Richardson Orbits For Real Semisimple Lie Groups

Alfred G. NoëlMathematics Department
The University of Massachusetts
Boston, Massachusetts 02125-3393

Conference on Nilpotent Orbits and Representation Theory, 2004
September 5 - 10, 2004
Fuji-Zakura So, Japan

1

2

PLAN OF THE TALK
• Richardson theorem for complex Lie groups• Connection with the Bala-Carter classification• Kostant-Sekiguchi correspondence• Extension of Bala-Carter to Real Groups• Connection with the work of E. Sommers• Richardson theorem for the real Lie groups• Tauvel’s Counter-Example• Trapa’s results (Classical groups)• Tauvel’s polarization result• Extension of Trapa’s Result to Exceptional groups• Conclusion

3• gC a semisimple complex Lie algebra• GC adjoint group of gC• q = l ⊕ u parabolic subalgebra of gC• Q cncted Lie subgroup of GC such that Lie(Q) = q
Theorem (Richardson 1974). There exists v ∈ usuch that Q.v is an open and dense subset of u.
In other words the pair (Q, u) is a prehomogeneous
space in the sense of Sato and Kimura.
Moreover there exists a unique nilpotent orbit OgCof GC in gC such that :OgC ∩ u is an open and dense subset of u
OgC ∩ u is single Q-orbit in u.
OgC is called a Richardson orbit.

4

For example the principal orbit is always Richardson
and corresponds to a Borel subgroup.
Variations of Richardson’s theorem have been extensively studied by Gerhard Röhrle and his collaborators.
The Bala-Carter Correspondence:[1976]. There is a 1-1 correspondence between nilpotent orbits of GC on gC and GC -classes of pairs (l, p),
where l is a Levi subalgebra of gC and p a distinguished parabolic subalgebra of [l, l].
A nilpotent E ∈ gC is distinguished if the only Levithat contains it is gC .
A parabolic subalgebra is distinguished if it contains
a distinguished nilpotent.
A nilpotent element is always distinguished in a minimal Levi that contains it.
The orbit OE of a distinguished nilpotent element E
is said to be distinguished also. And such an orbit isalways even.

5

Jacobson-Morozov Theorem
Let E be a nilpotent element of gC then there exists
a semisimple element H and a nilpotent element F
in gC such that:[H, E] = 2E, [H, F ] = −2F, [E, F ] = H
The triple (H, E, F ) is called a standard sl2-triple.
Jacobson-Morozov Parabolic Subalgebra
The standard triple (H, E, F ) defines a grading:
L i
gC =
gC
i∈Z

where giC = {Z ∈ gC : [H, Z] = iZ}.
Define q = l ⊕ u with l = g0C and u =

L i
gC .
i>0

l is a reductive subalgebra of gC and u is a nilpotent
vector space.
q is called the Jacobson-Morozov parabolic subalgebra of E. It is uniquely determined by E for any
two Jacobson-Morozov parabolic subalgebras of E
are conjugate under GE, the centralizer of E in GC ,C
and gE⊂ q (Kostant)C6Richardson Orbits : Bala-Carter Correspondence
Any distinguished orbit OE is uniquely determinedby the Jacobson-Morozov parabolic subalgebra q of
any of its representatives. It is the Richardson orbit
attached to q.
u(Bala-Carter) OE is distinguished ⇐⇒ dim l = dim [u,u]Such q is said to be distinguished: A Borel subalgebra is always distinguished and corresponds to theprincipal orbit.
Hence in one direction one can define an injective
map from distinguished orbits to conjugacy classesof distinguished parabolic subalgebras:
OE −→ [q]of any Jacobson-Morozov parabolic q of E.
The classification is completed by showing that any
distinguished parabolic subalgebra q = l ⊕ u is theJacobson-Morozov parabolic subalgebra of a distinguished nilpotent E (relative to a suitable standard
triple).

