A NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS

Heiko Dietrich,Wolfgang Globke,Marcos Origlia

doi:10.1017/s0004972721001283

Heiko Dietrich, Wolfgang Globke + Show 1 more

Open Access

https://doi.org/10.1017/s0004972721001283

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Abstract

AbstractA linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

Highlights

Let G be a complex linear algebraic group
The existence of étale representations implies the existence of left-symmetric products on Lie algebras and thereby that of left-invariant flat and torsion-free affine connections on the corresponding Lie groups, see Burde [4] for details and additional references
We take a look at the étale representations for reductive algebraic groups arising in the classification of nilpotent orbits in semisimple Lie algebras, in particular, the classifications of Vinberg [11] and Bala & Carter [1, 2]

Summary

INTRODUCTION

Let G be a complex linear algebraic group. A prehomogeneous module (G, ̺, V ) is a complex algebraic representation : G → GL(V ) such that V is finite-dimensional and G has a Zariski-open orbit in V. The points of the open orbit are said to be in general position in V In this case, V is a prehomogeneous vector space and dim G dim V. The existence of étale representations implies the existence of left-symmetric products on Lie algebras and thereby that of left-invariant flat and torsion-free affine connections on the corresponding Lie groups, see Burde [4] for details and additional references. Due to this relationship, Lie groups or Lie algebras admitting étale representations are called (locally) affinely flat. Additional examples with interesting properties were constructed by Burde et al [6]

Results

VINBERG’S CARRIER ALGEBRAS

BALA AND CARTER’S CONSTRUCTION

Relation to carrier algebras

GYOJA’S CONSTRUCTION