Page 124 - Textos de Matemática Vol. 44

114 X. LIU AND T.-Y. TAM
Lemma 3.2 ([10, p. 314–316], [28, p. 214])). The critical set of fC,X is KC,X = {k∈K:[C,Ad(k)X]=0}
= KCwKX w∈W
= KCwKX w∈WC \W/WX
where the second union is disjoint and over a complete set of double coset representatives.
Lemma 3.3 ([10, p. 317], [28, p. 216]). Let k = uxwv with u ∈ KC, v ∈ KX, and xw a representative of w in K. The Hessian Hk of fC,X at k ∈ K is given by
(3.2)
=
=
Hk (dLk (Z ), dLk (Z )) d2
2 fC,X (k exp tZ) dt t=0
− α(X)(w · α)(C)∥Fα(Ad (v)Z)∥2, α∈Σ+
∀Z ∈ k
where dLk : k → Tk(K) denotes the differential at the identity of the left trans- lationLk :K→KgivenbyLk(h)=khandFα :k→kα isanorthogonal projection. In particular, fC,X is a Morse-Bott function and its index at k is
(3.3) dim gα. α∈Σ+,α(X)(w·α)(C)>0
Remark 3.4. To explain (3.3), recall that the index of a symmetric bilinear form on a vector space is the maximal dimension of a subspace restricted on which the bilinear form is negative definite. For each α ∈ Σ+, define kvα = {Z ∈ k : Ad(v)Z ∈ kα}. Noting that Adv : k → k is nonsingular and that k = m⊕ kα, we have dimkvα = dimkα = dimgα and k = Ad(v−1)m⊕ kvα.
α∈Σ+ α∈Σ+ Restricting to each subspace dLkkvα, the Hessian Hk is either positive definite,
negative definite, or trivial, according to α(X)(w · α)(C) is less than, greater than, or equal to 0, respectively. Thus the index of fC,X is
dimkvα = dimgα. α∈Σ+,α(X)(w·α)(C)>0 α∈Σ+,α(X)(w·α)(C)>0
The following example shows the explicit expressions of the Hessian and index of fC,X for sln(C).