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

Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.


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 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 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 x r ∈ Z(G) for some power r ≥ 2. The homogeneous space G/C x possesses a canonical periodic r automorphism For this reason the pair (G/C x , σ x ) is called an r-symmetric space of G in [8].
In the special case of r = 2, they are the global Riemannian symmetric spaces of G in the sense of E. Cartan [11]. 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 [5,9,10,13,14].
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 method to decide its local type in some special cases [11,4,16], 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 §4.1. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G in §4.2, and of parabolic subgroups of G in §4. 3.
To be precise some notation 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 (1.1) K ∼ = (G 1 × · · · × G k × K Rad )/H 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 Φ G of their corresponding root systems tabulated below [12, p.57] G SU (n) Sp(n) Spin(2n + 1) Table 1. The types and centers of 1-connected simple Lie groups 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. i) The group K is semisimple; ii) K Rad = {e}; iii) The local type π of K agrees with the universal cover of K.
The paper is arranged as the following. Section 2 contains a brief introduction on the roots and weight systems of Lie groups, and obtains the local type π : C s x × C Rad x → 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 so called extended weight lattice Π 0 K , 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 an 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 §4.2 and §4.3 all the centralizers C exp(u) in an 1-connected exceptional Lie group G with u a multiple of a fundamental dominant weight of G.

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.

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 , · · · , L m } of m = 1 2 (dim K − n) hyperplanes in L(T ), namely, the set of singular hyperplanes through the origin in L(T ) [2, p.168]. Let l k ⊂ L(T ) be the normal line of the plane L k through the origin, 1 ≤ k ≤ m. Then the map exp carries l k onto a circle subgroup on T . Let ±α k ∈ l k be the non-zero vectors with minimal length so that exp(±α k ) = e, 1 ≤ k ≤ m.
Definition 2.1. The subset Φ K = {±α k ∈ 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. [1,12]. 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 lattice in L(T ) spanned by all simple roots is called the root lattice, and is denoted by Λ r K . The subset of Λ r K consisting of the sums of the simple roots α 1 , · · · , α h is denoted by Λ r,+ K . We shall also put Φ + K = Λ r,+ K ∩ Φ K , whose elements are called the positive roots of K.
The set S(x 0 ) 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(x 0 ) (i.e. belongs to Λ r,+ K [12, p.47]). If G is a simple Lie group, elements in Φ G has at most two lengths. Let β ∈ Φ + G (resp. γ ∈ Φ + G ) be the unique maximal short root (resp. unique maximal long root) relative to the partial order ≺ on Φ + G . From [12, p.66, Table  2] one gets 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
If α ∈ Φ K is a root, the map α * is called the inverse root of α ([12, p.67]). Definition 2.5. Assume that G is a semisimple Lie group. The weight lattice of G is the subset of L(T ) whose elements are called weights. Its subset Lemma 2.6. Let G be a semisimple Lie group with Cartan matrix A, and let Ω G = {ω 1 , · · · , ω n } be the set of fundamental dominant weights relative to the regular point x 0 . Then i) Ω G = {ω 1 , · · · , ω n } is a basis for Λ G over Z; ii) the fundamental dominant weights ω 1 , · · · , ω n can be expressed in term of the simple roots {α 1 , · · · , α n } as Proof. With the assumption that G is semisimple, we have that h = n by (2.1), and that the Weyl chamber F (x 0 ) is a convex cone with vertex 0 ∈ L(T ).
Properties i) and ii) are known. By (2.4) each weight ω i ∈ Ω G is perpendicular to all the roots α j (i.e. ω i ∈ L j ) with j = i. This shows iii).
For a simple Lie group G we shall adopt the convention that its fundamental dominant weights ω 1 , · · · , ω n are ordered by the order of their corresponding simple roots pictured as the vertices in the Dynkin diagram of G [12, p.58]. Let Π G ⊆ Ω G = {ω 1 , · · · , ω n } be the subset of minimal weights with respect to the partial order ≺ (see §2.1) on Ω G . By considering the center Z(G) as a finite subgroup of T one has the relation Explicitly, for each simple Lie group G with type Φ G the set Π G of minimal weights is presented in Table 2 below:

Computing in the fundamental Weyl cell
Let G be an 1-connected simple Lie group with maximal short root β ∈ Φ + G . The fundamental Weyl cell of G is the simplex in F (x 0 ) defined by In view of property iii) of Lemma 2.6, a vector u ∈ ∆ if and only if there is a subset I u = {k 1 , · · · , k r } ⊆ {1, · · · , n} so that (2.7) u = λ k1 ω k1 + · · · + λ kr ω kr with λ ks > 0 and β(u) ≤ 1.
Let I u be the complement of I u in {1, · · · , n}.
ii) α * (u) = 1 implies that β * (u) = 1, and that there is an k ∈ {1, 2} so that kβ − α is a sum of the simple roots α i with i ∈ I u .
On the other side, express α ∈ Λ r,+ G in term of the simple roots as With respect to the expression (2.7) we have Since Φ + G ⊂ Λ r,+ G the relation (2.8) has been shown by (2.9) and (2.10). By (2.10) α * (u) = 0 implies that k i = 0, i ∈ I u . This shows i). Let α ∈ Φ + G 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. Case 1. If α is short we get from α ≺ β that where the first inequality ≥ comes from (2.8), and the second follows from This is possible unless k i = 0 for all i ∈ I u . This shows ii) when α is short. Case 2. If α is long we get from α ≺ γ that γ − α = k 1 α 1 + · · · + k n α n with k i ∈ Z + .

The local type of a centralizer
Let G be an 1-connected simple Lie group with maximal short root β ∈ Φ + G . and Dynkin diagram Γ G . The extended Dynkin diagram of G with respect to −β is denoted by Γ G . For a u ∈ ∆ given as that in (2.7) denote by T u (resp. T β u ) for the identity component of the subgroup of T : where S 1 is the circle group {exp(2πit) ∈ C | t ∈ [0, 1]}, and where α : T → S 1 is the homomorphism whose tangent map at the group unit is the inverse root 3)). Let Γ u ⊆ Γ G (resp. Γ β u ⊆ Γ G ) be the subdiagram obtained by deleting all the vertices α k with k ∈ I u , as well as the edges adjoining to it, from Γ G (resp. Γ G ).
Since each x ∈ G is conjugate in G to an element of the form exp(u) with u ∈ ∆, and since the isomorphism type of a subgroup of G remains invariant under conjugation, the study the isomorphism type of a centralizer C x , x ∈ G, can be reduced to the cases x = exp(u), u ∈ ∆ . Geometrically, the path γ u (t) = exp(tu), t ∈ [0, 1], is a minimal geodesic on G joining the unit e to x.
Let C s x and C Rad x 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 [12, 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 u / ∈ Ω G ⊔ {0}. Then the centralizer C x is a compact, connected and proper subgroup of G. Moreover, Proof. According to Borel [3, Corollary 3.4, p.101] the centralizer C x in an 1connected Lie group G is always connected. Furthermore, with the assumption that u / ∈ Ω G ⊔ {0}, the group C x must be a proper subgroup of G. To show i) and ii) assume that the Cartan decomposition of L(G) is where L α is the root space belonging to the root α ∈ Φ + G ( [12, 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 In view of (2.13) the set Φ u = {±α | α ∈ Ψ u } can be identified with the root system of the semisimple part C s x of C x . If u ∈ ∆ with β(u) < 1 we have by Lemma 2.7 that Ψ u = {α ∈ Φ + G | α * (u) = 0} and that a) the set of simple roots α i with i ∈ I u is a base of Φ u [12, p.47].
Consequently, from the definition of the subgroup T u one gets The relation Γ C s x = Γ u is shown by a). For the dimension reason dim T u + rankC s x = dim T we get from b) that T u = C Rad x . This finishes the proof of i). Similarly, if u ∈ ∆ with β(u) = 1 we have by Lemma 2.7 that Ψ u = {α ∈ Φ + G | α * (u) = 0, 1} and that c) any element in Φ u is a linear combination of the simple roots α i with i ∈ I u and −β with coefficients all nonnegative or nonpositive.
As a result, we get from c) and the definition of the subgroup With the assumptions I u = ∅ and u / ∈ Ω G , we conclude by c) that the set {α i , −β | i ∈ I u } is a base of Φ u [12, p.47] and therefore, Γ C s x = Γ β u . Again, for dimension reason we get from d) that T β u = C Rad x . This completes the proof.
For an 1-connected simple Lie group G with rank n assume that the expression of the maximal short root β ∈ Φ + G in terms of the simple roots is β = m 1 α 1 + · · · + m n α n .
The set of vertices of the Weyl cell ∆ is clearly given by 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 Siebenthal [4, §7]: Corollary 2.9. For an 1-connected simple Lie group G with rank n, the set of centralizers {C exp(ui) | 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 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 §4.2 with the table in [4, §7].
In [16] M. Reeder gives a description of the Lie algebra L(C exp(u) ) under the assumption that m · u ∈ Λ e G 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 suggested 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 [15]. If {u 1 , · · · , u k } ⊂ ∆ 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 [15], implies the next Cartan decomposition of the Lie algebra of C H More precisely, a base for the root system of the group C H is either In addition, Theorem 4.3 in §4 is applicable to determine the isomorphism type of the identity component of the group C H .

