The isomorphism type of the centralizer of an element in a Lie group

Journal of Algebra 376 (2013) 25–45

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Journal of Algebra www.elsevier.com/locate/jalgebra

The isomorphism type of the centralizer of an element in a Lie group Haibao Duan ∗,1 , Shali Liu Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Chinese Academy of Sciences, China

a r t i c l e

i n f o

Article history: Received 18 January 2012 Available online 30 November 2012 Communicated by Gus I. Lehrer MSC: 22E15 53C35

a b s t r a c t Let G be a compact 1-connected simple Lie group, and let x ∈ G be a group element. We determine the isomorphism type of the centralizer C x in term of a minimal geodesic joining the group unit e ∈ G to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G (Borel and De Siebenthal, 1949 [4]), and the isomorphism types of parabolic subgroups of G. © 2012 Elsevier Inc. All rights reserved.

Keywords: Lie groups Centralizer Homogeneous spaces

1. Introduction Let G be a compact connected semisimple Lie group with a given element x ∈ G. The centralizer C x of x and the adjoint orbit M x through x are the subspaces of G

C x = { g ∈ G | gx = xg },







M x = gxg −1 ∈ G  g ∈ G ,

respectively. The map G → G by g → gxg −1 is constant along the left cosets of C x in G, and induces a diffeomorphism from the homogeneous space G /C x onto the orbit space M x ∼ =

f x : G /C x → M x

* 1

by [ g ] → gxg −1 .

Corresponding author. E-mail addresses: [email protected] (H. Duan), [email protected] (S.L. Liu). The author’s research is supported by 973 Program 2011CB302400 and NSFC 11131008.

0021-8693/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2012.10.028

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Table 1 The types and centers of 1-connected simple Lie groups. G

SU (n)

Sp(n)

Spin(2n + 1)

Spin(2n)

G2

F4

E6

E7

E8

ΦG Z (G )

A n−1 Zn

Bn Z2

Cn Z2

Dn Z 4 , n = 2k + 1 Z 2 ⊕ Z 2 , n = 2k

G2 {e }

F4 {e }

E6 Z3

E7 Z2

{e }

E8

The manifolds G /C x arising in this fashion have offered many important subjects in geometry, as shown in the next two examples. Example 1.1. Let Z (G ) be the center of G and let x ∈ G be an element with xr ∈ Z (G ) for some power r  2. The homogeneous space G /C x possesses a canonical periodic r automorphism





σx : G /C x → G /C x by σx [ g ] = xgx−1 . For this reason the pair (G /C x , σx ) can be called an r-symmetric space of G. In the special case of r = 2, they are the global Riemannian symmetric spaces of G in the sense of E. Cartan [12]. Example 1.2. If the minimal geodesic joining the group unit e ∈ G to x is unique, the centralizer C x is a parabolic subgroup of G. The corresponding homogeneous space G /C x is called a flag manifold of G, which is the focus of the classical Schubert calculus [5,10,11,14,15]. To investigate the geometry and topology of the homogeneous space G /C x it is often necessary to determine explicitly the isomorphism type of the centralizer C x in term of x ∈ G. However, in the existing literatures one merely finds certain method to decide its local type in some special cases [12, 4,17], see Remark 2.10. The purpose of this paper is to give an explicit procedure for calculating the centralizer C x in term of a minimal geodesic joining the unit e to x, see Theorem 4.3 in Section 4.1. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G in Section 4.2, and of parabolic subgroups of G in Section 4.3. To be precise some notations are needed. For a compact connected Lie group K the identity component of the center Z ( K ) of K will be denoted by K Rad , and will be called the radical part of K (this is always a connected torus subgroup of K ). According to Cartan’s classification on compact Lie groups, up to isomorphism, the group K admits a canonical presentation of the form





K∼ = G 1 × · · · × G k × K Rad / H

(1.1)

in which (i) each G t is one of the 1-connected simple Lie groups, 1  t  k; (ii) the denominator H is a finite subgroup of Z (G 1 ) × · · · × Z (G k ) × K Rad . It is also known that all 1-connected simple Lie groups G, together with their centers, are classified by the types ΦG of their corresponding root systems tabulated in Table 1 [13, p. 57]. Definition 1.3. In the presentation (1.1) the group G 1 × · · · × G k is called the semisimple part of K , and is denoted by K s . The obvious quotient (i.e. covering) map π : K s × K Rad → K is called the local type of K . Corollary 1.4. Let K be a compact connected Lie group. The following statements are equivalent. (i) The group K is semisimple; (ii) K Rad = {e }; (iii) The local type π of K agrees with the universal cover of K .

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27

The paper is arranged as follows. Section 2 contains a brief introduction on the roots and weight systems of Lie groups, and obtains the local type π : C xs × C xRad → C x of a centralizer C x in term of x ∈ G in Theorem 2.8. In Section 3 we introduce for each compact connected Lie group K the socalled extended weight lattice Π K0 , together with two deficiency functions on it. They are applied in Theorem 3.7 to specify the isomorphism type of a subgroup K in a semisimple Lie group G. Summarizing results in Theorems 2.8 and 3.7, an explicit procedure for calculating the isomorphism type of a centralizer C x in a 1-connected Lie group G is given in Theorem 4.3 of Section 4.1. Finally, to demonstrate the use of Theorem 4.3 we determine in Sections 4.2 and 4.3 all the centralizers C exp(u ) in a 1-connected exceptional Lie group G with u a multiple of a fundamental dominant weight of G. 2. The local type of a centralizer In this paper K denotes a compact connected Lie group, and the notion G is reserved for the compact semisimple ones. Equip the Lie algebra L ( K ) of K with an inner product ( , ) so that the adjoint representation acts as isometries on L ( K ). Fixing a maximal torus T on K the Cartan subalgebra of K is the linear subspace L ( T ) of L ( K ). The dimension n = dim T is called the rank of the group K . 2.1. The root system of a compact connected Lie group The restriction of the exponential map exp : L ( K ) → K to the subspace L ( T ) defines a set S ( K ) = { L 1 , . . . , Lm } of m = 12 (dim K − n) hyperplanes in L ( T ), namely, the set of singular hyperplanes through the origin in L ( T ) [2, p. 168]. Let lk ⊂ L ( T ) be the normal line of the plane L k through the origin, 1  k  m. Then the map exp carries lk onto a circle subgroup on T . Let ±αk ∈ lk be the nonzero vectors with minimal length so that exp(±αk ) = e, 1  k  m. Definition 2.1. The subset Φ K = {±αk ∈ L ( T ) | 1  k  m} of L ( T ) is called the root system of K . Remark 2.2. We note that the root system Φ K by Definition 2.1 is dual to those that are commonly used in literatures, e.g. [1,13]. In particular, the symplectic group Sp(n) is of the type B n , while the spinor group Spin(2n + 1) is of the type C n . The planes in S ( K ) divide L ( T ) into finitely many convex regions, called the Weyl chambers of K . Fix a regular point x0 ∈ L ( T ), and let F (x0 ) be the closure of the Weyl chamber containing x0 . Assume that L (x0 ) = { L 1 , . . . , L h } is the subset of S ( K ) consisting of the walls of F (x0 ). Then

h  n, Let

where the equality holds if and only if K is semisimple.

(2.1)

αi ∈ Φ K be the root normal to the wall L i ∈ L (x0 ) and pointing toward x0 .

Definition 2.3. The subset S (x0 ) = {α1 , . . . , αh } of the root system Φ K is called the system of simple roots of K relative to x0 . The Cartan matrix of K (relative to x0 ) is the h × h matrix defined by

A = (b i j )h×h ,

b i j = 2(ai , α j )/(α j , α j ).

(2.2)

The lattice in L ( T ) spanned by all simple roots is called the root lattice, and is denoted by ΛrK . The

subset of ΛrK consisting of the sums of the simple roots put

α1 , . . . , αh is denoted by ΛrK,+ . We shall also

Φ K+ = ΛrK,+ ∩ Φ K , whose elements are called the positive roots of K .

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The set S (x0 ) of simple roots defines a partial order ≺ on L ( T ) by the following rule: for two vectors u , v ∈ L ( T ) we say v ≺ u if and only if the difference u − v is a sum of elements in S (x0 ) (i.e. r ,+ belongs to Λ K [13, p. 47]). If G is a simple Lie group, elements in ΦG have at most two lengths. Let β ∈ ΦG+ (resp. γ ∈ ΦG+ ) be the unique maximal short root (resp. unique maximal long root) relative to the partial order ≺ on ΦG+ . From [13, p. 66, Table 2] one gets Lemma 2.4. Let G be a simple Lie group. We have β = γ unless G = G 2 , F 4 , B n or C n . Moreover, if G = G 2 , F 4 , B n or C n we have β ≺ γ and the lengths of the three vectors γ , β, δ = γ − β are given in the table below.

G2 F4 Bn Cn

γ 2

β 2

δ 2

6 2 2 4

2 1 1 2

2 1 1 2

2.2. The weight system of a semisimple Lie group A nonzero vector

α ∈ ΛrK gives rise to a linear map

α ∗ : L ( T ) → R by α ∗ (x) = 2(x, α )/(α , α ). If

(2.3)

α ∈ Φ K is a root, the map α ∗ is called the inverse root of α [13, p. 67].

Definition 2.5. Assume that G is a semisimple Lie group. The weight lattice of G is the subset of L ( T )

   ΛG = x ∈ L ( T )  α ∗ (x) ∈ Z for all α ∈ ΦG , whose elements are called weights. Its subset

   ΩG = ωi ∈ L ( T )  α ∗j (ωi ) = δi , j , α j ∈ S (x0 )

(2.4)

is called the set of fundamental dominant weights of G relative to x0 , where δi , j is the Kronecker symbol. Lemma 2.6. Let G be a semisimple Lie group with Cartan matrix A, and let ΩG = {ω1 , . . . , ωn } be the set of fundamental dominant weights relative to the regular point x0 . Then (i) ΩG = {ω1 , . . . , ωn } is a basis for ΛG over Z; (ii) the fundamental dominant weights ω1 , . . . , ωn can be expressed in term of the simple roots {α1 , . . . , αn } as

⎛

ω1 ⎞ ⎜ ω2 ⎟

⎜ . ⎟= A ⎝ . ⎠ .

ωn

⎛

α1 ⎞ ⎜ α2 ⎟ −1

⎜ . ⎟; ⎝ . ⎠ .

(2.5)

αn

(iii) for each 1  i  n the half-line {t ωi ∈ L ( T ) | t ∈ R+ } is the edge of the Weyl chamber F (x0 ) opposite to the wall L i .

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29

Table 2 The set ΠG of minimal weights of a simple Lie group G.

ΦG

An

Bn

Cn

Dn

E6

E7

ΠG

{ωi }1i n

{ωn }

{ω1 }

{ω1 , ωn−1 , ωn }

{ω1 , ω6 }

{ω7 }

Proof. With the assumption that G is semisimple, we have that h = n by (2.1), and that the Weyl chamber F (x0 ) is a convex cone with vertex 0 ∈ L ( T ). Properties (i) and (ii) are known. By (2.4) each weight ωi ∈ ΩG is perpendicular to all the roots α j (i.e. ωi ∈ L j ) with j = i. This shows (iii). 2 For a simple Lie group G we shall adopt the convention that its fundamental dominant weights

ω1 , . . . , ωn are ordered by the order of their corresponding simple roots pictured as the vertices in the Dynkin diagram of G [13, p. 58]. Let ΠG ⊆ ΩG = {ω1 , . . . , ωn } be the subset of minimal weights

with respect to the partial order ≺ (see Section 2.1) on ΩG . By considering the center Z (G ) as a finite subgroup of T one has the relation





exp ΠG {0} = Z (G ),

see [13, p. 72, Exercise 13].

(2.6)

Explicitly, for each simple Lie group G with type ΦG the set ΠG of minimal weights is presented in Table 2. 2.3. Computing in the fundamental Weyl cell Let G be a 1-connected simple Lie group with maximal short root β ∈ ΦG+ . The fundamental Weyl cell of G is the simplex in F (x0 ) defined by

  

= u ∈ F (x0 )  β ∗ (u )  1 . In view of property (iii) of Lemma 2.6, a vector u ∈ if and only if there is a subset I u = {k1 , . . . , kr } ⊆

{1, . . . , n} so that

u = λk1 ωk1 + · · · + λkr ωkr

with λks > 0 and β(u )  1.

(2.7)

Let I u be the complement of I u in {1, . . . , n}.

/ ΩG one has Lemma 2.7. If u ∈ is nonzero with u ∈ 0  α ∗ (u )  1

for any positive root α ∈ ΦG+ .

(2.8)

Moreover (i) (ii)

α ∗ (u ) = 0 implies that α is a sum of the simple roots αi with i ∈ I u ; α ∗ (u ) = 1 implies that β ∗ (u ) = 1, and that there is a k ∈ {1, 2} so that kβ − α is a sum of the simple roots αi with i ∈ I u .

Proof. Let d : T × T → R be the distance function on T induced from the metric on L (G ). Since G is 1-connected, u ∈ implies that d(e , exp(u )) = u , see [6,7]. It follows that u  u − α for any α ∈ ΛrG . In particular,

α ∗ (u ) =

2(α , u )

(α , α )

 1 for all nonzero α ∈ ΛrG,+ .

(2.9)

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On the other side, express

α ∈ ΛrG,+ in term of the simple roots as

α = k1 α1 + · · · + kn αn with ki ∈ Z+ . With respect to the expression (2.7) we have

α ∗ (u ) =



2

(α , α )

 λi ki (αi , ωi ) =



1

(α , α )

i∈I u

 λi ki (αi , ai )  0.

(2.10)

i∈I u

Since ΦG+ ⊂ ΛG the relation (2.8) has been shown by (2.9) and (2.10). By (2.10) α ∗ (u ) = 0 implies that ki = 0, i ∈ I u . This shows (i). Let α ∈ ΦG+ be with α ∗ (u ) = 1. The proof of (ii) will be divided into two cases, depending on whether α is a short or a long root. r ,+

Case 1. If

α is short we get from α ≺ β that β − α = k1 α1 + · · · + kn αn with ki ∈ Z+ .

Consequently, ∗

1  β (u ) = α (u ) +



1

∗

(β, β)

 λi ki (αi , ai )  1,

i∈I u

where the first inequality  comes from (2.8), and the second follows from

α ∗ (u ) = 1 and



1

(β, β)

 λi ki (αi , ai )  0.

i∈I u

This is possible unless ki = 0 for all i ∈ I u . This shows (ii) when Case 2. If

α is short.

α is long we get from α ≺ γ that

γ − α = k1 α1 + · · · + kn αn with ki ∈ Z+ . The relation

1  γ ∗ (u ) = α ∗ (u ) +

and the assumption

1



(γ , γ )

 λi ki (αi , ai )  1

i∈I u

α ∗ (u ) = 1 force that

γ ∗ (u ) = 1 and γ − α =



k i αi ,

k i ∈ Z+ .

(2.11)

i∈I u r ,+

Write γ = β + δ with δ = γ − β (note that δ ∈ ΛG ). From we get that

γ ∗ (u ) = 1 and β 2 = δ 2 by Lemma 2.4

γ 2 = β ∗ (u ) + δ ∗ (u ),

β 2

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31

where β ∗ (u ), δ ∗ (u )  1 by (2.9). Again by Lemma 2.4 this is possible if and only if when G = F 4 , B n , C n and

γ ∗ ( u ) = β ∗ ( u ) = δ ∗ ( u ) = 1.

