the group of infinite-order finite upper triangular matrices. It uses G-quasi-invariant measures on some G-spaces. The criterion for the irreducibility and equivalence of the constructed representations is given. This criterion allows us to generalize Ismagilov's conjecture on the irreducibility of an analog of regular representations of infinite-dimensional groups.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

On generalized Cartan subspaces

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and G _k (resp. H _k ) the set of k-rational points of G (resp. H). The variety G _k /H _k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L ²(G _k /H _k ) of square integrable functions decomposes into several series, one for each H _k -conjugacy class of Cartan subspaces of G _k /H _k .     

Krull–Schmidt reduction of principal bundles in arbitrary characteristic

Let M be an irreducible projective variety defined over an algebraically closed field k, and let E_G be a principal G-bundle over M, where G is a connected reductive linear algebraic group defined over k. We show that for E_G there is a naturally associated conjugacy class of Levi subgroups of G. Given a Levi subgroup H in this conjugacy class, the principal G-bundle E_G admits a reduction of structure group to H. Furthermore, this reduction is unique up to an automorphism of E_G.     

Rigid G2-representations and motives of type G2

We consider a family of motives associated to the rigid local system whose monodromy is dense in the simple algebraic group of type G₂ and which has a local monodromy of order 7 at ∞. We prove an explicit Hilbert irreduciblity theorem for the associated étale realizations and deduce that the specialized motives at the points of irreducibility have motivic Galois group G₂.     

Rigid geometric structures,isometric actions,and algebraic quotients

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on [(M)\tilde]{\widetilde{M}} are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π ₁(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.     

Decompositions and reductions of snarks

According to M. Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol. 234 (1976), pp. 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors. The problem of what “nontrivial” means is implicitly or explicitly present in most papers on snarks, and is the main motivation of the present paper. Our approach to the discussion is based on the following observation. If G is a snark with a k-edge-cut producing components G₁ and G₂, then either one of G₁ and G₂ is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G₁ and G₂ whose order does not exceed that of G. The two situations lead to a definition of a k-reduction and k-decomposition of G. Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively. The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity. The present paper is devoted to a detailed investigation of these invariants. The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6. A number of problems that have arisen in this research conclude the paper. © 1996 John Wiley & Sons, Inc.     

Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.     

Complete reducibility,Külshammer’s question,conjugacy classes: A D4 example

Let k be a nonperfect 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 a new example of subgroup H of G of type D₄ in characteristic 2 such that H is G-completely reducible but not G-completely reducible over k (or vice versa). This is new: all known such examples are for G of exceptional type. We also find a new counterexample for Külshammer’s question on representations of finite groups for G of type D₄. A problem concerning the number of conjugacy classes is also considered. The notion of nonseparable subgroups plays a crucial role in all our constructions.     

Reality properties of conjugacy classes in<Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

LetG be an algebraic group over a fieldk. We callg εG(k) real ifg is conjugate tog ⁻¹ inG(k). In this paper we study reality for groups of typeG ₂ over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG ₂ over special fields and construct examples to show that reality fails for semisimple elements inG ₂ over ℚ and ℚ_p. We show that semisimple elements are real forG ₂ overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG ₂ overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.     

Bundles on non-proper schemes: representability

Let X be a proper scheme over a field k which satisfies Serre’s condition S ₂ and G a reductive group over k. We prove that the functor of principal G-bundles, defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.     

On simple groups and simple singularities

We give a proof of a characteristicp version of Brieskorn’s theorem, namely, that ifG is a simply connected simple algebraic group of typeA, D orE over an algebraically closed fieldk whose characteristic is very good forG, then the categorical quotient morphismG→G//G _ad yields, when restricted to a general slice through a pointP in the subregular unipotent orbit inG, a miniversal deformation of the rational double point overk of the same type asG.     

Levi decompositions of a linear algebraic group

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G.     

Proof of Springer’s hypothesis

LetG be a reductive group over a finite fieldk of a characteristicp. Π:G _k → AutU is an irreducible representation ofG in “a general position”. Springer formulated a conjecture about values of the character of Π on unipotent elements. This conjecture is proved in the article.     

Set intersection representations for almost all graphs

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found that hold for almost all graphs. Specifically, if θ_k(G) is defined to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels, there exist positive constants c_k and c′_k such that almost every graph G on n vertices satisfies Changing the representation only slightly by defining θ;_odd (G) to be the minimum number of labels with which G can be represented using the rule that two vertices are adjacent if and only if they share an odd number of labels results in quite different behavior. Namely, almost every graph G satisfies Furthermore, the upper bound on θ_odd(G) holds for every graph. © 1996 John Wiley & Sons, Inc.     

A criterion for the strongly semistable principal bundles over a curve in positive characteristic

Let X be an irreducible smooth projective curve over an algebraically closed field k of positive characteristic and G a simple linear algebraic group over k. Fix a proper parabolic subgroup P of G and a nontrivial anti-dominant character λ of P. Given a principal G-bundle E_G over X, let E_G(λ) be the line bundle over E_G/P associated to the principal P-bundle E_G→E_G/P for the character λ. We prove that E_G is strongly semistable if and only if the line bundle E_G(λ) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles.