1.2. SIMPLE LIE ALGEBRAS AND GROUPS 5

G2

F4

E8

E7

Bn

A

n

Cn

E6

Dn

Figure 1.1. The list of Dynkin diagrams

1.2.3. The Borel (resp., opposite Borel) subalgebra b+ (resp., b−) of g relative

to h is spanned by h and vectors eα, α ∈ Φ+ (resp., eα, α ∈ Φ−). Let n+ = [b+, b+]

and n− = [b−, b−] be the corresponding maximal nilpotent subalgebras of g. The

connected subgroups that correspond to g, h, b+, b−, n+, n− will be denoted by G,

H, B+, B−, N+, N−.

The weight lattice P consists of elements γ ∈

h∗

such that γ(hi) ∈ Z for

i ∈ [1,r]. The basis of fundamental weights ω1,...,ωr in P is defined by ωj(hi) =

ωj|αi = δij, i, j ∈ [1,r]. Every weight γ ∈ P defines a multiplicative character

a →

aγ of H as follows: let a = exp(h) for some h ∈ h, then aγ = eγ(h). Positive

simple roots, fundamental weights and entries of the Cartan matrix are related via

(1.9) αi =

r

j=1

ajiωj = 2ωi +

r

j=1

j=i

ajiωj.

The Weyl group W is defined as a quotient W = NormGH/H. For each of the

embeddings ρi: SL2 → G, i ∈ [1,r], defined in (1.7), put

¯i s = ρi

0 −1

1 0

∈ NormG H

and si = ¯iH s ∈ W . All si are of order 2, and together they generate W . We

can identify si with the elementary reflection sαi , αi ∈ Π, in h0 ∗ defined by (1.5).

Consequently, W can be identified with the Coxeter group generated by reflections

s1,...,sr. The Weyl group acts on the weight lattice. For any w ∈ W and γ ∈ P the

action γ → wγ is defined by expanding w into a product of elementary reflections

and applying the reflections one by one to γ.

A reduced decomposition of an element w ∈ W is a representation of w as a

product w = si1 · · · sil of the smallest possible length. A reduced decomposition is

not unique, but the number l of reflections in the product depends only on w and

is called the length of w and denoted by l(w). The sequence of indices (i1,...,il)

that corresponds to a given reduced decomposition of w is called a reduced word for

w. The unique element of W of maximal length (also called the longest element of

W ) will be denoted by w0. Braid relations for ¯i s guarantee that the representative

¯ w ∈ NormG H can be unambiguously defined for any w ∈ W by requiring that