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A construction of pseudo-reductive groups with non-reduced root system

Michael Bate, Gerhard Röhrle, Damian Sercombe, David I. Stewart

Abstract

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type $BC_n$; these exist only in characteristic $2$. We also give a formula for the dimensions of their irreducible modules.

A construction of pseudo-reductive groups with non-reduced root system

Abstract

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type ; these exist only in characteristic . We also give a formula for the dimensions of their irreducible modules.
Paper Structure (7 sections, 15 theorems, 34 equations, 1 figure)

This paper contains 7 sections, 15 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Set-up
  3. Abstract complements in semidirect products
  4. Pinning.
  5. Definition of G
  6. Irreducible representations of DG and G
  7. An alternative construction

Key Result

Proposition 2.1

Let $V = k^{2n}$ be the natural $\mathrm{Sp}_{2n}$-module and let $\rho:V \rtimes \mathrm{Sp}_{2n} \to \mathrm{Sp}_{2n}$ be the natural projection. If $\mathop{\mathrm{char}}\nolimits(k)=2$ then there exists a $k$-subgroup $\mathrm{SO}_{2n+1} \hookrightarrow V \rtimes \mathrm{Sp}_{2n}$ that maps ont

Figures (1)

  • Figure 3.1: Illustration of the structure of the group $G_{K'/k,V^{(2)},V',V"}$ with root system $BC_2$ and Dynkin diagram $\dynkin[scale=2,labels={a_2,\!\!\!\!\!\! a_1,2 a_2}]{A}[2]{oo}$. Each layer indicates a subgroup generated by root groups of certain lengths. Above each root appears the vector space given by the $k$-points of the corresponding root group. Cartan subgroups are also indicated.

Theorems & Definitions (29)

  • Proposition 2.1
  • Lemma 2.2
  • Corollary 2.4
  • Definition 3.1: cf. CGP15
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 19 more