Arithmetic and metric aspects of open de Rham spaces
aa r X i v : . [ m a t h . AG ] J u l Arithmetic and metric aspects of open de Rham spaces
Tamás Hausel
IST Austria [email protected]
Michael Lennox Wong
Universität Duisburg–Essen [email protected]
Dimitri Wyss
Sorbonne Université [email protected]
July 12, 2018
Abstract
In this paper we determine the motivic class—in particular, the weight polynomialand conjecturally the Poincaré polynomial—of the open de Rham space, defined andstudied by Boalch, of certain moduli of irregular meromorphic connections on the triv-ial bundle on P . The computation is by motivic Fourier transform. We show that theresult satisfies the purity conjecture, that is, it agrees with the pure part of the conjec-tured mixed Hodge polynomial of the corresponding wild character variety. We alsoidentify the open de Rham spaces with quiver varieties with multiplicities of Yamakawaand Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkählermetrics on them, which in the -dimensional cases are expected to be of type ALF. Contents Motivic classes of open de Rham spaces 20
A Addendum to Section 6 58
A.1 Notation and duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.2 Representations of doubled quivers and duals . . . . . . . . . . . . . . . . . . . . . . . 58A.3 The symplectic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.4 Group actions and moment maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.5 General formulae for the symplectic form and moment map . . . . . . . . . . . . . . . 60
B Details of proof of Proposition 7.2.4 61C Additional proofs 64
C.1 Proof of Lemma 4.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64C.2 Proof of (5.1.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
For k, n ∈ Z > , let µ = ( µ , . . . , µ k ) ∈ P kn be a k -tuple of partitions of n . Let G := GL n ( C ) . Anadjoint orbit O ⊂ g := gl n ( C ) has type µ = ( µ , . . . , µ l ) ∈ P n , if the multiset of multiplicitiesof the eigenvalues of any element in O is { µ i } . Let ( O , . . . , O k ) be a generic k -tuple ofsemisimple adjoint orbits in g of type µ in the sense of Definition 3.1.6. Consider [Bo1, p.141]the variety M ∗ µ = { ( A , A , . . . , A k ) | A i ∈ O i , A + · · · + A k = 0 } // G , (1.1.1)2onstructed as an affine GIT quotient by the diagonal conjugation action of G .A point of M ∗ µ represented by ( A , A , . . . , A k ) yields a logarithmic connection k X i =1 A i dzz − a i on the trivial rank n bundle on P with poles at distinct points a i ∈ P \ {∞} and residues in O i . We call M ∗ µ the tame open de Rham space , as it sits inside as an open set in the full modulispace of flat rank n connections P with logarithmic singularities and prescribed adjoint orbitof residues around a i .Defining the conjugacy class C i = exp(2 πi O i ) ⊂ G we get a generic ( C , . . . , C k ) k -tupleof semisimple conjugacy classes of type µ . We define the tame character variety of type µ asthe GIT quotient M µ B := { M ∈ C , . . . , M k ∈ C k | M · · · M k = n } // G , (1.1.2)where n is the n × n identity matrix.The character variety parametrizes isomorphism classes of representations of the fun-damental group of P \ { a , . . . , a k } to G , with monodromy around a i in C i . One has theRiemann–Hilbert monodromy map ν a : M ∗ µ → M µ B , taking the flat connection P ki =1 A i dzz − a i to the representation given by its monodromy alongloops in P \ { a , . . . , a k } . The study of the symplectic geometry of this monodromy map,and its relations to isomonodromic deformations was initiated in [Hit2].Although this monodromy map is not algebraic, we have the following: Conjecture 1.1.3 (Purity conjecture) . The map ν ∗ a : H ∗ ( M µ , rB , Q ) → H ∗ ( M ∗ µ , r , Q ) preservesmixed Hodge structures and is an isomorphism on the pure parts.As the mixed Hodge structure of M ∗ µ is known to be pure [HLRV2, Proposition 2.2.6],the conjecture implies that the pure part of the cohomology of M µ B is isomorphic with thefull cohomology of M ∗ µ . The consistency of [HLRV2, Conjecture 1.2.1] on the mixed Hodgepolynomial of M µ B , with Conjecture 1.1.3 above was tested by checking in [HLRV2, Theorem1.3.1] that the pure part of the conjectured mixed Hodge polynomial of M µ B agrees with theweight polynomial of M ∗ µ .The proof of the purity of H ∗ ( M ∗ µ ; Q ) in [HLRV2, Proposition 2.2.6] proceeds by recalling[CB] the identification of M ∗ µ as a certain star-shaped Nakajima quiver variety. In particular, M ∗ µ aquires a natural complete hyperkähler metric of ALE type. For example, for the star-shaped affine Dynkin diagrams of ˜ D , ˜ E , ˜ E and ˜ E together with an imaginary root weobtain the corresponding ALE gravitational instantons of Kronheimer [Kr2]. The work in [Hit2] was generalized to allow meromorphic connections with irregular singu-larities by [Bo1]. The aim of the present paper is to extend the above circle of ideas to the caseof meromorphic connections with irregular singularities. Consider the following analogue3f (1.1.1). Fix k, n, s ∈ Z > and we will take µ and ( O , . . . , O k ) as in Section 1.1. In addition,for ≤ i ≤ s , let m i ∈ Z > and consider the truncated polynomial ring R m i := C [ z ] / ( z m i ) ,and the group GL n (R m i ) of invertible matrices over R m i . Let T ≤ G be the maximal torus ofdiagonal matrices and t denote its Lie algebra. We will fix an element of the form C i = C im i z m i + C im i − z m i − + · · · + C i z , (1.2.1)with C ij ∈ t , further assuming that C im i has distinct eigenvalues. An element g ∈ GL n (R m i ) acts on C i by conjugation, viewing both g and C i as matrices over the ring of Laurent poly-nomials over z ; however, we will truncate any terms with non-negative powers of z . Wedenote the GL n (R m i ) -orbit of C i under this action by O ( C i ) (it is explained in Section 2.1how we may view C i as an element of the dual of the Lie algebra of GL n (R m i ) , and O ( C i ) as its coadjoint orbit). Observe that G = GL n ( C ) sits in each GL n (R m i ) as a