7

The Kostant-Sekiguchi Correspondence
In 1998, we extended the Bala-Carter correspondence
to real reductive Lie groups. The Kostant-Sekiguchi
correspondence played a key rôle in the project. Somedefinitions: g = k ⊕ p : Cartan decomposition Θ : Cartan involution for g
 G adjoint group of g
 K ⊆ G :maxl compact Lie(K) = k
 gC = kC ⊕ pC : Complexification
 σ conjugation of gC wrt g
 KC ⊆ GC : Lie(KC ) = kC
 e nilpotent in pC
KC preserves pC .
 {x, e, f }: normal sl2-triple: x ∈ kC : e, f ∈ pC
Moreover e is KC -conjugate to a nilpotent e0 inside of a normal triple (x0, e0, f 0) with σ(e0) = f 0[Sekiguchi]. The triple (x0, e0, f 0) will be called a
Kostant − Sekiguchi or KS-triple .
Every nilpotent E 0 in g is G-conjugate to the element
E of a triple (H, E, F ) in g with the property that
θ(H) = −H and θ(E) = −F [Sekiguchi]. Such a
triple will be called a KS-triple also.

8Define a map c from the set of KS-triples of g to theset of KS- triples of gC as follows:x = c(H) = i(E − F )e = c(E) = 21 (H − i(E + F ))f = c(F ) = 12 (H + i(E + F ))The triple (x, e, f ) is called the Cayley transform of(H, E, F ). It is easy to verify that the triple (x, e, f )is a KS-triple and that x ∈ ik.
The Kostant-Sekiguchi correspondence gives a one to
one map between the set of G-conjugacy classes ofnilpotents in g and the KC -conjugacy classes of nilpotents in pC . This correspondence sends the zero orbit
to the zero orbit and the orbit through the nilpositiveelement of a KS-triple to the one through the nilpositive element of its Cayley transform. [Sekiguchi]
The correspondence was also proved independently
by D. Djoković (published ) and D. R. King (Classicalgroups) (unpublished manuscript )Related orbits are diffeomorphic [Vergne,1995]

9

Classification of Real Nilpotent Orbits q = l ⊕ u : Θ-stable parabolic in gC
A nilpotent e ∈ pC is noticed if the only (Θ, σ)stable Levi containing e is gC . And any e ∈ pC is
noticed in any minimal (Θ, σ)-stable Levi containingit.
Criterion:
Lemma 1 (N). Retain the above notations. Thena nilpotent element e in pC is noticed if and only
if dim g(0)∩ kC = dim g(2)∩ pC .
CC
Theorem [N 1997]. For any triple (gC , q, w),
where w ⊆ u∩pC is a certain L∩KC - module with adense orbit, there exists a sl2 normal triple (x, e, f )such that e is noticed, q is the Jacobson-Morozov for(x, e, f ) and w = g2C ∩ pC .
( Definition of w is technical refer to paper )Classification Theorem [N 1997].
There is a 1-1 correspondence between KC - nilpotent orbits on pC and KC -classes of certain triples(l, ql , wl )For details see:(N.) Nilpotent Orbits and Theta-Stable Parabolic
Subalgebras AMS Representation Theory Volume 2
Pages 1-32 (1998).

10What Failed?
1. Distinguished nilpotent orbits are even. There
are non-even noticed nilpotent orbits.
2. Distinguished parabolic subalgebras are defineduby: dim l = dim [u,u]u∩pCThe obvious extension dim l ∩ kC = dim [u∩k ,u∩p]CC
does not work. There are noticed nilpotent elements
whose theta-stable Jacobson-Morozov parabolic subalgebras do not satisfy the above formula.
We shall give the following example.

Let g be sl(7, R) . Then gC = sl(7, C), kC = so(7, C),and pC is the space of 7 × 7 complex symmetric matrices. The Cartan involution θ is defined as θ(X) =−X T for X ∈ g. Let
3 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 −1 0 0 0 0
H = 0 0 0 −3 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 −1 0
0 0 0 0 0 0 0
and

11

 √

0 3 0 0 0 0 0
0 0 2 0 0 0 0
0 0 0 √3 0 0 0
E = 0 0 0 0 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
Choose F = E T . Then θ(H) = −H, θ(E) = −F .
Hence (H, E, F ) is a KS-triple, and x = i(E − F )is in kC . In fact under the Kostant-Sekiguchi map(H, E, F ) corresponds to a normal triple (x, e, f ),with e = 12 (E + F + iH) and f = 12 (E + F − iH).
A simple computation shows that dim g(0)∩ kC =
C
dim g(2)∩ pC = 5. Hence the triple (x, e, f ) is noC
ticed.
From the theory of the classification of the real nilpotent orbits of sln and by Sekiguchi
GC .e = GC .E,
and the triple (x, e, f ) can be associated to the partition [4, 2, 1] of 7 and the corresponding weighted
Dynkin diagram is

