CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF
HOLOMORPHIC TYPE FOR E6(−14)

arXiv:1707.04436v1 [math.RT] 14 Jul 2017

HAIAN HE
Abstract. The author classifies Klein four symmetric pairs of holomorphic type for noncompact Lie group E6(−14) , which gives a class of pairs (G, G′ ) of real reductive Lie group
G and its reductive subgroup G′ such that there exist irreducible unitary representations π
of G, which are admissible upon restriction to G′ .

1. Introduction
1.1. Notations. Denote by R and C the real field and the complex field respectively. For any
positive integer n, denote by In the n×n identity matrix. For a Lie group G with its Lie algebra
g, denote by AutG and Autg the automorphism group of G and g respectively. Also, denote
by Intg the subgroup of Autg, which contains all the inner automorphisms of g. For f ∈ AutG,
write Gf := {g ∈ G | f (g) = g}. Similarly, for f ∈ Autg, write gf := {X ∈ g | f (X) = X}
and g−f := {X ∈ g | f (X) = −X}. Moreover, if two elements g1 and g2 in the group G are
conjugate by an element in the subgroup H of G, i.e., g2 = h−1 g1 h for some h ∈ H, then write
g1 ∼H g2 ; else, write g1 ≁H g2 . For g ∈ G, denote by [g] the conjugacy class of g in G. If
g1 , g2 , · · · , gn are n elements in a group G, then denote by hg1 , g2 , · · · , gn i the subgroup of G
generated by g1 , g2 , · · · , gn .
1.2. Outlines. Riemannian symmetric pairs were classified by Élie Joseph CARTAN in [2]
and [3], and the general semisimple symmetric pairs were classified by Marcel BERGER in
[1]. Jingsong HUANG and Jun YU studied semisimple symmetric spaces from a different
point of view in [7]; that is, by determining the Klein four subgroups in the automorphism
groups of compact Lie algebras. As an extension of the work of [7], Jun YU classified all
the elementary abelian 2-groups in the automorphism groups of compact Lie algebras, where
groups of rank 2 are just Klein four subgroups. However, the classification of Klein four
subgroups in the automorphism groups of non-compact simple Lie algebras is not clear. In this
acticle, the author will study the Klein four symmetric pairs for the connected simple Lie group
E6(−14) := Inte6(−14) ; namely, to find some pairs (E6(−14) , G′ ) with G′ the reductive subgroup
of the fixed points under the action of some Klein four subgroup of the automorphism group
of E6(−14) .
The motivation to study the Klein four symmetric pairs for E6(−14) comes from the branching
problem raised by Toshiyuki KOBAYASHI in [13, Problem 5.6]. Concretely, one may want
to classify the pairs (G, G′ ) of real reductive Lie groups satisfying the following condition:
there exist an infinite dimensional irreducible unitary representation π of G and an irreducible
unitary representation τ of G′ such that
0 < dimC Homg′ ,K ′ (τK ′ , πK |g′ ) < +∞

where K is a maximal compact subgroup of G, g is the complexified Lie algebra of G, πK |g′
is the restriction of underlying (g, K)-module of π to the complexified Lie algebra g′ of G′ ,
K ′ is a maximal compact subgroup of G′ such that K ′ ⊆ G′ ∩ K, and τK ′ is the underlying
(g′ , K ′ )-module of τ . For symmetric pairs (G, G′ ), the problem was solved in [14, Theorem
5.2]. Unfortunately, it is unknown for general cases, even for Klein four symmetric pairs.
2010 Mathematics Subject Classification. 17B10; 22E46.
Key words and phrases. admissible representation; holomorphic type; Klein four symmetric pair; reductive
Lie group.
1

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HAIAN HE

Here is a sufficient condition. If there exist an infinite dimensional irreducible unitary
representation π of G and an irreducible unitary representation τ of G′ such that
0 < dimC HomG′ (τ, π|G′ ) < +∞
where π|G′ is the restriction of π to G′ . This is true when G is a simple Lie group of Hermitian
type and (G, G′ ) is a symmetric pair of holomorphic type, which the author will tell the details
in this article. As a generalization of this result, if G is a simple Lie group of Hermitian type
and (G, G′ ) is a Klein four symmetric pair of holomorphic type, this is also true. More general,
for a simple Lie group of Hermitian type G and its reductive subgroup G′ , if the Lie algebra of
G′ contains the center of the Lie algebra of a maximal compact subgroup of G, then it holds.
It is not easy to give a complete list of all such pairs (G, G′ ), but one may just consider Klein
four symmetric pairs. Hence, in this article, the author focus on the case when G = E6(−14) .
The article is organized as follows. At the beginning, the author briefly introduces Klein
four symmetric pairs. After that, the author explains how Klein four symmetric pairs of
holomorphic type is related to representations of Lie groups. Finally, the author classifies Klein
four symmetric pairs of holomorphic type for e6(−14) , and applies the results to representation
theory.
2. Klein Four Symmetric Pairs
2.1. Finite groups in Autg0 . Let g0 be a real semisimple Lie algebra. For each element
σ ∈ Autg0 , define Θ(σ) : Autg0 → Autg0 given by f 7→ σf σ −1 whose differential is σ. In
particular, if θ is a Cartan involution of g0 , then Θ(θ) is a Cartan involution of Autg0 , and the
centralizer ZAutg0 (θ) := {σ ∈ Autg0 | σθ = θσ} is a maximal compact subgroup of Autg0 .
Conversely, Suppose that K is a maximal compact subgroup of Autg0 . Take a Cartan
involution θ′ of g0 , and ZAutg0 (θ′ ) is a maximal compact subgroup of Autg0 . Hence, there exists
an element f ∈ Autg0 such that K = f −1 ZAutg0 (θ′ )f = ZAutg0 (f −1 θ′ f ), and θ := f −1 θ′ f is
also a Cartan involution of g0 . Therefore, any maximal compact subgroup of Autg0 is given
by ZAutg0 (θ) for some Cartan involution θ of g0 .
Proposition 1. If H is a finite subgroup of Autg0 , then there exists a Cartan involution θ of
g0 , which centralizes H.
Proof. It is clear that the finite group H is included in a maximal compact subgroup of Autg0 ,

