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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