Arithmetic and metric aspects of open de Rham spaces

In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank n$n$ bundle on P1$\mathbb {P}^1$ . The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four‐dimensional cases are expected to be of type ALF.

In the paper [27] , the Hodge structure on the cohomology of character varieties of representations of the fundamental group of a Riemann surface was studied, using arithmetic harmonic analysis. It resulted in a conjecture [ [24], where a more geometric purity conjecture [24,Remark 1.3.1] appeared. First, we recall this conjecture.

Tame purity conjecture
For , ∈ ℤ >0 , let = ( 1 , … , ) ∈  be a -tuple of partitions of . Let G ∶= GL (ℂ). An orbit  ⊂ ∶= (ℂ) for the adjoint action of G has type ∈  if the partition given by the multiset of multiplicities of the eigenvalues of any element in  is . Let ( 1 , … ,  ) be a generic -tuple of semisimple adjoint orbits in of type in the sense of Definition 3.1.6. Then the variety where is the × identity matrix. The character variety parameterizes isomorphism classes of -dimensional representations of the fundamental group of ℙ 1 ⧵ { 1 , … , }, with monodromy around in  . One has the Riemann-Hilbert monodromy map The proof of the purity of * ( * ; ℚ) in [24,Proposition 2.2.6] proceeds by recalling [15] the identification of  * as a certain star-shaped Nakajima quiver variety. In particular,  * acquires a natural complete hyperkähler metric of ALE type. For example, for the star-shaped affine Dynkin diagrams of̃4,̃6,̃7, and̃8 together with an imaginary root, we obtain the corresponding asymptotically locally Eucledian (ALE) gravitational instantons of Kronheimer [36].

Irregular purity conjecture
The tame moduli spaces and the Riemann-Hilbert monodromy map above were generalized to allow irregular singularities in [7]. The aim of the present paper is to extend the above purity ideas to the case of meromorphic connections with irregular singularities. Consider the following analog of (1.1.1). Fix , , ∈ ℤ >0 and we will take and ( 1 , … ,  ) as in Section 1.1. In addition, for 1 ⩽ ⩽ , let ∈ ℤ >1 and consider the truncated polynomial ring R ∶= ℂ[ ]∕( ), and the group GL (R ) of invertible matrices over R . Let T ⊂ G be the maximal torus of diagonal matrices and denote its Lie algebra. We will fix an element of the form with ∈ , further assuming that has distinct eigenvalues. An element g ∈ GL (R ) acts on by conjugation, viewing both g and as matrices over the ring of Laurent polynomials over ; however, we will truncate any terms with nonnegative powers of . We denote the GL (R )-orbit of under this action by ( ) (it is explained in Section 2.1 how we may view as an element of the dual of the Lie algebra of GL (R ), and ( ) as its coadjoint orbit). Observe that G = GL (ℂ) sits in each GL (R ) as a subgroup and so acts on each ( ); in fact, given an element

2.2)
the G-action is by conjugation on each term . Now, for 1 ⩽ ⩽ , we set ∶= − 1, and write ∶= ( 1 , … , ) for the tuple. We may then construct the (irregular) open de Rham space as the smooth affine GIT quotient One likewise has an interpretation of  * , as a moduli space of meromorphic connections (see Section 3.2), this time with poles of higher order, on the trivial rank vector bundle over ℙ 1 . The class of ( , ) yields a connection for a set of distinct poles { 1 , … , , 1 , … , } ∈ ℙ 1 ⧵ {∞}. The definition of the corresponding wild character variety  , , which is the space of monodromy data for moduli spaces of irregular connections, is a little more involved than in the logarithmic case (see [8,Equation (2)]). For the poles of higher order , in addition to the topological monodromy, one must also take into account the Stokes data, which distinguish analytic isomorphism classes of locally defined connections from formal ones. However, even in this irregular case, the wild character variety retains certain similarities with the character variety defined above at (1.1.2): it is a smooth affine variety defined only in terms of G, certain algebraic subgroups of G, and orbits in them, and may be viewed as a space of "Stokes representations" [7, §3], though we will not be working with this space directly. There is again a Riemann-Hilbert monodromy map [8,Corollary 1] ∶  * , →  , (1.2.3) in this irregular case, which takes a connection to its monodromy data. An explanation of why the wild character variety takes the form that it does and a detailed description of the monodromy map may be found at [7, §3]. We then have the following.

Main results and layout of the paper
To formulate our main result, we fix integers g ⩾ 0 and > 0. is the modified Macdonald symmetric function defined in [22, (11)]; see [24, §2.3.4] for more details. Finally, we let where ℎ ( ) are the complete symmetric functions, (1 ) ( ) are the Schur symmetric functions in the corresponding variables, and ⟨⋅, ⋅⟩ is the extended Hall pairing and Log is the plethystic logarithm. The notation is explained in more detail in Section 5.2; see also [24, §2.3] for detailed explanations of the formalism. Note that ℍ , (− , ) as defined is a rational function in ℚ( , ), but conjecturally, because of Conjecture 1.3.2 below, it is a polynomial function in ℚ [ , ] with positive integer coefficients.
One of our main results is the computation of the weight polynomial of the open de Rham space  * , . Thus, if * ( * , , ℚ) were pure, our main Theorem 1.3.1 would be a consequence of our purity conjecture Conjectures 1.2.4 and 1.3.2. However, we were unable to prove that * ( * , , ℚ) is always pure and we only state it as Conjecture 5.2.7.
The proof of Theorem 1.3.1 is first performed, as Theorem 4.3.1, in the case of = 0, that is, only irregular punctures. In this case, we can proceed by motivic Fourier transform, as in [52], and the result will be a motivic extension of Theorem 1.3.1. The general case -Theorems 5.1.6 and 5.2.3 -is then proved via the arithmetic harmonic analysis technique of [24].
As an analog of Crawley-Boevey's result [15] for the irregular case, in Section 6, we prove that the open de Rham spaces  * , are isomorphic to quiver varieties with multiplicities. These varieties have been considered by Yamakawa [54] in the rank 2 case and their defining equations, in terms of certain preprojective algebras, have been studied by Geiss-Leclerc-Schröer [23] in general. Then, in Section 6.4.2, we consider the star-shaped nonsimply laced affine Dynkin diagrams that correspond to open de Rham spaces of dimension 2. Here, we discuss the main results of the paper in these special toy cases.
Finally, in Section 7, we prove the existence of natural complete hyperkähler metrics on  * , when the irregular poles have order 2. Existence of such metrics was discussed in [10, §3.1]), but as there seems to be no complete proofs given in the literature, we provide the details here. In the (real) four-dimensional toy example cases, for example, those appearing in Section 6.4.2, we expect the resulting metrics to be of type ALF, i.e. asymptotically locally flat.

Groups, Lie algebras, and their duals over truncated polynomial rings
Let us fix a perfect base field and an integer ⩾ 1. We will set G ∶= GL ( ) and denote by ∶= ( ) its Lie algebra. Furthermore, T ⊂ G will denote the standard maximal torus consisting of the invertible diagonal matrices, ⊂ its Lie algebra, and reg ⊂ the subset of elements with distinct eigenvalues. Fix another integer ⩾ 1 and let R ∶= [[ ]]∕( ) = [ ]∕( ). Then we may also consider these groups over R , G ∶= GL (R ) = { g 0 + g 1 + ⋯ + −1 g −1 | g 0 ∈ G, g 1 , … , g −1 ∈ } , (2.1.1) and we define T and similarly. We will regard G and T as algebraic groups over : from the description above, G = G × −1 as a -variety; writing out the components of each under the group law in G , it is easy to see that the operation is well defined on tuples, and so, one indeed gets an algebraic group over . Of course, is a vector space over . Observe that GL (R ) is not reductive; its unipotent radical G 1 ⊂ G and Lie algebra 1 are, respectively, where denotes the × identity matrix. There is a semidirect product decomposition where we identify G with the subgroup of those elements satisfying in the notation of (2.1.1); we will often refer to G identified as such as the subgroup of constant elements in G . We thus obtain a direct sum decomposition this decomposition is preserved by the adjoint action of G but not of G . We will write T 1 ∶= T ∩ G 1 and 1 ∶= ∩ 1 . It will be convenient to identify the dual vector space ∨ with via the trace residue pairing. This means that for ∈ and ∈ − , we set Under this identification, the dual ∨ of the subgroup G ⊆ G corresponds to the subspace −1 ⊂ − and ( 1 ) ∨ to those elements in − having zero residue term, that is, 1 = 0. We write for the natural projections. The latter projection may be identified with so we are simply truncating the residue term. The adjoint and coadjoint actions of G on and ∨ will both be denoted by Ad and are defined by the same formula: for g ∈ G , What we mean in the latter case is that we consider g, g −1 and as matrix-valued Laurent polynomials in and we truncate all terms of nonnegative degree in after multiplying. With this convention, we have ⟨ Let L ⊂ G be the subgroup of block diagonal matrices, with blocks of sizes 0 , … , , that is, Let denote its Lie algebra. The center of is given by We will write ( ) reg for the subset of ( ) for which the are pairwise distinct. The center of L satisfies Z(L ) = ( ) ∩ G, and can be described as the subset of ( ) with all ∈ × . An important special case is when = (1, … , 1), so one has = and ( ) reg = reg . We will also use the notation as above for these groups, namely, L , ∶= L (R ) L 1 , ∶= L , ∩ G 1 , ∶= (R ) 1 , ∶= , ∩ 1 .

