ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES

(UNFINISHED INCOMPLETE VERSION)

E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

C ONTENTS
1. Recall from fm 1
2. Recall from pems 2
3. The gradient vector field of the real part of f H 3
4. Singularities 5
5. Lagrangean spheres 7
6. Lefschetz fibrations 9
6.1. Real Lagrangean thimbles 10
6.2. The height function f H and graphs 13
7. Real thimbles 15
7.1. Graphs in FHµ × FHµ∗ 15
References 17

1. R ECALL FROM FM
Let g be a complex semisimple Lie algebra with Cartan–Killing form
of g, 〈X , Y 〉 = tr (ad (X ) ad (Y )) ∈ C, and G a connected Lie group with Lie
algebra g .
Fix a Cartan subalgebra h ⊂ g and a compact real form u of g. Associ-
ated to these subalgebras there are the subgroups T = 〈exp h〉 = exp h and
U = 〈exp u〉 = exp u. Denote by τ the conjugation associated to u, defined
by τ (X ) = X if X ∈ u and τ (Y ) = −Y if Y ∈ i u. Hence if Z = X +i Y ∈ g with
X , Y ∈ u then τ (X + i Y ) = X − i Y . In this case, H τ : g × g → C defined by
(1.1) H τ (X , Y ) = −〈X , τY 〉
is a Hermitian form on g (see [SM1, lemma 12.17]). We write the real and
imaginary parts of H as
H (X , Y ) = (X , Y ) + i Ω (X , Y ) X , Y ∈ g.
The real part (·, ·) is an inner product and the imaginary part of Ω is a
symplectic form on g. Indeed, we have
0 6= i H (X , X ) = H (i X , X ) = i Ω (i X , X ) ,
that is, Ω (i X , X ) 6= 0 for all X ∈ g, which shows that Ω is nondegenerate.
Moreover, d Ω = 0 because Ω is a constant bilinear form.
1
2 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

The fact that Ω (i X , X ) 6= 0 for all X ∈ g guarantees that the restriction

of Ω to any complex subspace of g is also nondegenerate.
Now, the tangent spaces to O (H0 ) are complex vector subspaces of g.
Therefore, the pullback of Ω by the inclusion O (H0 ) ,→ g defines a sym-
plectic form on O (H0 ). With this choice of symplectic form, we have:

Theorem [GGSM1, Thm. 2.2] Let h be the Cartan subalgebra of a com-

plex semisimple Lie algebra. Given H0 ∈ h and H ∈ hR with H a regular
element. The height function f H : O (H0 ) → C defined by

f H (x) = 〈H , x〉 x ∈ O (H0 )

has a finite number (= |W |/|WH0 |) of isolated singularities and gives O (H0 )

the structure of a symplectic Lefschetz fibration.

The data W : X → C of a manifold together with a complex function is

commonly known in the literature as a Landau–Ginzburg model and the
function W is called the superpotential. When the manifold is considered
with a Kähler structure such data is called an A-side LG model, whereas
when the manifold is given a symplectic structure and its Lagrangeans
are considered such data is called a B -side LG model. In such terminology
we are thus providing a large class of examples of A-side LG models.

2. R ECALL FROM PEMS

Given a regular element H0 ∈ g, consider the set Θ of simple roots that
have H0 in their kernel. Let pΘ be the parabolic subalgebra determined
by Θ, with corresponding parabolic subgroup P Θ . The quotient F Θ :=
G/P θ by the parabolic subgroup is the flag manifold determined by H0 .
Another regular element in g will correspond to the same flag manifold if
it is anihilated by the same set of roots Θ, so for questions regarding the
isomorphism with T ∗ F Θ we denote the regular element by HΘ instead of
H0 .
We proved that the adjoint orbit of a regular element HΘ is isomorphic
to the cotangent bundle of the flag manifold F Θ . We obtain an ¡ isomor-
phism ι : O (HΘ ) → T FΘ observing that O (HΘ ) = k∈K Ad (k) HΘ + n+
∗ S ¢
Θ ,
then taking for each X ∈ n+Θ , the correspondence:

Ad (k) (HΘ + X ) 7→ 〈Ad (k) X , ·〉

where Ad (k) n− Θ is identified with the tangent space Tkb Θ FΘ .

Let µ be the moment map of the action a : G × T ∗ FΘ → T ∗ FΘ . We show
that µ : T ∗ FΘ → Ad (G) HΘ is the inverse of the map ι : Ad (G) HΘ → T ∗ FΘ ,
satisfying
µ∗ ω = Ω,
where Ω is the canonical simplectic form of T ∗ FΘ and ω the (real) Kirillov–
Kostant–Souriaux form on Ad (G) HΘ .
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
3

We then compactify the total space of T ∗ FΘ to the trivial product F Θ ×

F Θ∗ as:
∼
O (HΘ ) → T ∗ FΘ ,→ T ∗ FΘ = F Θ × F Θ∗ .

Let w 0 be the principal involution of the Weyl group W , that is, the el-
ement of highest length as a product of simple roots. We showed that
the right action R w 0 : FH0 → FH0∗ is anti-symplectic with respect to the
Kähler forms on FH0 and FH0∗ given by the Borel metric and canonical
complex structures. The graph of R w 0 is the orbit of K by the diagonal
action. This orbit is the zero section of T ∗ FH¡0 under the identification
with O (H0 ) ≈ G · (H0 , −H0 ). Therefore, graph R w 0 is a real Lagrangean
¢

submanifold of the product.

We obtained further examples of real Lagrangean graphs by compos-
ites (either on the left or on the right) of R w 0 with symplectic maps.

Theorem[GGSM2][Thm. 6.3] For k 1 , k 2 ∈ K and for m ∈ T :

¡ ¢
• graph k 1 ◦ R w 0 ◦ k 2 corresponds to a Lagrangean submanifold of
O (HΘ ),
¡ and ¢
• graph m ◦ R w 0 corresponds to a Lagrangean submanifold of O (HΘ ) .

3. T HE GRADIENT VECTOR FIELD OF THE REAL PART OF f H

The field Z (x) = [x, [τx, H ]] is defined over the whole algebra g and is
tangent to the adjoint orbits, since the tangent space to Ad (G) x at x is the
image of ad (x). Assume here that both H and H0 are regular and belong
to the Weyl chamber h+ R . The field Z is gradient, not with respect to the
inner product coming from g (the real part of H ), but with respect to the
Riemannian metric m on the adjoint orbit O (H0 ), which does not extend
naturally to g.
The metric m is defined as follows: the tangent space T x O (H0 ) is the
image of ad (x), which is the sum of the eigenspaces associated to the
nonzero eigenvalues of¡ x. ¢This happens because ad (x) is conjugate to
ad (H0 ) (the formula ad φx = φ◦ad (x)◦φ−1¡ holds true for any automor-
phism φ ∈ Aut (g), in particular for φ = Ad g , g ∈ G). Now, ad (H0 ) is
¢

diagonalizable and its image is the sum of the root spaces, which are the
eigenspaces of the nonzero eigenvalues of ad (H0 ) (since H0 is regular).
By conjugation the same is true for ad (x), x ∈ O (H0 ). As a consequence,
the restriction of ad (x) to its image is an invertible linear transformation.
Taking this into account, define

m x (u, v) = ad (x)−1 u, ad (x)−1 v ,

¡ ¢

where (·, ·) is the inner product given by the real part of H (·, ·), and the
inverse of ad (x) is just the inverse of its restriction to the tangent space.
The form m x (·, ·) is a well defined Riemannian metric on O (H0 ).
4 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

Remark 3.1. The realification gR of g is a real semisimple Lie algebra. Its

Cartan–Killing form 〈·, ·〉R is given by 〈·, ·〉R = 2ℜ〈·, ·〉 (see [SM1]). Conse-
quently, the inner product (·, ·) is given by
1
(X , Y ) = − 〈X , τY 〉R
2
where τ is conjugation with respect to u, which is a linear transformation
of gR (over R).

Returning to the field Z (x), define the height function h H : O (H0 ) → R

by
h H (x) = (x, H ) .
Given A ∈ g, the tangent vector [A, x] is given by
d
Ad e t A x.
¡ ¢
[A, x] =
d t |t =0
Therefore,
d ¡ ¡ t A¢ ¢
(3.1) (d h H )x ([A, x]) = Ad e x, H = ([A, x], H ) .
d t |t =0
On one hand,
m x ([A, x], Z (x)) = −m x (ad (x) A, ad (x) [τx, H ])
= − (A, [τx, H ]) ,
by definition of m x . On the other hand, by lemma 3.3 below,
(A, [τx, H ]) = (A, ad (τx) H ) = − (ad (x) A, H ) .
Thus,
m x ([A, x], Z (x)) = (ad (x) A, H ) = − ([A, x], H ) .
Combining this with (3.1) we arrive at
(d h H )x ([A, x]) = −m x ([A, x], Z (x)) .
In conclusion:

Proposition 3.2. Z (x) = − grad h H with respect to the metric m x .

Lemma 3.3. Consider the inner product (·, ·) := ℜH (·, ·), then
• the conjugation τ is an isometry for this inner product, and
(ad (X ) Y , Z ) = − (Y , ad (τX ) Z ) .
• if τX = X , that is, if X ∈ u, then ad (X ) is antisymmetric for (·, ·),
• if τY = −Y , that is, Y ∈ i u, then ad (Y ) is symmetric for (·, ·).

Proof. If X ∈ g then H (ad (X ) Y , Z ) = −H (Y , ad (τX ) Z ), and it follows

that the same relation holds true for the inner product (·, ·) . In fact,
H (τX , Y ) = −〈τX , τY 〉 = −〈τY , τX 〉 = H (τY , X ) = H (X , τY ),
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
5

which means that (τX , Y ) = (X , τY ). For the second item:

H ([X , Y ], Z ) = −〈[X , Y ], τZ 〉 = 〈Y , [X , τZ ]〉
= 〈Y , τ[τX , Z ]〉 = −H (Y , [τX , Z ]) .


Remark 3.4. (Z as a field on g) We show that considered on the entire vec-
tor space g the vector field Z (x) = [x, [τx, H ]] is not gradient with respect
to (·, ·). Take the differential form αx (v) = (v, Z (x)). Then d α (v, w) =
vα (w) − wα (v) − α [v, w], where the last term vanishes if v and w are
regarded as constant vector fields on g. The expression for Z then gives

(d α)x (v, w) = (w, [v, [τx, H ]])+(w, [x, [τv, H ]])−(v, [w, [τx, H ]])−(v, [x, [τw, H ]]) .

Evaluating this expression on x = H1 ∈ h, we obtain

(d α)x (v, w) = (w, [H1 , [τv, H ]]) − (v, [H1 , [τw, H ]])
= (w, ad (H ) ad (H1 ) τv) − (v, ad (H ) ad (H1 ) τw) .

We have

(w, ad (H ) ad (H1 ) τv) = τw, ττ−1 ad (H ) ad (H1 ) τv = (τw, ad (τH ) ad (τH1 ) v)

¡ ¢

= (ad (τH1 ) ad (τH ) τw, v) = (ad (τH ) ad (τH1 ) τw, v)

where the last equality comes from the fact that ad (H ) commutes with
ad (H1 ). Therefore,

(d α)x (v, w) = (ad (τH ) ad (τH1 ) τw, v) − (ad (H ) ad (H1 ) τw, v) .

But, setting τH = −H and τ (H1 ) = H1 , then the right hand side becomes
−2 (ad (H ) ad (H1 ) τw, v) which does not vanish identically on v, w. Thus,
d α 6= 0, implying that the vector field is not gradient on g.

4. S INGULARITIES
We have verified that the set of singularities of Z on the orbit O (H0 )
is O (H0 ) ∩ h, which is the orbit of H0 ∈ h by the Weyl group. We now
recall the proof that these singularities are nondegenerate. To see this,
let x = w H0 be one of the singularities. Then the differential of Z at x is
given by

d Z x (v) = [v, [τx, H ]] + [x, [τv, H ]] = [x, [τv, H ]]

= −ad (x) ad (H ) (τv) .