(2.12)

Assume, apart from (2.7), that

u = λ1 ω1 + · · · + λn ωn . If G = F 4 the system (2.12) becomes

2 λ 1 + 3 λ2 + 2 λ3 + λ 4 = 2 λ1 + 4 λ2 + 3 λ3 + 2 λ4

= 2 λ1 + 2 λ2 + λ 3 = 1 . It implies that λ1 = 12 ; λ2 = λ3 = λ4 = 0 and consequently, I u = {2, 3, 4}. The proof of (ii) for G = F 4 is completed by (2.11) and

γ = 2β − α2 − 2α3 − 2α4





see [13, p. 66, Table 2] .

If G = B n the system (2.12) gives

λ1 + 2λ2 + · · · + 2λn−1 + λn = 2λ1 + · · · + 2λn−1 + λn = 2λ2 + · · · + 2λn−1 + λn = 1. It implies that λ1 = 0 and consequently I u ⊇ {1}. The proof of (ii) for G = B n is completed by (2.11) and

γ = 2β − α1





see [13, p. 66, Table 2] .

Finally, if G = C n the system (2.12) turns to be

λ1 + · · · + λn = λ1 + 2λ2 + · · · + 2λn = λ1 = 1. It implies that λ1 = 1, λ2 = λ3 = λ4 = 0 and consequently u = completes the proof of (ii) for G = C n . 2

ω1 . This contradiction to u ∈/ ΩG

2.4. The local type of a centralizer Let G be a 1-connected simple Lie group with maximal short root β ∈ ΦG+ , and denote the Dynkin G . For a diagram of G by ΓG . The extended Dynkin diagram of G with respect to −β is denoted by Γ β u ∈ given as that in (2.7) denote by T u (resp. T u ) for the identity component of the subgroup of T :

 i∈I u



ker  αi : T → S 1







: T → S 1 resp. ker β





ker  αi : T → S 1





,

i∈I u

where S 1 is the circle group {exp(2π it ) ∈ C | t ∈ [0, 1]}, and where  α : T → S 1 is the homomorphism whose tangent map at the group unit is the inverse root α ∗ : L ( T ) → R of α ∈ ΦG (i.e. (2.3)). Let

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Γu ⊆ ΓG (resp. Γu ⊆ ΓG ) be the subdiagram obtained by deleting all the vertices αk with k ∈ I u , as G ). well as the edges adjoining to it, from ΓG (resp. Γ Since each x ∈ G is conjugate in G to an element of the form exp(u ) with u ∈ , and since the β

isomorphism type of a subgroup of G remains invariant under conjugation, the study of the isomorphism type of a centralizer C x , x ∈ G, can be reduced to the cases x = exp(u ), u ∈ . Geometrically, the path γu (t ) = exp(tu ), t ∈ [0, 1], is a minimal geodesic on G joining the unit e to x. Let C xs and C xRad be respectively the semisimple part and the radical part of the centralizer C x . In view of the fact that the semisimple part of a group is classified by its Dynkin diagram [13, p. 56], the next result specifies the local type of the centralizer C exp(u ) in term of u ∈ . Theorem 2.8. Let G be a compact and 1-connected Lie group and let x = exp(u ) ∈ G be with u ∈ but / ΩG {0}. Then the centralizer C x is a compact, connected and proper subgroup of G. u∈ Moreover, (i) if β(u ) < 1, then ΓC xs = Γu , C xRad = T u ; β

β

(ii) if β(u ) = 1, then ΓC xs = Γu , C xRad = T u . Proof. According to Borel [3, Corollary 3.4, p. 101] the centralizer C x in a 1-connected Lie group G / ΩG {0}, the group C x must be a is always connected. Furthermore, with the assumption that u ∈ proper subgroup of G. To show (i) and (ii) assume that the Cartan decomposition of L (G ) is

L (G ) = L ( T ) ⊕



Lα ,

α ∈ΦG+

where L α is the root space belonging to the root α ∈ ΦG+ [13, p. 35]. According to [2, p. 189], for x = exp(u ) with u ∈ L ( T ) the Cartan decomposition of the Lie algebra L (C x ) is

L (C x ) = L ( T ) ⊕



Lα ,

where Ψu =







α ∈ ΦG+  α ∗ (u ) ∈ Z .

(2.13)

α ∈Ψu

In view of (2.13) the set Φu = {±α | α ∈ Ψu } can be identified with the root system of the semisimple part C xs of C x . If u ∈ with β(u ) < 1 we have by Lemma 2.7 that Ψu = {α ∈ ΦG+ | α ∗ (u ) = 0} and that (a) the set of simple roots

αi with i ∈ I u is a base of Φu [13, p. 47].

Consequently, from the definition of the subgroup T u one gets (b) T u ⊆ C xRad . The relation ΓC xs = Γu is shown by (a). For the dimension reason dim T u + rank C xs = dim T we get from (b) that T u = C xRad . This finishes the proof of (i). Similarly, if u ∈ with β(u ) = 1 we have by Lemma 2.7 that Ψu = {α ∈ ΦG+ | α ∗ (u ) = 0, 1} and that (c) any element in Φu is a linear combination of the simple roots coefficients all nonnegative or nonpositive. β

αi with i ∈ I u and −β with

As a result, we get from (c) and the definition of the subgroup T u ⊂ T that β

(d) T u ⊆ C xRad .

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

33

With the assumptions I u = ∅ and u ∈ / ΩG , we conclude by (c) that the set {αi , −β | i ∈ I u } is a base of β β Φu [13, p. 47] and therefore, ΓC xs = Γu . Again, for dimension reason we get from (d) that T u = C xRad . This completes the proof. 2 For a 1-connected simple Lie group G with rank n assume that the expression of the maximal short root β ∈ ΦG+ in terms of the simple roots is

β = m1 α1 + · · · + mn αn . The set of vertices of the Weyl cell is clearly given by

  (β, β)  V G = 0, X i = ωi ∈  1  i  n . mi (αi , αi ) 

Let us put FG = {u i ∈ | 1  i  n} with

ui =

1

X 2 i Xi

if αi is short and mi = 1; otherwise.

According to Theorem 2.8, for two vectors u , u  ∈ one has L (C exp(u ) ) ⊇ L (C exp(u  ) ) if either I u ⊆ I u  and β(u ), β(u  ) < 1, or I u = I u  and β(u  )  β(u ) = 1. Since a maximal connected subgroup of maximal rank of G must be the centralizer of some element in G [4, Theorem 5], we get from Theorem 2.8 the classical result due to Borel and De Siebenthal [4, §7]: Corollary 2.9. For a 1-connected simple Lie group G with rank n, the set of centralizers {C exp(u i ) | 1  i  n} contains all the isomorphism types of maximal subgroups of maximal rank of G. Remark 2.10. In the classical paper [4] Borel and De Siebenthal intended to find all maximal subgroups of maximal rank of compact connected Lie groups. For the 1-connected simple Lie groups they give the answers only up to local types. As application of our Theorem 4.3, the isomorphism types of these groups will be determined in Theorem 4.4, compare Table 4 in Section 4.2 with the table in [4, §7]. In [17] M. Reeder gives a description of the Lie algebra L (C exp(u ) ) under the assumption that m · u ∈ ΛeG for some multiple m. We emphasis that Theorem 2.8 is not obvious, in view of the crucial use of Lemma 2.7 in its proof. In a recent Web discussion J. Newman raised the problem of finding an algorithm for computing the isomorphism type of the centralizer C H of a finite subgroup H of a simple Lie group G [16]. If {u 1 , . . . , uk } ⊂ is a set of vectors in the cell and if H is the subgroup of G generated by exp(u i ), 1  i  k, then the proof of Theorem 2.8, together with a comment of A. Knutson in the discussion [16], implies the next Cartan decomposition of the Lie algebra of C H

L (C H ) = L ( T ) ⊕

 α (u 1 )=···=α (uk )=0



Lα ⊕

Lα ,

α ∈ ΦG+ .

