Centralizers of Commuting Elements in Compact Lie Groups

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in G, followed by some explicit examples. We conclude by showing that as a result of a compact, connected, simply connected Lie group G having a finite number of subgroups, each conjugate to the centralizer of any element in G, that there is a uniform bound on an irredundant chain of commuting elements.

Certainly the choice of a liftx is unique up to an element in Ker(π) ∼ = π 1 (G) which is identified as a subgroup of the center of the simply connected covering. Extending this for a c-pair: for k ∈ Ker(π), [x,ỹ] = [kx, kỹ] = c because k ∈ Ker(π) commutes with every element inG and is also invariant under the choice of x, y. We may define conjugation byg ∈G to beg[x,ỹ]g −1 = [gxg −1 ,gỹg −1 ] satisfying π(gxg −1 ) = gπ(x)g −1 = gxg −1 . This lift is independent of the choice of c ∈ CG and thus our c-pair is well-defined.
For completeness, we recall some definitions found in [6] on Dynkin diagrams and root/coroot systems that we will use throughout the paper. Let Φ be a reduced irreducible root system for a compact connected Lie group G, and let ∆ = {a 1 , . . . , a n } be a choice of simple roots for G. Let d be the highest root of Φ with respect to ∆. Setã = −d and let∆ = ∆ ∪ {ã} be the extended set of simple roots. Then ∆ ∨ is the set of coroots a ∨ inverse to each root a ∈ ∆. If we define A to be the unique alcove containing the origin in the positive Weyl chamber associated to ∆ then there is a bijection between the walls of A and∆. Therefore∆ is the set of nodes for the extended Dynkin diagramD(G). For each element c ∈ CG the differential w c ∈ W of the action of the center on the alcove is a linear map normalizing∆ ⊂ t and the action of w c on the nodes ofD(G) is a diagram automorphism. Given a maximial torus T ⊂ G, denote Lie(T ) = h and the exponential map identifies T with h/Q ∨ where Q ∨ = Za ∨ i is the lattice associated to the coroots dual to a choice of simple roots a i ∈ ∆ for G. Denote the affine Weyl group by W af f . The alcove is defined over the maximal torus T ⊆ G as A = h/W af f (Φ) ⊆ h where W af f (Φ) acts simply transitively on the set of alcoves in the vector space V ; thus there is an induced action of the center CG on A. x

II Component Group of the Centralizer of Commuting Pairs
The work in [4] gave an explicit characterization of the moduli space of c-pairs in terms of the extended coroot diagram of a simply connected group G and the action of the Weyl group on that diagram. This beneficial relationship between the root/coroot system and holonomy plays a crucial role. Throughout, we assume that G is a compact connected Lie group, unless otherwise defined. Note that if G is disconnected, then there is the appearance of a c − "1−chain" coming from a component which may be a finite group of a certain order. A group G is reductive if any representation is irreducible. Notice that when a group is compact, it is equivalent to being reductive. We will use the following theorem by Borel ([3], Theorem 5) Theorem II.0.2 Let G be a compact, connected and simply connected Lie group. Then Z G (x) is connected.
The following demonstrates the relationship between the conjugacy classes for commuting pairs (x, y) and flat G-bundles over T 2 : Proposition II.0.3 Assume that G is a compact, connected and simply connected Lie group. For any maximal Proof. Fix generators (γ x , γ y ) for π 1 (T 2 , x). Notice that we have a representation ρ : x, ρ(γ y ) = y and that these images define the commutator in G. In fact, the representation determines the commutator in the following sense. Let T be the maximal torus in G. Then for some g ∈ G, gxg −1 ∈ T and gyg −1 ∈ T since every element in G can be conjugated into the maximal torus. We want to show that both x, y ∈ T. To do this, define conjugation by g ∈ G for the pair (x, y) by g(x, y)g −1 = (gxg −1 , gyg −1 ) = (x ′ , y ′ ) where (x, y) ∈ T × T and (x ′ , y ′ ) ∈ T ′ × T ′ . The fact that G is simply connected implies that Z G (x) is connected ( II.0.2).
Thus we may restrict to the connected component of the identity Z 0 (x). Since x ∈ Z G (x) we must show that T ⊆ Z G (x) because this would imply that both x, y ∈ T. By definition of the representation, the image [x, y] = 1 so that y ∈ T is conjugate to x which implies we may project y to an element ζ y ∈ W. If we conjugate the pair (x, y) To prove the converse, notice that W acts by simultaneous conjugation on S ×S so that if g ∈ W is a reflection, then if γ x , γ y generates π 1 (T 2 ) and ρ(γ x , γ y ) = [gxg −1 , gyg −1 ] = 1 then the holonomy determines the commutator and vice versa.