Deficiency functions and their properties
According to (1.1) a centralizer C exp(u) with u ∈ ∆ admits the presentation Moreover, its local type π : G 1 × · · · × G k × C Rad 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 subgroup 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 × · · · × G k × C Rad exp(u) .

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 tours 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 Definition 3.1. With respect to the partition (3.3) the reduced weight system of the semisimple group G is the subset of its weight lattice Λ G : where Π Gi is the set of minimal weights of the simple group G i (see Table 2). The integer valued function δ G : Π 0 G → Z defined by δ G (θ) = the order of the element exp(θ) in the group Z(G), θ ∈ Π 0 G , is called the deficiency function of G.
Let Λ e G = exp −1 (e) be the unit lattice of G. In the Euclidean space L(T ) one has three lattices Λ r G , Λ e G and Λ G that are subject to the relations Immediate, but useful properties of the function δ G are Corollary 3.2. Let G be a semisimple Lie group with center Z(G). Then the exponential map exp : Gi , i = 1, 2, and where l.c.m. means the least common multiple of the indicated set of integers.
Proof. Property i) comes from the fact that the exponential map exp induces an one to one correspondences Λ G /Λ e G ∼ = Z(G).

The item ii), together with the relation exp(Π
Example 3.3. Let G be an 1-connected semisimple Lie group with reduced weight system Π 0 G . Properties i) and ii) of Corollary 3.2 is sufficient to evaluate the function δ G : Π 0 G → Z. a) If G is simple with Cartan matrix A, then the fundamental dominant weights ω 1 , · · · , ω n can be expressed by the simple roots as (by (2.5)) For all 1-connected simple Lie groups the expressions (3.5) can be found in [12, p.69], from which we can read off the function δ G : Π 0 G → Z and tabulate its values in the third column of the table below: Table 3. The deficiency function δ G : Π 0 where the values δ Gj (θ i ) with θ i ∈ Π 0 Gi are given in Table 3.
In general, assume that G is semisimple with local type The tangent map of π at the group unit of It is in this sense that the reduced weight system Π 0 G of G possesses two deficiency functions: where δ G =: δ G 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.

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 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 The unit lattice Λ e K Rad ⊂ L(K Rad ) of the radical part K Rad is obviously a subset of Λ e K . Let Λ e K Rad (Q) be the vector space spanned by the elements in Λ e K Rad over the rationals Q, regarded as a subset of L(T ).
Definition 3.5. The reduced weight lattice of K is the subset of L(T ) Again, the tangent map of π at the group unit induces a canonical identification Π 0 K s ×K Rad = Π 0 K and therefore, in analogue to the semisimple cases, the set Π 0 K possesses two deficiency functions: . The next result generalizes Lemma 3.4 from the semisimple Lie groups 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

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 : Therefore, one gets from Theorem 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

The isomorphism types of centralizers in an 1-connected Lie group
To avoid case by case discussion we assume in this section that G is an 1connected 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 §4.2 and §4.3 the isomorphism type of those centralizers C exp(u) with u ∈ ∆ a multiple of a fundamental dominant weight.