The tangent space to O (H0 ) at x is

T x O (H0 ) =
X X
gα = (gα ⊕ g−α ) .
α∈Π α>0
6 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

then τv = −
P P
If v = α∈Π a α X α α∈Π a α X −α .
Consequently,
à !
X
d Z x (v) = ad (x) ad (H ) a α X −α
α∈Π
a α α (x) α (H ) X −α .
X
=
α∈Π

In particular, let α be a positive root. Then, gα + g−α (regarded as a real

vector space) is invariant by d Z x . Furthermore, with respect to the basis
{X α , X −α , i X α , i X −α }, the restriction of d Z x to this subspace is given by
the matrix
 
0 1
 1 0
α (x) α (H ) 

,
 0 −1 
−1 0
which has eigenvalues ±α (x) α (H ) with associated eigenspaces

V−α(x)α(H ) = spanR {X α − X −α , i (X α + X −α )} = (gα + g−α ) ∩ u,

Vα(x)α(H ) = spanR {X α + X −α , i (X α − X −α )} = (gα + g−α ) ∩ i u.

Therefore, T x O (H0 ) = α∈Π gα decomposes into T x O (H0 ) = Vx+ ⊕ Vx− ,

P
where Vx+ (unstable space) is the sum of eigenspaces with positive eigen-
values and Vx− (stable space) is where d Z x has negative eigenvalues. The
dimension of T x O (H0 ) over R is 2|Π|, whereas dimR V ± = |Π|.

Proposition 4.1. The subspaces Vx+ and Vx− are Lagrangean with respect
to the symplectic form Ω = ℑH .

Proof. Vα(x)α(H ) and V−α(x)α(H ) are isotropic subspaces, since they are con-
tained in either u or i u and both are subspaces where the Hermitian form
H takes real values. On the other hand, if α 6= β are positive roots, then
gα + g−α is orthogonal to gβ + g−β with respect to the Cartan–Killing form,
and since these subspaces are τ-invariant they are also orthogonal with
respect to H . Therefore, H assumes real values on Vx+ and on Vx− as
well, hence these subspaces are isotropic, and by dimension count they
are Lagrangian. 

The subspaces Vx+ and Vx− are the tangent subspaces to the unstable
and stable submanifolds of Z with respect to the fixed point x. These
submanifolds are denoted by Vx+ and Vx− , respectively.
We now investigate the stable manifold of Z for the case x = H0 . If α > 0
then α (H0 ), α (H ) and α (H0 ) α (H ) are all positive. It follows that

V H+0 = i u ∩ V H−0 = u ∩
X X
(gα + g−α ) , (gα + g−α ) .
α>0 α>0

³ ´
Conjecture: VH−0 = z + V H−0 ∩ O (H0 ).
S
z∈h+
R
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
7

This conjecture is corroborated by the lemma below. The proof of the

P
lemma uses the following observations: u∩ α>0 (gα + g−α ) is a real vector
space with basis
{A α = X α − X −α , i S α = i (X α + X −α ) : α > 0}.
Moreover, for H ∈ h and α > 0 the following relations hold:
• [H , A α ] = α (H ) (X α + X −α ).
• [H , S α ] = α (H ) i (X α − X −α ).
• 〈A α , S α 〉 = 0, and 〈A α , A α 〉 = 〈A α , A α 〉 = 2 since, 〈X α , X −α 〉 = 0.
• If β 6= α then 〈A α , A β 〉 = 〈S α , S β 〉 = 〈A α , S β 〉 = 〈S α , A β 〉 = 0.
Lemma 4.2. For x = z + y with y ∈ u and z ∈ h+ R , H τ Z (x) , y is real < 0.
¡ ¢

Proof. First of all, τx = y −z. Thus, Z (x) = [y +z, [y −z, H ]] = [y +z, [y, H ]].
Consequently,
¡ ¢ ¡ ¢ ¡ ¢
Z (x) , y = [y + z, [y, H ]], y = [y, H ], [z, y]
¡ ¢
= − [H , y], [z, y] .
Set y = α>0 (a α S α + b α A α ) with a α , b α ∈ R. Then, by the above relations
P

α (H ) (a α (X α − X −α ) + b α i (X α + X −α )) ,
X
[H , y] =
α>0
and similarly with z in place of H . Still using the above relations, we ob-
tain ¡ 2 2
α (H ) α (H0 ) a α
X ¢
〈[H , y]], [z, y]〉 = 2 + bα
α>0
which is > 0¡because¢ α (H¡ ) , α (H0 ) > 0¢ and a α , b α ∈ R. This finishes the
proof, since Z (x) , y = − [H , y], [z, y] . 
5. L AGRANGEAN SPHERES
We construct Lagrangean spheres inside regular fibres, which are our
candidates for vanishing cycles. The correct dimension of the desired
spheres is n − 1 real, that is, half of the dimension of the regular fibre.
Here n is the complex dimension of the adjoint orbit, and the real dimen-
sion of the flag FΘ where Θ = Θ (H0 ) = {α ∈ Σ : α (H0 ) = 0}. The number of
Lagrangean spheres to be found equals |W |, that is, the number of singu-
larities.
Here we assume that H0 ∈ cla+ and that H ∈ a+ , hence H is regular.
Recall that the symplectic form Ω on the orbit O (H0 ) is the restriction of
the imaginary part of the Hermitian form of g
H τ (X , Y ) = −〈X , τY 〉.
On the other hand, the real part is the inner product defined by
1
B τ (X , Y ) = −Re〈X , τY 〉 = − 〈X , τY 〉R ,
2
R
where 〈·, ·〉 is the Cartan–Killing form of the realification of g. Thus,
H τ (X , Y ) = B τ (X , Y ) + i Ω (X , Y )
8 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

and the equality Ω (X , Y ) = B τ (X , i Y ) holds since H τ (X , Y ) is Hermitian.