subgroup and soacts on each O ( C i ) ; in fact, given an element Y i = Y im i z m i + Y im i − z m i − + · · · + Y i z ∈ O ( C i ) , (1.2.2)the G -action is by conjugation on each term Y ip . Now, for ≤ i ≤ s , we set r i := m i − , andwrite r := ( r , . . . , r s ) for the tuple. We may then define the (irregular) open de Rham space by M ∗ µ , r := ( A , . . . , A k , Y , . . . , Y s ) ∈ k Y j =1 O j × s Y i =1 O ( C i ) : k X j =1 A j + s X i =1 Y i = 0 (cid:30)(cid:30) G . One likewise has an interpretation of M ∗ µ , r as a moduli space of meromorphic connec-tions (see Section 3.2), this time with higher pole orders, on the trivial rank n vector bundleover P . The class of ( A j , Y i ) yields a connection k X j =1 A j dzz − a j + s X i =1 m i X p =1 Y ip dz ( z − b i ) p for a set of distinct poles { a , . . . , a k , b , . . . , b s } ∈ P \ {∞} .The definition of the corresponding wild character variety M µ , rB , which is the space ofmonodromy data for moduli of irregular connections, is a little more involved than in thelogarithmic case. For the higher order poles b i , in addition to the topological monodromy,one must also take into account the Stokes data, which distinguishes analytic isomorphismclasses of locally defined connections from formal ones. However, even in this irregularcase, the wild character variety retains certain similarities with the character variety definedabove at (1.1.2): it is an affine variety defined only in terms of G , certain algebraic subgroupsof G , and orbits in them, and may be viewed as a space of “Stokes representations” [Bo1, §3].As we will not be working with this space directly, we refer the reader to [HMW, Definition2.1.1] for a precise definition. The definition in [HMW, Definition 2.1.1] involves some choices; the precise choices required to correspondto the above space of meromorphic connections are as follows. First, note that while k denotes the numberof simple poles both here and in [HMW], here we written s for the number of irregular poles instead of m in [HMW]. ( C , . . . , C k ) in [HMW, Definition 2.1.1] is taken as in Section 1.1. The tuple r will have the samemeaning here as there. Finally, in the tuple ξ , we take ξ α = exp(2 π √− C α ) . ν : M ∗ µ , r → M µ , rB (1.2.3)in this irregular case, which takes a connection to its monodromy data. The monodromy map ν is described in great detail in [Bo1, §3], where it is also explained why the wild charactervariety takes the form that it does. We then have Conjecture 1.2.4 (Irregular purity conjecture) . The irregular Riemann–Hilbert map ν (1.2.3)induces a map ν ∗ : H ∗ ( M µ , rB , Q ) → H ∗ ( M ∗ µ , r , Q ) which preserves mixed Hodge structuresand is an isomorphism on the pure parts. To formulate our main result define the Cauchy kernel Ω k ( z, w ) := X λ ∈P H λ ( z, w ) k Y i =1 ˜ H λ ( z , w ; x i ) ∈ Λ( x , . . . , x k ) ⊗ Z Q ( z, w ) , and let H µ ,r ( z, w ) =( − rn ( z − − w ) (cid:10) Log (Ω k + r ) , h µ ( x ) ⊗ · · · ⊗ h µ k ( x k ) ⊗ s (1 n ) ( x k +1 ) ⊗ · · · ⊗ s (1 n ) ( x k + s ) (cid:11) . The notation is explained in Section 5.2. One of our main results is the computation of theweight polynomial of open de Rham space M ∗ µ , r . Theorem 1.3.1. P k,i ≥ ( − i dim C (cid:0) Gr Wk H ic ( M ∗ µ , r , C ) (cid:1) q k = q d µ , r / H ˜ µ ,r (0 , q / ) The main conjecture of [HMW, Conjecture 1.2.2] claims that
Conjecture 1.3.2. X i,j,k ≥ dim C (cid:0) Gr W k H ic ( M µ , rB , C ) (cid:1) q k t i = q d µ , r / H ˜ µ ,r ( − q − / t − , q / ) . Thus, if H ∗ ( M ∗ µ , r , Q ) were pure, our main Theorem 1.3.1 would be a consequence of ourpurity conjecture Conjecture 1.2.4 and Conjecture 1.3.2. However, we were unable to provethat H ∗ ( M ∗ µ , r , 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 k = 0 , i.e.,only irregular punctures. In this case, we can proceed by motivic Fourier transform, as in[W], and the result will be a motivic extension of Theorem 1.3.1. The general case—Theorem5.1.6 and Corollary 5.2.3—is then proved via the Fourier transform over finite fields, whichwas introduced in [HLRV1].As an analogue of Crawley-Boevey’s result [CB] for the irregular case, in Section 6 weprove that the open de Rham spaces M ∗ µ , r are isomorphic to quiver varieties with multi-plicities. These varieties have been considered by Yamakawa [Y2] in the rank case andtheir defining equations, in terms of certain preprojective algebras, have been studied byGeiss–Leclerc–Schröer [GLS] in general. Then, in Section 6.4.2, we consider the star-shapednon-simply laced affine Dynkin diagrams which correspond to open de Rham spaces of di-mension . Here we discuss the main results of the paper in these special toy cases.Finally, in Section 7 we study some natural complete hyperkähler metrics on M ∗ µ , r whenthe irregular poles have order . In the (real) four-dimensional toy example cases, e.g., thoseappearing in Section 6.4.2, we expect the resulting metrics to be of type ALF.5 cknowledgements We would like to thank Gergely Bérczy, Roger Bielawski, Philip Boalch,Sergey Cherkis, Andrew Dancer, Brent Doran, Eloïse Hamilton, Frances Kirwan, BernardLeclerc, Emmanuel Letellier, Alessia Mandini, Maxence Mayrand, Daisuke Yamakawa fordiscussions related to the paper. At various stages of this project the authors were supportedby the Advanced Grant “Arithmetic and physics of Higgs moduli spaces” no. 320593 of theEuropean Research Council, by grant no. 153627 and NCCR SwissMAP, both funded by theSwiss National Science Foundation as well as by EPF Lausanne and IST Austria. In the finalstages of this project, MLW was supported by SFB/TR 45 “Periods, moduli and arithmeticof algebraic varieties”, subproject M08-10 “Moduli of vector bundles on higher-dimensionalvarieties”. DW was also supported by the Fondation Sciences Mathématiques de Paris, aswell as public grants overseen by the Agence national de la recherche (ANR) of France aspart of the « Investissements d’avenir » program, under reference numbers ANR-10-LABX-0098 and ANR-15-CE40-0008 (Défigéo).