α (u 1 )=···=α (uk )=1

More precisely, a base for the root system of the group C H is either (i) {αi , −β | i ∈ I u 1 ∩ · · · ∩ I uk } if β(u i ) = 1 for all i, or (ii) {αi | i ∈ I u 1 ∩ · · · ∩ I uk } if β(u i ) < 1 for some i. In addition, Theorem 4.3 in Section 4 is applicable to determine the isomorphism type of the identity component of the group C H .

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3. Deficiency functions and their properties According to (1.1) a centralizer C exp(u ) with u ∈ admits the presentation





Rad C exp(u ) ∼ = G 1 × · · · × G k × C exp (u ) / H .

(3.1)

Rad Moreover, its local type π : G 1 × · · · × G k × C exp (u ) → C exp(u ) can be read from the expression of u in (2.7) by Theorem 2.8. To complete this work it remains for us to decide in term of u the finite Rad subgroup H ⊆ Z (G 1 ) × · · · × Z (G k ) × C exp (u ) appearing as the denominator in (3.1). The main idea to do so is to introduce the reduced weight system of C exp(u ) , as well as two deficiency functions on it, which play the role to clarify the difference between the group C exp(u ) and its local type G 1 × · · · × Rad G k × C exp (u ) .

3.1. Deficiency functions for semisimple Lie groups We begin by introducing the reduced weight system and the deficiency functions for semisimple Lie groups, and demonstrate their use in specifying the isomorphism type of such a group. Assume that G is a semisimple Lie group with local type π : G 1 × · · · × G k → G. Fix a maximal torus T i on each G i and take T = π ( T 1 × · · · × T k ) as the fixed maximal torus on G. Then L ( T ) = L ( T 1 ) ⊕ · · · ⊕ L ( T k ) and the tangent map of π at the group unit induces a partition

ΩG = ΩG 1 · · · ΩG k

(3.2)

where ΩG (resp. ΩG i ) is the set of fundamental dominant weights of G (resp. of G i ) with respect to a fixed regular point (x1 , . . . , xk ) ∈ L ( T ), xi ∈ L ( T i ). Definition 3.1. With respect to the partition (3.2) the reduced weight system of the semisimple group G is the subset of its weight lattice ΛG :

   ΠG0 = θ1 ⊕ · · · ⊕ θk ∈ ΛG  θi ∈ ΠG i {0} , where ΠG i is the set of minimal weights of the simple group G i (see Table 2). The integer-valued function δG : ΠG0 → Z defined by

δG (θ) = the order of the element exp(θ) in the group Z (G ),

θ ∈ ΠG0 ,

is called the deficiency function of G. Let ΛeG = exp−1 (e ) be the unit lattice of G. In the Euclidean space L ( T ) one has three lattices ΛrG , ΛeG and ΛG that are subject to the relations

ΛrG ⊆ ΛeG ⊆ ΛG .

(3.3)

Immediate, but useful properties of the function δG are Corollary 3.2. Let G be a semisimple Lie group with center Z (G ). Then the exponential map exp : L ( T ) → T satisfies exp(ΠG0 ) = Z (G ). Moreover,

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

35

Table 3 The deficiency function δG : ΠG0 → Z of 1-connected simple Lie groups. G

ΠG0 = ΠG  {0}

δG (θ), θ ∈ Λ0G

SU (n + 1)

{ω1 , . . . , ωn }  {0} {ωn }  {0} {ω1 }  {0} {ω1 , ω2n−1 , ω2n }  {0} {ω1 , ω2n−1 , ω2n }  {0} {ω1 , ω6 }  {0} {ω7 }  {0} {0 }

{ (nn++11,k) }1kn  {1} {2}  {1} {2}  {1} {2, 2, 2}  {1} {2, 4, 4}  {1} {3, 3}  {1} {2}  {1} {1}

Sp(n) Spin(2n + 1) Spin(4n) Spin(4n + 2) E6 E7 G2, F4, E8

(i) the value δG (θ) is the least positive multiple so that δG (θ) · θ ∈ ΛeG ; (ii) if G = G 1 × G 2 , δG 1 ×G 2 (θ1 ⊕ θ2 ) = l.c.m. {δG 1 (θ1 ), δG 2 (θ2 )}, where θi ∈ ΠG0 , i = 1, 2, and where l.c.m. means the least common multiple of the indicated set of integers. i

Proof. Property (i) comes from the fact that the exponential map exp induces a one-to-one correspondence ΛG /ΛeG ∼ = Z ( G ). The item (ii), together with the relation exp(ΠG0 ) = Z (G ), follows from exp(ΠG i {0}) = Z (G i ) by (2.6), and Z (G 1 × G 2 ) = Z (G 1 ) × Z (G 2 ). 2 Example 3.3. Let G be a 1-connected semisimple Lie group with reduced weight system ΠG0 . Properties (i) and (ii) of Corollary 3.2 are sufficient to evaluate the function δG : ΠG0 → Z. (a) If G is simple with Cartan matrix A, then the fundamental dominant weights ω1 , . . . , ωn can be expressed by the simple roots as (by (2.5))

ωi = ri,1 α1 + · · · + ri,n αn with ri,k ∈ Q, A −1 = (ri, j )n×n .

(3.4)

Since ΛrG = ΛeG for the 1-connected Lie group G, the value δG (ωi ) with ωi ∈ ΠG is the least positive integer so that δG (ωi ) · r i ,k ∈ Z for all 1  k  n. For all 1-connected simple Lie groups the expressions (3.4) can be found in [13, p. 69], from which we can read off the function δG : ΠG0 → Z and tabulate its values in the third column of Table 3. (b) If G is 1-connected with local type G = G 1 × · · · × G k , by (ii) of Corollary 3.2 the function δG : ΠG0 → Z is evaluated by

  δG (θ1 ⊕ · · · ⊕ θk ) = l.c.m. δG 1 (θ1 ), . . . , δG k (θk ) , where the values δG j (θi ) with θi ∈ ΠG0 are given in Table 3. i

In general, assume that G is semisimple with local type

π : G s = G 1 × · · · × Gk → G . The tangent map of π at the group unit of  T = T 1 × · · · × T k induces the canonical identifications L ( T ) = L ( T ), ΠG0 = ΠG0 s . It is in this sense that the reduced weight system ΠG0 of G possesses two deficiency functions:

δG ,  δG : ΠG0 → Z,

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H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

where  δG =: δG s . The next result tells that, comparison between these two functions enables one to specify ker π , which determines the isomorphism type of G. For a finite group H write H + for the set of all nontrivial elements in H . xp : L ( T) → T be the exponential map Lemma 3.4. Let G be semisimple with local type π : G s → G, and let e of the maximal torus  T on G s . Then







δG (θ) > δG (θ) = 1, θ ∈ ΠG0 , ker π + = e xp(θ) ∈ G s   where the order of an element e xp(θ) ∈ ker π + is  δG (θ), θ ∈ ΠG0 . Proof. Since ker π ⊆ Z (G s ) = e xp(ΠG0 s ) by Corollary 3.2, any element of ker π + has the form e xp(θ) for some θ ∈ ΠG0 . δG (θ) > 1 and δG (θ) = 1 are clearly On the other hand, for an element θ ∈ ΠG0 the statements  equivalent to e xp(θ) ∈ ker π + . 2 3.2. Deficiency functions for connected Lie groups We need to extend the deficiency functions to all compact connected Lie groups, so that an analogue of Lemma 3.4 holds for such a group. Assume therefore that K is a compact connected Lie group with local type

π : K s × K Rad → K ,

K s = G 1 × · · · × Gk .