We can then search for an isotropic submanifold by taking a subspace
V ⊂ g where H τ takes real values, and check whether the intersection
V ∩ g is indeed a submanifold.
Two examples of subspaces where H τ takes real values are: i) the com-
pact real form u, where H τ is negative definite and ii) the symmetric part
i u, where H τ is positive definite.
The intersection u ∩ O (H0 ) is empty because the eigenvalues of ad (X ),
for X ∈ O (H0 ) are real whereas those of ad (Y ), for Y ∈ u are imaginary.
The latter happens because ad (Y ) is anti-symmetric with respect to the
Cartan–Killing form of u, see lemma 3.3. On the other hand, the inter-
section i u ∩ O (H0 ) is the flag FΘ itself, since it is the orbit of the compact
group U = exp u.
Therefore, FΘ is an isotropic submanifold, in fact Lagrangean, and any
submanifold of FΘ is isotropic as well. Moreover, the function f H (x) =
〈H , x〉 takes real values on FΘ = i u ∩ O (H0 ). Since by hypothesis H is
regular, it follows that the restriction f HΘ of f H to FΘ is a Morse func-
tion. The origin H0 is a singularity and the hypothesis that H0 ∈ cla+ im-
plies that H0 is an atractor (with negative definite Hessian). Therefore,
¡ ¢−1 ¡ Θ ¢
the levels f HΘ f H (x) of f HΘ around H0 are codimension 1 spheres in
¡ ¢−1 ¡ Θ ¢
FΘ . These levels are isotropic submanifolds. Clearly f HΘ f H (x) ⊂
¡ ¢−1 ¡ ¡ ¢−1 ¡
f H (x) = dim FΘ − 2, it follows that
¢ ¢
fH f H (x) and since dim f H
¡ Θ ¢−1 ¡ Θ ¢
for x around H0 the spheres f H f H (x) are Lagrangean cycles at the
¡ ¢−1 ¡ ¢
levels f H f H (x) of the Lefschetz fibrations.
We can now carry out the analogous construction around other singu-
larities w H0 , w ∈ W . We use the following notation
(ι) Given a root α > 0, let

uα = (gα ⊕ g−α ) ∩ u and i uα = (gα ⊕ g−α ) ∩ i u.

Taking a Weyl basis X β ∈ gβ , with β a root, these subspaces are

generated by:
• uα = spanR {A α = X α − X −α , i S α = i (X α + X −α )} and
• i uα = spanR {i A α = i (X α − X −α ) , S α = X α + X −α }.
(ιι) For w ∈ W , let Πw = Π+ ∩ w −1 Π− be the set of positive roots that
are taken to negative roots by w.
(ιιι) For w ∈ W define the real vector subspace
X X
Vw = hR ⊕ uα ⊕ i uα .
α∈Πw α∈Π+ \Πw

When w = 1 the subspace V1 = i u, since Π1 = ;. The subspaces Vw ,

1 6= w ∈ W , will replace i u in the constructions of spheres around the
singularities w H0 .

Lemma 5.1. H τ and the Cartan–Killing form 〈·, ·〉 take real values in Vw .
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
9

Proof. Both H τ and 〈·, ·〉 are real in each of the components of Vw (pos-
P P
itive definite in hR and α∈Π+ \Πw i uα and negative definite in α∈Πw uα ).
Moreover hR , uα and uβ are orthogonal with respect to H τ and to 〈·, ·〉 if
α 6= β. 
Therefore, the restriction of the imaginary part Ω of Hτ to Vw vanishes
identically. On the orbit O (H0 ) we define a distribution ∆w (x) ⊂ T x O (H0 )
by
∆w (x) = Vw ∩ T x O (H0 ) .
By lemma 5.1, the subspaces ∆w (x) are isotropic with respect to the sym-
plectic form Ω (restricted to the obit). The goal is to prove that this dis-
tribution is integrable (at least around the singularity w H0 ). Once this
is accomplished, the integral submanifold passing through w H0 will be
a Lagrangean submanifold (for Ω). Consequently, a ball around the sin-
gularity w H0 , inside the integral submanifold will be our candidate to a
Lagrangean thimble.
Remark 5.2. A priori a distribution obtained by intersecting a fixed sub-
space with the tangent spaces of an embedded submanifold (such as
our distribution ∆w ) might not even be continuous (that is, admit local
parametrizations by continuous fields). As an example, consider the case
of the circle S 1 = {x ∈ R2 : |x| = 1}. The horizontal line {(t , 0) : t ∈ R} con-
tains the tangent space at (0, 1) however, it intersects in dimension zero
the tangent spaces of points near (0, 1). For a continuous distribution the
dimension does not decrease around a point.
At the singularity w H0 (or any other singularity) the distribution is
∆w (w H0 ) =
X X
uα ⊕ i uα .
α∈Πw α∈Π+ \Πw

This is due to the fact that the tangent space at w H0 is given by

T w H0 O (H0 ) =
X
gα
α∈Π
which intersects Vw at uα and i uα , showing that
1
dimR ∆w (w H0 ) = dimR O (H0 ) .
2
Hence, ∆w (w H0 ) is a Lagrangean subspace. It follows that dimR ∆w (w H0 ) ≥
dim ∆w (x), x ∈ O (H0 ), since the subspaces ∆w (x) are isotropic for Ω.
We will parametrize ∆w around w H0 by Hamiltonian fields.
Given X ∈ g define the real height function (with respect to the inner
product B τ ) f X : O (H0 ) → R by
f X (x) = B τ (X , x) .
Denote by ham f X the Hamiltonian field of f X with respect to Ω and by
grad f X its gradient with respect to B τ (both Ω and B τ are restricted to
O (H0 )). By definition, if v ∈ T x O (H0 ) then
d f X x (v) = Ω v, ham f X (x) = B τ v, grad f X (x) .
¡ ¢ ¡ ¢ ¡ ¢
10 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

The formula Ω (X , Y ) = B τ (X , i Y ), guaranties that

Ω v, ham f X (x) = B τ v, i ham f X (x) = B τ v, grad f X (x) .
¡ ¢ ¡ ¢ ¡ ¢

Since this equality holds for all v ∈ T x O (H0 ) it follows that

ham f X (x) = −i grad f X (x) ,
for all x ∈ O (H0 ).

Proposition 5.3. A basis for ∆w (w H0 ) is given by ham f X (i H0 ) with X be-

longing to
{i A α , S α : α ∈ Πw } ∪ {A α , i S α : α ∈ Π+ \ Πw }
where A α = X α − X −α and S α = X α + X −α .
Moreover, these Hamiltonian fields are tangent to the distribution ∆w .

Proof. ???? 

6. L EFSCHETZ FIBRATIONS
Let k ∈ K . Recall that we proved:
¡ ¢
Proposition
¡ ¢ ¡ 6.1. [GGSM2, ¢ Prop 6.2] The tangent space to graph k ◦ R w 0
at x, y = x, k ◦ R w 0 (x) is given by

{(A, Ad (k) A)∼ x, k ◦ R w 0 (x) : A ∈ u}

¡ ¢

where (A, Ad (k) A)∼ is the vector field on FH0 × FH0∗ = F¡H0 ,H ∗ ¢ induced by
0
(A, Ad (k) A) ∈ u × u (u = Lie algebra of K ).