Coadjoint orbits
Coadjoint orbits for groups of the form G will play a prominent role in this paper, so here we will set some notation and record some results that will be used later.

Conventions and variety structure
By a diagonal element or formal type of order , we will mean an element in ∨ of the form By permuting diagonal entries if necessary, may be written for some partition ( 0 , … , ) of , with the ∈ − R distinct. In fact, we will make the stronger assumption that for all 0 ⩽ ≠ ⩽ This assumption is stronger than what is required, for example, in [6, Main assumption], but many of our arguments will rely crucially on (2.2.3). (a) The coadjoint orbit ( ) is an affine variety.
commutes. In particular, ( ) is a homogeneous space for G and is a categorical quotient.
We will give a description of the centralizers G ( ) for certain in Lemma 2.2.8(d) below.
Proof. To see (a), we note that ( ) will be the set of elements satisfying the appropriate minimal polynomial over R , which may then be written down as equations over . To be explicit about this, let ∈ ∨ be written as in (2.1.4); we consider as an element of by writing Similarly, for 1 ⩽ ⩽ , we consider ∈ R by writing Then the minimal polynomial as mentioned gives defining equations for ( ) in the sense that the following matrix over R is zero if and only if ∈ ( ): Of course, this can be expanded and the coefficient of each power gives a matrix equation over ; the individual entries of all of these matrices give the algebraic equations for ( ). The fact that these are indeed defining equations for ( ) can be proved in the same way one proves the theorem that asserts that a matrix is diagonalizable if and only if its minimal polynomial has distinct roots; for this, we need to use the assumption (2.2.3).
For (b), notice that both G and ( ) are smooth over and for every closed point ∈ ( ), one has dim −1 ( ) = dim Z G ( ). Since is surjective, it is thus faithfully flat by [

Computational lemmata
Fix a tuple = ( 0 , … , ) ∈ ℕ +1 with ∑ = and consider the group L ⊂ G as in (2.1.8). Let od ⊆ denote the subspace consisting of matrices with zeros on the diagonal blocks (of course, the superscript can be read, "off diagonal"), so that we have an obvious L -invariant decomposition = ⊕ od . (2.2.5) In the case = (1, … , 1), we set od ∶= od , so that this is the space of matrices with zeroes along the main diagonal.
∈ G 1 . We may write where the ∈ are given by Using the fact that = − + … , we have the explicit formula be given as in (2.2.2) with ∈ ( ) reg and suppose g ∈ G is such that Ad g − ∈ ∨ , −1 . Then g ∈ L , and Ad g = .
Proof. The first two statements can be seen by writing everything out as block matrices. For part (c), by (2.1.2), we may write g = g 0 for some g 0 ∈ , ∈ G 1 . Thus, Ad g can be obtained by applying Ad g 0 to the expression in (2.2.7). The − term is then Ad g 0 ; the hypothesis is that this is . Then part (b) implies that g ∈ ∩ G = L . Already this shows that Ad g 0 = , since all ∈ ( ). Now, the −( −1) term in Ad g − is then The assumption is that this lies in ; by L -invariance, [ 1 , ] ∈ and again by (b), 1 ∈ . By induction, we may assume 1 , … , ∈ . We will show that +1 ∈ . Then from (2.2.7), the −( − −1) -term of Ad g − is ) .
The induction hypothesis implies that the only commutator that does not vanish is [ +1 , ], and then, by assumption, this lies in , and again, we conclude by (b). This shows that all ∈ and hence ℎ ∈ L 1 , ; as g 0 ∈ L , g = g 0 ∈ L , . Furthermore, as above, we can show that all commutators in (2.2.7) vanish, and we have already noted Ad g 0 = ; hence Ad g = .
The first assertion in part (d) follows from (c). The second assertion uses the same argument, and one simply needs to observe that we are omitting the residue term when computing in 1 , and hence, no condition is imposed on −1 .

Lemma 2.2.9.
(a) The restriction of the multiplication map yields isomorphisms In other words, every ∈ G 1 has unique factorizations = od T = 1,od rc with od ∈ G od , T ∈ T 1 , 1,od ∈ G 1,od , rc ∈ G 1,rc , and the map taking to any one of these factors is a morphism.
where the action of T on G × G od is given by Proof. To prove part (a), consider elements Then the product expression = imposes the relations for 1 ⩽ ⩽ − 1. Assuming that ∈ G 1 is arbitrary, the equation 1 = 1 + 1 allows us to take 1 ∈ od , 1 ∈ to be the off-diagonal and diagonal parts of 1 , respectively. The relations (2.2.10) allow us to continue this inductively, so that all ∈ od and ∈ , noting that at each stage, one has an algebraic expression in terms of the . This gives the first factorization. The second one is obtained in exactly the same way, except that the conditions are that −1 = 0, but there is no condition on −1 , which makes up for it.
Part ( by part (a). Taking the quotient by T = T × T 1 then gives the desired isomorphism. One obtains the indicated T-action on G × G od by identifying G × G 1 with G via multiplication. The final statement follows since G → G∕T is a Zariski locally trivial principal T-bundle, as T is a special group [48, §4.3]. □

Grothendieck rings with exponentials
Following [14] and [12], in this section, we introduce Grothendieck rings with exponentials, a naive notion of motivic Fourier transform and convolution. Similar techniques were used in [52] to compute the motivic classes of Nakajima quiver varieties. Throughout this section, will denote an arbitrary field; by a variety, we mean a separated scheme of finite type over ; and by a morphism of varieties, we will mean a -morphism.

Definitions
The Grothendieck ring of varieties, denoted by KVar, is the quotient of the free abelian group generated by isomorphism classes of varieties modulo the relation (ii) For a variety and 1 ∶ × 1 → 1 the projection onto 1 , the relation The class of ( , ) in KExpVar will be denoted by [ , ]. We define the product of two generators [ , ] and [ , g] as where • + g • ∶ × → 1 is the morphism sending ( , ) to ( ) + g( In general, * is a morphism of rings and ! a morphism of additive groups.

Realization morphisms
The rings KVar and KExpVar and their localizations M , E M , although easy to define, are quite hard to understand concretely. To circumvent this, one usually considers realization morphisms to simpler rings. If = is a finite field, there is a ring homomorphism KVar → ℤ sending the class of a variety ∕ to the number of -rational point of . More generally for a variety over , we can construct a realization from KExpVar to the ring Map( ( ), ℂ) of complex valued functions on ( ) by sending the class of [ , ] to the function where Ψ ∶ → ℂ × is a fixed nontrivial additive character and the fiber over . Under this realization, for a morphism ∶ → , the operations ! and * correspond to summation over the fibers of and composition with , respectively.
If is a field of characteristic 0, whose transcendence degree over ℚ is at most the one of ℂ∕ℚ, we can embed into ℂ and consider any variety over as a variety over ℂ. By the work of Deligne [17,18], the compactly supported cohomology * ( , ℂ) of any complex algebraic variety carries two natural filtrations, the weight and the Hodge filtration. Taking the dimensions of the graded pieces, we obtain the compactly supported mixed Hodge numbers of ) .
From these numbers, we define the E-polynomial as This way we obtain a morphism (see, e.g., [27,Appendix]) It is not hard to see that ( ; , ) = , and thus, this realization extends to a morphism ] .
These two realizations of KVar are related by the following theorem of Katz. We refer to [27,Appendix] for the precise definition of a strongly polynomial count variety over ℂ. In particular, in the situation of Theorem 2.3.4, the -polynomial of is a polynomial in one variable, which we also call the weight polynomial

Computational tools
We now introduce several tools for our computations in KVar and KExpVar, which are mostly inspired by similar constructions over finite fields through the realization (2.3.1).

Fibrations
We

Character sums
When computing character sums over finite fields, one has the following crucial identity: where is a finite-dimensional vector space over and ∈ ∨ a linear form. To establish an analogous identity in the motivic setting, we let be a finite-dimensional vector space over and a variety. We replace the linear form above with a family of affine linear forms, that is, a morphism g = (g 1 , g 2 ) ∶ → ∨ × , where is an -variety. Then we define to be the morphism Finally, we put = g −1 1 (0).

Fourier transforms
We now define the naive motivic Fourier transform for functions on a finite-dimensional -vector space and the relevant inversion formula. All of this is a special case of [14, Section 7.1].