Taking a maximal torus T i on each factor group G i the quotient homomorphism π then carries the maximal torus  T = T 1 × · · · × T k × K Rad on K s × K Rad onto the maximal torus T = π ( T ) on K . Moreover, the tangent map of π at the group unit induces an identification

L ( T ) = L(T ) = L(T 1 ) ⊕ · · · ⊕ L(T k )

 

L K Rad



where L ( K Rad ) is the Lie algebra of the radical part K Rad . Let ΠG0 1 ×···×G ⊂ L ( T ) be the reduced weight k system of the semisimple part G 1 × · · · × G k . The unit lattice Λe Rad ⊂ L ( K Rad ) of the radical part K Rad is obviously a subset of ΛeK . Let Λe Rad (Q) K K be the vector space spanned by the elements in Λe Rad over the rationals Q, regarded as a subset of K L ( T ). Definition 3.5. The reduced weight lattice of K is the subset of L ( T )

   Π K0 = ω ⊕ γ ∈ L ( T )  ω ∈ ΠG01 ×···×G k , γ ∈ ΛeK Rad (Q) . The deficiency function of K is the integer-valued map δ K : Π K0 → Z defined by

δ K (ω ⊕ γ ) = the least multiple so that δ K (ω ⊕ γ ) · (ω ⊕ γ ) ∈ ΛeK . Again, the tangent map of

π at the group unit induces a canonical identification Π K0 s × K Rad = Π K0

and therefore, in analogue to the semisimple cases, the set Π K0 possesses two deficiency functions:

 δK ,

δ K : Π K0 → Z,  δ K =: δ K s × K Rad .

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

37

Clearly, the one  δ K : Π K0 → Z depends only on the local type of K in the sense that

   δ K (θ1 ⊕ · · · ⊕ θk ⊕ γ ) = l.c.m. δG 1 (θ1 ), . . . , δG k (θk ), δ K Rad (γ ) , where θi ∈ ΠG0 , γ ∈ Λe Rad (Q). The next result generalizes Lemma 3.4 from the semisimple Lie groups i K to all compact connected ones. Lemma 3.6. If K is compact connected Lie group with local type π : K s × K Rad → K . Then







δ K (θ) > δ K (θ) = 1, θ ∈ Π K0 , ker π + = e xp(θ) ∈ K s × K Rad   where the order of an element e xp(θ) ∈ ker π + is  δ K (θ), θ ∈ ΠG0 . 3.3. The isomorphism type of a subgroup Let K be a compact, connected subgroup of a semisimple Lie group G with inclusion h : K → G and local type π : K s × K Rad → K . Assume that h carries a maximal torus T  of K into that T of G, and let h∗ : L ( T  ) → L ( T ) be the tangent map of h at the group unit. 0 e e 1 Since h is monomorphic, we have h− ∗ (ΛG ) = Λ K . It follows that the condition δ K (θ) = 1, θ ∈ Π K , e is equivalent to h∗ θ ∈ ΛG . Therefore, one gets from Lemma 3.6 that Theorem 3.7. Let K be a compact, connected subgroup of a semisimple Lie group G with inclusion h : K → G and local type π : K s × K Rad → K . Then







δ K (θ) > 1, h∗ (θ) ∈ ΛeG , θ ∈ Π K0 . ker π + = e xp(θ) ∈ K s × K Rad  

(3.5)

4. The isomorphism types of centralizers in a 1-connected Lie group To avoid case by case discussion we assume in this section that G is a 1-connected simple Lie group. Summarizing results in Sections 2 and 3, our main result is presented in Theorem 4.3, which gives an explicit procedure for calculating the isomorphism type of a centralizer C exp(u ) . As applications we determine in Sections 4.2 and 4.3 the isomorphism type of those centralizers C exp(u ) with u ∈ a multiple of a fundamental dominant weight. 4.1. The procedure for calculating a centralizer C x For a group element x = exp(u ) ∈ G with u ∈ given as that in (2.7), Theorem 2.8 specifies the local type π of the centralizer C x , hence the deficiency function  δC x : ΠC0x → Z, see discussion in Section 3.2. In order to apply the formula (3.5) to compute the group ker π we need to know the expressions of h∗ (θ), θ ∈ ΠC0x , in term of simple roots (or equivalently, the fundamental dominant weights) of the group G, where h : C exp(u ) → G is the inclusion. A) be the Cartan matrix (resp. the extended Cartan matrix) of the group G with Let A (resp.  respect to the system {α1 , . . . , αn } of simple roots (resp. the extended system {α1 , . . . , αn , −β} of simple roots). As in (2.7) for a vector u ∈ assume that I u = {k1 , . . . , kr } and let I u = { j 1 , . . . , jn−r } be the complement of the ordered sequence I u in {1, . . . , n}. Let A u (resp.  A u ) be the matrix obtained A) by deleting all the i-th columns and rows with i ∈ I u . from the matrix A (resp.  s Assume that the set ΩC s of fundamental dominant weights of the semisimple part C exp (u ) is

{ω1 , . . . , ωs }, where

exp(u )

s=n−r

if β(u ) < 1,

As direct consequences of Theorem 2.8 we have

and n − r + 1 if β(u ) = 1.

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H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

Lemma 4.1. The tangent map h∗ of h at the group unit satisfies that (i) if β(u ) < 1 then

⎛ ⎜ ⎝

h∗ (ω1 )

.. .

⎛

⎞

⎟ −1 ⎜ ⎠ = Au ⎝

h∗ (ωn −r )

⎞

α j1

⎟ ⎠

.. .

α jn−r

and





h∗ Λe Rad (Q) = {a1 ωk1 + · · · + ar ωkr | ai ∈ Q}; C exp(u )

(ii) if β(u ) = 1 then

⎛ ⎜ ⎜ ⎝

h∗ (ω1 )

.. .

h∗ (ωn −r ) h∗ (ωn −r +1 )

⎛ α j1 .. ⎜ ⎟ −1 ⎜ . ⎟= ⎠ Au ⎝ ⎞

α jn−r

⎞ ⎟ ⎟ ⎠

−β

and











h∗ Λe Rad (Q) = a1 ωk1 + · · · + ar ωkr  Σ ai β ∗ (ωki ) = 0, ai ∈ Q . C exp(u )

Proof. The formula of h∗ (ωi ) comes from (2.5). The expressions of h∗ (Λe Rad (Q)) follow from the C exp(u )

β

β

Rad relation C exp (u ) = T u or T u by Theorem 2.8, as well as the definition of the groups T u and T u in Section 2.4. 2

In general a centralizer C exp(u ) may not be semisimple. As a result its extended weight system

ΠC0exp(u) might contain the infinite factor Λe Rad (Q) ⊂ΠC0exp(u) , see Definition 3.5. This raises the question Cx

whether the deficiency function  δC x : ΠC0x → Z can be effectively calculated. The next result allows us to reduce the determination of ker π + by dealing with the finite set



H u = θ ∈ ΠC0s

exp(u )

  δC s

   (θ) > 1, h∗ (θ) ∈ h∗ Λe Rad (Q) mod ΛrG , exp(u ) C

(4.1)

exp(u )

which, in practice, can be easily decided from the concrete expressions of h∗ (θ) (with θ ∈ ΠC0s and h∗ (Λe Rad (Q)) C exp(u ) ΛrG = ΛeG .