6.1. Real Lagrangean thimbles. For Lefschetz fibrations on an adjoint

orbit O (H0 ) we can obtain Lagrangean subvarieties as graphs of symplec-
tic maps on the corresponding flag. The idea of our construction is based
on the following generalities.
The total space N of our Lefschetz fibration is a Hermitian manifold
with metric M (·, ·), complex structure J and Kähler form Ω (·, ·). Let f =
f 1 +i f 2 be the (C-valued) function that defines the Lefschetz fibration and
let V be a Lagrangean submanifold which contains a singularity x of f .
Let g = g 1 + i g 2 be the restriction of f to V .
Take the following gradient fields
• F 1 = grad f 1 , F 2 = grad f 2 , G 1 = gradg 1 and G 2 = gradg 2 .
Since f is a holomorphic function, d f (J v) = i d f (v) if v is a tangent
vector. This means
d f 1 (J v) + i d f 2 (J v) = i d f 1 (v) − d f 2 (v)
hence d f 2 (v) = −d f 1 (J v). That is, M (F 2 , v) = −M (F 1 , J v) = −M (J F 1 , v)
which shows that
F 2 = −J F 1 F1 = J F2 .
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
11

From these equalities it follows that x is a singularity of f if and only if x

is a singularity of f 1 and f 2 . The Hessians of f 1 and f 2 at the singularity x
are related as follows: if A and B are vector fields, then
Hess f 1 (A, B ) = B A f 1 = B M (F 1 , A)
= B M (F 2 , J A) = B (J A) f 2
= Hess f 2 (J A, B ) .
If f has isolated singularities, then both f 1 and f 2 are Morse functions.
The relation between the Hessians shows that every singularity has in-
dex 0 (= dimension of the positive definite part minus dimension of the
negative definite part). In fact, if Hess f 1 (A, A) > 0 then
Hess f 1 (J A, J A) = Hess f 2 (−A, J A) = −Hess f 2 (J A, A) = −Hess f 1 (A, A)
hence the positive definite and negative definite parts have the same di-
mension.
To obtain the relation between F i and G i we observe that, since V is a
Lagrangean submanifold, the space tangent to N at a point y ∈ V decom-
poses into
Ty N = Ty V ⊕ J Ty V y ∈ V,
as J T y V is the orthogonal complement with respect to the metric M (·, ·),
of T y V . Indeed, if u, v ∈ T y V then
M (u, J v) = −Ω (u, v) = 0
therefore the subspaces T y V and J T y V are orthogonal and have the same
dimension, thus are complementary. Consequently, the following rela-
tion between F i and G i holds on points of V .
¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢
Proposition
¡ ¢ 6.2. If y ∈ V then F 1 y = G 1 y −JG 2 y and F 2 y = G 2 y −
JG 1 y .
¡ ¢
Proof. For the case of F 1 take the decomposition F 1 y = u +¡ J v ∈ ¢ Ty V ⊕
J T y V . Since g 1 is the restriction of f 1 to V , it follows that d f 1 y (w) =
¡ ¢
d g 1 y (w) if w ∈ T y V . Therefore, for w ∈ T y V
¡ ¢ ¡ ¢ ¡ ¡ ¢ ¢
d g 1 y (w) = d f 1 y (w) = M F 1 y , w
= M (u + v, w) = M (u, w)
¡ ¢
and we see that u = G 1 y . Now take J w ∈ J T y V . So,
¡ ¢ ¡ ¢ ¡ ¢
d f 1 y (J w) = − d f 2 y (w) = − d g 2 y (w)
¡ ¡ ¢ ¢
= −M G 2 y , w
¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢
and M F 1 y , J w = −M G 2 y , w , that is,
¡ ¡ ¢ ¢
M (v, w) = M (J v, J w) = −M G 2 y , w .
¡ ¢
Since w is arbitrary, it follows that v = −G 2 y . Thus, for y ∈ V
F 2 = −J F 1 = −J (G 1 − JG 2 ) = G 2 − JG 1 .
12 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN


The expressions from proposition 6.2 show that if G 2 = 0, then F 1 = G 1
and, consequently F 1 is tangent to V . It follows that
Corollary 6.3. If the imaginary part is constant on the Lagrangean subva-
riety, then grad f 1 is tangent to V .
Consequently, we obtain the following method to construct stable and
unstable manifolds of grad f 1 at a singularity x (in the case of Morse func-
tions).
Proposition 6.4. Let V be a Lagrangean submanifold that contains a sin-
gularity x of the function f = f 1 + i f 2 that defines a Lefschetz fibration.
Suppose
¡ ¢ that f 2 is constant on V and that the restriction of the Hessian
Hess f (x) to the tangent subspace T x V is negative definite (respectivelly
positive definite). Then, the stable (respectivelly unstable) manifold of g 1
in V is an open subset of the stable (respectively unstable) manifold of f 1 .
Proof. The Hessian of g 1 is the restriction to T x V of the Hessian of f 1 . The
hypothesis guaranties that the fixed point x is an attractor (respectively
repeller) of G 1 = gradg 1 . Consequently, in the negative definite case, the
stable manifold of G 1 is and open subset V that contains x. In this open
set F 1 coincides with G 1 , since by hypothesis f 2 is constant on V , that
is, G 2 = 0. Therefore, the stable manifold of G 1 is contained in the stable
manifold of F 1 . A similar argument handles the positive definite case. 
Since the levels of a Morse function in the neighborhood of an attract-
ing or repelling singularity are spheres (by the Morse lemma), this propo-
sition has the following consequence.
Corollary 6.5. In the setup of proposition 6.4 take a level g 1−1 {c} = f 1−1 (c)∩
V with c near g 1 (x) = f 1 (x) and c < g 1 (x) in the negative definite case and
c > g (x) in the positive definite case. Then, g 1−1 {c} is a sphere of dimension
dimV − 1.
The sphere g 1−1 {c} in this corollary is a Lagrangean submanifold of the
level f −1 {c} (since in proposition 6.4 we took the hypothesis that g 2 is
constant).
The next idea is to construct a Lagrangean thimble having as boundary
the sphere g 1−1 {c} contained in the Lagrangean submanifold V . For this
observe ¡ ¢that for any y ∈ N , the symplectic orthogonal of the fibre Φ y =
f −1 { f y¡ }¢ is generated by F 1 = grad f 1 and J grad f 1 = −grad f 2 = −F 2 . In
fact, F 1 y is the metric orthogonal of T¡y Φ y¡ since
¢ ¢ it is gradient.
¡ ¡ ¢ However,
Φ y is a complex submanifold, thus Ω F 1 y , v = M F 1 y , J v = 0 se
¢