so H ⊆ ZAutg0 (θ) for some Cartan involution θ of g0 ; namely, θ centralizes H.
2.2. Klein Four Subgroups. From now on, the author focuses on real simple Lie groups G
and real simple Lie algebras g0 .
Definition 2. Let G (respectively, g0 ) be a real simple Lie group (respectively, Lie algebra),
and let σ and τ be two non-identity involutive automorphisms of G (respectively, g0 ). Then
{σ,τ }
the subgroup G{σ,τ } := Gσ ∩ Gτ (respectively, g0
:= gσ0 ∩ gτ0 ) is called a Klein four symmetric subgroup (respectively, subalgebra) of G (respectively, g0 ), and (G, G{σ,τ } ) (respectively,
{σ,τ }
(g0 , g0
)) is called a Klein four symmetric pair. Two Klein four symmetric pairs (G1 , G′1 )
(respectively, (g1 , g′1 )) and (G2 , G′2 ) (respectively, (g2 , g′2 )) are said to be isomorphic if there exists a Lie group (respectively, Lie algebra) isomorphism f : G1 → G2 (respectively, f : g1 → g2 )
such that f (G′1 ) = G′2 (respectively, f (g′1 ) = g′2 ).
Remark 3. The subgroup hσ, τ i of AutG or Autg0 generated
by the set {σ, τ } is a Klein
\ γfour
\
{σ,τ }
hσ,τ i
g0 .
Gγ or g0
= g0
:=
subgroup, and it is obvious that G{σ,τ } = Ghσ,τ i :=
γ∈hσ,τ i

γ∈hσ,τ i

Remark 4. According to the classification of Klein four symmetric pair for compact exceptional
Lie algebras u0 [7, Table 4], the conjugacy classes of Klein four symmetric subalgebras in u0
are in one-to-one correspondence to the conjugacy classes of Klein four subgroups in Autu0 .
However, it is not true for classical Lie algebras. For example, there are two non-conjugate
Klein four subgroups Γ1 and Γ2 in Autsu(4), such that su(4)Γ1 ∼
= su(4)Γ2
= 2sp(1) ∼
= 2so(2) ∼
by [7, Table 3].

CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF HOLOMORPHIC TYPE FOR E6(−14)

3

2.3. Klein Four Symmetric Pairs of Holomorphic Type. Take a Cartan involution θ
for a real simple Lie group G, which defines a maximal compact subgroup K. Let τ be an
involutive automorphism of G, which commutes with θ. Use the same letter τ to denote its
differential, and then τ stabilizes k0 and also the center Z(k0 ) = RZ. Because τ 2 = 1, there
are two possibilities: τ Z = Z or τ Z = −Z. Recall [12, Definition 1.4] that the symmetric pair
(G, Gτ ) or (g0 , gτ0 ) is said to be of holomorphic (respectively, anti-holomorphic) type if τ Z = Z
(respectively, τ Z = −Z), in which case τ may be said to be of holomorphic (respectively,
anti-holomorphic) type for convenience.
According to the classification of the symmetric pairs of holomorphic type and anti-holomorphic
type in [12, Table 3.4.1 & Table 3,4,2], there does not exist a symmetric pair which is of both
holomorphic type and anti-holomorphic type, so the author remarks that whether a symmetric pair is of holomorphic type or anti-holomorphic type does not depends on the choice of
the Cartan involution which commute with the given involutive automorphism defining the
symmetric pair.
Definition 5. Suppose that G (respectively, g0 ) is a real simple Lie group (respectively, Lie
algebra) of Hermitian type. Let τ and σ be two involutive automorphisms of G such that
{τ,σ}
) is called a
τ σ = στ . If both τ and σ are of holomorphic type, then (G, G{τ,σ} ) or (g0 , g0
Klein four symmetric pair of holomorphic type.
Unlike symmetric pairs of holomorphic type, the classification of Klein four symmetric pairs
of holomorphic type is unknown. In the last section, the author will classify Klein four symmetric pairs of holomorphic type for E6(−14) which is a simple Lie group of Hermitian type.
2.4. Admissible Representations. Suppose that G is a real simple Lie group, and G′ is a
reductive subgroup of G. Let π be a unitary representation of G on a Hilbert space. Recall
the definition of G′ -admissible representations given in [9] and [10]. The restriction π|G′ from
G to G′ is called G′ -admissible if the restriction π|G′ splits into a discrete Hilbert direct sum
of irreducible representations:
M
∼d
π|G′ =
n (τ )τ,
c′ π
τ ∈G

c′ denotes the unitary dual of G′ .
with each multiplicity nπ (τ ) a nonnegative integer, where G
The point here is that there is no continuous spectrum in the branching law and that each
multiplicity is finite. Here is a special but important example of G′ -admissible restrictions,
which was shown in [4], [5], and [6]: If K is a maximal compact subgroup of G, any irreducible
unitary representation π of G is K-admissible when restricted to K.
Suppose that G is a real simple Lie group of Hermitian type; that is, G/K carries a structure
of a Hermitian symmetric space where K is a maximal compact subgroup of G. Equivalently,
the center Z(k0 ) of Lie algebra k0 of K has dimension 1. The classification of simple Lie algebras
g0 of Hermitian type is given as follows:
su(p, q), sp(n, R), so(m, 2)(n 6= 2), e6(−14) , e7(−25) .
Such a Lie algebra g0 satisfies the condition that a Cartan subalgebra h0 of k0 becomes a
Cartan subalgebra of g0 . Moreover, there exists a characteristic element Z ∈ Z(k0 ) such that
g = l ⊕ p+ ⊕ p− is a decomposition with respect
of Z on the complexified Lie
√
√ to the eigenspaces
algebra g corresponding to the eigenvalue 0, −1, and − −1 respectively. Similarly, Remove
the subscript, and then h denotes the complexification of h0 . Choose a positive system Φ+
for (g, h) and its simple system ∆. Denote by Φ+
k the set of the positive roots of (k, h), and
+
+
+
+
write Φp := Φ \ Φk . Set ∆k := Φk ∩ ∆ and ∆p := ∆ \ ∆k . Obviously, ∆p contains exactly 1
element because Z(k) has complex dimension 1.
Suppose that V is a simple (g, K)-module, and then set V p+ = {v ∈ V | Y v = 0 for any Y ∈
+
p }. Since K normalizes p+ , V p+ is a K-submodule. Further, V p+ is either zero or an
irreducible finite-dimensional representation of K. A (g, K)-module V is called a highest
weight module if V p+ 6= {0}.

