Topology of character varieties of Abelian groups
Abstract.
Let be a complex reductive algebraic group (not necessarily connected), let be a maximal compact subgroup, and let be a finitely generated Abelian group. We prove that the conjugation orbit space is a strong deformation retract of the GIT quotient space . As a corollary, we determine necessary and sufficient conditions for the character variety to be irreducible when is connected and semisimple. For a general connected reductive , analogous conditions are found to be sufficient for irreducibility, when is free abelian.
Key words and phrases:
Character varieties, Abelian groups, Commuting elements in Lie groups, Reductive group actions2010 Mathematics Subject Classification:
Primary 14L30, 14P25; Secondary 14L17, 14L24, 22E46Contents
1. Introduction
The description of the space of commuting elements in a compact Lie group is an interesting algebrogeometric problem with applications in mathematical physics, remarkably in supersymmetric YangMills theory and mirror symmetry ([7, 28, 47, 48]). Some special cases of this problem and related questions have recently received attention as can be seen, for example, in the articles [1, 4, 5, 16, 22, 36, 42].
Let be a compact Lie group and view the module as a free Abelian group of rank , for a fixed integer . The space of commuting tuples of elements in can be naturally identified with the set of group homomorphisms from to , by evaluating a homomorphism on a set of free generators for . From both representationtheoretic and geometric viewpoints, one is interested in homomorphisms up to conjugacy, so the quotient space , where acts by conjugation, is the main object to consider. Since every compact Lie group is isomorphic to a matrix group, it is not difficult to see that this orbit space is a semialgebraic space, but many of its general properties remain unknown.
In this article, we also consider the analogous space for a complex reductive affine algebraic group . More generally, in many of our results we replace the free Abelian group by an arbitrary finitely generated Abelian group .
In this context, it is useful to work with the geometric invariant theory (GIT) quotient space, denoted by , and usually called the character variety of (see Section 2 for details). The character varieties are naturally affine algebraic sets, not necessary irreducible.
Here is our first main result:
Theorem 1.1.
Let a complex reductive algebraic group, a maximal compact subgroup of , and a finitely generated Abelian group. Then there exists a strong deformation retraction from to .
We remark that in [16], the analogous result was shown for a free (nonAbelian) group of rank ; that is, the free group character variety deformation retracts to . In the same article, the special case , and of Theorem 1.1 was also shown.
More recently, Pettet and Souto in [36] have shown that, under the hypothesis of Theorem 1.1, deformation retracts to . This motivated our related but independent proof of Theorem 1.1, which is very different from the argument in [16].
In fact, there are great differences between the free Abelian and free (nonAbelian) group cases. For instance, the deformation retraction from to is trivial, although the deformation from to is not. Moreover, in the case of free Abelian groups the deformation is not determined in general by the factorwise deformation (as happens in the free group case), since by a recent result, any such deformation cannot preserve commutativity (see [45]). We conjecture that the analogous result is valid for right angled Artin groups, a class of groups that interpolates between free and free Abelian ones (see Section 4.3).
Returning to the situation of a general finitely generated group , the interest in the spaces and is also related to their differentialgeometric interpretation. Consider a differentiable manifold whose fundamental group is isomorphic to (when , we can choose to be an dimensional torus). By fixing a base point in and using the standard holonomy construction in the differential geometry of principal bundles, one can interpret as the space of pointed flat connections on principal bundles over , and as the moduli space of flat connections on principal bundles over (see [47]).
The use of differential and algebrogeometric methods to study the geometry and topology of these spaces was achieved with great success when is a closed surface of genus (in this case is nonAbelian), via the celebrated NarasimhanSeshadri theorem and its generalizations, which deal also with noncompact Lie groups (see, for example [3, 26, 35, 43, 44]). Indeed, the character varieties introduced above can be interpreted as a moduli space of polystable Higgs bundles over a compact Kähler manifold with the fundamental group of the manifold, or central extension thereof (which yields an identification in the topological category, but not in the algebraic or complex analytic ones). Some of the multiple conclusions from this approach was the determination of the number of components for many spaces of the form for a closed surface and real reductive (not necessarily compact or complex) Lie group (see for example [8, 18]). We remark that these character varieties are also main players in mirror symmetry and the geometric Langlands programme (see, for example [19, 25, 29]).
In contrast, for the case of , the number of path components of has only recently found a satisfactory answer (for general compact semisimple and arbitrary ), see [28].
Using this determination, and also a very recent result by A. Sikora [42], one of the main applications of Theorem 1.1 is the following theorem. By the classification of finitely generated Abelian groups, any such group can be written as where is called the rank of and is the finite group of torsion elements (all of those having finite order). We say that is free if is trivial.
Theorem 1.2.
Let be a semisimple connected algebraic group over , and let be the rank of the Abelian group . Then is an irreducible variety if and only if:
is free, and
Either , or and is simply connected, or and is a product of ’s and ’s.
An interesting consequence of this result is a sufficient condition for irreducibility of for a general connected reductive (see Corollary 5.14), generalizing the result in [42, Prop. 2.6].
Theorem 1.2 should be considered in relation to an analogous problem in a different context. Let be the algebraic set of commuting tuples of complex matrices. Determining the irreducibility of is a surprisingly difficult linear algebra problem, related to the determination of canonical forms for similarity classes of general tuples of matrices (see [20]). In fact, as a consequence of deep theorems by Gerstenhaber and Guralnick (see [21, 23]) was shown to be reducible when , and when and ; moreover, is irreducible when (for all ) and when and . The remaining cases: and strictly between 8 and 32, are still open (as far as we know). These results yield corresponding statements for , when or .
We finish the Introduction with a summary of the article. The proof of Theorem 1.1 is divided into three main steps. The first step, carried out in Section 2, consists in obtaining the identifications:
Here, for a subset , we let , and the superscript refers to the subset of representations with closed orbits (called polystable). These identifications hold, in fact, for any finitely generated group . For the second step, in Section 3, we restrict to a fixed Abelian . It consists in showing that one can replace the polystable representations by “representations” into , the semisimple part of , and that we have , where .
Finally in Section 4, the proof is completed by constructing a strong deformation retraction from to with a certain number of desired properties. This last part of the proof is inspired by the methods and results of [36], although is selfcontained. Note also that (except for the partial results on right angled Artin groups) we do not need to use their generalized Jordan decomposition, since in our GIT quotient framework, we can work directly with polystable representations.
In Section 5, besides proving Theorem 1.2, we also study two different characterizations of a special component , usually called the identity component. This, together with known results on the compact group case, provides a final application of Theorem 1.1 to the simple connectivity and to the cohomology ring of the character varieties , in a few examples, such as when is or .
2. Quotients and Character Varieties
An affine algebraic group is called reductive if it does not contain any nontrivial closed connected unipotent normal subgroup (see [6, 27] for generalities on algebraic groups). Since we do not consider Abelian varieties in this paper, we will abbreviate the term affine algebraic group to simply algebraic group.
Let be a complex reductive algebraic group (we include the possibility that is disconnected), and let be a complex affine variety, that is, we have an action , which is a morphism between affine varieties. We use the term affine variety to mean an affine algebraic set, not necessarily irreducible. All of our algebraic groups and varieties will be considered over . In particular, the group is also a Lie group.
This action induces a natural action of on the ring of regular functions on . The subring of invariant functions is finitely generated since is reductive, and defines the affine GIT (geometric invariant theory) quotient as the corresponding affine variety, usually denoted as:
(2.1) 
See [12] or [34] for the details of these constructions and an introduction to GIT. If , we denote by or by its orbit in , and by its extended orbit in which is, by definition, the union of the orbits whose closure intersects . As shown in [12], is exactly the space of classes . In the context of GIT, one usually considers the Zariski topology. However, being interested also in the usual topology on Lie groups, we will always equip our variety with a natural embedding into an affine space . This induces on a natural Euclidean topology. When we need to distinguish the two kinds of topological closures, we use a label in the overline; we will mean the Euclidean topology, when no explicitly reference is made.
We observe that every orbit is the image of the action map, which is algebraic, so is constructible and thus contains a Zariski open subset , which is dense in its Zariski closure . Hence, is also open and dense, in the Euclidean topology, inside . Therefore, the Euclidean and Zariski closures of orbits coincide:
2.1. The polystable quotient
If the orbit of is closed, , we say that is polystable. Denote the subset of polystable points by . Since for any , , it is clear that is a space. It is not, in general, an affine variety.