Definition 2.3.7.
Let ∶ × ∨ → and ∨ ∶ × ∨ → ∨ be the obvious projections. The naive Fourier transformation  is defined as Here ⟨ , ⟩ ∶ × ∨ → denotes the natural pairing. We will often write  instead of  when no ambiguity will arise from doing so.
Of course, the definition is again inspired by the finite field version, where one defines for any function ∶ → ℂ the Fourier transform at ∈ ∨ by Notice that  is a homomorphism of groups, and thus, it is worth spelling out the definition in the case when = [ , ] is the class of a generator in KExpVar . Letting ∶ → be the structure morphism, we simply have We have the following version of Fourier inversion.

Convolution
Finally, we introduce a motivic version of convolution.
As expected, the Fourier transform interchanges product and convolution product. Proof. As both  and * are bilinear, it is enough prove the identity for two generators [ , ], [ , g] ∈ KExpVar with respective structure morphisms ∶ → , ∶ → . Using (2.3.8), we can then directly compute On the other hand, using the natural isomorphism Remark 2.3.12. We will use the convolution product to study equations in a product of varieties, that is, consider -varieties ∶ → say for = 1, 2. Then it follows from the definition of * that for any ∶ Spec → , the class of {( 1 , 2 ) ∈ 1 × 2 | 1 ( 1 ) + 2 ( 2 ) = } is given by ). Proposition 2.3.11 allows us to compute the latter by understanding the Fourier transforms  ( 1 ) and  ( 2 ) separately.

OPEN DE RHAM SPACES
In this section, we define open de Rham spaces as an additive fusion of coadjoint orbits, similar as in [29, §2]. They were first introduced in [7, Section 2] as certain moduli spaces of connections on ℙ 1 . We recall this viewpoint briefly in Section 3.2.

Additive fusion products of coadjoint orbits
As before, let be a fixed base field. Fix a diagonal element ∈ ∨ as in (2.2.1) and consider its G -coadjoint orbit ( ) ⊆ ∨ . In the case that = ℝ or ℂ, G is a real or complex Lie group, accordingly, and the coadjoint orbit  Now, for ∈ ℤ >0 , let ∈ ℤ >0 for 1 ⩽ ⩽ , and let ∈ ∨ be diagonal elements and ( ) their coadjoint orbits. We may form the product ∏ =1 ( ) that has the product symplectic structure. It also carries a diagonal G-action for which the moment map is In this case where = ℝ or ℂ, we can then form the symplectic (Marsden-Weinstein) quotient in the usual way. We now observe that if is any field, then both (3.1.1) and (3.1.2) are defined over .
Except possibly in Section 6, we will typically impose two more conditions on , regularity and genericity.
where ⟨, ⟩ is the pairing defined in (2.1.5). In other words, if is generic, there are no nontrivial subspaces 1 , … , ⊊ of the same dimension such that is invariant under for 1 ⩽ ⩽ and ∑ tr 1 | = 0.
Now, let be a regular generic tuple of formal types. We adapt the following notation so as to later (Section 5.2) make comparisons with [26]. First, we order 1 , … , in a way such that 1 = ⋯ = = 1 and +1 , … , + ⩾ 2, with + = . We write = ( 1 , … , ) ∈  for the -tuple of partitions of defined by the multiplicities of the eigenvalues of 1 for 1 ⩽ ⩽ . When one is talking about a meromorphic connection having a formal type of order at a pole with a semisimple leading order term (of course, here we have only discussed such types of poles), then it is standard terminology to refer to the number − 1 as the Poincaré rank of the pole. For our moduli spaces, since + + ∈ reg for 1 ⩽ ⩽ , we will write ∶= + − 1 for the Poincaré rank of and record these in the -tuple = ( 1 , … , ). Finally, we write = ∑ =1 and call this the total Poincaré rank. Remark 3.1.9. This notation is justified since the invariants we compute in Sections 4 and 5 will depend only on ( , ) and not on the actual eigenvalues of the formal types. We will always assume ⩾ 1 in this paper, in which case a generic always exists if is algebraically closed. This follows from [24, Lemma 2.2.2], because in our case when ⩾ 1 in loc. cit. both = = 1. If = is a finite field, one needs an additional lower bound on depending on and to make sure that the Zariski-open subvariety of defined by (3.1.7) has anrational point. We will not spell out an explicit bound here, as we are only interested in sufficiently large .
Proof. We take as in (3.1.2). Clearly, the scalars × ↪ G act trivially on −1 (0), hence the G-action factors though PGL n = G∕ × . We show first that this PGL n -action is free on −1 (0). Let ( 1 , … , ) ∈ −1 (0) and g ∈ G such that Ad g = for 1 ⩽ ⩽ . We show now that g is scalar, by looking at some nonzero eigenspace of g. Then clearly 1 will preserve for all and by the moment map condition The point is then that for each , there is a subspace ′ of the same dimension as such that (3.1.12) By the genericity of (Definition 3.1.6), this implies then = ′ = and hence g is scalar. For 1 ⩽ ⩽ , (3.1.12) follows simply because and are conjugate in G. To prove (3.1.12) for + 1 ⩽ ⩽ , we write = Ad ℎ for some ℎ ∈ G . By conjugating and g with the constant term ℎ 0 of ℎ, we can assume without loss of generality ℎ ∈ G 1 , that is, ℎ 0 = . Then = ∈ reg and thus g ∈ T. Next, considerh = ℎgℎ −1 , which satisfies Adh = . By Lemma 2.2.8(c), we haveh ∈ T and then by Lemma 2.2.8(e) ℎgℎ −1 = g. This implies that ℎ preserves for every 0 ⩽ ⩽ − 1 and hence we have tr | = tr(Ad ℎ )| = tr Ad ℎ| | = tr | . This proves (3.1.12) and hence PGL n acts freely on −1 (0).
In particular, all the G-orbits in −1 (0) are closed, and hence, they are in bijection with the points of the GIT quotient [20, Theorem 6.1]. Furthermore, as is a moment map, freeness of the PGL n action implies that 0 is a regular value of , which, in turn, implies smoothness of −1 (0) and hence of  * ( ). Looking at tangent spaces, we see that The formula now follows since dim ( ) = 2 − ( ) for 1 ⩽ ⩽ and dim ( We will see in Corollary 5.1.8 that  * , is nonempty and connected if , ⩾ 0. For more general (i.e., nongeneric) , the nonemptiness of  * ( ) has been determined in [28, Theorem 0.3].

Moduli of connections
We now work over the field = ℂ, otherwise adopting the notation of Section 2.1 for groups and Lie algebras. Let be a Riemann surface and fix a point ∈ . Let  and Ω be the sheaves of holomorphic functions and differentials on , respectively, and ,Ω the completions of their stalks at .
If ∈ ℤ >0 , we will let Ω( ⋅ ) denote the sheaf of meromorphic differentials with a pole of order ⩽ at ; we let Ω( * ) = ⋃ ∈ℤ >0 Ω( ⋅ ) be their union, that is, the sheaf of meromorphic differentials with a pole at of arbitrary order. Finally, we writeΩ( ⋅ ) ,Ω( * ) for their respective completions. If is a choice of coordinate centered at , one has isomorphismŝ Consider the space (Ω( * ) ∕Ω ) ⊗ ℂ . It is clear that one has a well-defined notion of the order of the pole of such an element. Further, under the isomorphisms (3.2.1), a given ∈ (Ω( * ) ∕Ω ) ⊗ ℂ with a pole of order has a unique representative in −1 [ −1 ] ⊆ (( )) of the form

Definition 3.2.3. A formal type of order
at is an element ∈ (Ω( * ) ∕Ω ) ⊗ ℂ with a pole of order . We will call such a formal type of order ⩾ 2 regular if, upon some choice of coordinate at , in the expression (3.2.2), one has ∈ reg .

Definition 3.2.4.
Let be a holomorphic vector bundle over and ∇ a meromorphic connection with a pole (only) at . Choose any holomorphic trivialization of in a neighborhood of and let be the connection matrix of ∇ with respect to this trivialization; if ∇ has a pole of order at , then yields an element of Ω( ⋅ ) ⊗ . Let be a formal type at . We say that ( , ∇) has formal type at if there exists a formal gauge transformation g ∈ ( ) such that the class of Ad g − g ⋅ g −1 ∈ (Ω( * ) ∕Ω ) ⊗ agrees with that of under the inclusion With these definitions, the open de Rham spaces admit the following moduli description. Set = ℙ 1 with an effective divisor = 1 1 + 2 2 + ⋯ + and for 1 ⩽ ⩽ , a formal type of order at . By "forgetting" in (3.2.2), we obtain a tuple = ( 1 , … , ) of diagonal elements in ∨ (depending on a fixed coordinate on ℙ 1 ). Remark 3.2.6. More generally, fixing the polar divisor and the local parameters , one may construct a moduli space ( ) of meromorphic connections ( , ∇) on ℙ 1 , where is a degree 0 vector bundle (though not necessarily trivial) and ∇ has formal type at . An analytic construction is given by [7,Proposition 4.5], which is generalized to higher genus curves in [6]. Algebraic constructions of these moduli spaces are given in [31]. After [50], such moduli spaces are typically referred to as "de Rham moduli spaces." Forgetting the connection, these spaces yield, a fortiori, families of vector bundles of degree 0. As the only semistable bundle of degree 0 on ℙ 1 is the trivial bundle, and since the semistable locus of a family is always Zariski open, the moduli spaces  * ( ) sit as open subvarieties It is for this reason that we call them "open" de Rham spaces.