Let us fix a perfect base field K and n ≥ an integer. We will set G := GL n ( K ) and its Liealgebra g := gl n ( K ) . Furthermore, T ⊂ G will denote the standard maximal torus consistingof the invertible diagonal matrices, t ⊂ g its Lie algebra and t reg ⊂ t the subset of elementswith distinct eigenvalues. Fix another integer m ≥ and let R m := K [[ z ]] / ( z m ) = K [ z ] / ( z m ) .Then we may also consider these groups over R m , G m := GL n (R m ) = (cid:8) g + zg + · · · + z m − g m − | g ∈ G , g , . . . , g m − ∈ g (cid:9) , (2.1.1) g m := gl n (R m ) = (cid:8) X + zX + · · · + z m − X m − | X i ∈ g (cid:9) , and we define T m and t m similarly. We will regard G m and T m as algebraic groups over K : from the description above, G m = G × g m − as a K -variety; writing out the componentsof each z i under the group law in G m , it is easy to see that the operation is well-defined ontuples and so one indeed gets an algebraic group over K . Of course, g m is a vector spaceover K .Observe that GL n (R m ) is not reductive; its unipotent radical G m ≤ G m and Lie algebra g m are, respectively, G m = GL n (R m ) = { n + zb + z b + · · · + z m − b m − | b i ∈ g } , g m = gl n (R m ) = { zX + z X + · · · + z m − X m − | X i ∈ g } , where n denotes the n × n identity matrix.There is semi-direct product decomposition G m = G m ⋊ G , (2.1.2)where we identify G with the subgroup of those elements satisfying g = · · · = g m − = 0 ,
6n the notation of (2.1.1); we will often refer to G identified as such as the subgroup of “con-stant” elements in G m . We thus obtain a direct sum decomposition g m = g m ⊕ g ; (2.1.3)this decomposition is preserved by the adjoint action of G but not of G m . We will write T m := T m ∩ G m and t m := t m ∩ g m .It will be convenient to identify the dual vector space g ∨ m with z − m g m = (cid:26) z − m Y m + z − ( m − Y m − + · · · + z − Y (cid:12)(cid:12)(cid:12)(cid:12) Y i ∈ g (cid:27) (2.1.4)via the trace residue pairing. This means that for X ∈ g m and Y ∈ z − m g m we set h Y, X i := Res z =0 tr Y X = m X i =1 tr Y i X i − . (2.1.5)Under this identification, the dual g ∨ of the subgroup G ⊆ G m corresponds to the subspace z − g ⊂ z − m g m and ( g m ) ∨ to those elements in z − m g m having zero residue term. We write π res : g ∨ m → g ∨ , π irr : g ∨ m → ( g m ) ∨ (2.1.6)for the natural projections. The latter projection may be identified with z − m g m → z − m g m (cid:14) z − g m , (2.1.7)so we are simply truncating the residue term.The adjoint and coadjoint actions of G m on g m and g ∨ m will both be denoted by Ad andare defined by the same formula: for g ∈ G m ,Ad g X = gXg − , X ∈ g m Ad g Y = gY g − , Y ∈ g ∨ m . What we mean in the latter case is that we consider g , g − and Y as matrix-valued Laurentpolynomials in z and we truncate all terms of non-negative degree in z after multiplying.With this convention we have h Ad g Y, X i = (cid:10) Y, Ad g − X (cid:11) . Consider the tuple λ = ( λ , . . . , λ l ) ∈ N l +1 with P l +1 i =1 λ i = n . Let L λ ≤ G be the subgroupof block diagonal matrices, with blocks of size λ , . . . , λ l , i.e., L λ = { diag ( f , . . . , f l ) ∈ G : f i ∈ GL λ i ( K ) } . (2.1.8)Let l λ denote its Lie algebra. The centre of l λ is given by z ( l λ ) = { diag ( c I λ , . . . , c l I λ l ) : c , . . . , c l ∈ K } ⊆ t . We will write z ( l λ ) reg to be the subset of z ( l λ ) for which the c i are pairwise distinct. The centerof L λ satisfies Z(L λ ) = z ( l λ ) ∩ G , and can be described as the subset of z ( l λ ) with all c i ∈ K × .An important special case is when λ = (1 , . . . , , so one has l λ = t and z ( l λ ) reg = t reg . Wewill also use the notation as above for these groups, namely L λ,m := L λ (R m ) L λ,m := L λ,m ∩ G m l λ,m := l λ (R m ) l λ,m := l λ,m ∩ g m . .2 Coadjoint orbits Coadjoint orbits for groups of the form G m will play a prominent role in this paper, so herewe will set some notation and record some results that will be used later. By a diagonal element or formal type in g ∨ m , we will mean an element of the form C = C m z m + C m − z m − + · · · + C z ∈ t ∨ m = z − m t m , with C ℓ ∈ t ( K ) , ≤ ℓ ≤ m. (2.2.1)By permuting diagonal entries if necessary, C may be written C = diag ( c I λ , c I λ , . . . , c l I λ l ) (2.2.2)for some tuple ( λ , . . . , λ l ) of n , with the c i ∈ z − m R m distinct. In fact, we will make thestronger assumption that for all ≤ i = j ≤ lz m ( c i − c j ) ∈ R × m . (2.2.3)If we write c i = c im z m + · · · + c i z with c iℓ ∈ K , ≤ ℓ ≤ m , then this is equivalent to c im = c jm for all ≤ i = j ≤ l ; that is, all the leading coefficients are distinct. We fix such a diagonalelement C and write O ( C ) for the image of the orbit map η : G m → gl ∨ m g Ad g ( C ) Lemma 2.2.4.
Let C be a diagonal element satisfying (2.2.3).(a) The coadjoint orbit O ( C ) is an affine variety.(b) Let Z G m ( C ) denote the centralizer of C in G m . There is a G m -equivariant isomorphism φ : G m / Z G m ( C ) ∼ −→ O ( C ) such that the diagram G m { { ①①①①①①①①①① η (cid:30) (cid:30) ❁❁❁❁❁❁❁❁❁ G m /Z G m ( C ) φ / / O ( C ) commutes. In particular O ( C ) is a homogeneous space for G m and η is a categoricalquotient.We will give a description of the centralizers Z G m ( C ) for certain C in Lemma 2.2.8(d)below. 8 roof. To see (a), we note that O ( C ) will be the set of elements satisfying the appropriate“minimal polynomial” over R m , which may then be written down as equations over K . Tobe explicit about this, let Y ∈ g m be written as in (2.1.4); we consider z m Y as an element of g m by writing z m Y = Y m + zY m − + · · · + z m − Y . Similarly, we take z m c i ∈ R m writing z m c i = c im + zc im − + · · · + z m − c i . Then the minimal polynomial as mentioned gives defining equations for O ( C ) in the sensethat the following matrix over R m is zero if and only if Y ∈ O ( C ) : ( z m Y − z m c ) · · · ( z m Y − z m c l ) . Of course, this can be expanded and the coefficient of each power z i gives a matrix equationover K ; the individual entries of all of these matrices give the algebraic equations for O ( C ) .The fact that these are indeed defining equations for O ( C ) can be proved in the same way oneproves the theorem which asserts that a matrix is diagonalizable if and only if its minimalpolynomial has distinct roots; for this we need to use the assumption (2.2.3).For (b) notice that both G and O ( C ) are smooth over K and for every closed point A ∈O ( C ) one has dim η − ( A ) = dim Z G m ( C ) . Thus η is faithfully flat by [Mat, Theorem 23.1] and φ an isomorphism by [Mi, Proposition 7.11]. Fix a tuple λ = ( λ , . . . , λ l ) ∈ N l +1 with P λ i = n and consider the group L λ ≤ G as in(2.1.8). Let l od λ ⊆ g denote the subspace consisting of matrices with zeros on the diagonalblocks (of course, the superscript can be read, “off diagonal”), so that we have an obvious L λ -invariant decomposition g = l λ ⊕ l od λ . (2.2.5)In the case λ = (1 , . . . , , we set g od := l od λ , so that this is the space of matrices with zeroesalong the main diagonal.Let b = n + P m − i =1 z i b i ∈ G m . We may write b − = + m − X i =1 z i w i , where the w i ∈ g are given by w i = i X p =1 ( − p X ( λ ,λ ,...,λ p ) ∈ N p λ + ··· + λ p = i b λ · · · b λ p . (2.2.6)Suppose A = P mj =1 z − j A j ∈ g ∨ m . Then we haveAd b A = A + m − X p =1 z − m + p p X i =1 [ b i , A m + i − p ] + p − X i =1 p − i X j =1 [ b i , A m − p + i + j ] w j . (2.2.7)9 emma 2.2.8. (a) If X ∈ g and Y ∈ z ( l λ ) , then [ X, Y ] ∈ l od λ .(b) If X ∈ g , Y ∈ z ( l λ ) reg and [ X, Y ] ∈ l λ (or equivalently, by (2.2.5), [ X, Y ] = 0 ), then X ∈ l λ .(c) Let C = m X i =1 z − i C i ∈ z ( l λ ) ∨ m ⊆ t ∨ m be given as in (2.2.2) with C m ∈ z ( l ) reg and suppose g ∈ G m is such that Ad g C − C ∈ l ∨ λ,m − . Then g ∈ L λ,m and Ad g C = C .(d) Let C be as in (c). Then the centralizers of C and C := π irr ( C ) ∈ ( t m ) ∨ are Z G m ( C ) = L λ,m Z G m ( C ) = G , rc λ,m := b = + m − X j =1 z j b j ∈ G m (cid:12)(cid:12)(cid:12)(cid:12) b , . . . , b m − ∈ l λ . The superscript rc stands for “regular centralizer” which name is justified by the state-ment.(e) Let g ∈ Z (L λ ) and h ∈ G m such that hgh − ∈ L λ,m . Then hgh − = g . Proof.
The first two statements can be seen by writing everything out as block matrices. Forpart (c), by (2.1.2), we may write g = g b for some g ∈ G , b ∈ G m . Thus, Ad g C can beobtained by applying Ad g to the expression in (2.2.7). The z − m term is then Ad g C m ; thehypothesis is that this is C m . Then part (b) implies that g ∈ l λ ∩ G = L λ . Already this showsthat Ad g C = C , since all C i ∈ z ( l λ ) .Now, the z − ( m − term in Ad g C − C is then Ad g [ b , C m ] . The assumption is that this liesin l λ ; by L λ -invariance, [ b , C m ] ∈ l λ and again by (b), b ∈ l λ . By induction, we may assume b , . . . , b r ∈ l λ . We wish to show that b r +1 ∈ l λ . Then from (2.2.7), the z − ( m − r − -term ofAd g C − C is Ad g r +1 X i =1 [ b i , C m + i − r − ] + r X i =1 r − i +1 X j =1 [ b i , C m − r + i + j − ] w j . The induction hypothesis implies that the only commutator that does not vanish is [ b r +1 , C m ] and then by assumption, this lies in l λ , and again we conclude by (b). This shows that all b i ∈ l λ and hence h ∈ L λ,m ; as g ∈ L λ , g = g b ∈ L λ,m . Furthermore, as above, we canshow that all commutators in (2.2.7) vanish, and we have already noted Ad g C = C ; henceAd g C = C .The first assertion in part (d) follows from (c). The second assertion uses the same argu-ment, and one simply needs to observe that we are omitting the residue term when comput-ing in g m and hence no condition is imposed on b m − .Part (e) is proved with an inductive argument similar to that of (c), using (2.2.6).Next, we study “regular semisimple” coadjoint orbits. We take C ∈ t ∨ m , let C := π irr ( C ) ∈ ( t m ) ∨ with C m ∈ t reg . We will write O ( C ) for the G m -coadjoint orbit through C and O ( C ) for the G m -coadjoint orbit through C . These are coadjoint orbits for different groups. 10et G od m := b = + m − X j =1 z j b j ∈ G m (cid:12)(cid:12)(cid:12)(cid:12) b , . . . , b m − ∈ g od G , od m := b = + m − X j =1 z j b j ∈ G od m (cid:12)(cid:12)(cid:12)(cid:12) b m − = 0 Observe that G , od m ⊆ G od m are subvarieties of G m , though not subgroups, isomorphic toaffine spaces of dimensions ( m − n ( n − and ( m − n ( n − , respectively. Lemma 2.2.9. (a) The restriction of the multiplication map yields isomorphisms G od m × T m ∼ −→ G m G , od m × G , rc m ∼ −→ G m . In other words, every b ∈ G m has unique factorizations b = b od b T b = b , od b rc with b od ∈ G od m , b T ∈ T m , b , od ∈ G , od m , b rc ∈ G , rc m , and the map taking b to any one ofthese factors is a morphism.(b) The morphism G , od m → O ( C ) b Ad b C is an isomorphism. In particular, O ( C ) ∼ = A ( m − n ( n − .(c) There is an isomorphism Γ : (cid:16) G × G od m (cid:17) (cid:14) T → O ( C ) [ g, b ] Ad gb C, where the action of T on G × G od m is given by ( g , b ) t = ( g t , Ad t − b ) . In particular, G × G od m → (G × G od m ) / T is a Zariski locally trivial principal T -bundle. Proof.