A 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 δ Cx : Π 0 Cx → Z, see discussion in Section §3.2. In order to apply the formula (3.6) to compute the group ker π we need to know the expressions of h * (θ), θ ∈ Π 0 Cx , in term of simple roots (or equivalently, the fundamental dominant weights) of the group G, where h : C exp(u) → G is the inclusion.
Let A (resp. A) be the Cartan matrix (resp. the extended Cartan matrix) of the group G with respect to the system {α 1 , · · · , α n } of simple roots (resp. the extended system {α 1 , · · · , α n , −β} of simple roots). As in (2.7) for a vector u ∈ ∆ assume that I u = {k 1 , · · · , k r } and let I u = {j 1 , · · · , j n−r } be the complement of the ordered sequence I u in {1, · · · , n}. Let A u (resp. A u ) be the matrix obtained from the matrix A (resp. A) by deleting all the i th columns and rows with i ∈ I u .
Assume that the set Ω C s exp(u) of fundamental dominant weights of the semisim- As direct consequences of Theorem 2.8 we have follows from the relation C Rad exp(u) = T u or T β u by Theorem 2.8, as well as the definition of the groups T u and T β u in §2.4.
In general a centralizer C exp(u) may not be semisimple. As a result its extended weight system Π 0 C exp(u) might contain the infinite factor Λ e C Rad x (Q) ⊂Π 0 C exp(u) , see Definition 3.5. This raises the question whether the deficiency function δ Cx : Π 0 Cx → Z can be effectively calculated. The next result allows us to reduce the determination of ker π + by dealing with the finite set which, in practice, can be easily decided from the concrete expressions of h * (θ) (with θ ∈ Π 0 C s

exp(u)
) and h * (Λ e C Rad exp(u) (Q)) by Lemma 4.1. Note that with the assumption that G is 1-connected, one has Λ r G = Λ e G .
Theorem 4.2. Let π be the local type of the centralizer C exp(u) . Then Proof. The exponential map exp of the local type C s exp(u) × C Rad exp(u) of C exp(u) will be written as exp 1 × exp 2 , where exp 1 and exp 2 are the exponential maps of the factors C s exp(u) and C Rad exp(u) , respectively. Let be the restriction of π on the second factor C Rad exp(u) . By the definition of the radical part K Rad of a Lie group K (see §1) the map π 2 (hence the composition that exp(θ−γ θ ) ∈ ker π + by Theorem 3.7. Furthermore, the element exp Consequently, Conversely, for any pair (θ, γ) ∈ Π 0 and the injectivity of h • π 2 that δ C s exp(u) (θ) > 1. This completes the proof.
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 ∈ ∆. ∈ Ω G the isomorphism type of C exp(u) can be obtained by the procedure below: Step 1. Apply Theorem 2.8 to get the local type of C exp(u) in the form G 1 × · · · × G k × C Rad x with each G i an 1-connected and semisimple Lie group. Accordingly, write the reduced weight system of the semisimple part C s exp(u) as Step 2. Apply Lemma 4.1 to get the expressions of the vectors h * (θ) (with θ ∈ Π 0 Gi ) and the subspace h * (Λ e C Rad exp(u) (Q)) in Λ G (Q). Accordingly, specify the finite set Step 3. The group C exp(u) is isomorphic to 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:

The maximal subgroups of maximal rank in a Lie group
Let G be an 1-connected exceptional Lie group with rank n. From the expression of the maximal short root β given in [12, p.66], the set F G (see §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 ∈ F G , that are presented in the third column of the table.
In view of the explicit presentation of F G in the second column of the table we note that elements in F G are of the form u i = ωi pi with p i > 0 an integer, 1 ≤ i ≤ n. According to Borel and Siebenthal [4, Theorem 6] we have (4.4) the centralizer C exp(ui) with u i = ωi pi ∈ F G is a maximal subgroup of maximal rank of G if and only if p i is a prime.
Carrying on discussion in Corollary 2.9 and Remark 2.10 we determine the isomorphism types of the centralizer C exp(ui) for all u i ∈ F G . In order to make the generators of ker π explicit, for a product K = K 1 × K 2 of two groups we write exp 1 × exp 2 instead of exp, where exp (resp. exp i , i = 1, 2) is the exponential map of the group K (resp. of K i , i = 1, 2).   Proof. It suffices to establish the results in fifth columns of the table. These will be done by applying the steps 2 and 3 entailed in Theorem 4.3. The calculations will be divided into five cases in accordance to G = G 2 , F 4 , E 6 , E 7 and E 8 .
, the centralizer C exp(u) are parabolic. The proofs of these cases will be postponed to the next section.
In the remaining cases the centralizers C exp(u) are always semisimple. Therefore, the constraint h * (θ) ∈ h * (Λ e C Rad exp(u) (Q)) mod Λ r G on the set H u (see (4.1)) is equivalent to h * (θ) ≡ 0 mod Λ r G . It is more convenient for us to express h * (θ) with θ ∈ Π C exp(u) in term of the weights of G (instead the simple roots). For the transitions from weights to roots we refer to the table in [12, p.69].
, the local type of the centralizer C exp(u) is Spin(9). Accordingly, assume that the set of fundamental dominant weights of C exp(u) is It follows that H u = ∅. Consequently, ker π = {0}. ii) If u = ω2 4 , the local type of the centralizer C exp(u) is SU (2) × SU (4). Accordingly, assume that the set of fundamental dominant weights of C exp(u) is It follows that the set H u consists of the single element ω 1 1 ⊕ ω 2 2 whose deficiency in the group SU (2) × SU (4) is 2.
Consequently, ker π = Z 4 with generator exp See in the proof of Theorem 4.6.
, the local type of the centralizer C exp(u) is Spin (16). Accordingly, assume that the set of fundamental dominant weights of C exp(u) is It follows that the set H u consists of the single element ω 1 7 whose deficiency in the group Spin(16) is 2.

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 homogenous space G/C exp(u) is a smooth projective variety, called a flag manifold of G [5,9,10,13,14].
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 [9]. Theorem 4.3 is ready to apply 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 an 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 4 below: G Iu the local type of C exp(u) ker π (generator of ker π) Proof. Again, the proof will be divided into five cases in accordance to G = G 2 , F 4 , E 6 , E 7 and E 8 .
Since the isomorphism type of the parabolic subgroup C exp(λui) with u i ∈ F G and λ ∈ (0, 1) is irrelevant with the parameter λ, we can take u = 1 2 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(tω i ) ∈ G | t ∈ R} on G by Theorem 2.8. As a result we have by Lemma 4.1 that For u = ω1 4 (resp. ω2 4 ) the local type of the centralizer C exp(u) is SU (2) × S 1 . Accordingly, assume that the set of fundamental dominant weights of the semisimple part SU (2) is Ω = {ω 1 1 }. Applying Lemma 4.1 we get the expression of h * (ω) with ω ∈ Π SU(2) by the simple roots of G 2 : and get It follows that the set H u consists of the single element ω 1 1 whose deficiency in the group SU (2) × S 1 is 2.
, the local type of the centralizer C exp(u) is Spin(7)×S 1 . Accordingly, assume that the set of fundamental dominant weights of the semisimple part Spin (7) is Ω = {ω 1 1 , ω 1 2 , ω 1 3 }. Applying Lemma 4.1 we get the expressions of h * (ω) with ω ∈ Π Spin(7) = {ω 1 1 } by the simple roots of F 4 : and get It follows that the set H u consists of the single element ω 1 1 whose deficiency in the group Spin(7) is 2.
iii) If u = ω4 4 , 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 = {ω 1 1 , ω 1 2 , ω 1 3 }. Applying Lemma 4.1 we get the expressions of h * (ω) with ω ∈ Π Sp(3) = {ω 1 3 } by the simple roots of F 4 : and that It follows that the set H u consists of a single element ω 1 3 whose deficiency in the group Sp(3) is 2.
and get It follows that the set H u contains 9 elements in which the one ω 1 1 ⊕ ω 2 1 has deficiency 10 in the group SU (2) × SU (5).