MOTIVIC CLASSES OF OPEN DE RHAM SPACES
Throughout this section , will always be an algebraically closed field of 0 or odd characteristic. The main result in this section is Theorem 4.3.1, a formula for the motivic class of  , in the localized Grothendieck ring M for any -tuple = ( 1 , … , ) ∈ ℕ . By definition,  , is an additive fusion of coadjoint orbits, and thus, the computation of [ , ] can be split up in two parts.
First, we determine in Theorem 4.2.1 the motivic Fourier transform of the composition under the assumption ⩾ 2. In particular, we prove that  (( )) ∈ KExpVar is supported on semisimple conjugacy classes, which is not true for = 1. The motivic convolution formalism then allows us to deduce the formula for [ , ] from these local computations.

Some notation for partitions
For ∈ ℤ >0 , we denote by  the set of partitions of . For = ( 1 ⩾ 2 ⩾ ⋯ ⩾ ) ∈  , we use the following abbreviations: Further we write ( ) for the multiplicity of ∈ ℤ >0 in and Then if L ≅ ∏ =1 GL denotes the subgroup of block diagonal matrices in GL , we have since for any ∈ ℤ >0 , we have, for example, by [21, Proposition 1.1]

Fourier transform at a pole
In this section, we compute the Fourier transform  (( )) ∈ KExpVar of a coadjoint orbit res ∶ ( ) → ∨ of an element ∈ ∨ , say in the notation of (2.1.6), where we use the language of Section 2.3. Assuming ⩾ 2, we can give an explicit formula for  ([( )]), but to do so, we will first need to introduce some more notation.
Proof. By the formula (2.3.8), we have where for the second equality sign, we used the definition of ⟨, ⟩, see (2.1.5). By Lemma 2.2.9(c), the natural map G × G od → ( ) is a Zariski-locally trivial -bundle, and thus, we can rewrite this in E M as Now, notice that for all (g, , ) ∈ G × G od × , we have Thus, we finally obtain We will simplify this by applying Lemma 2.3.6. Consider the decomposition G od = G od It follows from Lemma 4.2.6 below that there are functions such that ⟨Ad , Ad g −1 ⟩ = ⟨ℎ 1 (g, , + ), − ⟩ + ℎ 2 (g, , + ) for all g ∈ G, ∈ and = ( + , − ) ∈ G od + ⊕ G od − . More explicitly, ℎ 1 and ℎ 2 are given by Now assume first is even. In this case, Together with dim G od − = 2 ( 2 − ), the theorem follows when is even. If is odd, we have If we decompose −1 2 = + into strictly lower and upper diagonal parts, then Lemma 4.2.6 implies that ℎ 2|ℎ −1 1 (0) is affine linear in and independent of if and only if commutes with Ad g −1 . As before, we can now apply Lemma 2.3.6 and see that (4.2.5) also holds for odd, which finishes the proof of (4.2.2).
Finally, we see from (4.2.4) and the description of ℎ −1 (0) ⊂ G × × G od + in both the even and odd case, that the structure morphism ℎ −1 (0) → , which is the projection onto , has image in the semisimple elements of .
can appear at most once in each summand on the right-hand side of (2.2.6). This is the crucial observation in the proof of Lemma 4.2.6. Proof. It follows directly from the observation above that depends linearly on for ⌊ +1 2 ⌋ ⩽ ⩽ − 1.

Lemma 4.2.6. For ∈ , the function
For ∈ G od , using the notation (2.2.6), we have where we use the convention 0 = 0 = . We start by looking at the dependence of ⟨Ad , ⟩ when varying −1 . The terms in (4.2.7) containing −1 are given by As ∈ reg , the commutator [ −1 , ] can take any value in od ; thus, tr[ −1 , ] is independent of −1 if and only if ∈ ( od ) ⟂ = .
Assume from now on ∈ . We show now inductively that is independent of ⌊ +1 2 ⌋ , … , −2 if and only if 1 , 2 , … , ⌊ −2 2 ⌋ all commute with . To do so, fix ⌊ +1 2 ⌋ ⩽ ⩽ − 2 and assume that 1 , … , −2− commute with . Consider the element The point now is that the -parts of the explicit formulas for ⟨Ad , ⟩ and (4.2.8) To write a formula for Res 0 tr( ′ ′−1 ), we write ′−1 = + ′ 1 + ⋯ + −1 ′ −1 . Then we can use a similar expression as (4.2.7) to conclude that all the terms containing in Next, we want to study the dependence of the difference (4.

Motivic classes of open de Rham spaces
In this section, we compute the motivic class of the generic open de Rham space [ * , ] ∈ M as defined in Definition 3.1.8.
We start by simplifying [ * , ] in the following standard way.
is a principal T -bundle. Notice that is G-equivariant with respect to the free G-action on G given by diagonal left multiplication. By restriction, we obtain a G-equivariant principal T -bundle → −1 (0). Taking the (affine GIT-)quotient by G, we obtain a principal × ⧵ T -bundle G ⧵ →  * , . Also, → G ⧵ is a principal G-bundle, as it is the restriction of G → G ⧵ G . As the groups T , G and × ⧵ T are special [48, §4.3], all the principal bundles here are Zariski locally trivial and we get Since the motivic class of ∏ =1 ( ) relative to is the convolution of the individual classes [( )], we have by Proposition 2.3.11 the equality Notice that the last product is relative to ; hence, we have by Theorem 4.2.1 for every , with the notation = ∑ , Z for the -fold product Z × ⋯ × Z and ∑ for the function taking ( 1 , … , ) ∈ Z to ∑ ( ). By Fourier inversion (Proposition 2.3.9), we thus get . ] as an element of KExpVar. We start by taking a closer look at Z = {(g, ) ∈ × | Ad g −1 ∈ }. If we put = ∩ , we have an isomorphism Next, we need to fix some notation to describe combinatorially. To parameterize the eigenvalues of elements in define for any ∈ ℕ the open subvariety • ⊂ as the complement of Furthermore, we need some discrete data. A set partition of is a partition is ordered, and hence, we have Proof. For ∈ ℕ, let S be the symmetric group on elements. Then the lemma follows from the fact that the subgroup ∏ ⩾1 S ( ) of S acts simply transitively on the fibers of . □ Lemma 4.3.9. The following relation holds in KExpVar Proof. For ∈ • , the map | ×{ } ∶  → from Lemma 4.3.8 is injective and its images are exactly the elements in with eigenvalues given by . Combining this with (4.3.7), we see Applying this reasoning − 1 times and then using Lemma 4.3.8, we obtain a trivial covering of degree Now, for a fixed 1 ⩽ ⩽ , we have by definition | 1 | = ⋯ = | |. Thus, by our genericity assumption (3.1.7), the numbers = ∑ =1 ⟨ 1 , ⟩ satisfy the assumptions of Lemma 4.3.11 below, and we deduce Proof. We use induction on . For = 1, we have 1 = 0 and 1 • = 1 ; hence, the statement is clear. For the induction step, consider • as a subvariety of −1 • , which implies the formula. □

TAME POLES AND FINITE FIELDS
Here, the notation will be the same as in Sections 3 and 4, especially Definition 3.1.8 and Section 4.1. In this section, we replace the motivic computations over an algebraically closed field with arithmetic ones over a finite field . This allows us to study the number of -rational points of  * , for any pair ( , ) with ⩾ 1. First, we derive in Theorem 5.1.6 a closed formula for | * , ( )| using techniques similar to those of Section 4. Then, in Theorem 5.2.3, we give a second description of | * , ( )| in terms of symmetric functions, which allows us to identify the -polynomial of  * , with the pure part of the conjectural mixed Hodge polynomial of the corresponding character variety [26, Conjecture 0.2.2]. This gives strong numerical evidence for the purity Conjecture 1.1.3, which was one of the main motivations of this paper.