)

exp(u )

by Lemma 4.1. Note that with the assumption that G is 1-connected, one has

Theorem 4.2. Let π be the local type of the centralizer C exp(u ) . Then







ker π + = e xp(θ − γθ )  θ ∈ H u , where γθ ∈ Λe Rad (Q) is an arbitrary element satisfying C exp(u )

h∗ (θ) ≡ h∗ (γθ )

mod ΛrG .

(4.2)

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

39

Rad s Proof. The exponential map e xp of the local type C exp (u ) × C exp(u ) of C exp(u ) will be written as Rad s exp1 × exp2 , where exp1 and exp2 are the exponential maps of the factors C exp (u ) and C exp(u ) , respectively. Let

Rad π2 : C exp (u ) → C exp(u ) Rad Rad be the restriction of π on the second factor C exp of a Lie (u ) . By the definition of the radical part K group K (see Section 1) the map π2 (hence the composition h ◦ π2 ) is injective. For a θ ∈ H u let γθ ∈ Λe Rad (Q) be a vector with h∗ (θ) ≡ h∗ (γθ ) mod ΛrG . We get from C exp(u )

   s s δC exp(u) (θ − γθ ) = l.c.m. δC exp (θ), δC Rad (−γθ )  δC exp (θ) > 1 (u ) (u ) exp(u )

and

h∗ (θ − γθ ) ∈ ΛrG



since h∗ (θ) ≡ h∗ (γθ ) mod ΛrG



that e xp(θ − γθ ) ∈ ker π + by Theorem 3.7. Furthermore, the element e xp(θ − γθ ) is independent of the choice of γθ since if γθ ∈ Λe Rad (Q) is a second one with h∗ (θ) ≡ h∗ (γθ ) mod ΛrG , then the relation C exp(u )

h∗ (γθ ) = h∗



γθ



mod ΛrG

and the injectivity of h ◦ π2 imply that

exp2 (γθ ) = exp2



γθ

 



Rad in C exp (u ) .

Consequently,

e xp(θ − γθ ) = exp1 (θ) × exp2 (−γθ )

      s Rad = exp1 (θ) × exp2 −γθ = e xp θ − γθ in C exp (u ) × C exp(u ) .

Conversely, for any pair (θ, γ ) ∈ ΠC0s

exp(u )

× Λe Rad (Q) (= ΠC0exp(u) ) with e xp(θ − γ ) ∈ ker π + , we get C exp(u )

from h∗ (θ − γ ) ∈ ΛrG that h∗ (θ) ≡ h∗ (γ ) mod ΛrG , and from

   s δC exp(u) (θ − γ ) = l.c.m. δC exp (θ), δ (− γ ) >1 Rad C (u ) exp(u )

and the injectivity of h ◦ π2 that δC s

exp(u )

(θ) > 1. This completes the proof. 2

Summarizing the results in Theorems 2.8, 4.2 and Lemma 4.1, we obtain the next explicit procedure for calculating C exp(u ) in term of u ∈ . Theorem 4.3. Let G be a 1-connected simple Lie group with fundamental Weyl cell . For a vector u ∈ with / ΩG the isomorphism type of C exp(u ) can be obtained by the procedure below: u∈

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H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

Step 1. Apply Theorem 2.8 to get the local type of C exp(u ) in the form G 1 × · · · × G k × C xRad with each G i a 1-connected and semisimple Lie group. Accordingly, write the reduced weight system of the semisimple part s C exp (u ) as

ΠC0s

exp(u )

   = θ1 ⊕ · · · ⊕ θk  θi ∈ ΠG i {0} .

Step 2. Apply Lemma 4.1 to get the expressions of the vectors h∗ (θ) (with θ ∈ ΠG0 ) and the subspace i h∗ (Λe Rad (Q)) in ΛG (Q). Accordingly, specify the finite set C exp(u )



H u = θ ∈ ΠC0s

exp(u )

  δC s

exp(u )

   (θ) > 1, h∗ (θ) ∈ h∗ ΛeC Rad (Q) mod ΛrG . exp(u )

Step 3. The group C exp(u ) is isomorphic to G 1 × · · · × G k × C xRad / ker π with







ker π + = e xp(θ − γθ ) ∈ C xs × C xRad  θ ∈ H u , where γθ ∈ Λe Rad (Q) is an element satisfying h ∗ (θ) ≡ h∗ (γθ ) mod ΛrG . C exp(u )

Finally, concerning the structure of ker π as a group, the next observation from the relation (4.2) will be repeatedly used in the forthcoming calculation:

if the set H u contains p − 1 elements and if θ0 ∈ H u is an element with δC s

exp(u )

(θ0 ) = p,

then ker π is the cyclic group Z p of order p generated by the element e xp(θ0 − γθ0 ) ∈ G 1 × · · · × G k × C xRad .

(4.3)

4.2. The maximal subgroups of maximal rank in a Lie group Let G be a 1-connected exceptional Lie group with rank n. From the expression of the maximal short root β given in [13, p. 66], the set FG (see Section 2.4) is determined and presented in the second column of Table 4 below. Applying Theorem 2.8 one obtains the local types of the centralizers C exp(u ) , u ∈ FG , that are presented in the third column of the table. In view of the explicit presentation of FG in the second column of the table we note that elements ω in FG are of the form u i = p i with p i > 0 an integer, 1  i  n. According to Borel and De Siebenthal i [4, Theorem 6] we have

the centralizer C exp(u i ) with u i =

ωi pi

∈ FG is a maximal subgroup of maximal rank of G

if and only if p i is a prime.

(4.4)

Carrying on discussion in Corollary 2.9 and Remark 2.10 we determine the isomorphism types of the centralizer C exp(u i ) for all u i ∈ FG . In order to make the generators of ker π explicit, for a product K = K 1 × K 2 of two groups we write exp1 × exp2 instead of exp, where exp (resp. expi , i = 1, 2) is the exponential map of the group K (resp. of K i , i = 1, 2). Theorem 4.4. Let G be a 1-connected exceptional Lie group. The isomorphism types of the centralizers C exp(u ) with u ∈ FG are given by the third and fourth columns of Table 4 below, in which those are of maximal in G are specified by (4.4) (compare with the table in [4, §7]).

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

41

Table 4 The isomorphism types of the centralizer C exp(u ) with u ∈ FG . G

u ∈ FG

Local type of C exp(u )

ker π (generator)

G2

ω1

SU (2) × SU (2)

Z2 (exp1 (ω11 ) × exp2 (ω12 ))

SU (3)

Z3 (exp1 (ω11 ))

Spin(9)

{0}

4

SU (2) × SU (4)

Z2 (exp1 (ω11 ) × exp2 (ω22 ))

3

SU (3) × SU (3)

Z3 (exp1 (ω11 ) × exp2 (ω22 ))

Sp(3) × SU (2)

Z2 (exp1 (ω31 ) × exp2 (ω12 ))

Spin(10) × S 1

Z4 (exp1 (ω51 ) × exp2 (− 94 ω1(6) ))

SU (2) × SU (6)

Z2 (exp1 (ω11 ) × exp2 (ω32 ))

SU (3) × SU (3) × SU (3)

Z3 (exp1 (ω21 ) × exp2 (ω12 ) × exp3 (ω13 ))

2

ω2 3

F4

ω1 2

ω2 ω3 ω4 2

E6

ω1 2

ω2 2

,

ω6

,

ω3

,

ω6

2 2

ω4 3

E7

ω1 2

2

ω2 2

ω3 3

ω4 4

ω7 2

E8

, ω35

,

ω5 2

Spin(12) × SU (2)

Z2 (exp1 (ω51 ) × exp2 (ω12 ))

SU (8)

Z2 (exp(ω41 ))

SU (3) × SU (6)

Z3 (exp1 (ω11 ) × exp2 (ω42 ))

SU (2) × SU (4) × SU (4)

Z4 (exp1 (ω11 ) × exp2 (ω12 ) × exp2 (ω33 ))

