24 - Orbits in internal Chevalley modules

Summary

INTRODUCTION

Let G be a simple, connected algebraic group over an algebraically closed field F of characteristic p ≥ 0. Let P = LQ be a parabolic subgroup of G, where Q is the unipotent radical of P and L is a Levi subgroup of P. Here L acts on Q via conjugation. This induces an L-action on consecutive subquotients of the lower central series of Q. Provided with a suitable F-vector space structure these quotients can be regarded as L-modules. They are called internal Chevalley modules for L.

There exists a unique parabolic subgroup P⁻ of G such that P∪P⁻ = L. We refer to P⁻ as the opposite of P and Q⁻ = R_u(P⁻) is called the opposite unipotent radical of P. The internal Chevalley modules that occur in Q⁻ are dual to the ones in Q.

In this note we describe some results from regarding the structure of orbits for the action of L on these modules and give some information on the associated stabilizers for arbitrary characteristic.

(1.1) A motivation for this is a result of R. Richardson asserting that L has only finitely many orbits on each of its internal Chevalley modules. We show that there is a close connection between the L-orbits on Q⁻/(Q⁻)′ and (P, P)-cosets of G. For details and further information we refer to.

We say that p is a ‘very bad’ prime for G, if p occurs as a structure constant in Chevalley's commutator relations for G. In this situation there are degeneracies in these relations affecting the structure of orbits. We assume throughout these notes that p is not very bad for G.