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HAIAN HE

Α2

Α6

Α5

Α4

Α3

Α1

Figure 1. Dynkin diagram of E6 .
Recall the definition of unitary highest weight representation. If π is a unitary representation
of G on a Hilbert space H, and HK is the underlying (g, K)-module, then π is called a unitary
p
highest weight representation if HK+ 6= {0}; namely, HK is a highest weight (g, K)-module.
Proposition 6. Let G be a real simple Lie group of Hermitian type, and G′ a reductive
subgroup of G. Let K be a maximal compact subgroup of G such that K ′ := G′ ∩ K is a
maximal compact subgroup of G′ . Suppose that the Lie algebra g′ of G′ contains the center
Z(k0 ) of the Lie algebra k0 of K. If π is an irreducible unitary highest weight representation of
G, the restriction π |G′ is G′ -admissible.
Proof. Let λ be the highest weight of the underlying (g, K)-module VK of π, and then each
weight in VK is of the form λ − β − nα for some non-negative integer n, where β is an integer
linear combination of elements in Φ+
k and α is the unique simple root in ∆p . It is obvious that
the weight vectors with any fixed weight of VK form a finite-dimensional space. Take an element
Z ∈ Z(k) such that α(Z) = 1, and the weight vectors with the weight λ − β − nα lie in the
eigenspace of Z with the eigenvalue λ(Z)−n; namely, the eigenvalue of Z on each weight vector
only depends on n. Because Φ+
k is a finite set, each eigenspace is of finite-dimensional. Notice
that K ′ is compact, VK is decomposed as a direct sum of K ′ -representations. By assumption,
Z is in the complexified Lie algebra of K ′ , and hence the vectors in the same K ′ -type have
the same eigenvalue of Z; namely, each K ′ -type lies in some eigenspace of Z. It follows that
VK is K ′ -admissible, and thus the restriction π|K ′ is K ′ -admissible. By [10, Theorem 1.2], the

restriction π|G′ is G′ -admissible.
Remark 7. Proposition 6 is a generalization of the result [11, Theorem 7.4(4)] which deals
with the case when (G, G′ ) is a symmetric pair, but the proof for [11, Theorem 7.4(4)] does
not involve any property of symmetric pairs, so the method of the proof can also be applied
to the generalized case when (G, G′ ) is a pair of reductive Lie groups. However, in the proof
of Proposition 6, the author avoids constructing holomorphic sections of holomorphic vector
bundles as in the proof for [11, Theorem 7.4(4)].
Corollary 8. If (G, G{τ,σ} ) is a Klein four symmetric pair of holomorphic type, then any
unitary highest weight representation π of G is G′ -admissible.
Proof. Since τ σ = στ , by Proposition 1, there exists a maximal compact subgroup K of G,
which is fixed by both τ and σ. Moreover, both gτ0 and gσ0 contains the center Z(k0 ) of k0
{τ,σ}
. The conclusion follows
because both τ and σ are of holomorphic type. Thus, Z(k0 ) ⊆ g0
from Proposition 6.

3. Klein Four Symmetric pairs of holomorphic type for E6(−14)
3.1. Elementary abelian 2-subgroups in Aute6(−78) . Let g = e6 , the complex simple Lie
algebra of type E6 . Fix a Cartan subalgebra of g and a simple root system {αi | 1 ≤ i ≤ 6},
the Dynkin diagram of which is given in Figure 1. For each root α, denote by Hα its coroot,
and denote by Xα the normalized root vector so that [Xα , X−α ] = Hα . Moreover, one can
normalize Xα appropriately such that
√
√
SpanR {Xα − X−α , −1(Xα + X−α ), −1Hα | α : positive root} ∼
= u0 := e6(−78)

CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF HOLOMORPHIC TYPE FOR E6(−14)

5

is a compact real form of g by [8]. It is well known that
Autu0 /Intu0 ∼
= Autg/Intg
which is just the automorphism group of the Dynkin diagram.
The author follows the constructions of involutive automorphisms of u0 in [7]. Let τ be the
specific involutive automorphism of the Dynkin diagram defined by
τ (X±α1 ) = X±α6 ,
τ (Hα1 ) = Hα6 ,
τ (X±α2 ) = X±α2 ,
τ (Hα2 ) = Hα2 ,
τ (X±α3 ) = X±α5 ,
τ (Hα3 ) = Hα5 ,
τ (X±α4 ) = X±α4 .
τ (Hα4 ) = Hα4 ,
√
√
√
Let σ1 = exp( −1πHα2 ), σ2 = exp( −1π(Hα1 +Hα6 )), σ3 = τ , σ4 = τ exp( −1πHα2 ), where
exp represents the exponential map from u0 to Autu0 . Then σ1 , σ2 , σ3 , and σ4 represent all
conjugacy classes of involutions in Autu0 , which correspond to real forms e6(2) , e6(−14) , e6(−26) ,
and e6(6) .
Let G = Autu0 , and G0 = Intu0 be the identity component of G. From [7], it is known that
(G0 )σ3 ∼
= F4(−52) , the compact Lie group of type F4 , and there exist involutive automorphisms
2
1
∼
∼
τ1 and τ2 of F4(−52) such that fτ4(−52)
= so(9), where f4(−52) denotes
= sp(3) ⊕ sp(1) and fτ4(−52)
the compact Lie algebra of type F4 .
By [7], τ1 , τ2 , σ3 τ1 , and σ3 τ2 represent all conjugacy classes of involutive automorphisms in
Gσ3 except for σ3 and there are conjugacy relations in G: τ1 ∼G σ1 , τ1 ∼G σ2 , σ3 τ1 ∼G σ4 ,
and σ3 τ2 ∼G σ3 . Moreover, ((G0 )σ3 )τ1 ∼
= Sp(3) × Sp(1)/h(−I3 , −1)i. Let i, j, and k denote
the fundamental quaternion
units,
and
= σ3 , x1 = τ1 =