To prove part (b), consider elements x := + m − X j =1 z j x j , y := + m − X j =1 z j y j , b := + m − X j =1 z j b j ∈ G m Then the product expression xy = b imposes the relations b j = x j + y j + j − X ℓ =1 x ℓ y j − ℓ (2.2.10)for ≤ j ≤ m − . Assuming that b ∈ G m is arbitrary, the equation b = x + y allowsus to take x ∈ g od , y ∈ t to be the off-diagonal and diagonal parts of b , respectively. Therelations (2.2.10) allow us to continue this inductively, so that all x j ∈ g od and y j ∈ t , noting11hat at each stage, one has an algebraic expression in terms of the b j . This gives the firstfactorization. The second one is obtained in exactly the same way, except that the conditionsare that x m − = 0 , but there is no condition on y m − , which makes up for it.Part (b) follows directly from Lemma 2.2.8(d) and part (a).For part (c), note that Lemma 2.2.4(b), together with Lemma 2.2.8(d), gives an isomor-phism φ : G m / T m ∼ −→ O ( C ) . Observe that G m = G ⋉ G m and hence, as a variety, one has G m = G × G m = G × G od m × T m , by part (a). Taking the quotient by T m = T × T m then gives the desired isomorphism. Oneobtains the indicated T -action on G × G od m by identifying G × G m with G m via multiplication.The final statement follows since G → G / T is a Zariski locally trivial principal T -bundle, as T is a special group [Se, §4.3]. Following [CL] and [CLL], in this section we introduce Grothendieck rings with exponen-tials, a naive notion of motivic Fourier transform and convolution. Similar techniques wereused in [W] to compute the motivic classes of Nakajima quiver varieties. Throughout thissection, K will denote an arbitrary field, by a variety we mean a separated scheme of finitetype over K , and by a morphism of varieties we will mean a K -morphism. The
Grothendieck ring of varieties , denoted by
KVar , is the quotient of the free abelian groupgenerated by isomorphism classes of varieties modulo the relation X − Z − U, for X a variety, Z ⊂ X a closed subvariety and U = X \ Z . The multiplication is given by [ X ] · [ Y ] = [ X × Y ] , where we write [ X ] for the equivalence class of a variety X in KVar .The
Grothendieck ring with exponentials
KExpVar is defined similarly. Instead of varietieswe consider pairs ( X, f ) , where X is a variety and f : X → A = Spec ( K [ T ]) is a morphism.A morphism of pairs u : ( X, f ) → ( Y, g ) is a morphism u : X → Y such that f = g ◦ u . Then KExpVar is defined as the free abelian group generated by isomorphism classes of pairsmodulo the following relations.(i) For a variety X , a morphism f : X → A , a closed subvariety Z ⊂ X and U = X \ Z the relation ( X, f ) − ( Z, f | Z ) − ( U, f | U ) . (ii) For a variety X and pr A : X × A → A the projection onto A the relation ( X × A , pr A ) . The class of ( X, f ) in KExpVar will be denoted by [ X, f ] . We define the product of twogenerators [ X, f ] and [ Y, g ] as [ X, f ] · [ X, g ] = [ X × Y, f ◦ pr X + g ◦ pr Y ] , f ◦ pr X + g ◦ pr Y : X × Y → A is the morphism sending ( x, y ) to f ( x ) + g ( y ) . Thisgives KExpVar the structure of a commutative ring with identity [pt , .Denote by L the class of A in KVar , resp. ( A , in KExpVar . The localizations of
KVar and
KExpVar with respect to the the multiplicative subset generated by L and L n − , where n ≥ are denoted by M and E xp M .For a variety S there is a straightforward generalization of the above construction toobtain the relative Grothendieck rings KVar S , KExpVar S , M S and E xp M S . For example, gen-erators of KExpVar S are pairs ( X, f ) where X is an S -variety (i.e., a variety with a morphism X → S ) and f : X → A a morphism. The class of ( X, f ) in KExpVar S will be denoted by [ X, f ] S or simply [ X, f ] if the base variety S is clear from the context.The usual Grothendieck ring KVar S can be identified with the subring of KExpVar S gen-erated by pairs ( X, , see [CLL, Lemma 1.1.3].For a morphism of varieties u : S → T we have induced maps u ! : KExpVar S → KExpVar T , [ X, f ] S [ X, f ] T u ∗ : KExpVar T → KExpVar S , [ X, f ] T [ X × T S, f ◦ pr X ] S . In general, u ∗ is a morphism of rings and u ! a morphism of additive groups. The rings
KVar and
KExpVar and their localizations M , E xp M although easy to define, arequite hard to understand concretely. To circumvent this, one usually considers realizationmorphisms to simpler rings.If K = F q is a finite field and S a variety over F q we can construct a realization from KExpVar S to the ring Map( S ( F q ) , C ) of complex valued functions on S ( F q ) by sending theclass of [ X, f ] to the function s ∈ S ( F q ) X x ∈ X s ( F q ) Ψ (cid:0) f ( x ) (cid:1) , (2.3.1)where Ψ : F q → C × is a fixed non-trivial additive character. Under this realization, for amorphism u : S → T , the operations u ! and u ∗ correspond to summation over the fibers of u and composition with u , respectively.If K is a field of characteristic , whose transcendence degree over Q is at most the oneof C / Q , we can embed K into C and consider any variety over K as a variety over C . By thework of Deligne [De1, De2] the compactly supported cohomology H ∗ c ( X, C ) of any complexalgebraic variety X carries two natural filtrations, the weight and the Hodge filtration. Tak-ing the dimensions of the graded pieces we obtain the compactly supported mixed Hodgenumbers of X h p,q ; ic ( X ) = dim C (cid:0) Gr Hp Gr Wp + q H ic ( X, C ) (cid:1) . From these numbers we define the
E-polynomial as E ( X ; x, y ) = X p,q,i ≥ ( − i h p,q ; ic ( X ) x p y q . (2.3.2)This way we obtain a morphism (see for example [HRV, Appendix]) KVar → Z [ x, y ] [ X ] E ( X ; x, y ) . (2.3.3)13t is not hard to see that E ( L ; x, y ) = xy and thus this realization extends to a morphism M → Z [ x, y ] (cid:20) xy ; 1(1 − ( xy ) n ) , n ≥ (cid:21) . These two realizations are related by the following Theorem of Katz. We refer to [HRV,Appendix] for the precise definition of a strongly polynomial count variety over C . Theorem 2.3.4. [HRV, Theorem 6.1.2(3)] If X over C is strongly polynomial count withcounting polynomial P X ( t ) ∈ Z [ t ] , then E ( X ; x, y ) = P X ( xy ) . In particular in the situation of Theorem 2.3.4, the E -polynomial of X is a polynomial inone variable, which we also call the weight polynomial E ( X ; q ) = E ( X ; q , q ) . We now introduce several tools for our computations is
KVar and
KExpVar , which aremostly inspired by similar constructions over finite fields through the realization (2.3.1).
Fibrations.