v ∈ T y Φ y . It follows that
Ω J F1 y , v = M J F1 y , J v = M F1 y , v = 0
¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢

if v ∈ T y Φ y , which shows that F 1 y e J F 1 y generate the symplectic or-

¡ ¢ ¡ ¢

thogonal of Φ y .
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
13

Consequently we obtain the following Lagrangean thimble for f .

Proposition 6.6. In the setup of proposition 6.4 take c near f 1 (x) = g 1 (x).
We have that
g 1−1 c, g 1 (x) = f 1−1 c, f 1 (x) ∩ V
£ ¤ £ ¤

(in the negative definite case) or g 1−1 g 1 (x) , c = f 1−1 f 1 (x) , c ∩ V (in the
£ ¤ £ ¤

positive definite case) is homemorphic to a closed ball in RdimV . This ball

is a Lagrangean thimble.

Proof. In the negative definite case g 1−1 [c, g 1 (x)] is the Lagrangean thim-
ble obtained by parallel transport of the Lagrangean sphere g −1 {c} along
the line segment [c, g (x)] ⊂ R. In fact, if s ∈ [c, g (x)] and z ∈ g −1 {s} then
the horizontal lift of the vector d /d t is a multiple of F 1 (z). This hap-
pens because the horizontal ¡ lift ¡ W
¢ is a vector ¢ = aF 1 (z)+b J F 1 (z), a, b ∈ R,
which satisfies d f z (W ) = d f 1 z (W )+i
¡ d¢ f 2 z (W ) = d /d t , that
¡ is,¢d f z (W )
is real and therefore coincides with d f 1 z (W ). This implies d f 2 z (W ) =
0, that is,

0 = M (F 2 ,W ) = −M (J F 1 (z) , aF 1 (z) + b J F 1 (z))

= −bM (J F 1 (z) , J F 1 (z))

consequently, b = 0. In the negative definite case the coeficient a > 0,

since f 1 grows in the direction of F 1 .
£ Therefore,
¤ the parallel transport of a point of g 1−1 {c} along the segment
c, g 1 (x) follows the trajectories of F 1 (reparametrized). Such trajectories
converge£ to x, thus,
¤ the ¤ parallel transports of s ∈ [c, g (x)] is the
£union of
−1 −1
ball g 1 c, g 1 (x) = f 1 c, f 1 (x) ∩ V .
The same argument works in the positive definite case, with −F 1 in
place of F 1 . 
Definition 6.7. A Lagrangean thimble inside a stable or unstable of sub-
manifold constructed as in proposition 6.6, will be called a real Lagrangean
thimble, since it is obtained by lifting of a real horizontal curve.

6.2. The height function f H and graphs. The goal here is to analyze the
behavior of the height function f H (x) = 〈x, H 〉 on Lagrangean graphs.
The case of interest here are the graphs of the composites m ◦ R w 0 with
m in the torus T = exp (i hR ). Such graphs all pass through the singulari-
ties of f H . In fact, in the product FH0 ×FH0∗ these singularities are given by
w H0 , w w 0 H0∗ = (w H0 , −w H0 ). Since
¡ ¢

m ◦ R w 0 (w H0 ) = Ad (m) w w 0 H0∗ = w w 0 H0∗

¡ ¢

¡ ¢
we see that these pairs ¡ ¢ belong to graph m ◦ R w 0 .
The Hessian Hess f H at the singularities is calculated considering ev-
erything from the point of view of the adjoint orbit O (H0 ) = Ad (G) H0 .
In this case the field Ae induced by A ∈ g is linear Ae = ad (A). Therefore
14 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

Ae f H (x) = 〈[A, x], H 〉 and the second derivative is Be Ae f H (x) = 〈[A, [B, x]], H 〉.
Thus, if x = w H0 is a singular point, then
¡ ¢¡ ¢
(6.1) Hess f H Ae (x) , Be (x) = −〈[B, w H0 ], [A, H ]〉 = −〈[w H0 , B ], [H , A]〉.
The goal now is to find the ¡ restriction
¢ of this Hessian to the tangent
spaces to the graphs graph m ◦ R w 0 , m ∈ T at the singular points. These
tangent spaces were described in proposition 6.1 using the realization of
the homogenous space as an orbit inside the product FH0 ×FH0∗ = F¡H0 ,H ∗ ¢ .
0
Such description must be translated to the viewpoint where the homo-
geneous space is the adjoint orbit O (H0 ) = Ad (G) H0 . This translation
will be made in the next proposition. First recall that from the point of
view of the open orbit G · (H0 , −H0 ) ⊂ FH0 × FH0∗ the singular points are
w H0 , w w 0 H0∗ = (w H0 , −w H0 ), w ∈ W .
¡ ¢

¡ ¢
Proposition 6.8. Let m ∈ T = exp (i hR ) and consider graph m ◦ R w 0 as a
Lagrangean submanifold of O (H0 ) = Ad (G)·H0 . Then the tangent space to
graph m ◦ R w 0 at the singularity w H0 , w ∈ W , is generated by the vectors
¡ ¢