4
The next corollary follows immediately because the fundamental group of the centralizer Z(x 1 ) is trivial, and for a commuting n-tuple in a simply connected group, the component group is contained in the fundamental group of the semisimple subgroup π 1 (DZ(x 1 )).
Corollary II.0.4 When G is of type A n , C n every commuting n-tuple can be conjugated into the maximal torus T in G so that the moduli space has the form M G = T × · · · × T n /W.
The corollary can also be seen directly as follows. If (x 1 , . . . , x n ) is a commuting n-tuple such that [x 1 , and [x 1 ,x i ] = 1, ∀ 2 < i ≤ n, choosingx 1 in the alcove over the torus implies thatx 2 projects to a Weyl element and therefore conjugates back into the maximal torus; every other element has trivial commutator and thus can be conjugated to the maximal torus. This also works when (x i , x j ) for 1 ≤ i < j ≤ n is an arbritrary n-tuple because the lifts [x i ,x j ] = c ij ∈ CG and for the cases of type A n , C n the center is generated by one cyclic element. Hence only one pair in the n-tuple determines what happens to the other elements in the n-tuple.
Corollary II.0.5 When G is simply connected, the component group π 0 (Z(S)) is a subgroup of Z/n i Z where n i ≤ 6 and corresponds to the coroot integer for x 1 ∈ G which is associated to the node in the extended Dynkin diagram D(G).
If π 1 (DZ(x 1 )) is not central then it defines a possibly non-trivial diagonal subgroup CZ(x 1 ) which acts simultaneously on the components of the centralizer Z(x 1 ). If x 2 is chosen so thatx 2 ∈ A does not lie in the fixed point space under this diagonal action, the component group is trivial. However, ifx 2 lies somewhere in the fixed space under 5 the action of the center, there will be a nontrivial component group π 0 (Z(x 1 , x 2 )). The structure of the centralizer itself will differ depending on where the element x 2 lies. Regardless of which compact, connected Lie group G we are working with, whether or not there is a component group from the choice of second element relies on the diagonal Proposition II.0.6 Let G be simply connected and let ∆ = {a 1 , . . . , a n } be a choice of simple roots. Let ∆ x = {ã, a 1 , . . . , a k }, k ≤ n, be a choice of simple roots for Z G (x) and let h(x) ⊆ h be the real linear span of the coroots dual to the roots in ∆ x . Then there is an exact Proof. By definition of the fundamental group, n i = gcd(g k+1 , . . . , g n ) knowing that all the coroot integers for both the classical and exceptional groups are less than or equal to six, n i ≤ 6. Dividing each of the coroot integers in any group G by n i we may define a new integer g ′ r = g r /n i for r > k. By definition, this element will have order and thus is a generator for the cokernel.
Proposition II.0.7 For an arbitrary compact, connected simple group G and for a commuting n-tuple x = (x 1 , . . . , x n ), the component group of the centralizer of the n-tuple can be defined in terms of the roots as Proof. If G is not simply connected, then under complexification of roots which annihilate x we must determine how Stab W (x) is defined with respect to this smaller subset of roots.
If x corresponds to some node a i in the extended Dynkin diagram such that its kernel consists of all the roots in L not in Q ∨ i.e. L/Q ∨ . Therefore the roots which annihilate x are the same as those annihilatingx.
Let W(Φ(x)) be a subgroup in W defined by a subroot system when viewed as characters which annihilate x. The faithful action of W af f on h/Q ∨ yields a split exact sequence 1 → P ∨ /Q ∨ → W af f → W → 1 and since the 6 kernel is central, W af f = P ∨ /Q ∨ × W is a direct product because the action of the Weyl group is trivial on the center. Restrict the Weyl group to L : W L af f = L/Q ∨ ×W. The torus action of L/Q ∨ on h/Q ∨ provides the quotient x as the unique lift to the alcove. Therefore we may define Stab L/Q ∨ (x) = Stab W aff (x) in the sense that the roots which annihilatex can be used to define a subset S ⊆ L/Q ∨ , where S = π 0 (Z G (x 1 , . . . , x n )). This allows for a component group larger than the fundamental group and therefore it is not necessarily cyclic. Since S ⊆ CG it induces a well-defined cyclic permutation on the vertices in the alcove and its fixed space h S may be something other than the barycenter.