We first show that, considering Euclidean topologies, there is a natural homeomorphism between the polystable quotient and the GIT quotient .
Consider the canonical projection , and the GIT projection . Define also to be the canonical map which sends a orbit to its extended orbit . These maps form the following diagram, whose commutativity is clear:
(2.2)  
In [32, 31], is shown to be a closed mapping on invariant Euclidean closed sets (see also [39], page 141).
Theorem 2.1.
Let be a complex reductive algebraic group, and a complex affine variety with the natural Euclidean topology coming from . Then is a homeomorphism.
Proof.
With respect to the Euclidean topologies, and the induced topologies on the quotients, the map is continuous. This follows from the identification of with the composition of continuous maps
We will show that is a bijection and its inverse is continuous. It is a standard fact (see [12, 34]) that every extended orbit equivalence class contains a unique closed orbit, and this closed orbit lies in the closure of all of the others in that class. In particular, for every whose orbit is closed and . So, and . Thus, which shows is surjective. there exists a representative
Since two orbits and are contained in if and only if , no two distinct closed orbits are identified in the GIT quotient . Thus is injective.
Finally, let us show that is a closed map. Suppose that is a closed set. Then, by definition is invariant. By commutativity of the diagram (2.2) we have
Let be the Euclidean closure of in . By definition of subspace topologies, since is closed in , we have .
Now let us show that , the set of limit points of sequences in , is invariant. Let . If in , then clearly is in . Otherwise, there is a sequence converging to . But acts by polynomials and hence is continuous in the Euclidean topology. Therefore, limits to and is in for all and since is invariant. Therefore, is in the limit set and hence in .
Obviously, . Let us show the reverse inclusion. If , then, as above, there is a whose orbit is closed and . In particular, , so there exists such that . Thus, since is invariant and (Euclidean) closed, . Note that here we are using the fact, observed above, that the Euclidean and Zariski closures of orbits coincide. Therefore , and so . . We conclude that indeed . This implies is an extended orbit for some
Since is closed, invariant in and is a closed map for invariant sets as mentioned above, we conclude that is a closed set. This shows that is a closed map. Being bijective, it is also an open map. So, the inverse of is continuous, and hence is a homeomorphism. ∎
Remark 2.2.
Since is a complete metric space, the same holds for as well. The metric can be explicitly given as follows. Let generate the ring of invariants , define to be the mapping , and let be the Euclidean metric on . For define . Thus is welldefined since is invariant; and is nonnegative, symmetric and satisfies the triangle inequality because does. It is not definite however. Since if the Zariski closure of and intersect (even if they are not equal orbits), this problem is exactly fixed upon restricting to .
2.2. Polystable and compact quotients for character varieties
Let be a finitely generated group, be a Lie group, and consider the representation space , the space of homomorphisms from to . For example, when is a free group on generators, the evaluation of a representation on a set of free generators provides a homeomorphism with the Cartesian product
where we consider the compactopen topology on with given the discrete topology.
In general, by choosing generators for (for some ), we have a natural epimorphism . This allows one to embed and consider on the Euclidean topology induced from the manifold .
The Lie group acts on by conjugation of representations; that is, for and . Let be the quotient space by this action, and let denote the identification space , where two orbits are equivalent if and only if they are members of a chain of orbits whose closures pairwise intersect.
Two main classes of examples are important for us, for both of which the induced topology on will be Hausdorff. When is a compact Lie group, this is the usual orbit space (which is semialgebraic and compact) since all such orbits are closed, so . When is a reductive algebraic group over each equivalence class is indeed an extended orbit in the sense defined just before subsection 2.1(see [34]). So, in this case the quotient can be identified with the GIT quotient considered in (2.1):
and is called the character variety of . In either case, the spaces will be semialgebraic sets and thus CWcomplexes in the natural Euclidean topologies we consider in this paper.
Recall that denotes the subset of consisting of representations whose orbit is closed. Applying the map from Proposition 2.1 to character varieties, we obtain the following proposition.
Proposition 2.3.