de Rham spaces and finite fields
Let be a finite field of characteristic coprime to and large enough (see Remark 3.1.9) so that there exists a regular generic tuple = ( 1 , … , ) of formal types over for a given pair ( , ) with ⩾ 1. Then also  * , is defined over and we can prove a version of Theorem 4.3.1 that also includes tame poles.
For a variety defined over , we sometimes abbreviate | ( )| = | |. We write ′ ( ) ⊂ ( ) for the matrices in ( ) whose eigenvalues are in and for ∈  , we write ′ ( ) = ′ ( ) ∩ ( ). The proof of Theorem 4.2.1 then implies that the Fourier transform of the count function # ∶ ∨ ( ) → ℤ, ↦ | −1 res ( )| associated to the coadjoint orbit res ∶ ( ) → ∨ is supported on ⨆ ∈ ′ ( ). Given such an ∈ ′ ( ), the -version of formula (4.2.2) reads denotes the -rank. This is a shorthand notation for the value at 1 of the Green function [19, Theorem 7.1], and we can be even more explicit. If we write ( ) ∈  for the partition defined by the cycles of ∈ S , we have where ⊂ denotes the locus fixed by ∈ . If is regular, then  (1 ) is supported on ′ ( ).
Proof. The proof consists of writing out Equation (2.5.5) in [24] explicitly. In their notation, we have Here, L ≅ ∏ =1 GL denotes the centralizer of and W L ≅ ∏ =1 S the normalizer of the standard maximal torus T in L. We have L = dim G − dim L = 2 − ( ) and since T is split over also G L = (−1) (−1) = 1. The formula then becomes ) .
Proof. The argument is similar to the proof of Theorem 4.3.1. First, by repeating the proof of Lemma 4.3.3 with  * , replaced by  * , , we see that where is the moment map defined in (3.1.2). Using Fourier inversion and convolution over finite fields, we further have For the last equation, we used that  (# + ) is supported on ⨆ ∈ ′ ( ) for all 1 ⩽ ⩽ and our assumption ⩾ 1. We now parameterize ′ ( ) using the same notation as in Section 4.3, that is, we have a surjective map with each fiber of having cardinality where we abbreviated Ω = 1 2 ( 2 ( + ) − 2 − ∑ ( )) + ( ) 2 ( − ). Now multiplying out the products over and , we are left with computing sums of the form

⟩)
, with ∈  . Because of the genericity assumption (3.1.7) and (the finite field version of) Lemma 4.3.11, this sum always equals (−1) −1 ( − 1)! independently of the 's. With this simplification, we finally deduce Summing up over all ∈  finishes the proof. □ Theorem 5.1.6 immediately implies that | * , ( )| is a rational function in as we vary the finite field . Because it is the count of an algebraic variety, we deduce that it is, in fact, a polynomial in , namely, its weight polynomial by Theorem 2.3.4. By analyzing the combinatorics of (5.1.7), we obtain the following. Proof. Let ∈ ℕ and write T for the standard maximal torus of GL . The lemma boils down to the following combinatorial identity: Here, the first equality follows from (5.1.2), while the second one can be deduced from [40, (2.14') and Example I. We prove this inequality by induction on . Let ∈  be such that ( ) is minimal and suppose ∈ . We may assume that +1 < since otherwise we could interchange and +1 and the resulting ′ would still minimize ( ). Hence, Clearly, if ≠ or ≠ , we have |̂∩ | = | ∩ | and so and taking the minimum, we see that the inequality (5.1.11) holds. The equality cases follows from a direct inspection.
By [38, (2.2)], St is supported on semisimple elements and its value on ∈ ′ ( ) is Thus, in total, we obtain Comparing this with Equations (5.1.1) and (5.1.4), we finally see where the product is over the boxes in the Young diagram of and and are the arm length and the leg length of the given box. We fix integers g ⩾ 0 and > 0. Let now ∈  , = ( 1 , … , ) ∈ ℤ >0 and ∶= 1 + ⋯ + be as in Definition 3.1.8. We let ℍ , ( , ) Let now  , B denote a generic wild character variety of type , and g = 0 as studied in [26]. The main conjecture [

QUIVERS WITH MULTIPLICITIES AND THEIR ASSOCIATED VARIETIES
The greater part of this section is devoted to a generalization of the theorem of Crawley-Boevey [15, Theorem 1], which realizes an additive fusion product of coadjoint orbits for GL ( ), where is a fixed perfect ground field (the primary examples the reader should have in mind are where = ℂ or is a finite field), as a quiver variety associated to a star-shaped quiver. The generalization we will achieve is to replace the GL ( )-coadjoint orbits with those for the nonreductive group GL (R ) of Section 2.1 and to replace quivers with "quivers with multiplicities" or "weighted quivers" that are defined in Section 6.1 below. The main theorem is stated as Theorem 6.4.2. It states that an open de Rham space as defined in Section 3.1 may be realized as a variety associated to a weighted star-shaped quiver for an appropriate choice of multiplicities.
The main novelty that enters when one introduces multiplicities is that the groups one obtains are no longer reductive. As in the usual quiver case, one wants to define a variety associated to a quiver with multiplicities as an algebraic symplectic quotient. In the approach we take here, we simply define this quotient as the spectrum of the appropriate ring of invariants. Of course, this makes good sense as an affine scheme; however, without the reductivity hypothesis, we are not guaranteed that this ring of invariants is a finitely generated -algebra. For the star-shaped quivers described in Section 6.1.2, we show that we do, in fact, get finite generation. Essentially, what we show is that the preimage of the moment map is a trivial principal bundle for the unipotent radical of the group (Proposition 6.2.1), the proof of which occupies Section 6.2.
Let us also mention that there seem to be several difficulties in applying the general theory of nonreductive GIT as developed by Bérczy, Doran, Hawes, and Kirwan [3] directly to our situation: The varieties we consider are affine and the unipotent radical does not seem to admit a suitable -grading. In Section 6.3, we explain how a certain symplectic quotient of a "leg"-shaped quiver (out of which one builds a star-shaped quiver) yields a coadjoint orbit for the group GL (R ), generalizing Boalch's explanation of Crawley-Boevey's theorem [11,Lemma 9.10]. This was done in the holomorphic category for "short legs" in [54, Lemma 3.7]; ours is an algebraic version, along the lines described above.
Quivers with multiplicities have been introduced in [54] for purposes quite similar to ours (namely, the description of moduli spaces of irregular meromorphic connections), and indeed, the quivers of interest in that paper are a special case of those considered here [54, §6.2]. We would like to point out that the quotients referred to there are either taken in the holomorphic category (when restricted to the stable locus, in the sense defined there) or simply set-theoretic [54,Definition 3.3]. The results of Sections 6.2 and 6.3 show that these quotients indeed make sense algebraically.
Finally, we put together the results in Section 6.4, relating the varieties constructed in this section to the open de Rham spaces of Section 3. Quivers with multiplicities are also discussed by [23] in order to generalize certain classical results known for finite-dimensional algebras associated to symmetric Cartan matrices to the case where the Cartan matrix is only symmetrizable. It is well known that given a simply laced affine Dynkin diagram, one can associate a two-dimensional quiver variety (depending on some parameters), which will, in fact, be a gravitational instanton (i.e., carry a complete hyperkähler metric). In a parallel generalization, the construction above allows us to construct quiver varieties for nonsimply laced Dynkin diagrams; this is explained in Section 6.4.2. In the next section, we will see that these also admit complete hyperkähler metrics (see Theorem 7.3.3 and Remark 7.3.6(i)).