 then set x0 
 (I3 , −1), x2 = (iI3 , i),
−1 0 0
−1 0 0
x3 = (jI3 , j), x4 = ( 0 −1 0  , 1), and x5 = ( 0 1 0  , 1).
0
0 1
0 0 −1
For a integer pair (r, s) with r ≤ 2 and s ≤ 3, define
and

Fr,s := hx0 , x1 , · · · , xs , x4 , x5 , · · · , xr+3 i

′
:= hx1 , x2 , · · · , xs , x4 , x5 , · · · , xr+3 i.
Fr,s



−I2 0
, and η3 =
Sp(4)/h−I
i.
Let
η
=
iI,
η
=
On the other hand, (G0 )σ4 ∼
=
4
1
2
0
I2


−1 0
. By [7], η1 , η2 , η3 , σ4 η1 , σ4 η2 , and σ4 η3 represent all conjugacy classes of involu0 I3
tive automorphisms in Gσ4 except for σ3 and there are conjugacy relations in G: η1 ∼G η3 ∼G
σ1 , η2 ∼G σ2 , σ4 η1 ∼G σ4 η2 ∼G σ4 , and
, y3 =
 σ4 η3 ∼G σ3 . Sety0 = σ4 , y1= iI4 , y2 = jI4
0
1
0
0
1
0
0
0






 0 −1 0 0 
0 I2
−I2 0
, and y6 =  1 0 0 0 .
, y5 = 
, y4 =



0 0 0 1 
0 0 1 0
I2 0
0
I2
0 0 1 0
0 0 0 −1
For a integer pair (u, v, r, s) with u + v ≤ 1 and r + s ≤ 2, define
and

Fu,v,r,s := hy0 , y1 , · · · , yu+2v , y3 , y4 , · · · , y2+2s , y3+2s , y5+2s , · · · , y1+2r+2s i

′
Fu,v,r,s
:= hy1 , y2 , · · · , yu+2v , y3 , y4 , · · · , y2+2s , y3+2s , y5+2s , · · · , y1+2r+2s i.
According to [15, Proposition 6.3 & Proposition 6.5], each elementary abelian 2-subgroup in
′
′
Autu0 is conjugate to one of the groups in the one family of Fr,s , Fr,s
, Fu,v,r,s , and Fu,v,r,s
,
′
′
and all groups in the families of Fr,s , Fr,s , Fu,v,r,s , and Fu,v,r,s are pairwisely non-conjugate.

3.2. Elements conjugate to σ2 . In this part, the author needs to determine which involutive
automorphisms are conjugate to σ2 in G because the noncompact dual g0 = e6(−14) of u0
corresponding to σ2 is of Hermitian type.
′
Firstly, consider the elements in Fr,s and Fr,s
. In fact, one only needs to check the elements
in the largest group F2,3 = hx0 , x1 , x2 , x3 , x4 , x5 i.

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HAIAN HE

Lemma 9. The elements in F2,3 conjugate to σ2 in Autu0 are exactly x4 , x5 , and x4 x5 .
Proof. Let H = Sp(3) × Sp(1)/h(−I3 , −1)i for convenience. Consider the following four points.
• Obviously, x0 ∼G σ3 ≁G σ2 . Moreover, x0 is an outer automorphism while xi for 1 ≤ i ≤ 5
′
are all inner automorphisms. Thus if ω ∈ F2,3
= hx1 , x2 , x3 , x4 , x5 i, then x3 ω is an outer
automorphism which cannot be conjugate to the inner automorphism σ2 . Hence the author
′
only needs to check the elements in the biggest group F2,3
= hx1 , x2 , x3 , x4 , x5 i.
• Let S1 = {x1 , x2 , x3 , x1 x2 , x1 x3 , x2 x3 , x1 x2 x3 }. Firstly, x1 ∼G σ1 by [15]. Secondly, x2 is
conjugate to τ1 in F4(−52) by [7], and it follows that x2 ∼G σ1 . Thirdly, since it is well known
that ±i, ±j, and ±k are pairwisely conjugate in the group ring of the quoternion group, it
is immediate that x2 ∼H x3 ∼H x2 x3 ∼H x1 x2 ∼H x1 x3 ∼H x1 x2 x3 . In particular, it is
conclude that all elements in S1 are conjugate to σ1 in Autu0 .
• Let S2 = {x4 , x5 , x4 x5 }. Firstly, x4 is conjugate to τ2 in F4(−52) by [7], and it follows that
x4 ∼G σ2 . Secondly, it is obvious that x4 ∼H x5 ∼H x4 x5 . In particular, it is conclude that
all elements in S2 are conjugate to σ2 in Autu0 .


1 0 0
• Consider ω1 ω2 where ω1 ∈ S1 and ω2 ∈ S2 . Firstly, x1 x4 = ( 0 1 0  , 1) in Sp(3) ×
0 0 −1
Sp(1)/h(−I3 , −1)i, which is conjugate to τ1 in F4(−52) by [7], and it follows that x1 x4 ∼G σ1 .
Similary, one has x1 x5 ∼G x1 x4 x5 ∼G σ1 . Secondly, because i is conjugate to −i in the
group ring of the quoternion group, it is easy to see that x2 x4 ∼H x1 x2 x4 ∼H x2 , and hence
x2 x4 ∼G x1 x2 x4 ∼G σ1 . In similar way, one can show that ω1 ω2 ∼G σ1 for all ω1 ∈ S1 and
ω 2 ∈ S2 .
Based on the four points, the conclusion holds.