2

0

1

1

0

2

12

o

o

o

o

o

o

One sees thatdim g(1)= 4, dim g(1)∩ kC = dim g(1)∩ pC = 2.
C
C
C
But
(1)
dim[g(1)∩k,g∩ pC ] = 3
CCCSo dim l ∩ kC = 5
while
(2)

gC ∩pC
u∩pC
(1)
)=4
=
dim(g
∩
p
⊕
dim [u∩k ,u∩p
(1)
(1)
C
]
C
[g ∩k ,g ∩p ]
C
C
C

C

C

C

13

It turns out the formula works for Jacobson-Morozov
parabolic subalgebras attached to even noticed nilpotent orbits.
Theorem 1 (N). Let e be an even nilpotent element of pC . Let q be a θ-stable Jacobson-Morozov
parabolic subalgebra of e relative to a triple (x, e, f )defined as above. Then q = l ⊕ u and dim l ∩ kC
u∩pC
= dim [u∩k ,u∩p
] if and only if e is noticed.
C

C

Proof. We
may assume that q is defined as above.
L
Let u0 =g(i). Then
C
i≥4

while

dim l ∩ kC = dim g(0)∩ kC ,
C

dim g(2)∩ pC = dim u ∩ pC - dim u0 ∩ pC .
C
By definition we have[u ∩ kC , u ∩ pC ] ⊂ u0 ∩ pC .
By the representation theory of sl2,
if z ∈ g(i)∩ pC and i ≥ 4 thenCz = [e, z 0] for some z 0 in g(i−2)∩ kC ⊂ u ∩ kC .
CHenceu0 ∩ pC ⊂ [u ∩ kC , u ∩ pC ].

14

The conclusion follows at once.
For exceptional Lie algebras one obtains the following:
Proposition 1 (N). Let gC be an exceptional simple complex Lie algebra. Let q = l⊕u be a θ-stable
Jacobson-Morozov parabolic subalgebra of e relative to a normal triple (x, e, f ) defined as above.
u∩pC
If e is noticed then dim l ∩ kC = dim [u∩k ,u∩p].
CC

Proof. (Use Djoković’s data)
3. The prehomogeneous space w helps deal with thefollowing cases:
KC .e ←→ (l, ql , w1l )KC .e0 ←→ (l, ql , w2l )15

A connection with the work of E. Sommers
Let GeC centralizer of e in GC . GeC is not connected
in general. Often it is required to compute the finite
group
GeC
A(e) = e(GC )◦
where (GeC )◦ is the identity component.
E. Sommers (1997) gave a unified description of the
conjugacy classes of A(e).
A pseudo-Levi subalgebra l of gC is defined to be thecentralizer in gC of a semisimple element z of GC .
A nilpotent element e of gC is called distinguished
if the conditions x ∈ gC semisimple and [x, e] = 0
imply that x is in the center of gC .
Theorem 2. (Sommers). There is a bijection Φ
between GC conjugacy classes of pairs (l, e), wherel is a pseudo-Levi subalgebra and e is a distinguished nilpotent in l, and GC conjugacy classesof pair (e, c̄), where e is a nilpotent element in gC
and c̄ is a conjugacy class of A(e).
In fact (e, 1) always goes to the Bala-Carter Levi l.
[Sommers] Nilpotent Orbits and the Affine Flag
Manifold. PhD. thesis MIT Sept 1997.

16

We have extended Sommers’s result to reductive real
Lie groups.
Let e be a nilpotent element in pC andKe
Ak (e) = (K eC)◦ .

Give a unified description ofC

the elements of Ak (e).
A pseudo-Levi subalgebra l of gC is said to be elliptic if it is the centralizer in gC of an elliptic elementof KC .
A nilpotent element e of l ∩ pC is called noticed if
the conditions z ∈ l ∩ kC semisimple and [z, e] = 0
imply that z is in the center of l.
Theorem 3. (King, N). There is a one to one
correspondence between KC -conjugacy classes ofpairs (e, z̄), where e is a nilpotent in pC , z̄ ∈ Ak (e)and KC -conjugacy classes of triples (l, ql , w) wherel is an elliptic pseudo-Levi subalgebra in which e
is noticed, ql is a noticed parabolic of l for e, andw is a certain prehomogeneous space.
In fact (e, 1) always goes to this (Θ, σ)-stable Levi lin our extension of the Bala-Carter classification .
[King &N] Component Groups of Centralizersof Nilpotents in Complex Symmetric Spaces Journal of Algebra, (232) 94-125, 2000.

