Page 125 - Textos de Matemática Vol. 44

CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES 115
Example 3.5. Let g = sln(C) be viewed as a real semisimple Lie algebra. The KillingformofgisgivenbyB(X,Y)=RetrXY forallX,Y ∈guptoascalar multiple of 4n. Let the Cartan involution θ on g be defined by θ(X) = −X∗ for all X ∈ g. Then k = su(n), K = SU(n), and p consists of Hermitian matrices in g. Let a ⊂ p be the subspace of (real) diagonal matrices. The root space decomposition of g with respect to a is
g = (a ⊕ ia) ⊕ CEij , i̸=j
where Eij is the matrix with 1 at the (i,j)-entry and 0 elsewhere. The root systemisΣ={ei−ej :1≤i̸=j≤n},whereei ∈a∗ sendsA∈ato the i-th diagonal entry of A. The Weyl group W is isomorphic to the group Pn of permutation matrices. Let a+ ⊂ a be the fundamental Weyl chamber consisting of all diagonal matrices whose diagonal entries are in descending order.ThesetofpositiverootsisthenΣ+ ={ei−ej :1≤i<j≤n}.Foreach α = ei − ej ∈ Σ+, kα = {cEij − cEji : c ∈ C}. Pick C = diag(c1, . . . , cn), X = diag(x1, . . . , xn) ∈ a. The centralizers KC (resp., KX ) of C (resp., X) in K consists of all matrices in SU(n) that commute with C (resp., X). The critical set of fC,X is thus KC,X = KCPnKX. For each k = UPV with U ∈ KC, P∈Pn,andV∈KX,theHessianoffC,X atkis
− (w · α)(C)α(X)∥Fα(Ad (v)Z)∥2 α∈Σ+
=−8n ((PCP−1)ii−(PCP−1)jj)(xi−xj)·|(VZV−1)ij|2, i<j
The index of fC,X at k is
dimR CEij i<j,((PCP−1)ii−(PCP−1)jj)(xi−xj)>0
Z ∈su(n).
= 2 · |{(i, j) : 1 ≤ i < j ≤ n and ((P CP −1)ii −(P CP −1)jj )(xi −xj ) > 0}| The following lemma of Atiyah is crucial.
Lemma 3.6 ([1, p. 4]). Let f : M → R be a Morse-Bott function on a compact connected manifold M. If neither f nor −f has a critical manifold of index 1, then f−1(c) is connected (or empty) for every c ∈ R.