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 £.
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