17

Richardson’s Theorem fails for symmetric spaces
Maintaining the above notations, a straightforward
extension of the Richardson’s Theorem would be:
Let q be a θ-stable parabolic subalgebra of gC with
Lie group Q ⊆ GC . Does Q ∩ KC admit an open
dense orbit in u ∩ pC ?
This is not true in general. Here is a counter-example
due to Patrice Tauvel.
Let g = so(4, 4). Then we can represent g by thefollowing Vogan diagram:
α3
α1
g

g


α2 

w
??
??
??α4
?
g

where α2 is the unique non-compact simple root andk = so(4) ⊕ so(4)g is of inner-type.

18

kC = CHα1 ⊕ CHα2 ⊕ CHα3 ⊕ CHα4 ⊕ CX±α1 ⊕CX±α3 ⊕ CX±α4 ⊕ CX±(α1+2α2+α3+α4)
pC = CX±α2 ⊕CX±(α1+α2)⊕CX±(α2+α3)⊕CX±(α2+α4)⊕
CX±(α1+α2+α3)⊕CX±(α1+α2+α4)⊕CX±(α2+α3+α4)⊕
CX±(α1+α2+α3+α4)
Let q be the Borel subalgebra defined by the above
system of simple roots then clearly q is θ-stable for
q = (q ∩ kC ) ⊕ q ∩ pC )dim(q ∩ kC ) = dim(q ∩ pC ) = 8
u ∩ pC = q ∩ pC
Let z ∈ u ∩ pC . Then:[z, X(α1+2α2+α3+α4)] = 0
Hence dim[z, q ∩ kC ] ≤ 7 < dim(u ∩ pC ).
Consequently [z, q ∩ kC ] 6= u ∩ pC .

19

Two good cases
Let (x, e, f ) to be a normal sl2-triple with x ∈ ike and f ∈ pC . Let q be the θ-stable JacobsonMorozov parabolic subalgebra of e relative to thetriple (x, e, f ).
Retain the above notations. Let Q be the connected
subgroups of GC with Lie algebra q.
Proposition 2. Q ∩ KC .e is a dense open subset
L (i)gC ∩ pC . Moreover if e is even, that is g(i)
of
C
i≥2

= 0 for i odd, then Q ∩ KC .e = u ∩ pC .
Proof. The
z → Adz (e) is a morphism fromLmap(i)
Q ∩ KC togC ∩ pC and its differential is the map:
i≥2
L (i)
− ade : q ∩ kC →gC ∩ pC .
i≥2This map is surjective. Thus the given morphism isdominant and separable. Since the image of such a
map is open in its closure, the Q ∩ KC -orbit of e isL (i)a non-empty open dense subset ofgC ∩ pC .
i≥2

L (i)
If e is even thengC ∩ pC = u ∩ pC . Hence,i≥2

Q ∩ KC .e = u ∩ pC .



20

Borel-de Siebenthal parabolic subalgebras
A complex
L∞ Lie kalgebra gkC is said to be graded if
gC = k=−∞ gC where gC is a vector subspace of gC
and [giC , gjC ] = gCi+j for all integers i and j.
We shall need the following theorem of Vinberg:
Theorem 4. Let GC be a complex semisimple
LieL kgroup with graded Lie algebra gC = k gC , and let
G0C be the analytic subgroup of GC with Lie algebra
g0C . Then the adjoint action of G0C on g1C has only
finitely many orbits. Hence one of them must be
open.

