WEYL CHAMBERS IN g* 9
Henceforth G denotes a compact connected Lie group with Lie alge-
bra Q.
Since G is compact, there exists an Ad-invariant inner product (£,77) 1— » £ rj
on g. So Ad9 £ Adp 7 7 £ 7 7 for all £, 7 7 G g and g £ G. Infinitesimally, we have,
for arbitrary ^,r/,(Gg,
1.3 adc£ 7 7 + ^ adC77 = [C,£] 7 7 + £ [(,77] = 0 .
The inner product on g has a unique extension to a (non-degenerate) C-bilinear
form on its complexification g c = g ®R C, which we also denote by (£,77) •— £ 77.
Identity 1.3 generalizes to an identity on gc.
Weyl chambers in g*
Fix a maximal torus T and corresponding maximal Abelian subalgebra t. Equa-
tion 1.3 says that ad^ : g g is skew-symmetric with respect to the Ad-invariant
product. Its image and kernel are therefore orthogonal for any £ G t. This fact,
and the commutativity of t, easily show that the decomposition 1.2.8 is orthogonal
(explaining the notation
t-1).
If we define t = Ann
t1-
(Ann denotes the annihilator)
and
t1-
= Ann t, then we obtain the dual decomposition
1.4 g* = t e t J - .
Let (p : g » g* denote the isomorphism induced by the Ad-invariant inner
product. Then the orthogonality of the decomposition 1.2.8 implies
1.5 ip(i) = t .
The isomorphism tp is G-equivariant if we let G act on g* via the co-adjoint action,
g - fi = Ad*-i /x. It thus establishes an equivalence between the adjoint and co-
adjoint representations. In particular, by virtue of 1.5, tp maps a Weyl chamber
to C t to a connected component of t H g*eg, which we call a Weyl chamber in g*.
Here g* denotes the regular points of the co-adjoint action. The equivalence of the
two representations establishes analogues of 1.2.5, 1.2.6 and 1.2.7 for the co-adjoint
action:
1.6 Corollary.
1. For each fi £ $* there exists a maximal Abelian subalgebra t C g such that
/iG t = Ann[g,t].
2. Let W be a Weyl chamber in g*, i.e., a connected component of t Pi g*eg,
for some maximal Abelian subalgebra t C g. Then G^ = T for all \i G W,
where T is the maximal torus with Lie algebra t.
3. Each regular co-adjoint orbit intersects each Weyl chamber in g* in exactly
one point.
Previous Page Next Page