6.1
Definitions, notation, and a statement for star-shaped quivers 6.1.1 Definitions, notation, and a finite generation criterion Here, we discuss a generalization of the definition of quiver representations to allow for multiplicities at the vertices. This was first done in [54]. We follow the approach of [23, §1.4] and [55], using the greatest common divisor of neighboring multiplicities in the definition of a representation. Let be a field. For a positive integer ∈ ℤ ⩾1 , we will denote by R the truncated polynomial ring R ∶= [ ]∕( ), so that 1 = . Observe that if | , then there is a natural inclusion of -algebras R ↪ , which makes a free R -module of rank ∕ . As in Section 2.1, we will use the groups GL (R ) and GL 1 (R ) and their respective Lie algebras (R ) and 1 (R ). However, since the value of will vary, we will not shorten these to G or G , and so on. We do adopt the identifications via the trace-residue pairing (2.1.5).
As usual, a quiver = ( 0 , 1 , ℎ, ) is a finite directed graph, that is, 0 and 1 are finite sets and one has head and tail maps ℎ, ∶ 1 → 0 . By a set of multiplicities for , we will mean an element ∈ ℤ 0 ⩾1 . The pair ( , ) of a quiver and a set of multiplicities can be referred to as a quiver with multiplicities or a weighted quiver.
A dimension vector is also an element ∈ ℤ 0 ⩾1 . The space of representations of ( , ) for a given dimension vector is defined as where we have made the abbreviation R ∶= R ( ( ) , ℎ( ) ) . Here ( ( ) , ℎ( ) ) denotes the greatest common divisor of ( ) and of ℎ( ) . In the case that = 1 for all ∈ 0 , this is the usual definition of the space of representations Rep( , ) for with dimension vector .
The main difference between this and the case without multiplicities is that , is not reductive if there is some ∈ 0 with ⩾ 2.
As usual, we denote by the doubled quiver: that is, 1 ∶= 1 ⊔ ′ 1 , where ′ 1 ∶= { ′ ∶ ∈ 1 } and ℎ( ′ ) = ( ), ( ′ ) = ℎ( ) for all ∈ 1 . Then Thus, we have the usual situation of a cotangent bundle, which has a canonical symplectic form, and an action of a group on the base, for which the induced action on the cotangent bundle is Hamiltonian. We will be working with the corresponding moment map for the , -action. An explicit formula can be derived as in the case without multiplicities (see, e.g., [39, Lemma 3.1]) and is given by where we identify for each ∈ 0 the dual ∨ , with − , as in (2.1.4). We will usually drop the factor − in the computations below if no confusion arises. Definition 6.1.3. Let us fix an element ∈ ∨ , whose , -coadjoint orbit is a singleton. One defines the reduced quiver scheme associated to ( , ) at as . Remark 6.1.5. By definition,  is an affine scheme over . However, as , is not reductive, [ −1 ( )] , is not a priori a finitely generated -algebra; therefore, we cannot say that  is an affine variety (possibly nonreduced) without further justification.
In view of the preceding remark, we now give a criterion that is sufficient to conclude that  is an affine variety. Then in Section 6.1.2, we will describe a class of quivers with multiplicities and conditions on for which this criterion is fulfilled.
Observe that the group , is a semidirect product: there is a surjective morphism of algebraic groups is the usual group associated to the quiver and dimension vector . We will call the kernel 1 , . This surjection splits, with GL ( ) being the subgroup of "constant" group elements in GL (R ), and then taking the product of these inclusions (cf. Section 2.1). We may thus write With this decomposition, we may similarly decompose the Lie algebra , and its dual as direct sums: For ∈ ∨ , , we will write res ∈ ∨ and irr ∈ ( 1 , ) ∨ for its components. Likewise, the moment map (6.1.2) will also have components res and irr corresponding to ∨ and ( 1 , ) ∨ , respectively. Of course, each of these components is a moment map for the restriction of the action to the respective subgroup. irr ( irr ) is a trivial (left) principal 1 , -bundle over . Then  is an affine algebraic symplectic variety, and hence, we will often refer to it as a quiver variety. Remark 6.1.8. By "symplectic variety," we mean that it is an algebraic symplectic manifold along its smooth locus. We are making no claims about the possible singular locus.

Lemma 6.1.7. Suppose that there is a -invariant closed subvariety
Proof. We deal only with the finite generation of [ −1 ( )] , ; the symplectic structure arises as usual on the smooth locus [41,Theorem 1]. Since 1 , is normal in , , it is not hard to see that res is constant on 1 , -orbits in Rep( , , ). We have that ) .
Using the fact just mentioned, we obtain an isomorphism ) .
Now write ∶= −1 res ( res ) ∩ for the second factor and note that this is an affine algebraic -variety. From this, and since is reductive, this is a finitely generated -algebra. □ Remark 6.1.9. The idea in the proof comes from the procedure in the general theory of Hamiltonian reduction known as "reduction in stages." When taking a symplectic quotient by a group with a semidirect product decomposition, one obtains the same quotient by first reducing by the normal subgroup and then by the quotient group.
In the next subsection, we will show that the hypotheses of Proposition 6.1.14 imply those of Lemma 6.1.7, which then provides a proof of the proposition.

Proof of Proposition 6.1.14: Construction of a section
Here we prove the following. Simplified case: = 1 For notational simplicity, we will first assume that = 1, so that we have a quiver with a single leg. This allows us to drop the index from the notation in (6.1.10), (6.1.11), and (6.1.12). Thus, the doubled quiver is the following: where the vertices are labeled 0, … , from left to right, multiplicity vector = (1, , … , ) and dimension vector = ( , 1 , … , ) with > 1 > ⋯ > . To avoid having to write out things separately for the vertex 0, we will often write 0 ∶= . Also, The condition (6.1.13) on reads

Explicit description of the groups, Lie algebras, and their duals
Let us now be explicit about the groups that are involved. One has with respective Lie algebras One has direct sum decompositions as in (6.1.6) and the projection maps to each factor simply omits the "irregular" part or the "residue" term, respectively, as in (2.1.6) and (2.1.7). As before, we write res and irr for the images of under the respective projections to and to 1 , .

Explicit description of the space of representations Rep( , , ) and group actions
Explicitly, the space Rep( , , ) is given by ) .
Later, we will need to be even more explicit, and so, we will write with ∈ × −1 ( ), ∈ −1 × ( ). When we will need to evaluate moment maps or group actions, we will need to regard, for example, the product as an element of (R ) ∨ = − (R ). We do this by multiplying and as matrices of Laurent polynomials and truncating the terms of degree ⩾ 0. Explicitly, Of course, products of the form or g −1 (g ) −1 as in (6.2.6) are written similarly. Products such as g (g −1 ) −1 in (6.2.6) will be considered as multiplication of the relevant matrices with entries in R .

Explicit description of the moment maps
The moment map ∶ Rep( , , ) → ∨ , has the explicit expression We recall that ( 1 , ) ∨ has no component corresponding to the vertex 0. Thus, if we write out an explicit expression as in (6.2.7), we should ignore any contributions coming from the residue terms, because we are considering these equations in ( 1 , ) ∨ .

Definition of and its -invariance
We now proceed with defining the subvariety appearing in Lemma 6.1.7, where ( , , ) is as in Proposition 6.1.14 in the special case where = 1. We start by defining and define inductively a nested sequence of subvarieties of 0 by Remark 6.2.11. On the left side of the defining equation for , the matrix is a priori a ( )-valued polynomial in of degree ⩽ − 1, but the right-hand side is, in fact, a constant matrix. So, the defining condition is equivalent to the further conditions The fact that is an affine variety is thus clear. Furthermore, from this description, it is easy to see that is a -invariant subvariety of Rep( , , ) using (6.2.6) and the appropriate expressions in (6.2.7): the vanishing conditions (6.2.12) imposed by (6.2.10) are left unchanged. Remark 6.2.13. Equation (6.3.13) below implies that ∈ GL (R ) for 1 ⩽ ⩽ . In particular, the constant term, namely, 0 , must lie in GL ( ). This fact is crucial in the construction of the isomorphism (6.2.2); see Lemma 6.2.17 below.
We now establish some properties of the with respect to the action of 1 , and its subgroups. Let us consider each GL 1 (R ) for 1 ⩽ ⩽ as a subgroup of 1 , via the obvious inclusion in (6.2.5). We will also use the subgroups Observe that ⋅ ( , ) ∈ −1 since the condition for this to hold depends only on the components 1 , 1 , … , , , which are unchanged under the action of . We only need to check the condition for , which involves the terms +1 , +1 0 . It suffices to see that But since +1 ∈ GL 1 +1 (R ), the constant terms of +1 and ( +1 ) −1 are both +1 . □ Lemma 6.2.15. Let ∈ GL 1 (R ) and ( , ) ∈ . If ⋅ ( , ) ∈ , then = .

Construction of the isomorphism (6.2.2)
The condition we want is that all the nonconstant (with respect to ) terms vanish (Remark 6.2.11). But now, by Remark 6.2.13, 0 is invertible, and so starting with the coefficient of above, we may solve for 1 = − 1 ( 0 ) −1 so that this coefficient vanishes. It is then clear that we may successively solve for 2 , …, −1 algebraically as functions of and to eliminate the remaining nonconstant terms. This produces with the stated property. □ Corollary 6.2.18.

Conclusion in the case = 1
We can now put the of Corollary 6.2.19 into a 1 , -equivariant isomorphism −1 It was already noted in Remark 6.2.11 that is -invariant. These are the hypotheses of Lemma 6.1.7, and so, we may conclude in the case = 1.

The general case of legs
Now, consider the situation of Proposition 6.1.14, where the number of legs ∈ ℤ >0 in the quiver is arbitrary. Since the multiplicity at the central vertex 0 is 0 = 1, irr has no component at 0 and thus −1 where −1 irr ( irr ) denotes the moment map preimage for the th leg. Likewise, 1 , factors as a product of the groups (6.2.5) for each leg and the action on −1 irr ( irr ) is just the product action. Thus, if we set , is in (6.2.10) for the th leg, we get a 1 , -equivariant isomorphism Finally, the -invariance of follows from the corresponding statement for each leg (see Remark 6.2.11) since there we already included the action of GL ( ) at the central vertex. Applying Lemma 6.1.7 completes the proof of Proposition 6.1.14.