Let be a complex reductive algebraic group, and be a finitely generated group. Then, the natural map is a homeomorphism.
Let now be a fixed maximal compact subgroup of a complex reductive algebraic group . Then is the Zariski closure of . Over , all reductive algebraic groups arise as the Zariski closure of compact Lie groups . In particular, as discussed in [39], we may assume is a real affine variety by the PeterWeyl theorem, and thus is the complex points of the real variety .
Using the fact that is also isomorphic to the unique complexification of , one can show the following.
Proposition 2.4.
Let be a finitely generated group and . Then the inclusion mapping induces an injective mapping
such that is a CWsubcomplex of .
Proof.
Since is the complex points of the real variety , the real points of coincide with . In the same way, the set of real points of the affine variety is precisely . Since is compact and stable under , the result is a direct consequence of [17, Thm. 4.3]. ∎
Given a representation , the subset is the group algebraically generated by , where are the generators of .
Lemma 2.5.
Let be a finitely generated group and , Then .
Proof.
Suppose . Then the Euclidean closure of in is a compact subgroup of . This implies that the Zariski closure of coincides with the Zariski closure of in and hence it is a reductive algebraic group (precisely equal to the complexification of ). Thus, is a linearly reductive group (all its linear representations are completely reducible). In [38, Thm. 3.6], it is shown that is linearly reductive if and only if the orbit of in is closed. We conclude is closed, that is, . We note that Richardson’s result in [38] is stated for tuples of elements in . The case of closed invariant subsets of , such as our , is an easy consequence (see also [15]). ∎
Define the mapping as the composition
We want now to describe the image of .
Proposition 2.6.
The following diagram is commutative:
Consequently, as CW complexes.
Proof.
Since all mappings in the diagram are composites of natural inclusions and projections, they are continuous. The top triangle of maps is commutative by definition of . Note that is the cellular inclusion from Proposition 2.4. Then Proposition 2.4 implies that all equivalent valued representations are in fact equivalent (else would not be injective). Therefore, since , we conclude that is also injective. Therefore, the bottom triangle of maps is commutative. ∎
Now define
Clearly, as a subspace since all representations have closed orbits by Lemma 2.5, and conjugates of representations with closed orbits likewise have closed orbits (since ).
Proposition 2.7.
is an embedding and .
Proof.
Proposition 2.6 implies that is a continuous injection. We now show it is onto . First, let . For any , is a conjugate of a conjugate of and so ; thus . Conversely, let . Then by definition, there exists such that . Thus and ; we conclude . Therefore,
From Proposition 2.3, and so is Hausdorff (the latter is an algebraic subset of some , with the Euclidean subspace topology). On the other hand, is compact (a closed subset of the compact is a compact set, and a compact quotient of a compact space is compact). Since a continuous injection from a compact space to a Hausdorff space is an embedding, we are done. ∎
We record the following fact for later use.
Proposition 2.8.
for any reductive algebraic groups .
Proof.
Write as . Clearly, and under this identification, the action separates into an independent action on and action on . Thus, as orbit spaces . Moreover, since the GIT quotients are determined by orbit closures, we conclude our result simply by noting that ∎
3. Finitely generated Abelian groups
From now on, we let be a finitely generated Abelian group. By the classification of finitely generated Abelian groups, there are integers and such that
where denotes the cyclic group . In general, is an algebraic subvariety of , given by where generate and is a generator of for . Recall that an element in is called semisimple if for any finite dimensional rational representation of the element acts completely reducibly. Let denote the set of semisimple elements in . Then define
This is an abuse of notation (since is not a group) but a harmless one, in view of the next result. Since is preserved by conjugation, acts on by simultaneous conjugation. In what follows we will often abbreviate . Recall that a diagonalizable group is an algebraic group isomorphic to a closed subgroup of a torus (see [6]).
Proposition 3.1.
Let be a finitely generated Abelian group and let be the Zariski closure of . Then, the following are equivalent:
is a diagonalizable group
is a reductive group
In particular, .
Proof.
In [38], it is shown that the Zariski closure of in is linearly reductive if and only if the orbit of in is closed. Since being reductive is equivalent to linearly reductive (in characteristic 0), this shows the equivalence between (2) and (4) (which is in fact valid for any not necessarily Abelian).