A proof of the uniqueness and denseness of such an
open orbit is found in Knapp’s Lie Groups Beyond
an Introduction Proposition 10.1.
Let g be of inner-type, that is rank(k) = rank(g) and∆ a Vogan set of simple roots of gC . Then ∆ canbe partitioned into two disjoint sets: ∆kC the set ofcompact roots and ∆pC the set of imaginary noncompact roots. Let αp be a non-compact imaginarylPsimple root such that if β =ck αk is a positiveroot then 0 ≤ cp ≤ 2. Thus,

k=1

1
2
−1
0
gC = g−2
⊕
g
⊕
g
⊕
g
⊕
g
C
C
C
C
C

21

is a grading of gC where giC is the sum of the roots
spaces for roots whose coefficient of αp is i in an
expansion in terms of simple roots in ∆. Define l =g0C and u = g1C ⊕ g2C . Then q = l + u is a maximal
parabolic subalgebra of gC and is called a Borel-deSiebenthal parabolic subalgebra. Furthermore, pC =g1C ⊕ g−1
. Denote by Q the connected subgroup ofC
GC with Lie algebra q. Then u ∩ pC = g1C is a Q ∩KC -module under the adjoint action which we shallidentify with its diffrential ad : q ∩ kC → u ∩ pC
Proposition 3. Q ∩ KC has a unique open dense
orbit in u ∩ pC .
P roof. Observe that q ∩ kC = g0C ⊕ g2C and that g2C
acts trivially on g1C . Therefore, the adjoint action ofq ∩ kC on g1C is equivalent to that of g0C on g1C . The
proposition follows from Vinberg’s theorem.
22

Variations on a theorem of Peter Trapa
We say that a nilpotent orbit Ok of KC on pC is a
Richardson orbit if there exists a θ-stable parabolic
subalgebra q of gC with Levi decomposition q = l⊕usuch that Ok is the unique dense orbit admitted by
the saturation of KC on u ∩ pC . That is Ok ∩ (u ∩ pC )is open and dense in u ∩ pC . In the case where GC
is a classical complex Lie group Peter Trapa proves
the following theorem:
Theorem 5. Fix a special nilpotent orbit O of
GC on gC . Then there exists a real form G suchthat some irreducible component of O ∩ pC is a
Richardson orbit of KC on the nilpotent cone ofpC .
Source: (P. Trapa) Richardson Orbit for RealClassical Groups To appear in Journal of Algebra.
We shall see that the above theorem does not extend
to exceptional groups. It fails for the minimal orbitsof E7 and E8.
Definition 1. A nilpotent orbit of the real group
G on g is a Richardson orbit if its image under the
Kostant-Sekiguchi correspondence is a Richardson
orbit as defined above.

23

More of Trapa’s results
Using the explicit computation of Richardson orbits
in the classical cases, Trapa establishes the following:
Theorem 6. For the classical groups, the annihilator of any module of the form Aq(λ) is explicitly
computable.
The definition of Aq(λ) is found in Knapp and Voganbook “Cohomological Induction and Unitary Representations”. (Aq is related to a real form.)
Using the preceding theorem Trapa gives new examples of simple highest weight modules with irreducible associated varieties via the following:
Theorem 7. Fix GC complex semisimple (not necessarily classical). Suppose I is the annihilatorof an Aq module for some real form G of GC .
If AnnU(gC )(L(w1)) = I, then AV (L(w)) is irreducible , i.e. is the closure of a unique orbital
variety for gC . If GC is classical, this orbital variety is effectively computable

24

Tauvel’s result
Definition 2. q is a polarization of gC at e if
2dimq = dimgeC + dimgC and B(e, [q, q]) = 0
where geC is the centralizer of e in gC and B is the
Killing form of gC .
The next proposition could be seen as a version of
Richardson’s theorem for complex symmetric spaces.
Proposition 4. (P. Tauvel). Maintaining theabove notations, suppose that there exists z in pC
such that q is a polarization of gC at z. Theni. There exists a unique KC -nilpotent orbit Ok
in pC such that S = u ∩ pC ∩ Ok is open and dense
in u ∩ pC .
ii. S is a Q ∩ KC -orbit.
iii. if x ∈ S then [x, kC ∩q] = u∩pC , [x, pC ∩q] =u ∩ kC and q is a polarization of gC at x.
Source: (P. Tauvel), Quelques résultats sur lesalgèbres de Lie symétriques, Bull. Sci. math. 125
8 (2001), 641-665.