L is defined as follows. For Z G (x 1 ) the vector space is h and its coroot lattice is the entire Q ∨ . Because we are considering commuting elements, we choose as the associated lattice to the centralizer of the prior n − 1 elements. From the definition of these lattices, when they are quotiented out by the coroot lattice, they will either be a cyclic subgroup of the center whose order divides the order of the center or will be the entire center. Thus we have the above conclusion since S ⊆ L/Q ∨ ⊆ P ∨ /Q ∨ .

III Properties of Centralizers
In general, the component group of an ordered n-tuple is some subquotient of the Weyl group and lies in the In order to determine the component group for a non-simply connected group we note that the finite diagonal subgroup contained in the center of each centralizer Z(x 1 , . . . , x k ), for some k, at some point becomes the component group and therefore defines the singularities in the moduli space. For the classical groups, Z(x 1 ) will be a product of type A n , B n or D n and for the exceptional groups, Z(x 1 , x 2 ) will be of type A n , D n Therefore, it sufficies to consider the diagonal group action of the fundamental group π 1 (DZ(x 1 )) on groups of these types. Since the fundamental group is a subgroup of the center of the simply connected covering, 7 for type B n we only consider the Z/2Z action on the alcove given by flipping two vertices;the action of any higher order central cyclic group is trivial.
Definition III.0.8 Define the rank rk(x 1 , . . . , x n ) of an n-tuple to be the rank of Z(x 1 , . . . , x n ). An n-tuple has rank zero if and only if Z(x 1 , . . . , x n ) is a finite group. A c-pair (x, y) is in normal form with respect to the maximal torus T in the alcove A if x ∈ T is the image under the exponential map ofx ∈ t c and y ∈ N G (T ) projects to w c ∈ W. Note w c ∈ W is the differential action of c ∈ CG that, as a group of affine isometries of the Lie algebra t of the maximal torus T normalizes the alcove A.
Let∆ = {ã, a 1 , . . . , a n } be the set of extended simple roots for a Lie group G. Any closed subset of the extended simple roots for G gives a subdiagram of the extended diagram. We are interested in the subset of roots ∆ x that annihilate the n-tuple. In particular, the simple root system for the centralizer Z(x) for any x ∈ G is defined as by ∆ x = {a ∈ ∆ | α(x) ∈ Z} which has an associated Weyl group W(Φ(x)). Let x = (x 1 , . . . , x n ) be a commuting n-tuple. Any element x ∈ π 0 (Z(x)) can be represented by g ∈ Z G (x) since g normalizes Z G (x) and therefore via is an integral lattice defined with respect to T. Namely, Ker(exp) = Q ∨ , thus for λ − λ ′ ∈ Ker(π) this implies that r a − r ′ a ∈ Z and hence ∆(c) = ∆(c ′ ) if and only if r a − r ′ a ≡ 0 (mod Z) which we have since r a − r ′ a ∈ Q ∨ .
Therefore, ∆(c) depends only on the choice c ∈ CG ∼ = P ∨ /Q ∨ and any two elements in the Lie algebra t differ by an element in the coroot lattice Q ∨ . By definition of a c−pair of rank zero, c ∈ DZ(S c ). Thus the moduli space is precisely M = (T × T )/W (T, G). It certainly will not be true that for a general non-simply connected group that every element of a commuting n-tuple can be put inside the maximal torus.
We show that the fundamental group of the centralizer is finite cyclic by using diagram automorphisms.
Proposition III.0.10 Under a cyclic permutation of the vertices in the extended diagram of the type A n where the permutation is given by the fundamental group π 1 (DZ(x 1 )) = Z k , the quotient space has the form where n + 1 = kl and π 1 (DZ(x 1 )) ∼ = Z k , with 1 ≤ k ≤ 6.
Proof. Since any inner automorphism of type A n is dihedral, it is either a rotation or a reflection. Consider the cyclic permutation τ given by rotation. (note: this is an element of a group of affine automorphisms of a vector space which normalizes the alcove of a root system on that vector space. Such automorphisms are equivalent to diagram automorphisms of the extended Dynkin diagram of the root system.) If τ ∈ Z k has order n + 1 then the fixed point set is simply the barycenter and thus t τ = {0}. If n is odd then τ may have order k | (n + 1). If k = n+1 2 then either the barycenter is the only fixed point or the fixed point set is the join of type A 2k−1 or there is a rotation subgroup of τ of order exactly k which implies it is an involution of the extended diagram which fixes two vertices and thus the quotient coroot diagram is a product of type A 1 . The Z 2 action on the alcove over A 1 is simply to switch the two vertices leaving the barycenter fixed.