(1) Xeα (w H0 )−Adã (m) X −α (w H0 ) = [X α , w H0 ]−[Ad (m) X −α , w H0 ] with

α (w H0 ) < 0 and
(2) ifX α (w H0 )+Adã (m) i X −α (w H0 ) = i [X α , w H0 ]+i [Ad (m) X −α , w H0 ]
with α (w H0 ) < 0.
¡ ¢
Proof. By proposition 6.1 the tangent space to graph m ◦ R w 0 at the sin-
gularity (w H0 , −w H0 ) (seen as the orbit in the product) is generated by
³ ´
(A, Ad (m) A)∼ (w H0 , −w H0 ) = Ae (w H0 ) , Ad ã (m) A (−w H0 )

with A ∈ u.
The real compact form u is generated by i hR , A α = X α − X −α and Zα =
i (X α + X −α ) with α running through all roots. The field induced by an el-
ement of i hR vanishes at the singularity w H0 hence it suffices to consider
the fields induced by A α and Zα .
Choose a root α such that α (w H0 ) < 0. Then, in FH0 , Aeα (w H0 ) =
X α (w H0 ) and Zeα (w H0 ) = if
e X α (w H0 ) since Xe−α (w H0 ) = 0.
On the other hand, Ad (m) A α = Ad (m) X α −Ad (m) X −α and Ad (m) Zα =
Ad (m) i X α + Ad (m) i X −α given that both Ad (m) X ±α and Ad (m) i X ±α be-
long to g±α since Ad (m) g±α = g±α (because m ∈ T ).
Taking now the induced field on FH0∗ and using the fact that α (w H0 ) < 0
we obtain that Ad ã (m) X α (−w H0 ) = 0 on FH ∗ (since α (−w H0 ) > 0). There-
0
fore Ad
ã (m) A α (−w H0 ) = −Ad
ã (m) X −α (−w H0 ) and Adã (m) Z α (−w H0 ) =
i Ad
ã (m) X −α (−w H0 ).
Now, the isomorphism between G · (H0 , −H0 ) and O (H0 ) takes a field
induced by and element of u to and induced field. Moreover, the isomor-
phism associates (w H0 , −w H0 ) ∈ FH0 ×FH0∗ to w H0 ∈ O (H0 ). This way, the
³ ´
isomorphism takes Ae (w H0 ) , Ad
ã A H
(m) (−w 0 ) to Ae (w H0 )+Ad ã (m) A (w H0 )
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
15

(where the formere· means the field induced on FH0 and FH0∗ whereas the
latter the one induced on O (H0 )).
Hence the tangent space at the singularity w H0 ∈ O (H0 ) is generated
by Xeα (w H0 )− Adã (m) X −α (w H0 ) and if X α (w H0 )+ i Ad ã (m) X −α (w H0 ). 
¡ ¢
The generators of the tangent space at graph m ◦ R w 0 of the previous
proposition can also be described in the following simpler manner. Take
H1 ∈ hR such that m = e i H1 . Then, Ad (m) X α = e i α(H1 ) X α . This way, the
vector fields that provide the generators at w H0 become
• Xeα − Adã(m) X −α = Xeα − e −i α(H1 ) Xe−α with α (w H0 ) < 0 and
• ifX α + Adã X α + e −i α(H1 ) if
(m) i X −α = if X −α with α (w H0 ) < 0.
¡ ¢
It is now possible to calculate Hess f H at the singularity w H0 using
the formula (6.1). The elements of g which define the generating fields
belong to gα ⊕ g−α . Hence the Hessian vanishes at a pair of generators
coming from distinct roots, since, with respect to the Cartan-Killing form,
g±α is orthogonal to g±β if β 6= ±α. For the fields provided by a root α with
α (w H0 ) < 0, we obtain (in w H0 ):
• Hess f H Xeα − e −i α(H1 ) Xe−α , Xeα − e −i α(H1 ) Xe−α =
¡ ¢¡ ¢

−〈[w H0 , X α − e −i α(H1 ) X −α ], [H , X α − e −i α(H1 ) X −α ]〉.

The second term equals
−〈α (w H0 ) X α + e −i α(H1 ) α (w H0 ) X −α , α (H ) X α + e −i α(H1 ) α (H ) X −α 〉.
Now, 〈X α , X α 〉 = 〈X −α , X −α 〉 = 0 and 〈X α , X −α 〉 = 1 (Weyl basis)
therefore, the Hessian becomes
−2α (w H0 ) α (H ) e −i α(H1 ) .
X α + e −i α(H1 ) if X α + e −i α(H1 ) if
¡ ¢¡ ¢
• Hess f H if X −α , if X −α = −〈[w H0 , i X α +
e −i α(H1 ) i X −α ], [H , i X α + e −i α(H1 ) i X −α ]〉. That is,
〈α (w H0 ) X α − e −i α(H1 ) α (w H0 ) X −α , α (H ) X α − e −i α(H1 ) α (H ) X −α 〉.
Thus the Hessian equals
−2α (w H0 ) α (H ) e −i α(H1 ) .
• Hess f H Xeα − e −i α(H1 ) Xe−α , if X α + e −i α(H1 ) if
¡ ¢¡ ¢
X −α = −〈[w H0 , X α −
e −i α(H1 ) X −α ], [H , i X α + e −i α(H1 ) i X −α ]〉. That is,
−i 〈α (w H0 ) X α + e −i α(H1 ) α (w H0 ) X −α , α (H ) X α − e −i α(H1 ) α (H ) X −α 〉 = 0.
Summing up,
Proposition 6.9. The Hessian of f H restricted to the tangent space
³ ³ ´´
T w H0 graph e i H1 ◦ R w 0

is diagonalizable in the basis

³ ´ ³ ´
{ Xeα − e −i α(H1 ) Xe−α (w H0 ) , if
X α + e −i α(H1 ) if
X −α (w H0 ) : α (w H0 ) < 0}.
16 E. GASPARIM L. GRAMA, AND LUIZ A. B. SAN MARTIN

The diagonal elements are given by

−2α (w H0 ) α (H ) e −i α(H1 ) .
For example, the orbit of the compact group (the¡ zero ¢ section in the
identification with the cotangent bundle) is graph R w 0 . If w = 1 and
H1 = 0, then −2α (w H0 ) α (H ) e −i α(H1 ) = −2α (H0 ) α (H ) which is < 0 since
α (H0 ) < 0 implies that α < 0 and consequently α (H ) < 0. That is, the
Hessian is negative definite, which was to be expected given that the sin-
gularity H0 is a maximum of Re f H on the zero section.