′
. In fact, one only needs to check the
Secondly, consider the elements in Fu,v,r,s and Fu,v,r,s
elements in the largest group F0,1,0,2 = hy0 , y1 , y2 , y3 , y4 , y5 , y6 i. Similarly, exclude the outer
′
automorphisms, and one only needs to check the elements in F0,1,0,2
= hy1 , y2 , y3 , y4 , y5 , y6 i.
′
In order to find the elements in F0,1,0,2 which are conjugate to σ2 , the author uses a function
constructed in [15, Definition 6.1]. Define
′
µ : F0,1,0,2
−→ {±1}

given by
µ(y) =

(

−1, if y ∼G σ1 ,
1, if y ∼G σ2 or y = 1.

By [15, Lemma 2.13 & Lemma 6.6], there is another equivalent definition for the function
′
′
µ on F0,1,0,2
. For y ∈ F0,1,0,2
⊆ (G0 )σ4 ∼
= Sp(4)/h−I4 i, choose A ∈ Sp(4) representing y, and
then A2 = ǫA I4 for some ǫA = ±1. Obviously, the value ǫA does not depend on the choice of
A. Then define µ(x) := ǫA .
′
Lemma 10. The values of the function µ on F0,1,0,2
are given as follows:

(i) µ(y1 ) = µ(y2 ) = µ(y1 y2 ) = µ(y3 y4 ) = µ(y5 y6 ) = −1 and µ(y3 ) = µ(y4 ) = µ(y5 ) =
µ(y6 ) = 1.
′
(ii) For x, y ∈ F0,1,0,2
, if xy = yx, then µ(xy) = µ(x)µ(y). In particular, if x ∈ hy1 , y2 i,
y ∈ hy3 , y4 i, and z ∈ hy5 , y6 i, then µ(xyz) = µ(x)µ(y)µ(z).
Proof. The conclusion follows from the equivalent definition of µ immediately.



′
Lemma 11. There are totally 27 elements in F0,1,0,2
, which are conjugate to σ2 in G.
′
Proof. According to the definition of µ, elements conjugate to σ2 in F0,1,0,2
are exactly µ−1 (1)\
{1}. Thus, by Lemma 10, there are 27 such elements: y3 , y4 , y5 , y6 , y3 y5 , y3 y6 , y4 y5 , y4 y6 ,
y1 y3 y4 , y1 y5 y6 , y2 y3 y4 , y2 y5 y6 , y1 y2 y3 y4 , y1 y2 y5 y6 , y1 y3 y4 y5 , y1 y3 y4 y6 , y1 y3 y5 y6 , y1 y4 y5 y6 ,
y2 y3 y4 y5 , y2 y3 y4 y6 , y2 y3 y5 y6 , y2 y4 y5 y6 , y3 y4 y5 y6 , y1 y2 y3 y4 y5 , y1 y2 y3 y4 y6 , y1 y2 y3 y5 y6 , and
y1 y2 y4 y5 y6 .


CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF HOLOMORPHIC TYPE FOR E6(−14)

7

Proposition 12. There are 7 conjugacy classes of the elementary abelian 2-subgroups of
rank 3 in G, which contain elements conjugate to σ2 , represented by hx0 , x1 , x4 i, hx0 , x4 , x5 i,
hx1 , x2 , x4 i, hx1 , x4 , x5 i, hy0 , y3 , y4 i, hy1 , y3 , y4 i, and hy3 , y4 , y5 i respectively.
Proof. This follows from Lemma 9, Lemma 11, and [15, Proposition 6.3 & Proposition 6.5]
immediately.

3.3. Klein four symmetric subalgebras of e6(−14) . Let g0 be a non-compact simple Lie
algebra. For each involutive automorphism σ of√g0 , there exists a Cartan involution θ of g0 ,
which commutes
with σ. Denote by g := g0 + −1g0 the complexification of g0 , and write
√
u0 := gθ0 + −1g−θ
0 for the compact dual of g0 . Extend σ to the unique holomorphic involutive
automorphism of g, restrict it to u0 , and σ becomes an involutive automorphism of u0 . Thus,
any involutive automorphism σ of g0 gives a pair (u0 , σ), where u0 is a compact real form of
g and σ is regarded as an involutive automorphism of u0 . If σ ′ is conjugate to σ in Autg0 ,
which gives another pair (u′0 , σ ′ ), then there exists an element g ∈ Autg such that g · u0 = u′0
and g −1 σg = σ ′ ; namely, the two pairs (u0 , σ) and (u′0 , σ ′ ) are conjugate by elements in Autg.
Thus, up to conjugations of compact real forms of g, there exists a map
π : {Conjugacy classes of Autg0 } −→ {Conjugacy classes of Autu0 }

