Pseudo-reductive Groups

  title={Pseudo-reductive Groups},
  author={Brian Conrad and Ofer Gabber and Gopal Prasad},
Preface to the second edition Introduction Terminology, conventions, and notation Part I. Constructions, Examples, and Structure Theory: 1. Overview of pseudo-reductivity 2. Root groups and root systems 3. Basic structure theory Part II. Standard Presentations and Their Applications: 4. Variation of (G', k'/k, T', C) 5. Ubiquity of the standard construction 6. Classification results Part III. General Classification and Applications: 7. The exotic constructions 8. Preparations for classification… 

On Unipotent Radicals of Pseudo-Reductive Groups

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, let k' be a purely inseparable field extension of k of degree p^e and let G

Reductive Group Schemes (sga3 Summer School, 2011)

The aim of these notes is to develop the theory of reductive group schemes, incorporating some simplifications into the methods of [SGA3]. We assume the reader is familiar with the basic structure


1. Basic structure of reductive groups 2 2. The unipotent radicals 9 3. Central isogeny decomposition 14 4. Grothendieck’s covering theorem 24 5. Exponentiating root spaces 29 6. Dynamic description

Linear algebraic groups with good reduction

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has

Linear algebraic groups with good reduction

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has


We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose

Generically free representations III: exceptionally bad characteristic

In parts I and II, we determined which irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Abstract Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed



On Weil restriction of reductive groups and a theorem of Prasad

Abstract.Let G be a connected simple semisimple algebraic group over a local field F of arbitrary characteristic. In a previous article by the author the Zariski dense compact subgroups of G(F) were

Unipotent algebraic groups

On the theory of unipotent algebraic groups over an arbitrary ground field.- Notations, conventions and some basic preliminery facts.- Forms of vector groups groups of Russell type.- Decomposition


Let k be a field, and let G be an algebraic group over k, by which we mean a connected smooth k-group scheme (not necessarily affine). Recall that such a G is automatically separated, finite type,

On quasi-reductive group schemes

This paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpre- tation of the dual group (as a Chevalley group over Z) of

Instability in invariant theory

Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to

On finite group actions on reductive groups and buildings

Let H be a connected reductive group over a non-archimedean local field k and let F ⊂ Autk(H) be a finite group of order not divisible by p, the residual characteristic of k. Let G = (H F )◦ be the


§1.1. Motivation. The purpose of these notes is to explain the definition and basic properties of the Néron model A of an abelian variety A over a global or local field K. We also give some idea of

Finiteness theorems for algebraic groups over function fields

  • B. Conrad
  • Mathematics
    Compositio Mathematica
  • 2011
Abstract We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and

Introduction to Lie Algebras and Representation Theory

Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-