Coadjoint orbits
Here, we discuss the relationship between coadjoint orbits for the group GL (R ) for a fixed ⩾ 1 and varieties associated to quivers with multiplicities, where the underlying quiver is a single leg. It may thus help the reader to refer back to the diagram (6.2.3). What will be true is that coadjoint orbits of certain diagonal elements ∈ ∨ in (R ) ∨ (2.2.1) can be realized as symplectic reductions of the spaces Rep( , , ) that we considered in the case = 1 in Section 6.2. To be able to make a precise statement, we will first need to explain the conditions on the coadjoint orbits and set some notation.
We will take ∈ ∨ , and suppose that it is written in the form (2.2.2). We will make the further assumption that (2.2.3) holds. For such a , we wish to describe a quiver , which will be a leg, as well as some data on it, from which we will recover ( ). The quiver will be the same as in (6.2.3), having + 1 nodes and 2 arrows, and will have the same multiplicity vector , with multiplicity 1 at the vertex 0 and all other vertices receiving multiplicity . The dimension vector will be defined by taking ∶= , −1 ∶= + −1 , and so on, with ( 0 , … , ) given as in (2.2.2). The statement that we want is that the GL (R )-coadjoint orbit of as above is given by a symplectic reduction of Rep( , , ) by a subgroup of , . The subgroup in question is that we obtain by leaving out the group GL ( ) corresponding to the vertex 0, namely, We write , ,0 for its Lie algebra. The reason the vertex 0 in (6.2.3) was drawn empty is because we want to consider only the symplectic quotient by , ,0 . Of course, 1 , is a normal subgroup of , ,0 with quotient which is precisely the group associated to the underlying quiver with dimension vector , ignoring the multiplicities, where again we are omitting the group GL ( ) corresponding to the vertex 0. Thus, we want to take a symplectic quotient by the semidirect product ) .
For less burdensome notation, we will often abbreviate the left-hand side of the above to Rep ∕ ∕ , ,0 . Observe that the assumption (2.2.3), the arguments of Section 6.2, and Lemma 6.1.7 already show that Rep ∕ ∕ , ,0 is an affine symplectic variety. We will write ∶ −1 0 ( 0 ) → Rep ∕ ∕ , ,0 for the quotient map; since this is defined as a GIT quotient, this is a categorical quotient. The following is a slight generalization of [11, Proposition D.1] that will be important in the proof of the proposition. are conjugate in GL ( ).
Proof. Observe that = ker ⊕ im . Indeed, given ∈ , let ∶= ( ) −1 ∈ . Then ( − ) = 0 hence = ( − ) + ∈ ker + im , that is, = ker + im . Furthermore, the sum is direct, for if ∈ ker ∩ im , say = with ∈ and = = 0, then = 0 and hence = 0. Furthermore, the assumption that is invertible also implies that ker and im are freemodules. This can be seen via the Cauchy-Binet formula: for a subset ⊆ {1, … , } of size , one sets det to be the determinant of the × submatrix of taking the columns with indices in ; one defines det the same way, using columns instead of rows; then the formula states that where the sum is over all subsets of size . Since det ∈ × and is local, there must be some with det ∈ × (otherwise, all the terms in the sum would lie in the maximal ideal, and hence, det could not be a unit). Hence, there exists ∈ GL ( ) for which the submatrix of corresponding to is the identity matrix. That is, the matrix is in reduced row echelon form and one can find a basis of ker = ker as one does in a first-year linear algebra class. A similar argument shows that the columns of are linearly independent over and hence already give a basis of im . Now, we choose a basis of by taking the first − vectors as a basis of ker and the last vectors as the columns of . Then the fact that = ker ⊕ im implies that the matrix obtained in this way lies in GL ( ). Writing with respect to this basis gives the second matrix in (6.3.7). □ With this, the proof of Proposition 6.3.4 follows the idea of [11, Lemma 9.10], which explains the proof of [15, §3]. However, there one can rely on usual results of linear algebra over fields, while in our case, working with the orbits of the unipotent groups involved requires a little care, which makes the arguments somewhat longer. For this to define a GL (R )-equivariant morphism Φ ∶ Rep ∕ ∕ , ,0 → ( ), we need to verify three things: first, a priori,Φ takes values only in (R ) ∨ , so we need to see that it indeed takes values in ( ); second, we need to check thatΦ is , ,0 -invariant; finally, one wants to see thatΦ is GL (R )-equivariant. The latter two statements are easy to check simply from their definitions: (6.3.9) for , ,0 -invariance and (6.3.10) for GL (R )-equivariance. The first statement is a bit longer and so we will justify it in the next paragraph.

Verification that Φ and Ψ are mutually inverse
We first check that Φ • Ψ = ( ) . Now, Φ • Ψ is the morphism induced via L , -invariance from the mapΦ • Ψ ′ ∶ GL (R ) → ( ), which is, by (6.3.16), But this is precisely , as in Lemma 2.2.4(b), hence the induced map on the quotient must be the identity.
Below we will list the star-shaped nonsimply laced affine Dynkin diagrams, which correspond to open de Rham spaces in Theorem 6.4.2 of dimension 2. The simply laced star-shaped ones 4 ,̃6,̃7,̃8 correspond to open de Rham spaces with logarithmic singularities.
This case is special in that Boalch's [11,Theorem 9.11] identification with a quiver variety does not apply, as we have two irregular poles. In fact, there is no ALE space that is isomorphic with  * 2,(1,1) as the intersection form on 2 ( * 2,(1,1) ) is divisible † by 2 , whereas the intersection form of the 2 ALE space is not divisible by 2.
if ⊂ G is a maximal compact subgroup, then × acts by left and right multiplication (7.2.1) and these actions admit hyperkähler moment maps [16].
For coadjoint orbits in ∨ , hyperkähler metrics were first constructed by [34] for regular semisimple orbits and for general semisimple orbits in [5] and [32]. The coadjoint G-action can be restricted to and it can be shown (Proposition 7.2.4), in a manner similar to that for * G, that this action also admits a hyperkähler moment map. Now, if a hyperhamiltonian action of a compact group extends to a holomorphic action of its complexification, then the hyperkähler reduction can be understood as a holomorphic symplectic reduction [30, §3(D)]. Here, we will need the opposite direction:  * ( ) is given as an algebraic symplectic quotient; we will show that it, in fact, arises as a hyperkähler quotient. A special case of a version of the Kempf-Ness theorem due to Mayrand [44] gives sufficient conditions for algebraic symplectic quotients of the type we have been considering to be upgraded to hyperkähler quotients.

Holomorphic symplectic quotients to hyperkähler quotients
Let us begin by incorporating Mayrand's statement into the following, which will give us the criterion we will apply later to obtain the theorem.  Suppose that ( , g, , , ) is a hyperkähler manifold, with Kähler forms , , ∈ Ω 2 ( ) in the corresponding complex structures. We suppose that ( , ) is a (smooth) complex affine variety and refer to as such with the complex structure in mind. Suppose that is a complex reductive group with an algebraic action on for which the restriction to its maximal compact admits a hyperkähler moment map , , ∶ → ∨ . As usual, we will write ℝ ∶= ℂ ∶= + for the real and complex components of the moment map; we will assume that ℂ ∶ → ∨ = ∨ ⊕ ∨ is algebraic. Here = Lie( ) is a complex and = Lie( ) a real Lie algebra and ∨ means dual vector space over the respective field. Let ∈ ( ∨ ) be such that acts freely on the affine variety −1 ℂ ( ); thus, the algebraic symplectic quotient is smooth. Then, if there exists a -invariant, proper global Kähler potential for | −1 ℂ ( ) , which is bounded below, then there exists ℝ ∈ ∨ and, for the complex structures induced from , a natural biholomorphism where ∕ ∕ ∕ ( ℝ , ) ∶= ( −1 ℝ ( ℝ ) ∩ −1 ℂ (0))∕ denotes the hyperkähler quotient.
Proof. The Kähler potential produces a moment map̃ℝ for the -action with respect to [44,Proposition 4.1]. This must differ from the given ℝ by a constant, that is, there exists ℝ ∈ ∨ such that ℝ =̃ℝ + ℝ . Thus,̃− 1 ℝ (0) = −1 ℝ ( ℝ ). Then [44,Proposition 4.2] gives a homeomorphism at the last step of the sequence Furthermore, by the freeness of the action, there is a single orbit-type stratum and hence the homeomorphism is, in fact, a biholomorphism, again from [44,Proposition 4.2]. □

Hyperkähler moment maps on the factors
Here, we show that the two kinds of factors that appear in the relevant additive fusion product each admit hyperhamiltonian group actions.

Cotangent bundles
Let be a complex reductive group with Lie algebra and let ⩽ be a maximal compact subgroup. We recall that for * = × ∨ , there is an algebraic hamiltonian action of × given by  [5,32,34], once the coadjoint orbit is fixed, the family of such hyperkähler metrics is parameterized by an element 1 ∈ (see Subsection 7.4).