which is surjective but not necessarily injective.
If Γ is a finite abelian subgroup of Autg0 , there exists a Cartan involution θ ∈ Autg0
centralizing Γ by Proposition 1. In the same way, Γ becomes a finite abelian subgroup of
Autu0 . Thus, any finite abelian subgroup of Autg0 is obtained from a finite abelian subgroup
of u0 .
On the other hand, fix an involutive automorphism θ of u0 , and by holomorphic extension
and restriction, θ is a Cartan involution of a noncompact dual g0 of u0 . Let Θ : Autu0 → Autu0
be given by f 7→ θ−1 f θ, whose differential is θ on u0 . Suppose that Γ1 and Γ2 are two finite
abelian subgroups of Autu0 , which are centralized by θ. If Γ2 = g −1 Γ1 g for some g ∈ (Autu0 )Θ ,
then Γ1 and Γ2 is conjugate in Autg0 because (Autu0 )Θ is contained in Autg0 .
Thus, for a compact Lie algebra u0 , a pair (θ, Γ) with θ an involutive automorphism of u0
and Γ a Klein four subgroup of Autu0 such that θ ∈
/ Γ and θ centralizes
Γ, gives a Klein four
√
.
Moreover,
if two such
symmetric subalgebra of the noncompact Lie algebra g0 = uθ0 + −1u−θ
0
pairs are conjugate Autu0 , then they give a same Klein four symmetric pair up to isomorphism.
Here, the pairs (θ1 , Γ1 ) and (θ2 , Γ2 ) are said to be conjugate in Autu0 if there exists an element
g ∈ Autu0 such that θ2 = g −1 θ1 g and Γ2 = g −1 Γ1 g.
In order to find all the Klein four symmetric pair of holomorphic type for g0 = e6(−14) ,
according to the argument above, the author needs to find all the conjugacy classes of the
pairs (θ, Γ) with θ ∈ G involutive automorphisms and Γ ⊆ G Klein four subgroups such that
(i) θ ∈
/ Γ;
(ii) θ ∼G σ2 ;
(iii) θ centralizes Γ;
(iv) every element σ ∈ Γ is identity on the center
of uθ0 ; namely, each σ ∈ Γ gives a symmetric
√
θ
pair of holomorphic type for g0 = u0 + −1u−θ
0 .
Because of the items (i), (ii), and (iii), one knows immediately that each subgroup generated
by such θ and Γ must be contained in the some elementary abelian 2-subgroup of rank 3 in G,
i.e., one of the 7 subgroups listed in Proposition 12 up to conjugation.
On the other hand, notice that π is a surjection from the set of conjugacy classes of g0
to the set of conjugacy classes of u0 . According to the classification of symmetric pairs, it is
known that the cardinality of pre-image |π −1 ([σ1 ])| of [σ1 ] is 2, while that of [σ2 ] is 3. Write
σ1,1 and σ1,2 for the two representatives of the two conjugacy classes in Autg0 corresponding
to [σ1 ], and write σ2,1 , σ2,2 , and σ2,3 for the two representatives of the two conjugacy classes
σ
σ
in Autg0 corresponding to [σ2 ]. Here, g0 1,1 = su(4, 2) ⊕ su(2), g0 1,2 = su(5, 1) ⊕ sl(2, R),
σ2,1
σ2,2
σ
2,3
g0 = s0(8, 2) ⊕ so(2), g0 = so∗ (10) ⊕ so(2), and g0 = so(10) ⊕ so(2). According to the
classification of symmetric pairs of holomorphic type [14, Table C.2], σ1,1 , σ1,2 , σ2,1 , σ2,2 , and
σ2,3 give all the symmetric pairs of holomorphic type for g0 . Hence, taking the item (iii) into

8

HAIAN HE

consideration, there is no need to consider the elementary abelian 2-subgroups of rank 3 in G,
which contain outer automorphisms. Thus, a pair (θ, Γ) satisfies the above four requirements
if and only if θ ∼G σ2 and the group hθ, Γi generated by θ and the elements in Γ is conjugate
to one of following abelian elementary 2-subgroups of rank 3:
(i) hx1 , x2 , x4 i,
(ii) hx1 , x4 , x5 i,
(iii) hy1 , y3 , y4 i,
(iv) hy3 , y4 , y5 i.
In order to compute the explicit Klein four symmetric pairs of holomorphic type for g0 , for each
elementary abelian 2-subgroup of rank 3 listed above, the author needs to take three steps.
Step 1: Determine the conjugacy classes (θ, Γ) in Autu0 ;
Step 2: Compute uΓ0 which is the compact dual of gΓ0 ;
hθ,Γi
Step 3: Compute u0
which is a maximal compact subalgebra of g0 .
After the three steps, the Klein four symmetric subalgebras gΓ0 are totally determined.
Lemma 13. Every (θ, Γ) in Autu0 , such that θ ∼G σ2 and the group hθ, Γi generated by θ and
Γ is conjugate to one of the above 4 elementary abelian 2-subgroups of rank 3, is conjugate to
one of the following pairs.
(i) hx1 , x2 , x4 i:
• θ = x4 , Γ = hx1 , x2 i.
(ii) hx1 , x4 , x5 i:
• θ = x5 , Γ = hx1 , x4 i.
(iii) hy1 , y3 , y4 i:
• θ = y3 , Γ = hy1 , y4 i;
• θ = y3 , Γ = hy1 y3 , y4 i;
• θ = y3 , Γ = hy1 y3 , y3 y4 i;
(iv) hy3 , y4 , y5 i:
• θ = y5 , Γ = hy3 , y4 i;
• θ = y3 , Γ = hy4 , y5 i;
• θ = y3 , Γ = hy3 y4 , y5 i;
• θ = y3 , Γ = hy4 , y3 y5 i.
Proof. If the group hθ, Γi is conjugate to hx1 , x2 , x4 i, then the pair (θ, Γ) is conjugate to one
of (x4 , hx1 , x2 i), (x4 , hx1 x4 , x2 i), (x4 , hx1 , x2 x4 i), and (x4 , hx1 x4 , x2 x4 i). The author needs to
show that they are all conjugate. In fact, there is an automorphism f of the group hx1 , x2 , x4 i,
given by f (x1 ) = x1 x4 , f (x2 ) = x2 , and f (x4 ) = x4 . By Lemma 9, it is immediate that
f (x) ∼G x for all x ∈ hx1 , x2 , x4 i. Then by [15, Proposition 6.9], there exists an element
g ∈ G such that f (x) = g −1 xg for all x ∈ hx1 , x2 , x4 i, and this shows that (x4 , hx1 , x2 i) and
(x4 , hx1 x4 , x2 i) are conjugate. Similarly, both (x4 , hx1 x4 , x2 i) and (x4 , hx1 , x2 x4 i) are conjugate
to (x4 , hx1 , x2 i).
If the group hθ, Γi is conjugate to hx1 , x4 , x5 i, apply Lemma 9 and [15, Proposition 6.9] in
the same way, one shows that (θ, Γ) is conjugate to (x5 , hx1 , x4 i).
If hθ, Γi is conjugate to hy1 , y3 , y4 i, then the pair (θ, Γ) is conjugate to one of (y3 , hy1 , y4 i),
(y3 , hy1 y3 , y4 i), (y3 , hy1 , y3 y4 i), (y3 , hy1 y3 , y3 y4 i), (y4 , hy1 , y3 i), (y4 , hy1 y4 , y3 i), (y4 , hy1 , y3 y4 i),
(y4 , hy1 y4 , y3 y4 i), (y1 y3 y4 , hy1 , y3 i), (y1 y3 y4 , hy1 , y4 i), (y1 y3 y4 , hy3 , y4 i), and (y1 y3 y4 , hy1 y4 , y3 y4 i).
One may use Lemma 11 and [15, Proposition 6.9] to show that the pairs in each of the following
items are conjugate:
• (y3 , hy1 , y4 i), (y3 , hy1 , y3 y4 i), (y4 , hy1 , y3 i), (y4 , hy1 , y3 y4 i), (y1 y3 y4 , hy1 , y3 i), and (y1 y3 y4 , hy1 , y4 i);
• (y3 , hy1 y3 , y4 i), (y4 , hy1 y4 , y3 i), and (y1 y3 y4 , hy3 , y4 i);
• (y3 , hy1 y3 , y3 y4 i), (y4 , hy1 y4 , y3 y4 i), and (y1 y3 y4 , hy1 y4 , y3 y4 i).
Moreover, since the groups hy1 , y4 i, hy1 y3 , y4 i, and hy1 y3 , y3 y4 i are pairwisely non-conjugate
by [7, Table 4], the three items above correspond to three conjugacy classes exactly.
If If hθ, Γi is conjugate to hy3 , y4 , y5 i, in the same way, one shows that (θ, Γ) is conjugate
to at least one of (y5 , hy3 , y4 i), (y3 , hy4 , y5 i), (y3 , hy3 y4 , y5 i), and (y3 , hy4 , y3 y5 i). Moreover,

CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF HOLOMORPHIC TYPE FOR E6(−14)

9

(y5 , hy3 , y4 i), (y3 , hy4 , y5 i), (y3 , hy3 y4 , y5 i), and (y3 , hy4 , y3 y5 i) are pairwisely non-conjugate by
[7, Table 4] except that (y5 , hy3 , y4 i) and (y3 , hy4 , y3 y5 i) may be conjugate.

hx ,x i

By Lemma 13, each Klein four symmetric subalgebra of g0 is isomorphic to one of g0 1 2 ,
hx ,x i
hy ,y i
hy y ,y i
hy y ,y y i
hy ,y i
hy ,y i
hy y ,y i
hy ,y y i
g0 1 4 , g0 1 4 , g0 1 3 4 , g0 1 3 3 4 , g0 3 4 , g0 4 5 , g0 3 4 5 , and g0 4 3 5 , whose compact
duals are give according to [7, Table 4] as follows.
√
hy y ,y y i
hx ,x i
• u0 1 2 ∼
= 2su(3) ⊕ 2( −1R)
= u0 1 3 3 4 ∼
√
hy y ,y i
hy ,y i
hx ,x i
• u0 1 4 ∼
= su(4) ⊕ 2su(2) ⊕ −1R
= u0 3 4 5 ∼
= u0 1 4 ∼
√
hy ,y y i
hy ,y i
hy y ,y i
• u0 1 3 4 ∼
= su(5) ⊕ 2( −1R)
= u0 4 3 5 ∼
= u0 3 4 ∼
√
hy ,y i
• u 4 5 ∼
= so(8) ⊕ 2( −1R)
0

hθ,Γi

Finally, the author needs to determine all u0
listed in Lemma 13. It is very hard to compute
them directly because all xi and yj are inner automorphisms taken from Gσ3 and Gσ4 , but
both σ3 and σ4 are outer automorphisms. However, by [15, Proposition 6.9] again, one may
find the groups isomorphic to those hθ, Γi in Gσ1 or Gσ2 .
√
∼ (SU(6) × Sp(1)/h(e 2π 3 −1 I6 , 1), (−I6 , −1)i) ⋊
According to [7], it is known that Gσ1 
=
 

0 −I3
0
I3
hτ i, where σ1 = (I6 , −1) and τ (X, Y ) = (
X
, Y ) for (X, Y ) ∈
I3
0
−I3 0
complex conjugation
of X in su(6). Then
su(6) ⊕ sp(1),
with X
 in G,

 being the
 natural

√
−I
0
−I2 0
−1I3
0
4
√
, 1) ∼G
, i) ∼G σ1 and (
, 1) ∼G (
one has (
0
I2
0
I4
0
− −1I3
 √

−1I5
√0
(
, i) ∼G σ2 . By tedious computations, one has group homomorphisms
0
− −1I1
as follows.


−I2 0
0
0

 √
 0
I1
0
0 
−1I3
∼ h(I6 , −1), (
 , 1)i given
√0
, i), (
(i) hx1 , x2 , x4 i =

0
0 −I2 0 
0
− −1I3
0
0
0
I1


−I
0
0
0
2
 √

 0
−1I3
I1
0
0 
 , 1);
√0
by x1 7→ (I6 , −1), x2 7→ (
, i), and x4 7→ (
 0
0 −I2 0 
0
− −1I3
0
0
0
I1




−I
0
−I
0
4
2
, 1)i given by x1 7→ (I6 , −1),
, 1), (
(ii) hx1 , x4 , x5 i ∼
= h(I6 , −1), (
0
I
0
I4
 2



−I4 0
−I2 0
, 1);
, 1), and x5 7→ (
x1 x4 7→ (
0
I
0
I4
 √
 2  √

−1I3
−1I5
∼ h(I6 , −1), (
√0
√0
(iii) hy1 , y3 , y4 i =
, i), (
, i)i given by
0
− −1I3
0
− −1I
 √

 √ 1

−1I3
0
−1I5
0
√
√
y1 y3 7→ (I6 , −1), y1 y4 7→ (
, i), and y4 7→ (
, i);
0
− −1I3
0
− −1I1