We say that f : X → Y is a Zariski locally-trivial fibration if Y admits an opencovering Y = ∪ j U j such that f − ( U j ) ∼ = F × U j for some fixed variety F . In this case wehave the product formula [ X ] = [ F ][ Y ] (2.3.5)in KVar as we can compute directly [ X ] = X j [ f − ( U j )] − X j When computing character sums over finite fields, one has the followingcrucial identity X v ∈ V Ψ (cid:0) f ( v ) (cid:1) = ( q dim V if f = 00 otherwise,where V is a vector space over F q and f ∈ V ∨ a linear form. To establish an analogousidentity in the motivic setting, we let V be a finite-dimensional vector space over K and S a variety. We replace the linear form above with a family of affine linear forms, i.e., amorphism g = ( g , g ) : X → V ∨ × K , where X is an S -variety. Then we define f to be themorphism f : X × V → K ( x, v ) 7→ h g ( x ) , v i + g ( x ) . Finally we put Z = g − (0) . Lemma 2.3.6. [W, Lemma 2.1] With the notation above we have the relation [ X × V, f ] = L dim V [ Z, g | Z ] in KExpVar S . In particular, if X = Spec K and f ∈ V ∨ , we have [ V, f ] = 0 unless f = 0 .14 ourier transforms. We now define the naive motivic Fourier transform for functions on afinite dimensional K -vector space V and the relevant inversion formula. All of this is aspecial case of [CL, Section 7.1]. Definition 2.3.7. Let p V : V × V ∨ → V and p V ∨ : V × V ∨ → V ∨ be the obvious projections. The naive Fourier transformation F V is defined as F V : KExpVar V → KExpVar V ∨ φ p V ∨ ! (cid:0) p ∨ V φ · [ V × V ∨ , h , i ] (cid:1) . Here h , i : V × V ∨ → K denotes the natural pairing. We will often write F instead of F V when no ambiguity will arise from doing so.Of course, the definition is again inspired by the finite field version, where one definesfor any function φ : V → C the Fourier transform at w ∈ V ∨ by F ( φ )( w ) = X v ∈ V φ ( v )Ψ( h w, v i ) . Notice that F is a homomorphism of groups and thus it is worth spelling out the defini-tion in the case when φ = [ X, f ] is the class of a generator in KExpVar V . Letting u : X → V be the structure morphism we simply have F ([ X, f ]) = [ X × V ∨ , f ◦ pr X + h u ◦ pr X , pr V ∨ i ] . (2.3.8)We have the following version of Fourier inversion. Proposition 2.3.9. [W, Proposition 2.2] For every φ ∈ KExpVar V we have the identity F ( F ( φ )) = L dim V · i ∗ ( φ ) , where i : V → V is multiplication by − . Convolution. Finally, we introduce a motivic version of convolution. Definition 2.3.10. Let R : KExpVar V × KExpVar V → KExpVar V × V be the natural morphismsending two varieties over V to their product, and s : V × V → V the sum operation. The convolution product is the associative and commutative operation ∗ : KExpVar V × KExpVar V → KExpVar V ( φ , φ ) φ ∗ φ = s ! R ( φ , φ ) . As expected the Fourier transform interchanges product and convolution product. Proposition 2.3.11. For φ , φ ∈ KExpVar V we have F ( φ ∗ φ ) = F ( φ ) F ( φ ) . Proof. As both F and ∗ are bilinear, it is enough prove the identity for two generators [ X, f ] , [ Y, g ] ∈ KExpVar V with respective structure morphisms u : X → V , v : Y → V .Using (2.3.8) we can then directly compute F ([ X, f ] ∗ [ Y, g ]) = F ( s ! [ X × Y, f ◦ pr X + g ◦ pr Y ])= [ X × Y × V ∨ , f ◦ pr X + g ◦ pr Y + h s ◦ ( u × v ) ◦ pr X × Y , pr V ∨ i ]= [ X × V ∨ , f ◦ pr X + h u ◦ pr X , pr V ∨ i ][ Y × V ∨ , g ◦ pr Y + h v ◦ pr Y , pr V ∨ i ]= F [ X, f ] F [ Y, g ] . emark . We will use the convolution product to study equations in a product of vari-eties, i.e., consider V -varieties u i : X i → V say for i = 1 , . Then it follows from the definitionof ∗ , that for any v : Spec K → V the class of { ( x , x ) ∈ X × X | u ( x ) + u ( x ) = v } isgiven by v ∗ ([ X ] ∗ [ X ]) . Proposition 2.3.11 allows us to compute the latter by understandingthe Fourier transforms F ( X ) , F ( X ) separately. In this section we define open de Rham space as an additive fusion of coadjoint orbits, similaras in [HY, §2]. They were first introduced in [Bo1, Section 2] as certain moduli spaces ofconnections on P . We recall this viewpoint briefly in Section 3.2. Fix a diagonal element C ∈ g ∨ m as in (2.2.1) and consider its G m -coadjoint orbit O ( C ) ⊆ g ∨ m .In the case that K = R or C , G m is a real or complex Lie group, accordingly, and the coad-joint orbit O ( C ) admits a canonical symplectic form (the Kirillov–Kostant–Souriau form) forwhich the coadjoint action is Hamiltonian with moment map given by the inclusion µ O ( C ) : O ( C ) ֒ → g ∨ m [A, §§3.3.5, 3.3.8]. We may restrict the action on O ( C ) to that of the constantsubgroup G ≤ G m , using (2.1.2). The moment map µ res = π res ◦ µ O ( C ) : O ( C ) → z − g = g ∨ for the restricted action is the composition of the inclusion and the projection π res : g ∨ m → g ∨ (2.1.6), so simply takes the residue term Y m z m + Y m − z m − + · · · + Y z Y z . (3.1.1)Now, for d ∈ Z > , let m i ∈ Z > for ≤ i ≤ d , and let C i ∈ g ∨ m i be diagonal elements and O ( C i ) their coadjoint orbits. We may form the product Q di =1 O ( C i ) which has the productsymplectic structure. It also carries a diagonal G -action for which the moment map µ : Q di =1 O ( C i ) → g ∨ is ( Y , . . . , Y d ) d X i =1 Y i z . (3.1.2)In this case where K = R or C , we can then form the symplectic (Marsden–Weinstein) quo-tient in the usual way. We now observe that if K is any field, then both (3.1.1) and (3.1.2) aredefined over K . Definition 3.1.3. Let C = ( C , . . . , C d ) be a tuple of diagonal elements satisfying (2.2.3), sothat the product Q di =1 O ( C i ) is an affine variety (Lemma 2.2.4(a)). The open de Rham space M ( C ) is the affine GIT quotient M ∗ ( C ) = d Y i =1 O ( C i ) (cid:30)(cid:30) G := Spec (cid:0) K [ µ − (0)] G (cid:1) . Except possibly in Section 6 we will typically impose two more conditions on C , regular-ity and genericity. 