Specifically, if n + 1 = kl then every node in the extended diagram included in this k-orbit is nonzero which leaves the quotient coroot diagram as the join of k, (l − 1)-simplicies with the barycenter (since the barycenter is the only fixed space under the action of the full center) times the remaining torus and semidirect product with rotation group. In terms of extended roots in the diagram, if∆ = {ã, a 1 , . . . , a n } is the set of simple roots for A n then the quotient space A n /Z k is defined by the elements in the orbit,∆/Z k = {ã, a l , a 2l , . . . , a (k−1)l }. Thus the gaps between the nodes are of length (l − 1). Therefore, the fixed space will be given by What we have shown is that in A n the Stab τ (× k A l−1 ) = Z k . The fact that π 1 (DZ(x 1 )) ∼ = Z k where 1 ≤ k ≤ 6, follows directly from looking at the coroot integers for all the extended Dynkin diagrams.
Proposition III.0.11 Let G be a simple group of dimension n. The centralizer Proposition III.0.12 Given G 1 × F G 2 where G 1 , G 2 are subgroups of G, F ⊆ CG 1 and F ⊆ CG 2 and F ∈ This demonstrates that the coker of π is F and that π is not surjective. Therefore, Note also that by the definition of the centralizer of [a, b] ∈ G 1 × F G 2 , that the generalized Stiefel-Whitney class [12] is w 2 (a, c) = −w 2 (b, d) ∈ F. Hence w 2 : H 2 (T * ) → Z n defines an obstruction.
Corollary III.0.13 Following propsition III.0.12, if G 1 = T for some torus and G 2 is of type A r then so that AB = BAζ for ζ ∈ F. Thus they are equal up to an element in the finite group. Therefore we have ∈ A r /F and consider its liftÃ ∈ A r arbitrary. Then because the kernel is Ker(π) = F and from what we have already deduced, AB = BAζ for ζ ∈ F . Hence We Corollary III.0.14 Consider a subgroup in G of the form A k × F A r , for r + k = n + 1, then the centralizer of an It does not necessarily follow that π 0 (Z(x 1 , . . . , x n ) ⊂ π 1 (DZ(x 1 )) because DZ(x 1 ) is not necessarily connected.
The fact that for G of type D n that π 1 (D n ) = CD n ∼ = Z × Z and that the characteristic class for a principal G-bundle over T n lies in H 2 (T n ; π 1 (G)) ∼ = Z × Z means that there is a possibility that the component group for an n-tuple inside D n will not be finite cyclic.

IV Uniform Bound on Chains of Commuting Elements
Assume that G is a compact, connected Lie group. Define a chain of elements . Otherwise, the chain is said to be irredundant. Define an ordering of the n-tuple by the property dim Z(x, x) ≥ dim Z(x, x, y) for x = y. Thus for a chain of commuting elements x 1 , x 2 , . . . ∈ G if we consider a decreasing chain of the centralizers of these elements, for t i ∈ CG, the chain of strict inclusions will terminate once the chain of elements becomes redundant, for any further choices of x i ∈ G.. This is equivalent to saying that the chain of decreasing centralizers will terminate once the centralizer of an n-tuple Z(x 1 , . . . , x n ) becomes abelian; this will occur if the centralizer is either the Lemma IV.0.16 Let G be a compact, connected Lie group of rank n and A ∈ h be the alcove over the maximal torus. Then for anyx ∈ A c such that exp(x) = x ∈ T, it is sufficient to consider that eitherx is a vertex in the fixed subspace A c under the central action of c ∈ CG or thatx lies on an edge whose vertices are both central elements in order to get a subgroup of maximal rank in G.
Proof. The case when G is of type A n is unique in that the fixed space under the central action is the barycenter.
Thus the only choice forx ∈ A c is the barycenter and its centralizer is given byZ(x) = h ⋊ CG.
Every c-pair is conjugate to one in normal form which means thatx corresponds to the barycenter in A c andỹ projects to an element in the Weyl group W. Since a c-tuple is defined by a c-pair and then n − 2 commuting elements, we may always choosex to be the barycenter in A c and therefore it will lie in the connected component of the identity. Thus it follows from the commuting case that the maximal subgroup always corresponds to choosing x as a vertex in A c .
As a result of the commuting case, it's necessary to choosex ∈ A c which is connected. However, when working with C -tuples, we may choosex to lie either in A c or not. Elements in the connected component of the identity of the centralizer follow from the case for commuting n-tuples. The center CG is a finite group, thus the key is to always follow and extremal path when choosing elements in the fixed space A c .
Corollary IV.0.17 Let x = (x 1 , . . . , x n ) be an n-tuple for a compact, connected Lie group G. Assume that the n-tuple is irreducible. Then it is sufficient to only consider the semisimple part when determining the moduli space.
