Rational fixed points for linear group actions

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. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analog of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if is Cayley for some r≥0. Here denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov, and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper, we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.

We solve the question about the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups over the algebraic closures of finite fields of positive characteristic.

Degree 2 cohomological invariants of linear algebraic groups

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

Linearity for actions on vector groups

Local invariants and exceptional divisors of group actions

Let K be a differential field with algebraically closed field of constants CK . Let K P V ∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X ( K P V ∞ ) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X ( K P V ∞ ) = X ( K diff ) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic varietyXX. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms ofXX(existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results onpp-versality for a prime integerpp. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

Arithmetic purity of strong approximation for homogeneous spaces

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension≥2{\geq 2}, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.

We study the algebra of conformal endomorphisms CendG,G n of a finitely generated free module M n over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cend n acting irreducibly on M n generates an essential left ideal of CendG,G n if enriched with operators of multiplication on elements of H. In particular, we describe such subalgebras for the case where G is finite.

The purpose of this section is to give a brief informal introduction, with very few proofs, to the subject of linear algebraic groups, a far-reaching generalization of matrix theory and linear algebra.