4

0. Review of Semisimple Lie Algebras

Here are a few useful facts which emerge from the study of root systems

with a given simple system A:

(1) The set A

v

forms a simple system in

3v.

(2) For any a, (3 G $, the roots of the form a + kf3 form an unbroken

root string a — rj3,..., a — /?, a, a + /?,..., a + 5/3, which involves

at most four roots.

(3) If P is positive but not simple, there exists a simple root a for which

(/?,av)

0; thus safi G $

+

and ht sa{3 ht/?.

0.3. Weyl Groups

The natural symmetry group attached to a root system $ is its Weyl group

W', the (finite!) subgroup of GL(E) generated by all reflections sa with

a G $ (or just the simple reflections sa with a G A when A is a fixed simple

system). Evidently the root lattice Ar is stable under the action of W.

Abstractly, W is a finite Coxeter group, having generators sa (a G A)

and defining relations of the form

(sasp)rri(OL^

— 1. Moreover, W satisfies

the crystallographic restriction m(a,(3) G {2,3,4,6} when a ^ /?. (In the

Lie algebra setting there is a natural identification of W with a finite group

of automorphisms of f ) in g; this induces an action on I)*.) The Weyl group

of

3v

is naturally isomorphic to W. Moreover,

(wa)v

=

wav

when a G $

and w G W.

The subgroup of W fixing a point A G E is itself a reflection group, being

generated by those sa for which

(A,av)

= 0 (see for example [129, 1.12]).

Write £(w) = n if w = si • • • sn with Si simple reflections and n as small

as possible; such an expression is called reduced. Standard facts about the

length function on W include:

(1) The number of a G $

+

for which wa 0 is precisely £(w). In

particular, when a G A (equivalent to £(sa) — 1), we have sa/3 0

for all /3 ^ a in $

+

. Moreover, w is uniquely determined by the set

of a 0 for which wa 0.

(2) UweW, then £(w) =

£{w-x).

Thus £(w) = |$+ n w($~)|.

(3) There is a unique element wQ G W of maximum length | $

+

| , send-

ing $

+

to 3~. Moreover, £(w0w) — £(w0) — £(w) for all w G W\

(4) If a 0 and ^ G V K satisfy £{wsa) £(w), then w;a 0, while

£(wsa) £{w) implies wa 0. It follows that £(saw) £(w) ^

w~xa

0.

Given a subset I C A, the subgroup W\ it generates is called a "para-

bolic" subgroup of W. It is a Coxeter group in its own right and its length

function £\ agrees with the restriction of £.