REDUCTIVE GROUPS

5

1.5. Properties of root systems. Let 0 c Kbe a root system. Proofs of the pro-

perties reviewed below can be found in [7].

(a) If aG0and^aG0then A = ±1 , ±i, ±2. The root system 0 is called

reduced if for all a e 0 the only multiples of a in 0 are ±a. To every root system,

there belong two reduced root systems, obtained by removing for every a e 0 the

longer (or shorter) multiples of a.

(b) 0 is the direct sum of root systems 0' c V and 0" c V" if V = V ® V"

and 0 = 0' U 0". A root system is irreducible if it is not the direct sum of two

subsystems.

Every root system is a direct sum of irreducible ones.

(c) The only reduced irreducible root systems are the usual ones: A„ (n ^ 1),

Bn {n £ 2), Cn (n £ 3), Dn {n ^ 4), E6i E7, Es, FA, G2.

(d) For each dimension n there exists one irreducible nonreduced root system,

denoted by BCn (see below).

EXAMPLES,

(i) Bn. Take V = Qn with standard basis {eh •••, en}. Then Vy =

Q», with dual basis {e¥, ..., ei). We have Bn = {±ei±ej (i j) and ±e{

(1 g /, j ^ /i)}. If a = ±e,-±*y then a

v

= ±ey±e)\ if a = ±ef- then a

v

=

±2e{. The Weyl group W(Bn) consists of the linear transformations which

permute the coordinates and change their signs in all possible ways.

The

av

e

Vv

form an irreducible root system of type Cn. We have W(Cn) ^

W(Bn\

(ii) With the same notations we have BCn = {±ei±eJ (i j), ±eiy and ±2et

(1 ^ Uj ^ ")}• Then W(BCn) = W(Bn).

1.6. Weyl chambers. Let 0 = K be a root system. We now view it as a subset of

VR = K ®Q i?. A hyperplane H of KR is singular if it is orthogonal to an av. A

Weyl chamber C in KR is a connected component of the complement of the union

of the singular hyperplanes. To a Weyl chamber one associates an ordering of the

roots: a 0 o (x,

av

0 for all xeC.

a e 0 is simple (for this ordering) if it is not the sum of two positive roots. The

set of simple roots A is called a basis of 0. We have the following properties.

(a) The Weyl group W(0) acts simply transitively on the set of Weyl chambers.

(b) The sa (a e J) generate W(0). More precisely, (W, (sa)aG/i) is a Coxeter system

(see [7]).

(c) Every root is an integral linear combination of simple roots, with coefficients

all of the same sign.

(d) Say that A is connected if it cannot be written as a disjoint union A =

A' U A", where

(Af

+ A") f] A = 0 .

Then we have: 0 is irreducible o A is connected.

A connected A leads to a connected Dynkin graph. These are described in [7].

1.7. We collect a few facts about root data to be needed later. First, there is the

notion of direct sum of root data. This is clear and we skip the definition.

Next we have to say something about morphisms of root data. The following

suffices. For more general cases see [7]; see also 2.1 l(ii) and 2.12.

Let V = (X, 0, X", 0V) and W = {X\ 0', (X')y, (0')v) be two root data. A

homomorphism/: X' - X is called an isogeny ofW into ¥ if:

(a)/is injective and Im/has finite index in X,