 √


−I4 0
−1I5
√0
, 1), (
(iv) hy3 , y4 , y5 i ∼
, i)i given by y3 y4 7→
= h(I6 , −1), (
0
I2
0
− −1I1


 √

−I4 0
−1I5
√0
, i).
, 1), and y3 7→ (
(I6 , −1), y5 7→ (
0
I2
0
− −1I1
Lemma 14. The following isomorphisms of subalgebras in u0 hold.
√
hx ,x ,x i ∼
2su(2) ⊕ 4( −1R);
(i) u0 1 2 4 =
√
hx ,x ,x i
(ii) u0 1 4 5 ∼
= 4su(2) ⊕ 2( −1R);
√
hy ,y ,y i ∼
su(3) ⊕ su(2) ⊕ 3( −1R);
(iii) u0 1 3 4 =
√
hy ,y ,y i
(iv) u 3 4 5 ∼
= su(4) ⊕ 3( −1R).
0

Proof. Based on the group isomorphisms of elementary abelian 2-subgroups of rank 3 in Autu0 ,
make use of [15, Proposition 6.9], and the conclusions hold immediately by computations. 

10

HAIAN HE

Theorem 15. There are totally 8 Klein for symmetric pairs of holomorphic type for e6(−14)
up to conjugation:
√
(i) (e6(−14) , 2su(2, 1) ⊕ 2( −1R));
√
(ii) (e6(−14) , su(2, 2) ⊕ 2su(2) ⊕ −1R);
√
(iii) (e6(−14) , su(3, 1) ⊕ su(1, 1) ⊕ su(2) ⊕ −1R);
√
(iv) (e6(−14) , su(3, 2) ⊕ 2( −1R));
√
(v) (e6(−14) , su(2, 1) ⊕ su(3) ⊕ 2( −1R));
√
(vi) (e6(−14) , su(4, 1) ⊕ 2( −1R));
√
(vii) (e6(−14) , so(6, 2) ⊕ 2( −1R));
√
(viii) (e6(−14) , 2su(1, 1) ⊕ su(4) ⊕ −1R).
Proof. The conclusion follows from Lemma 13, Lemma 14, and the corresponding Klein four
symmetric subalgebras of e6(−78) given in [7, Table 4], which are also listed above.

3.4. Application to Representation Theory. As an application of Theorem 15, recall
Corollary 8 that Klein four symmetric pairs of holomorphic type are related to the pairs
(G, G′ ) of real reductive groups such that there exists a unitary highest weight representation
of G, which is G′ -admissible.
Let {ei }ni=1 be a orthonormal basis for Rn with respect to the standard inner product,
such that Pin(n) is defined as a subgroup contained in the Clifford algebra Cl(Rn ). Write
c = e1 e2 · · · en ∈ Pin(n), and it is known that c ∈ Spin(n) if and only if n is even; in this case,
c is in the center of Spin(n) and any noncompact dual Spin(m, n − m) for 1 ≤ m ≤ n − 1.
Theorem 16. There are totally 8 Klein for symmetric pairs of holomorphic type for E6(−14)
up to conjugation:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)

2π

√

2π

−1

√

2π

−1

√
−1

2π

√

−1

(E6(−14) , SU(2, 1)2 × U(1)2 /h(e 3 I3 , I3 , e 3 , 1), (I3 , e 3 I3 , e− 3 , 1)i);
√
√
(E6(−14) , SU(2, 2) × SU(2)2 × U(1)/h( −1I4 , −I2 , I2 , −1), (I4 , −I2 , −I2 , −1)i);
√
√
(E6(−14) , SU(3, 1) × SU(1, 1) × SU(2) × U(1)/h( −1I4 , −I2 , I2 , −1), (I4 , −I2 , −I2 , −1)i);
(E6(−14) , SU(3, 2) × U(1)2 );
2π

√

−1

2π

√

−1

2π

√
−1

2π

√

(E6(−14) , SU(2, 1) × SU(3) × U(1)2 /h(e 3 I3 , I3 , e 3 , 1), (I3 , e 3 I3 , e− 3
(E6(−14) , SU(4, 1) × U(1)2 );
(E6(−14) , Spin(6, 2) × U(1)2 /h(−1, −1, 1), (c, 1, −1)i);
√
√
(E6(−14) , SU(1, 1)2 × SU(4) × U(1)/h(−I2 , I2 , −1I4 , −1), (−I2 , −I2 , I4 , −1)i).

−1

, 1)i);

Proof. Since E6(−14) = Inte6(−14) is connected, the conclusion follows from Theorem 15 and
[7, Table 6].

Corollary 17. If (G, G′ ) is one of the pairs listed in Theorem 16, then any irreducible unitary
highest weight representation of G is G′ -admissible. In particular, there exist an infinite dimensional irreducible unitary representation π of G and an irreducible unitary representation
τ of G′ such that
0 < dimC Homg′ ,K ′ (τK ′ , πK |g′ ) < +∞

where K is a maximal compact subgroup of G, g is the complexified Lie algebra of G, πK |g′
is the restriction of underlying (g, K)-module of π to the complexified Lie algebra g′ of G′ ,
K ′ is a maximal compact subgroup of G′ such that K ′ ⊆ G′ ∩ K, and τK ′ is the underlying
(g′ , K ′ )-module of τ .
Proof. The first conclusion follows from Corollary 8 and Theorem 16 immediately, and the second statement follows from the correspondence between unitary representations of Lie groups
and underlying (g, K)-modules.

Acknowledgement
The author would like to express the thankfulness to professor Jun YU who gave some helpful
suggestions and supports to this article.

CLASSIFICATION OF KLEIN FOUR SYMMETRIC PAIRS OF HOLOMORPHIC TYPE FOR E6(−14)

11

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[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]

[12]

[13]
[14]

[15]

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Department of Mathematics, College of Sciences, Shanghai University, No. 99 Shangda Road,
Baoshan District, Shanghai, China P. R. 200444
E-mail address: hebe.hsinchu@yahoo.com.tw

