Page 123 - Textos de Matemática Vol. 44

Let
a = 0 −a :a∈R .
1 0 0 1
C=X= 0 −1 ∈a, Y= 1 0 ∈p.
CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES 113
Foreachw=mwM ∈W,thelinearmapAd(mw):a→adoesnotdependon the choice of mw ∈ M′ representing w. Therefore, w → Ad(mw) is a faithful representation of W on a. Thus we may regard w ∈ W as the linear map Ad(mw) : a → a and W as a group of linear operators on a. The Weyl group W alsoactsona∗ byw·α=α◦w−1 forallα∈a∗.Thisactionisinducedby the action of W on a via the Killing form of g.
For any C, X ∈ p, we consider the smooth function fC,X : K → R defined by
(3.1) fC,X (k) := B(C, Ad (k)X).
The C-numerical range WC(X,Y) with C,X,Y ∈ p is convex if every fibre
f−1 (c) with c ∈ R is connected (or empty) in K. Since p = ∪k∈KAd(k)a and C,X
B is Ad K-invariant, we can assume that C, X ∈ a.
Example 3.1. Let g = sl2(R) and G = SL2(R). Up to a multiple of 4, the
Killing form is given by B(X,Y) = trXY for all X,Y ∈ g. Let the Cartan
involution θ be defined by θ(X) = −X⊤ for all X ∈ g. Then ab
p = b −a :a,b∈R 0c
k=so(2)= −c0:c∈R
cosθ sinθ
K = SO(2)= −sinθ cosθ :θ∈R a 0
cosθ sinθ −1 For k = −sinθ cosθ , we have fC,X(k) = tr(CkXk
) = 2cos2θ. Thus the fibre fC,X(2) = 0 1 , 0 −1 is clearly not connected in K. In fact,
−1 1 0 −1 0
the C-numerical range of (X, Y ) is not convex. More precisely, WC (X, Y ) =
{(2cos2θ,2sin2θ) : θ ∈ R} is a circle.
For each X ∈ a, let KX and WX denote the centralizers of X in K and in W, respectively. It is obviously that M ⊂ KX, which guarantees that the notion KC wKX makes sense for w ∈ W . The following two lemmas show that fC,X is a Morse-Bott function.