Take Z G (x) and look at S max ⊆ Z G (x). Then take the centralizer of this small maximal torus and consider its maximal torus S ⊆ Z G (S max ). Then Z G (S max ) = S max which implies that S max is semisimple because it's a torus 13 so communtes with the maximal torus S. Thus there exists a g ∈ Z G (S max ) such that gS max g −1 = S. But by definition S = S max .
The following theorem states that there exists a uniform upper bound on irredundant chains of commuting elements in G.
Theorem IV.0.18 (Weak Theorem) Given a compact Lie group G there exists an integer m = m(G) such that any irredundant chain has fewer than m elements.
Theorem IV.0.19 (Strong Theorem) Given G a compact Lie group, there exists a finite number of subgroups is a finite group. Choosing x ∈ (CG) 0 then Z G (x) = G hence it suffices to choose x ∈ DG. If x ∈ T is a non-central element for a choice of a maximal torus T ⊂ G, then since T is abelian and x is regular, Z(x) = T is the smallest possible subgroup of maximal rank. We claim that there are a finite number of closed subgroups between G and T such that each is of the form Z(x i ) for some x i ∈ G.
Whenx corresponds to a vertex or an edge with two central vertices, the element yields a maximal rank subgroup.
Thus choose x 1 as image under the exponential map of a non-central vertex. Then H 1 = Z(x 1 ) = DZ(x 1 ) · C 0 Z(x 1 ).
If Z(x 1 ) is connected then we may choosex 2 as a vertex or a "central edge" which gives H 2 = Z(x 1 , x 2 ) where the rank of the semisimple part of the centralizer decreases by at least one. In other words, continuing to choose vertices or "central edges" such that the rank decreases, we are following a connected path and at some point there will be some H k such that the rank of the semisimple part of H k is zero. Thus H k is abelian and we have proven the claim.
If Z(x 1 ) is not connected, then we may restrict to the connected component of the identity which is still reductive and the above analysis holds as long as there is no choice of element x j which is fixed by the finite abelian subgroup of the center of H j−1 = Z(x 1 , . . . , x j−1 ). Otherwise, if some x j ∈ Z(x 1 , . . . , x j−1 ) is fixed by this diagonal subgroup ∆ then H j = Z(x 1 . . . , x j ) = DZ(x 1 , . . . , x j ) · CZ ⋊ ∆ is disconnected. If the semisimple part is not abelian, then we must choose elements in the semisimple part until it becomes abelian. At the point in which it does, say some H k−1 where 1 ≤ j < k − 1, choosing x k ∈ ∆ makes H k into a finite abelian group and thus we have again shown that there are only a finite number of normal subgroups H 1 , . . . , H k in G which are conjugate to the centralizer of some number of elements.
is a normal simple subgroup in DG which is characteristic in DG because it lies in the center of DG and the center is always characteristic for any group. H 1 is also closed since it is a semisimple subgroup in G compact. If x is a semisimple element, then it is conjugate to an element in the torus T which implies that the connected component Z 0 (x) contains a torus T ′ which is conjugate to T. In other words, Z 0 (x) = DZ 0 (x) · CZ 0 (x) where the semisimple part corresponds to a subsystem subgroup and the other is a commuting torus. To consider the remaining cases, we use the algorithm by Borel and de Siebenthal (Theorem 3 [5]) which determines all the closed subsystems of Φ(G) (all the roots of G) for closed subgroups of maximal rank. Take the extended Dynkin diagramD(G) and remove a finite collection of nodes, repeating the process with all the connected components of the resultant graph. These diagrams correspond to subdiagrams of the subroot systems and there is one conjugacy class of subsystem subgroups for each such subsystem. In particular, if G is connected, compact then it possesses only finitely many conjugacy classes of subsystem subgroups and hence the theorem is satisfied. It remains to prove what happens if G is disconnected because a direct product of connected subgroups will each have a finite number of subgroups each conjugate to Z(x i ) for some x i ∈ G. Note that there does not necessarily have to be any ordering on these H i . In fact, it would be virtually impossible to impose such a requirement because the H i are not necessarily subgroups of each other. If G is the semidirect product of a connected group U and a discrete group V then the above applies to the connected part and the only centralizer for any of the elements in the discrete subgroup is the entire discrete group. Thus there is a still finite number. If G is disconnected, then there is a finite number of subgroups of type Z G (x) when restricting to the connected component of the identity.
If G is a disconnected group where both the clopen sets U, V are connected, then there is still a finite number.
H 0 = G 0 . So G 0 ⊂ H G which implies that the components π 0 (H) < = π 0 (G). Thus we have shown that the Strong theorem implies the Weak theorem.