25

If e ∈ pC and GC · e is an even orbit then the
Jacobson-Morozov parabolic subalgebra of e is a polarization of gC at e.
This is due to the fact that :2 dim q = dim gC + dim g0C , dim geC = dim g0C + dim g1Cwith B(e, giC ) = {0} if i 6= −2.
Remember Trapa’s Theorem:
Let gC be a classical semisimple complex Lie algebra.
Fix a special nilpotent orbit O of GC on gC . Then
there exists a real form G such that some irreducible
component of O ∩ pC is a Richardson orbit of KC on
the nilpotent cone of pC .
We shall now try to extend this result to the exceptional cases. Trapa’s proof used the fact that in
the classical case nilpotent orbits are parametrizedby signed Young tableaux. This is not available
for exceptional groups. However, labelled Dynkin
Diagrams from the work of Dynkin and Kostant,and information from the work of Bala, Carter andDjoković is used instead.

26

The Exceptional Cases
From the previous remarks we need to consider only
non-even special orbits. Since there are none in G2
we conclude that Trapa’s result extends to G2.
Definition 3. An orbit O of GC on gC is polarizable
if for some irreducible component of KC · e of O ∩ pC
there exists a θ-stable parabolic subalgebra q of gCsuch that q is a polarization of gC at e.
Strategy
1. Find all special polarizable non-even nilpotent
orbits of the other complex exceptional simple Liegroups. In this case, Tauvel’s result implies Trapa’s
result.
2. It turns out that the non-polarizable special nilpotent orbits have low dimension. Let O be such anorbit, using information from Djoković’s classification we were able either to find an appropriate q or
to show that for all q = l ⊕ u, we can find e ∈ u ∩ pC
such that dim KC · e > dim O ∩ pC .

27

ALGORITHMIC SCHEME
Input:
1. Type of simple Lie algebra g (inner-type)
2. A Θ-stable system of roots ∆1 (Vogan)Computation:
3. n = # of non equivalent Θ-stable system of roots
W (G ){n = | W (KC ) |}C

4. Create list L = {∆1 . . . ∆n}{∆i = Sβ (∆i−1)} β’s non-compact imaginary roots.
5. Initialize storage Q
6. For each ∆i ∈ L do
For each subset S ⊂ ∆i do
Create the parabolic q associated with S
If q 6∈ Q then Insert (Q, q)
Output:7. Q consists of distinct q’s.
Source: (N.) Computing theta-stable parabolic subalgebras using LiE. Lectures Notes in Computer
Science, Springer-Verlag Volume 3039 (335-342) 2004.
It contains a complete description, implementation
details in LiE (pidgin), proof of correctness and complexity.

28

What happens when g is not of inner-type?
The algorithm is essentially the same. But the implementation will be more complex.
We only had to deal with the real form EIV of E6
for the minimal orbit case. The computation is quite
tractable for there is only one class of roots system ofinterest since that real form does not have any noncompact imaginary roots. It turns out that in thiscase one can find a parabolic q = l ⊕ u such thatu ∩ pC = C(Xα1+α2+2α3+2α4+α5+α6 − Xα1+α2+α3+2α4+2α5+α6 )29Results
Trapa’s Theorem extends to G2, F4 and E6. It fails
for the minimal orbit of E7 and for several low dimensional orbits in E8.
Theorem 8. (N.) Maintaining our previous notations, let gC be a simple complex Lie algebraother than E8 and fix a non-minimal special nilpotent orbit O of GC on gC . Then there exists a real
form G such that some irreducible component ofO ∩ pC is a Richardson orbit of KC on the nilpotent cone of pC .
Source: (N.) Some remarks on Richardson Orbits in Complex Symmetric spaces. Preprint (2004)

30

Conlusion
The last result has the flavor of a corresponding resultfor Admissible nilpotent orbits. Such orbits wereclassified by T. Ohta (1990 complete), and Schwartz(1987, partial ) for classical groups. Using some facts
from Ohta’s paper and Djoković’s classification weclassified the admissible orbits for exceptional real
Lie groups.
Sources:[N]Classification of Admissible Nilpotent Orbits
in Simple Exceptional Real Lie Algebras of Inner
type. AMS Journal of Representation Theory 5 2001
(455-493 )[N] Classification of Admissible Nilpotent Orbits
in the Simple Real Lie Algebras E6(6) and E6(−26).
AMS Journal of Representation Theory 5 2001 (494502)
If G is a split real form of a classical GC then all
real orbits that come from a special ( Lusztig’s sense)
complex orbits are admissible (Nevins). However thisfails for two orbits in E8(8) and for one orbit in E7(7).
Although it is true that at least one of such real orbits
will be admissible.
More conceptual arguments for these problems might be very enlightening.