E6 × S1

Z3 (exp1 (ω11 ) × exp2 (− 43 ω7 ))

2

Spin(16)

Z2 (exp(ω71 ))

3

SU (9)

Z3 (exp(ω31 ))

4

SU (8) × SU (2)

Z4 (exp1 (ω21 ) × exp2 (ω12 ))

6

SU (2) × SU (3) × SU (6)

Z6 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (ω53 ))

5

SU (5) × SU (5)

Z5 (exp1 (ω11 ) × exp2 (ω22 ))

4

Spin(10) × SU (4)

3

E 6 × SU (3)

Z4 (exp1 (ω41 ) × exp2 (ω32 ))

2

E 7 × SU (2)

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8

Z3 (exp1 (ω11 ) × exp2 (ω22 )) Z2 (exp1 (ω71 ) × exp2 (ω12 ))

Proof. Granted with Theorem 4.3 the proof of Theorem 4.4 is straightforward. As an illustration of the use of Theorem 4.3 we present here a proof for the cases concerning G = F 4 . The proofs of the other cases are devoted to the arXiv version [9] of the present paper. For G = F 4 the centralizers C exp(u ) with u ∈ FG are all semisimple. Therefore, the constraint h∗ (θ) ∈ h∗ (Λe Rad (Q)) mod ΛrG on the set H u (see (4.1)) is equivalent to h∗ (θ) ≡ 0 mod ΛrG . C exp(u )

It is more convenient for us to express h∗ (θ) with θ ∈ ΠC exp(u) in term of the weights of F 4 (instead the simple roots). Note that since the group F 4 is free of center, we have ΛrG = ΛG . Case 1. If u = ω21 , the local type of the centralizer C exp(u ) is Spin(9) by Theorem 2.8. Accordingly, assume that the set of fundamental dominant weights of C exp(u ) is {ω11 , ω21 , ω31 , ω41 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSpin(9) = {ω11 } by fundamental dominant weights of F 4

h∗





ω11 = −

ω1 2

.

It follows that H u = ∅. Consequently, ker π = {0}. Case 2. If u = ω42 , the local type of the centralizer C exp(u ) is SU (2) × SU (4) by Theorem 2.8. Accordingly, assume that the set of fundamental dominant weights of C exp(u ) is {ω11 }  {ω12 , ω22 , ω32 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSU(2) ΠSU(4) by the fundamental dominant weights of F 4

h∗





1

ω11 = ω1 − ω2 , 2

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H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

h∗ h∗ h∗





3

ω12 = ω3 − ω2 , 4



 2

1

ω2 = ω4 − ω2 , 2



 2

1

ω3 = − ω2 . 4

It follows that the set H u consists of the single element ω11 ⊕ ω22 whose deficiency in the group SU (2) × SU (4) is 2. Consequently, ker π = Z2 with generator exp1 (ω11 ) × exp2 (ω22 ). Case 3. If u =

ω3 3

, the local type of the centralizer C exp(u ) is SU (3) × SU (3). Accordingly, assume that

the set of fundamental dominant weights of C exp(u ) is Ω = {ω11 , ω21 }  {ω12 , ω22 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSU(3) ΠSU(3) by the fundamental dominant weights of F 4 :

h∗ h∗ h∗ h∗





2

ω11 = ω1 − ω3 , 3



 1

4

ω2 = ω2 − ω3 , 3



 2

2

ω1 = ω4 − ω3 , 3



 2

1

ω2 = − ω3 . 3

It follows that the set H u consists of the two elements ω11 ⊕ ω22 and ω21 ⊕ ω12 whose deficiencies in the group SU (3) × SU (3) are both 3. Consequently, ker π = Z3 with generator exp1 (ω11 ) × exp2 (ω22 ) by (4.3). Case 4. If u = ω24 , the local type of the centralizer C exp(u ) is Sp(3) × SU (2) by Theorem 2.8. Accordingly, assume that the set of fundamental dominant weights of C exp(u ) is Ω = {ω11 , ω21 , ω31 }  {ω12 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSp(3) ΠSU(2) by the fundamental dominant weights of F 4 :

h∗





3

ω31 = ω3 − ω4 ; 2

h∗





1

ω12 = − ω4 . 2

It follows that the set H u consists of the single element ω31 ⊕ ω12 whose deficiency in the group Sp(3) × SU (2) is 2. Consequently, ker π = Z2 with generator exp1 (ω31 ) × exp2 (ω12 ). This completes the proof of Theorem 4.4 for the case G = F 4 . 2 Remark 4.5. Based on concrete constructions of the 1-connected exceptional Lie groups, Yokota obtained also the isomorphism types of maximal subgroups of maximal rank in the recent book [18]. In comparison, our approach is free of the types of the Lie groups. In our sequel work [8] certain results of Theorem 4.4 are applied to determine the fixed set of the inverse involution G → G, g → g −1 on an exceptional simple Lie group G. 4.3. The isomorphism types of parabolic subgroups If u ∈ is a vector with β(u ) < 1, the centralizer C exp(u ) is a parabolic subgroup of G whose isomorphism type depends only on the subset I u ⊆ {1, . . . , n} by Theorem 2.8. The corresponding homogeneous space G /C exp(u ) is a smooth projective variety, called a flag manifold of G [5,10,11,14,15].

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

43

Table 5 The parabolic subgroup corresponding to a weight in a 1-connected exceptional Lie groups. G

Iu

Local type of C exp(u )

ker π (generator of ker π )

G2

{1} {2}

SU (2) × S

1

Z2 (exp1 (ω11 ) × exp2 (− 12 ω1 ))

SU (2) × S 1

Z2 (exp1 (ω11 ) × exp2 (− 12 ω2 ))

{1} {2} {3} {4}

Spin(7) × S 1

Sp(3) × S 1

Z2 (exp1 (ω31 ) × exp2 (− 12 ω4 ))

{1} {2} {3} {4} {5} {6}

Spin(10) × S 1

Z4 (exp1 (ω41 ) × exp2 (− 34 ω1 ))

{1} {2} {3} {4} {5} {6} {7}

Spin(12) × S 1

E6 × S1

Z3 (exp1 (ω11 ) × exp2 (− 43 ω7 ))

{1} {2} {3} {4} {5} {6} {7} {8}

Spin(14) × S 1

Z4 (exp1 (ω61 ) × exp2 (− 14 ω1 ))

SU (8) × S 1

Z8 (exp1 (ω11 ) × exp2 (− 38 ω2 ))

F4

E6

E7

E8

Z2 (exp1 (ω31 ) × exp2 (− 12 ω1 ))

SU (2) × SU (3) × S

1

Z6 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 56 ω2 ))

SU (2) × SU (3) × S

1

Z6 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 56 ω3 ))

Z2 (exp1 (ω31 ) × exp2 (− 12 ω2 ))

SU (6) × S 1 SU (2) × SU (5) × S

9 Z10 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 10 ω3 ))

1

Z6 (exp(ω11 ) × exp2 (ω12 ) × exp3 (ω23 ) × exp4 ( 16 ω4 ))

SU (2) × SU (3) × SU (3) × S 1

ω )) Z10 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 21 10 5

SU (2) × SU (5) × S 1 Spin(10) × S

Z4 (exp1 (ω11 ) × exp2 (− 94 ω6 ))

1

Z2 (exp1 (ω51 ) × exp2 (− 12 ω1 )) Z7 (exp1 (ω31 ) × exp2 (− 27 ω2 ))

SU (7) × S 1

Z6 (exp1 (ω11 ) × exp2 (ω42 ) × exp3 (− 56 ω3 ))

SU (2) × SU (6) × S 1 SU (2) × SU (3) × SU (4) × S

7 ω4 )) Z12 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (ω13 ) × exp3 (− 12

