Page 122 - Textos de Matemática Vol. 44

112 X. LIU AND T.-Y. TAM
The index of Hp, referred to as the index of f at p, is the maximal dimension of a subspace of Tp(M) on which Hp is negative definite, or equivalently the number of negative eigenvalues of the matrix associated with Hp.
A smooth function on a manifold is called a Morse function if its Hessian is nondegenerate at every critical point. A Morse-Bott function [3, 4] is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is nondegenerate in the normal direction. Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.
Now we return to the context of a real semisimple Lie algebra g with a Cartan decomposition g = k ⊕ p corresponding to a Cartan involution θ. Let a be a maximal abelian subspace of p. The commutative family ad a is simultaneously diagonalizable. Thus we have the root space decomposition g = g0 ⊕ gα,
where
Σ := {α∈a∗ :α̸=0and dimgα ̸=0},
gα := {X∈g:[H,X]=α(H)XforallH∈a}.
Foreachα∈Σ,thesetPα ={X∈a:α(X)=0}isasubspaceofaof codimension 1. The subspaces Pα (α ∈ Σ) divide a into finitely many open convex cones, called Weyl chambers. Fix a Weyl chamber, denote it by a+ and refer it as the fundamental Weyl chamber. The set of positive root is
Σ+ ={α∈Σ:α(H)>0forallH∈a+}. For each α ∈ Σ+, define
kα := {X∈k:(adH)2X=α(H)2XforallH∈a}, pα := {X∈p:(adH)2X=α(H)2XforallH∈a}.
Let m := {X ∈ k : ad(X)H = 0 for all H ∈ a} be the centralizer of a in k. It follows that [20, p. 107]
(1) k=m⊕ kα andp=a⊕ pα aredirectsumswhosecomponents α∈Σ+ α∈Σ+
are mutually orthogonal under B,
(2) gα ⊕ g−α = kα ⊕ pα for all α ∈ Σ+, and (3) dimgα = dimkα = dimpα for all α ∈ Σ+.
Foranyα∈Σ+,ifX∈gα thenθX∈g−α,X+θX∈kα andX−θX∈pα. Let W := M′/M be the Weyl group of G relative to a, where M′ and M are the normalizer and centralizer of a in K, respectively, i.e.,
M := {k∈K:Ad(k)H=HforallH∈a}, M′ := {k∈K:Ad(k)H∈aforallH∈a}.
α∈Σ