Hyperkähler metrics on open de Rham spaces
We first give a lemma describing coadjoint orbits for 2 as algebraic symplectic reductions of * .
where 1 denotes the tuple ( +1 1 , … , 1 ) of residue terms of the formal types. Each factor of acts via the left action on the corresponding * factor as in (7.2.1) and the factor acts diagonally: by the coadjoint action on  , for 1 ⩽ ⩽ and with the right action in (7.2.1) for the factors indexed by + 1 ⩽ ⩽ . We have thus expressed  * ( ) as an algebraic symplectic quotient and we can now apply Proposition 7. , if we first take the quotient of * × * by , one can see that it is a quotient of the form * ∕ ∕ 2 (for appropriate values of the moment map), which is precisely the reduction carried out at [1, pp. 88-89]. Hence,  * ( ) is isometric to the deformation of the 2 singularity, which was already observed at [9, p. 3], the hyperkähler metric on which was previously constructed via twistor methods in [46, §7], and proved to be ALF in [13, §5.3]. It is worth noting that this space can also be described via a slice construction [4, (3.1)] which, although an algebraic operation, also yields the metric [4, §4]. Furthermore, we expect the metrics on higher dimensional open de Rham spaces to exhibit "higher dimensional ALF behavior." Remark 7.3.6.
(i) It is true more generally, and not much harder to prove, that with no restriction on the order of the formal types, the spaces  * ( ) always admit complete hyperkähler metrics.
However, there is some extra choice involved. In general, a coadjoint orbit for G may be realized as an algebraic symplectic quotient of * × ( ′ ) ([7, Lemma 2.3(2), Lemma 2.4], cf. Remark 7.3.2), where ( ′ ) is a coadjoint orbit for the unipotent group G 1 . As such, ( ′ ) is an even-dimensional complex affine space, hence, upon some choice of coordinates, admits a flat hyperkähler metric. One can show that the coordinates can be chosen so that acts on pairs of coordinates with opposite weights, and hence with a hyperkähler moment map. (ii) The spaces * × ( ′ ) are (isomorphic to) what are known as "extended orbits" and these can be arranged into a moduli space by taking an additive fusion product (this is referred to as the "extended" moduli space in [7, Definition 2.6] [29, Definition 2.4]. This moduli space will admit an action of , where is the number of irregular poles, and  * ( ) can thus be realized as a hyperkähler quotient of the extended moduli space from this action. Indeed, in the last expression in (7.3.4), if we first take the quotient by , then we obtain the extended moduli space.

Details of proof of Proposition 7.2.4
Here, we will give the details of the proof of Proposition 7.2.4. The statements that need to be proved are: the existence of a hyperkähler moment map on the coajdoint orbit  and its complex part is simply the inclusion map into the dual of the complex Lie algebra, which is Lemma 7.4.3 below; and the existence of a Kähler potential with the appropriate properties, which is Lemma 7.4.6.
To proceed, we will need to fix notation, and so we will adopt that of [5, §3, L'espace des modules]. As such, will now denote a compact Lie group, which is, of course, the maximal compact subgroup of its complexification ℂ , whereas in Section 7.2, it denoted the complexification and a maximal compact subgroup; we hope that this will cause the reader no confusion. Of course, will denote the Lie algebra of and ℂ its complexification, and ⟨ , ⟩ will denote a Ad-invariant inner product on .
Let us recall how the coadjoint orbit  is identified with a moduli space of solutions to Nahm's equations. Let ⊂ be a maximal torus (which is, of course, compact) with Lie algebra , and respective complexifications ℂ and ℂ . As usual, using the invariant inner product, we identify ∨ = and ∨ ℂ = ℂ . Viewing  as a subset of ℂ , as it is a semisimple orbit, its intersection with ℂ is a singleton; we write this element as 2 + 3 with 2 , 3 ∈ . One chooses a third element 1 ∈ so that we have a triple = ( 1 , 2 , 3 ) ∈ 3 , so that for an appropriate > 0, we can make sense of the space Ω 1 ∇; as described at [5, §3, p. 265]. We will write an element ∇ + ∈ A as a quadruple = ( 0 , 1 , 2 , 3 ) of smooth maps ∶ (−∞, 0] → satisfying an asymptotic condition depending on : one has ∇ + = + 0 + ∑

=1
. We consider such which are solutions to Nahm's equations: for cyclic permutations ( ) of (1 2 3). The quotient of the space of such solutions by the group G , also defined at [5, §3, p.265], will be referred to as the moduli space = ( ) of solutions to Nahm's equations. We will often write [ ] for the G -orbit of a solution . The isomorphism ∼ →  is given by [5,Corollaire 4.5] (see also the definition before Equations (IIIa) and (IIIb)) as [ ] ↦ 2 (0) + 3 (0).  } .
Consider G + → the evaluation map at = 0 and G its preimage of 1 G . It is a normal subgroup of G + with quotient . Of course, we have a parallel statement for the Lie algebras. Furthermore, G + acts on the space of solutions to Nahm's equations inducing the adjoint action of on , as is easily seen via the map (7.4.2).
A for cyclic permutations ( ) of (1 2 3) (cf. [16,Equations (6)- (9)]; note that there is a slight difference in the complex structures there and in [5] accounting for the sign differences in the equations). These equations are obtained simply by linearizing Nahm's equations (7.4.1).
Let ∈ and choose a lift ( ) ∈ Lie(G + ), so that (0) = . A representative for the tangent vector ([ ]) to at [ ] generated by the infinitesimal action of is given by ) .
We evaluate using (7. The statement about the complex part of the moment map is obvious from the expressions (7.4.2) and (7.4.4). □ Lemma 7.4.6. The semisimple coajdoint orbit  admits a global -invariant Kähler potential (for the complex structure ) which is proper and bounded below.
As mentioned, the proof here is adapted from that of [44,Lemma 4.5].
Proof. A global Kähler potential ∶ → ℝ for the Kähler form is given by (see [ It is then sufficient to show that the is -invariant, proper, and bounded below. -invariance follows from that of the bilinear form ⟨ , ⟩ and lower-boundedness is obvious, so properness is essentially all that needs to be proved. Let = ( 1 , 2 , 3 ) ∈ ⊕3 . Then the existence and uniqueness theorem for systems of ordinary differential equations gives a unique solution = ( 0 ≡ 0, 1 , 2 , 3 ) to the reduced Nahm's equations + [ , ] = 0 (7.4.7) (this is just (7.4.1) with 0 = 0) with (0) = for = 1, 2, 3. This allows us to define a functioñ ∶ ⊕3 → ℝ by 4.8) which is (at the very least) continuous, again by the assertions of the existence and uniqueness theorem on the dependence on the initial conditions. Let us explain how this is related to . Consider the map ev ℂ ∶ ⊕3 → ℂ ↦ 2 + 3 .
As  is a semisimple (co)adjoint orbit,  is closed in ℂ and hence so is ev −1 ℂ (). We may then identify  with the subset of ∈ ev −1 ℂ (0) for which is gauge equivalent to an element of A .
We observe that this will be closed, as we are imposing an asymptotic condition on 1 . As the definition of is independent of the gauge equivalence class, one sees that under the identification of  with the described subset of ev −1 ℂ (), It now suffices to show that there is a closed subset ⊆ ⊕3 containing  (using the identification above) for which˜| is proper, since the restriction of a proper map to a closed subset is still proper.
For this, we will take which is closed in ⊕3 . Let ⊆ ⊕3 be the unit sphere (here, we may take the invariant inner product on in each factor); this is compact. Then ∩ is a compact subset of ⊕3 , and hence,h as a minimum value > 0. It is nonzero, for if ⋅ ∈ is such that˜( ) = 0, then assuming that  ≠ {0} (which we may of course do), then one finds ev ℂ ( ) = 0, and hence = 0; but this would contradict ∈ .
By uniqueness of the solutions to (7.4.7), for ∈ ℝ >0 , it is easy to see that ( ) = ( ). From this, we obtain for any ∈ ⊕3 , From this, one finds that the preimage of a bounded set in ℝ under˜| is bounded in . By continuity, the preimage of a closed set is closed, and so,˜| is proper. □

A C K N O W L E D G M E N T S
We would like to thank Gergely Bérczy, Roger Bielawski, Philip Boalch, Sergey Cherkis, Andrew Dancer, Brent Doran, Eloïse Hamilton, Frances Kirwan, Bernard Leclerc, Emmanuel Letellier, Alessia Mandini, Maxence Mayrand, András Némethi, Szilárd Szabó, and Daisuke Yamakawa for discussions related to the paper. We especially thank the referee for an extensive list of very careful comments. At various stages of this project, the authors were supported by the Advanced Grant "Arithmetic and physics of Higgs moduli spaces" no. 320593 of the European Research Council, by grant no. 153627 and NCCR SwissMAP, both funded by the Swiss National Science Foundation as well as by EPF Lausanne and IST Austria. In the final stages of this project, MLW was supported by SFB/TR 45 "Periods, moduli and arithmetic of algebraic varieties," subproject M08-10 "Moduli of vector bundles on higher-dimensional varieties." DW was also supported by the Fondation Sciences Mathématiques de Paris, as well as public grants overseen by the Agence national de la recherche (ANR) of France as part of the Investissements d'avenir program, under reference numbers ANR-10-LABX-0098 and ANR-15-CE40-0008 (Défigéo).

J O U R N A L I N F O R M AT I O N
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