1

ω )) Z15 (exp1 (ω21 ) × exp2 (ω22 ) × exp3 (− 16 15 5

SU (3) × SU (5) × S 1

Z4 (exp1 (ω11 ) × exp2 (ω52 ) × exp3 (− 34 ω6 ))

SU (2) × Spin(10) × S 1

ω )) Z14 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 11 14 3

SU (2) × SU (7) × S 1 SU (2) × SU (3) × SU (5) × S

Z30 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (ω13 ) × exp4 (− 11 ω )) 30 4

1

SU (4) × SU (5) × S 1

ω )) Z20 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 17 20 5

Spin(10) × SU (3) × S 1

7 ω6 )) Z12 (exp1 (ω41 ) × exp2 (ω12 ) × exp3 (− 12

E 6 × SU (2) × S 1

Z6 (exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 56 ω7 ))

E7 × S1

Z2 (exp1 (ω71 ) × exp2 (− 12 ω8 ))

In the case where the set I u is a singleton and β(u ) < 1, the vector u is an interior point on an edge of the cell from the origin 0. The homogeneous space G /C exp(u ) is also known as a generalized Grassmannian of G [10]. Theorem 4.3 can be used to determine the isomorphism types of all parabolic subgroups of in a given 1-connected Lie group G. This is demonstrated in the proof of the next result. Theorem 4.6. Let G be a 1-connected exceptional Lie group. For each u ∈ with β(u ) < 1 and with I u = {i } a singleton, the isomorphism type of the centralizer C exp(u ) is given in Table 5. Proof. Again for the sake of simplicity, as an illustration of the use of Theorem 4.3 we present here a proof of Theorem 4.6 for the case G = F 4 , and refer the proofs for the other cases to the arXiv version [9] of the present paper. Since the isomorphism type of the parabolic subgroup C exp(λu i ) with u i ∈ FG and λ ∈ (0, 1) is irrelevant with the parameter λ, we can take u = 12 u i as a representative for the case I u = {i }. With this convention the radical part of C exp(u ) is simply the circle subgroup S 1 = {exp(t ωi ) ∈ G | t ∈ R} on G by Theorem 2.8. As a result we have by Lemma 4.1 that





h∗ Λe Rad (Q) = {t ωi | t ∈ Q}. C exp(u )

44

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

Case 1. If u = ω41 , the local type of the centralizer C exp(u ) is Spin(7) × S 1 by Theorem 2.8. Accordingly, assume that the set of fundamental dominant weights of the semisimple part Spin(7) is Ω = {ω11 , ω21 , ω31 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSpin(7) = {ω11 } by the simple roots of F 4 :

h∗



1 1

ω



  1 r = α2 + 2α3 + α4 ≡ ω1 mod ΛG 3 2

2

and get





h∗ Λe Rad (Q) = {λω1 | λ ∈ Q}. C exp(u )

1 1 whose deficiency 1 1 1 ) × exp2 (− 2 1 ).

It follows that the set H u consists of the single element

ω

is 2. Consequently, ker π = Z2 with generator exp1 (ω

in the group Spin(7)

ω

ω

Case 2. If u = ω82 (resp. 43 ), the local type of the centralizer C exp(u ) is SU (2) × SU (3) × S 1 . Accordingly, assume that the set of fundamental dominant weights of the semisimple part SU (2) × SU (3) is Ω = {ω11 }  {ω12 , ω22 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSU(2) ΠSU(3) by the simple roots of F 4 :

h∗ h∗ h∗





1



2

 



1

ω11 = α1 ≡ ω2 mod ΛrG ,  2

2

2

1

3

3

1

2

3

3





1

ω1 = α3 + α4 ≡ ω2 mod ΛrG , 2 2





3



2

ω = α3 + α4 ≡ ω2 mod 3

ΛrG

(resp.

h∗ h∗ h∗





1



2

 



1

ω11 = α4 ≡ ω3 mod ΛrG ,  2

2

2

1

3

3

1

2

3

3





1

ω1 = α1 + α2 ≡ ω3 mod ΛrG , 2 2





3 2

ω = α1 + α2 ≡ ω3 mod 3

 ΛrG

)

and that





h∗ Λe Rad ( Q ) = {λω2 | λ ∈ Q } C exp(u )





resp. = {λω3 | λ ∈ Q } .

ω11 ⊕ ω12 has deficiency 6 in the group SU (2) × SU (3). Consequently, ker π = Z6 with generator exp1 (ω11 ) × exp2 (ω12 ) × exp3 (− 56 ω2 ) by It follows that the set H u consists of 5 elements in which the one

(4.3).

Case 3. If u = ω44 , the local type of the centralizer C exp(u ) is Sp(3) × S 1 . Accordingly, assume that the set of fundamental dominant weights of the semisimple part Sp(3) is Ω = {ω11 , ω21 , ω31 }. Applying Lemma 4.1 we get the expressions of h∗ (ω) with ω ∈ ΠSp(3) = {ω31 } by the simple roots of F 4 :

H. Duan, S.L. Liu / Journal of Algebra 376 (2013) 25–45

h∗





1

3

2

2



1

45



ω31 = α1 + α2 + α3 ≡ ω4 mod ΛrG 2

and that





h∗ Λe Rad (Q) = {λω4 | λ ∈ Q}. C exp(u )

It follows that the set H u consists of a single element

ω31 whose deficiency in the group Sp(3) is 2.

Consequently, ker π = Z2 with generator exp1 (ω31 ) × exp2 (− 12 ω4 ). This completes the proof of Theorem 4.6 for the case G = F 4 . 2 Acknowledgment

The authors are grateful to their referee for the improvements on the earlier version of this paper. References [1] N. Bourbaki, Groupes et algébres de Lie, Chapitres I–VIII, Éléments de Mathématique, Hermann, Paris, 1960–1975. [2] T. Bröker, T. tom Dieck, Representations of Compact Lie Groups, Grad. Texts in Math., vol. 98, Springer-Verlag, New York, 1985. [3] A. Borel, Semisimple Groups and Riemannian Symmetric Spaces, Hindustan Book Agency, New Delhi, 1998. [4] A. Borel, J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949) 200–221. [5] I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, Schubert cells and cohomology of the spaces G / P , Russian Math. Surveys 28 (1973) 1–26. [6] J. Cheeger, Pinching theorems for a certain class of Riemannian manifolds, Amer. J. Math. 91 (1969) 807–834. [7] P.J. Crittenden, Minimum and conjugate points in symmetric spaces, Canad. J. Math. 14 (1962) 320–328. [8] H. Duan, S.L. Liu, The fixed sets of the inverse involution on exceptional Lie group, in preparation. [9] H. Duan, S.L. Liu, The isomorphism type of the centralizer of an element in a Lie group, arXiv:1201.3394 [math.GR]. [10] H. Duan, X. Zhao, The Chow rings of generalized Grassmannians, Found. Comput. Math. 10 (3) (2010) 245–274. [11] W. Fulton, P. Pragacz, Schubert Varieties and Degeneracy Loci, Lecture Notes in Math., vol. 1689, Springer, 1998. [12] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001 (Chapter X, Section 5). [13] E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math., vol. 9, Springer-Verlag, New York, Berlin, 1972. [14] S. Kumar, Kac–Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math., vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. [15] V. Lakshmibai, N. Gonciulea, Flag Varieties, Hermann – Actualités Mathématiques, Hermann Editeurs Des Sciences et Des Arts, 2001. [16] J. Newman, A. Knutson, Computing centralizers in Lie groups, http://mathoverflow.net/questions/67469. [17] M. Reeder, Torsion automorphisms of simple Lie algebras, Enseign. Math. (2) 56 (1–2) (2010) 3–47. [18] I. Yokota, Exceptional Lie groups, arXiv:0902.0431 [math.DG].