16 efinition 3.1.4. A diagonal element C ∈ g ∨ m of order m ≥ is regular if C m ∈ t reg . A d -tuple C = ( C , . . . , C d ) is called regular if for all ≤ i ≤ d with m i ≥ , C i is regular.We now define genericity of the tuple C in a similar manner as in [HLRV1, 2.2.1]; ourmore explicit formulation will be used in later computations, e.g., in Lemma 4.3.11. Definefor I ⊂ { , , . . . , n } the matrix E I ∈ g by ( E I ) ij = ( if i = j ∈ I otherwise. (3.1.5) Definition 3.1.6. We call C generic if P di =1 tr C i = 0 and for every integer n ′ < n and subsets I , . . . , I d ⊂ { , . . . , n } of size n ′ we have d X i =1 (cid:10) C i , E I i (cid:11) = 0 . (3.1.7)In other words, there are no non-trivial subspaces V , . . . , V d ( K n of the same dimensionsuch that V i is invariant under C i for ≤ i ≤ d and P i tr C i | V i = 0 , if C is generic.Now, let C be a regular generic tuple of formal types. We adapt the following notationso as to later (Section 5.2) make comparisons with [HMW]. First, we order C , . . . , C d in away such that m = · · · = m k = 1 and m k +1 , . . . , m k + s ≥ , with k + s = d . We write µ = ( µ , . . . , µ k ) ∈ P kn for the k -tuple of partitions of n defined by the multiplicities of theeigenvalues of C i for ≤ i ≤ k .When one is talking about a meromorphic connection having a formal type of order m at a pole with a semisimple leading order term (of course, here we have only discussed suchtypes of poles), then it is standard terminology to refer to the number m − as the Poincarérank of the pole. For our moduli spaces, since C k + im k + i ∈ t reg for ≤ i ≤ s , we will write r i := m k + i − for the Poincaré rank of C i and record these in the s -tuple r = ( r , . . . , r s ) .Finally, we write r = P si =1 r i and call this the total Poincaré rank . Definition 3.1.8. For a regular generic C we write M ∗ µ , r instead of M ∗ ( C ) and refer to it asthe generic open de Rham space of type ( µ , r ) . If, furthermore, k = 0 we write M ∗ n, r for M ∗ ( C ) . Remark . This notation is justified since the invariants we compute in Sections 4 and 5will depend only on ( µ , r ) and not on the actual eigenvalues of the formal types.We will always assume s ≥ in this paper, in which case a generic C always exists if K is infinite [HLRV1, Lemma 2.2.2]. If K = F q is a finite filed one needs an additional lowerbound on q depending on n and d to make sure that the Zariski-open subvariety of A nd defined by (3.1.6) has an F q -rational point. We will not spell out an explicit bound here, aswe are only interested in sufficiently large q . Proposition 3.1.10. If non-empty, M ∗ µ , r is smooth and equidimensional of dimension d µ , r = dn − sn + r ( n − n ) − k X i =1 N ( µ i ) − n − , (3.1.11)where for µ = ( µ ≥ · · · ≥ µ l ) ∈ P n we define N ( µ ) = P lj =1 µ j .17 roof. We take µ as in (3.1.2). Clearly the scalars K × ֒ → G act trivially on µ − (0) , hencethe G -action factors though PGL n = G / K × . We show first that this PGL n -action is free on µ − (0) .Let ( A , . . . , A d ) ∈ µ − (0) and g ∈ G such that Ad g A i = A i for ≤ i ≤ d . We show nowthat g is scalar, by looking at some non-zero eigenspace V of g . Then clearly A i will preserve V for all i and by the moment map condition we deduce P i tr A i | V = 0 . The point is thenthat for each i there is a subspace V ′ i of the same dimension as V such that tr A i | V = tr C i | V ′ i . (3.1.12)By the genericity of C (Definition 3.1.6), this implies then V = V ′ i = K n and hence g is scalar.For ≤ i ≤ k , (3.1.12) follows simply because A i and C i are conjugate in G . To prove(3.1.12) for k + 1 ≤ i ≤ d , we write A i = Ad h C i for some h ∈ G m i . By conjugating A i and g with the constant term h of h we can assume without loss of generality h ∈ G m i ,i.e., h = . Then A im i = C im i ∈ t reg and thus g ∈ T . Next consider ¯ h = hgh − , whichsatisfies Ad ¯ h C i = C i . By Lemma 2.2.8(c) we have ¯ h ∈ T m i and then by 2.2.8(e) hgh − = g .This implies that h j preserves V for every ≤ j ≤ m i − and hence we have tr A | V =tr( Ad h C i ) | V = tr Ad h | V C i | V = tr C i | V . This proves (3.1.12) and hence PGL n acts freely on µ − (0) .In particular, all the G -orbits in µ − (0) are closed and hence they are in bijection with thepoints of the GIT quotient [Do, Theorem 6.1]. Furthermore, as µ is a moment map, freenessof the PGL n action implies that is a regular value of µ , which in turn implies smoothnessof µ − (0) and hence of M ∗ ( C ) . Looking at tangent spaces, we see that dim M ∗ ( C ) = dim d Y i =1 O ( C i ) − n . The formula now follows since dim O ( C i ) = n − N ( µ i ) for ≤ i ≤ k and dim O ( C s + i ) =( r i + 1)( n − n ) for ≤ i ≤ s .We will see in Corollary 5.1.8, that M ∗ µ , r is never empty if d µ , r ≥ . For more general,that is non-generic, C the non-emptyness of M ∗ ( C ) has been determined in [Hir]. We now work over the field K = C , otherwise adopting the notation of Section 2.1 for groupsand Lie algebras. In addition, we will let X be a Riemann surface and fix a point x ∈ X . Let m x ⊆ O X,x denote its maximal ideal, and b O x the completion of O x at m x . If Ω = Ω X denotesthe sheaf of holomorphic differentials on X , then Ω x the local module at x , b Ω x its completionwith respect to m x . If m ∈ Z > , we will let Ω( m · x ) denote the sheaf of meromorphicdifferentials with a pole of order ≤ m at x ; we let Ω( ∗ x ) = S m ∈ Z > Ω( m · x ) be their union,i.e., the sheaf of meromorphic differentials with a pole at x of arbitrary order. Then Ω( m · x ) x , Ω( ∗ x ) x will be the local modules and b Ω ( m · x ) x , b Ω ( ∗ x ) x their respective completions. Ofcourse, if z is a choice of coordinate centered at x one has isomorphisms b Ω ∼ −→ C [[ z ]] · dz b Ω ( m · x ) ∼ −→ z − m C [[ z ]] · dz b Ω ( ∗ x ) ∼ −→ C (( z )) · dz. (3.2.1)18onsider the space ( b Ω ( ∗ x ) x / b Ω x ) ⊗ C t . It is clear that one has a well-defined notion ofthe order of the pole of such an