Page 121 - Textos de Matemática Vol. 44

CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES 111
negative definite on k. Let K be the analytic subgroup of G with Lie algebra k. For C,X1,...,Xp ∈ p, the C-numerical range of (X1,...,Xp) is defined by
(2.2) WC(X1,...,Xp) = {(B(C,Ad(k)X1),...,B(C,Ad(k)Xp)) : k ∈ K}.
Since Ad(K)X is independent of the choice of connected G, so is (2.2). Li and Tam [19] proved that WC(X1,X2) is convex for all classical real simple Lie algebras except sl2(R). They also investigated WC(X1,X2,X3) case by case for each classical real simple Lie algebra. It would be nice if we can show the convexity results of [19] in a unified way.
Remark 2.1. Reductive Lie algebras have similar structures with semisimple ones [17, p. 446]. More precisely, a reductive Lie group is a 4-tuple (G, K, θ, B), where G is a Lie group, K is a compact subgroup of G, θ is a Lie algebra involution of the Lie algebra g of G, and B is an AdG-invariant, θ-invariant, nondegenerate symmetric bilinear form on g such that
(1) gisreductive,i.e.,g=z⊕[g,g],wherezisthecenterofgand[g,g]is semisimple,
(2) g = k⊕p, where k and p are the +1 and −1 eigenspaces of θ, respectively, and k is also the Lie algebra of K,
(3) k and p are orthogonal with respect to B, and B is negative definite on k and positive definite on p,
(4) the map K × exp p → G given by multiplication is a diffeomorphism onto,
(5) for every g ∈ G, the automorphism Adg of g, extended to the com- plexification gC of g, is contained in IntgC, and
(6) the semisimple connected subgroup Gss of G with Lie algebra [g, g] has finite center.
Thus the C-numerical range (2.2) is also well defined for reductive Lie algebras. 3. A third proof of Tam’s theorem
We begin with the notation of Morse function [22]. Let M be a smooth manifold and f : M → R a smooth function. A point p ∈ M is called a critical point of f if the differential map dfp : Tp(M) → Tf(p)(R) is trivial. If p is a critical point of f, the Hessian Hp of f at p is a symmetric bilinear form on Tp(M) defined by
Hp(v,w) = Vp(Wf), ∀v,w ∈ Tp(M),
where V and W are vector fields extended by v and w (i.e., Vp = v, Wp = w), respectively.