7. R EAL THIMBLES
7.1. Graphs in FHµ × FHµ∗ . The isomorphism between the open orbit in
FHµ × FHµ∗ (diagonal action) and the orbit G · (v 0 ⊗ ε0 ) of v 0 ⊗ ε0 ∈ V ⊗ V ∗
(representation of G) leads to a convenient description of the intersection
of graphs of anti-holomorphic functions FHµ → FHµ∗ with the open orbit.
We return to the anti-holomorphic functions considered earlier m ◦
R w 0 : FHµ → FHµ∗ with m ∈ T , the maximal torus. The submanifold de-
termined by graph R w 0 in R w 0 on FHµ × FHµ∗ is the orbit of the compact
¡ ¢

group K through (v 0 , ε0 ). This orbit stays inside G · (v 0 , ε0 ) and is iden-

tified with the K -orbit of v 0 ⊗ ε0 in V ⊗ V ∗ (by equivariance). The iso-
morphism with the adjoint orbit Ad (G) Hµ associates this K -orbit inside
V ⊗V ∗ with the intersection i u∩Ad(G) Hµ (the Hermitian matrices in the
case of sl (n, C) or else the zero section of T ∗ FHµ ). This set is formed
by the elements v ⊗ ε ∈ G · (v 0 ⊗ ε0 ) such that ker ε = v ⊥ (with respect
to the K -invariant Hermitian form (·, ·)µ ), since u ∈ K is an isometry of
(·, ·)µ and ker ε0 = v 0⊥ . The converse is true as well: if v ⊗ ε ∈ G · (v 0 ⊗ ε0 )
and ker ε = v ⊥ then v ⊗ ε ∈ graph R w 0 . In fact, if ker ε = v ⊥ and X ∈ u
¡ ¢
¢µ
then ρ µ (X ) is anti-Hermitian, thus ρ µ (X ) v, v is purely imaginary and
¡

since ker ε = v ⊥ , then ε ρ

¡ ¢
µ (X ) v is purely imaginary as well. Therefore,
〈M (v ⊗ ε) , X 〉 = ε ρ µ (X ) v is imaginary for arbitrary X ∈ u, which im-
¡ ¢

plies that M (v ⊗ ε) ∈ i u. ¡ ¢
Summing up, we obtain the following description of graph R w 0 re-
garded as a subset of G·(v 0 ⊗ ε0 ). Consider Φ−1 graph ¡ R w¢0 ⊂ G·(v 0 ⊗ ε0 ),
¡ ¡ ¢¢

which, abusing notation, we also denoted by graph R w 0 :

Proposition 7.1. graph R w 0 = {v ⊗ ε ∈ G · (v 0 ⊗ ε0 ) : ker ε = v ⊥ }.
¡ ¢

Consider now the graph of m ◦ R w 0 : FHµ → FHµ∗ with m ∈ T . In general

graph m ◦ R w 0 ⊂ FHµ × FHµ∗ is not contained in the open orbit and, con-
¡ ¢

sequently, intercepts this orbit in a noncompact subset. In any case, take

the subgroup
U m = { u, mum −1 ∈ U ×U : u ∈ U }.
¡ ¢

Then, graph m ◦ R w 0 is the orbit of K m through (v 0 , ε0 ). This happens

¡ ¢

because, if x = u · v 0 ∈ FHµ then R w 0 (x) = u · ε0 therefore

¡ ¢
x, m ◦ R w 0 (x) = (x, m · uε0 ) .
ADJOINT ORBITS AND REAL LAGRANGEAN THIMBLES(UNFINISHED INCOMPLETE VERSION)
17
¡ ¢ ¡ ¢
This means
¡ ¢ that graph
¡ ¢m ◦ R w 0 is formed by elements of the form x, m y
with x, y ∈ graph R w 0 , that is,
¡ ¢ ¡ ¡ ¢¢
graph m ◦ R w 0 = m 2 graph R w 0
∗
¡ ¢ ¡ ¢
where m 2 x, y = y, mx . Passing to¡ the realization ¢¢inside V ⊗ V we
realization of Φ graph m ◦ R w 0 , also denoted by
−1
¡
obtain¡ a geometric
¢
graph m ◦ R w 0 :

Proposition 7.2. graph m ◦ R w 0 = {v ⊗ρ ∗µ (m) ε ∈ G ·(v 0 ⊗ ε0 ) : ker ε = v ⊥ }.

¡ ¢

¡ ¢
Now we have the setup to prove that f H is real on graph m ◦ R w 0 . This
is essential to obtain real Lagrangean thimbles. With the realization of
G/Zµ as an orbit in V ⊗ V ∗ the proof that f H is real greatly simplifies.
Actually, this is not only true for elements m ∈ T , but for more general
linear transformations of V (or more precisely, of V ∗ ).
Before stating the result, observe that the function ¡f H is a priori
¢ defined
on the orbit G · (v 0 ⊗ ε0 ) and is given by f H (v ⊗ ε) = ε ρ µ (H ) v . From this
expression we see that f H extends to a linear functional of V ⊗V ∗ , that is,
it is defined on points outside the orbit G · (v 0 ⊗ ε0 ) as well.
Proposition 7.3. Let D : V → V be a linear transformation that is diago-
nalizable on a basis adapted to the root subspaces and consider the set
D 2 graph R w 0 = {v ⊗ D ∗ ε ∈ V ⊗ V ∗ : ker ε = v ⊥ }
¡ ¡ ¢¢

where D ∗ ε = ε ◦ D.
¡ Supose ¡ that
¢¢ D has real eigenvalues. Then, f H assumes
real values on D 2 graph R w 0 .
¡ ¡ ¢¢
Proof. If v ⊗ Dε ∈ D 2 graph R w 0 then
f H v ⊗ D ∗ ε = ε Dρ µ (H ) v = tr (v ⊗ ε) Dρ µ (H ) .
¡ ¢ ¡ ¢ ¡ ¢

On a basis adapted to the root subspaces, Dρ µ (H ) is diagonal with real

eigenvalues. If this basis is orthonormal, then v ⊗ ε has a Hermitian ma-
trix, and therefore real diagonal entries. Hence, the last term of the equal-
ity above is real, as is f H (v ⊗ D ∗ ε). 
Corollary 7.4. If m ∈ T satisfies m 2 = 1 then f H is real on graph m ◦ R w 0 .
¡ ¢

Proof. In fact, if m 2 = 1 then the eigenvalues of m are ±1 and since m ∈ T ,

ρ µ (m) = ±id on the root spaces. 

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