Now let be a generating set for and let , for a fixed . Since is Abelian, , the subgroup generated by , is an Abelian subgroup of . So, the Zariski closure of is an Abelian algebraic group (since commutation relations are polynomial). Now suppose that which means by definition that , and consider a linear embedding of , . Then, the matrices can be simultaneously conjugated by an element in to lie in some maximal torus of . Because is Zariski closed in , this means that for every we will have . Recall that the multiplicative Jordan decomposition is preserved by homomorphisms: for with and unipotent, we have and . Thus, for , we have which implies , and since is injective . So, consists only of semisimple elements of and by [6] this means that is a diagonalizable group, and hence reductive. Thus, (1) implies (3) and so (4). Conversely, let be reductive. Since it is also Abelian (as is Abelian) then, again by [6] it consists of semisimple elements. In particular, . So (4) implies (1) as well.∎
Remark 3.2.
The fact that is Abelian is crucial in Proposition 3.1. Indeed, if is not Abelian, neither of the inclusions or is true in general. Here are simple counterexamples. Let , the free group of rank , and . Let with are each not . Then both and are semisimple, so . However, by conjugating with goes to we see that the tuple limits to and taking a limit as , where and
satisfying and . If is an irreducible representation, then its orbit is closed; and if it were reducible, then could be made simultaneously uppertriangular. However, a simple computation shows this to be impossible. Thus, , but , so .
Next, define to be the set of all conjugates of the group , and consequently define
Clearly, .
Lemma 3.3.
If is a finitely generated Abelian group, then
Proof.
Since for any , it suffices to prove the reverse inclusion for finitely generated Abelian groups. This argument follows the one in [36, pg. 16]. Let be generated by and let . Let be the Zariski closure of the group generated by . Since only consists of semisimple elements, the proof of Lemma 3.1 tells us that is Abelian and consists of only semisimple elements. Since is algebraic it has a maximal compact subgroup . Since it is Abelian, is unique. Since each is conjugate to an element in , each is in some maximal compact subgroup. Therefore, each of them is in the unique maximal compact . However, all maximal compact subgroups are conjugate, so there exists a so that which in turn implies that for all . By definition, this implies and as such . ∎
Summarizing the last two sections, we have shown that when is a finitely generated Abelian group, we can replace the right inclusion, with the the left inclusion in the following diagram:
We will now show that there is a equivariant strong deformation retraction .
4. Deformation Retraction of Character Varieties
Recall that a strong deformation retraction (SDR) from a topological space to a subspace is a continuous map is the identity on , (2) for all and , and (3) . In short, it is a homotopy relative to between the identity on and a retraction mapping to . We are going to construct an explicit equivariant strong deformation retraction from to . such that (1)
Following PettetSouto [36], we start with a deformation in the case when . Let be the subgroup of diagonal matrices in , which is a maximal torus, identified in the usual way with a subgroup of . Consider the following deformation retraction from to the subset :
(4.1)  
where, for , and ,
The strong deformation retraction properties of are easily established. Note that is a homomorphism for every .
Suppose that is semisimple, which means it is diagonalizable. Since all maximal tori are conjugate, there is (depending on ) so that . Define the following map:
(4.2) 
by letting
We have the following properties of .
Lemma 4.1.
The map satisfies:

is well defined; that is, it does not depend on the choice of ;

for all and all ;

is a strong deformation retraction from to the set of conjugates of .
Moreover, for every , we have:

is equivariant;

if then ;

for all , and all
Proof.
Let . For any group , one can define the power map , by If and we say that is an th root of .
For later convenience, we here record the following fact about the power map on . Recall that denotes the subgroup of diagonal matrices in .
Lemma 4.2.
Let be the power map. Then .
Proof.
One inclusion is clear. If then so that by definition. For the converse, assume that , which means that , so is semisimple. Let be the multiplicative Jordan decomposition, so that is semisimple, is unipotent and . Then . Since powers of diagonalizable and unipotent matrices remain respectively diagonalizable and unipotent, is semisimple and is unipotent. By the uniqueness of the Jordan decomposition, we conclude that and . Since the exponential map is a diffeomorphism between nilpotent and unipotent matrices in , we have which implies that