CHAPTER 2

Overview

Let G be a simple algebraic group defined over an algebraically closed field K

of characteristic p 0. Let X be a connected reductive subgroup of G. Here we

give an overview of the work contained in this paper.

In Chapter 3 we cover the general theory we need for the proofs of the main

results. We start in §3.1 with a section of notation.

Much of our work revolves around finding G-reducible subgroups X of G. That

is, we are in the following situation:

We have a reductive subgroup X inside a parabolic subgroup P of G. Let L be a

Levi subgroup of P so that P = LQ is a semidirect product of L with the unipotent

radical of P . It will be necessary in this paragraph (and elsewhere) to regard X

as an abstract group. As X is reductive, it can have no intersection with Q. If

¯

X denotes the image of X under the canonical projection π : P → L we have

¯

X

isomorphic to X. In particular XQ =

¯

XQ and X ∩Q = {1}. So X is a complement

to Q in the semidirect product

¯

XQ. By standard results, X corresponds to a 1-

cocycle γ : X → Q. Up to Q-conjugacy, these are in natural bijection with a

set denoted by

H1(X,

Q) (defined in §3.2). Now Q is non-abelian in general, and

for general non-abelian Q finding

H1(X,

Q) would be a very hard computation.

However, by 3.2.1, Q admits a certain filtration Q = Q(1) ≥ Q(2) ≥ . . . so that

successive quotients, called ‘levels’, have the structure of L-modules. This allows

us to calculate H1(X, Q) via the better understood abelian cohomology groups

H1(X, Q(i)/Q(i + 1)).

In §3.2, we describe the ingredients of non-abelian cohomology that we need

for these calculations. To begin we remind the reader of the filtration of unipotent

radicals of parabolics by modules for Levi subgroups. This filtration has the useful

property that it is central. This will permit us to apply results from the theory

of Galois cohomology, in particular an exact sequence (Proposition 3.2.3) relating

cohomology in degrees 0, 1 and 2.

In general

H1(X,

Q) is not a vector space; just a pointed set, though in the

cases we study it will in fact be a variety with origin. Our first task is to create

an approximation of

H1(X,

Q) by way of a surjection ρ from a certain vector space

V to

H1(X,

Q); this happens in Definition 3.2.5 and Proposition 3.2.6. The map

ρ is sometimes strictly partially defined. This is due to the existence of non-split

extensions between X and subgroups of the radical. Such non-split extensions arise

by non-trivial values of second cohomology. We make this link more explicit in

Definition 3.2.11.

Now, we would like to use the map ρ as an explicit parameterisation of the

elements of

H1(X,

Q). Unfortunately ρ is very far from being injective in general,

though 3.2.14 gives us a suﬃcient condition for ρ to have that property. When ρ is

an isomorphism, this often leads to an easy observation that

H1(X,

Q) = K. This

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