element. Further, under the isomorphisms (3.2.1), a given C ∈ ( b Ω ( ∗ x ) x / b Ω x ) ⊗ C t with a pole of order m has a unique representative in z − t [ z − ] ⊆ t (( z )) of the form (cid:18) C m z m + · · · + C z (cid:19) dz (3.2.2)with C i ∈ t . Definition 3.2.3. A formal type of order m at x is an element C ∈ ( b Ω ( ∗ x ) x / b Ω x ) ⊗ C t with a poleof order m . We will call such a formal type C of order m ≥ regular if, upon some choice ofcoordinate z at x , in the expression (3.2.2), one has C m ∈ t reg . Definition 3.2.4. Let V be a holomorphic vector bundle over X and ∇ a meromorphic con-nection with a pole (only) at x . Choose any holomorphic trivialization of V in a neighbour-hood of x and let A be the connection matrix of ∇ with respect to this trivialization; if ∇ hasa pole of order m at x , then A yields an element of Ω( m · x ) x ⊗ g . Let C be a formal typeat x . We say that ( V, ∇ ) has formal type C at x if there exists a formal gauge transformation g ∈ G ( b O x ) such that the class of Ad g A − dg · g − ∈ ( b Ω ( ∗ x ) x / b Ω x ) ⊗ g agrees with that of C under the inclusion (cid:16)b Ω ( ∗ x ) x / b Ω x (cid:17) ⊗ t ֒ → (cid:16)b Ω ( ∗ x ) x / b Ω x (cid:17) ⊗ g . With these definitions, the open de Rham spaces admit the following moduli description.Set X = P with an effective divisor D = m a + m a + · · · + m d a d and for ≤ i ≤ d , a formaltype C i of order m i at a i . By ’forgetting’ dz in (3.2.2) we obtain a tuple C = ( C , . . . , C d ) ofdiagonal elements in g ∨ (depending on a fixed coordinate on P ). Proposition 3.2.5. [Bo1, Proposition 2.1], [HY, Proposition 2.7, Corollaries 2.14, 2.15] For C regular and generic, the open de Rham space M ∗ µ , r is isomorphic to the moduli spaceof meromorphic connections ∇ on the trivial bundle of rank n on P , where ∇ has polesbounded by D and prescribed formal type C i at a i . Remark . More generally, fixing the polar divisor D and the local parameters C , onemay construct a moduli space M ( C ) of meromorphic connections ( V, ∇ ) on P , where V isa degree vector bundle (though not necessarily trivial) and ∇ has formal type C i at a i . Ananalytic construction is given by [Bo1, Proposition 4.5], which is generalized to higher genuscurves in [BB]. Algebraic constructions of these moduli spaces are given in [IS].Following the tradition initiated in [Si], such moduli spaces are typically called “de Rhammoduli spaces”. Forgetting the connection, these spaces yield, a fortiori, families of vectorbundles of degree . As the only semi-stable bundle of degree on P is the trivial bundle,and since the semi-stable locus of a family is always Zariski open, the moduli spaces M ∗ ( C ) sit as open subvarieties M ∗ ( C ) ⊆ M ( C ) . It is for this reason that we call them “open” de Rham spaces, preferring this to the awkwardnotation-inspired name of “m-star spaces” which may nevertheless be a useful informal des-ignation. 19 Motivic classes of open de Rham spaces The main result is this section is Theorem 4.3.1, a formula for the motivic class of M n, r in thelocalized Grothendieck ring M for any d -tuple r = ( r , . . . , r d ) ∈ N d . By definition M n, r isan additive fusion of coadjoint orbits and thus the computation of [ M n, r ] can be split up intwo parts.First we determine in Theorem 4.2.1 the motivic Fourier transform of the composition O ( C ) ֒ → g ∨ m → g ∨ , under the assumption m ≥ . In particular we prove that F ( O ( C )) ∈ KExpVar g is supportedon semisimple conjugacy classes, which is not true for m = 1 . The motivic convolutionformalism then allows us to deduce the formula for [ M n, r ] from these local computations.Throughout this section K will always be an algebraically closed field of or odd char-acteristic. For n ∈ Z > , we denote by P n the set of partitions of n . For λ = ( λ ≥ λ ≥ · · · ≥ λ l ) ∈ P n we use the following abbreviations l ( λ ) := lN ( λ ) := l X i =1 λ i λ ! := l Y i =1 λ i ! (cid:18) nλ (cid:19) := n ! Q li =1 λ i ! . Further we write m k ( λ ) for the multiplicity of k ∈ Z > in λ and u λ := Y k ∈ Z > m k ( λ )! . The polynomial φ λ ∈ Z [ t ] is defined as φ λ ( t ) := l Y i =1 (1 − t )(1 − t ) · · · (1 − t λ i ) . Then if L λ ∼ = Q li =1 GL λ i denotes the subgroup of block diagonal matrices in GL n we have [L λ ] = ( − n L N ( λ ) − n φ λ ( L ) ∈ KVar . In this section we compute the Fourier transform F ( O ( C )) ∈ KExpVar g of a coadjoint orbit π res : O ( C ) → g ∗ of an element C ∈ t ∨ m , say in the notation of (2.2.1), where we use thelanguage of Section 2.3. Assuming m ≥ , we can give an explicit formula for F ([ O ( C )]) ,but to do so we will first need to introduce some more notation.For n ∈ Z > , we denote by P n the set of partitions of n . A semisimple element X ∈ g has type λ = ( λ ≥ · · · ≥ λ l ) ∈ P n if X has l distinct eigenvalues with multiplicities λ , . . . , λ l .We write g λ := { X ∈ g | X has type λ } and i λ : g λ ֒ → g .20 heorem 4.2.1. Let C ∈ g ∨ m be a regular diagonal element of order m ≥ . For any partition λ ∈ P n , in KExpVar g λ we have the formula i ∗ λ F ([ O ( C )]) = L n + ( m ( n − n )+( m − N ( λ ) )( L − n (cid:2) Z λ , φ C (cid:3) , (4.2.2)where Z λ := { ( g, X ) ∈ G × g λ | Ad g − X ∈ t } and φ C :Z λ → A ( g, X ) (cid:10) C , Ad g − X (cid:11) . Furthermore, the pullback of F ([ O ( C )]) to the complement g \ F λ g λ equals . Proof. By the formula (2.3.8) we have F ([ O ( C )]) = [ O ( C ) × g , (cid:10) π irr ◦ pr O ( C ) , pr g (cid:11) ] = [ O ( C ) × g , (cid:10) pr O ( C ) , pr g (cid:11) ] , where for the second equality sign we used the definition of h , i , see (2.1.5). By Lemma2.2.9(c) we can rewrite this in E xp M g as F ([ O ( C )]) = ( L − − n h G × G od m × g , D Γ ◦ pr G × G od m , pr g Ei . Now, notice that for all ( g, b, X ) ∈ G × G od m × g we have h Γ( g, b ) , X i = h Ad gb C, X i = (cid:10) Ad b C, Ad g − X (cid:11) . Thus we finally obtain F ([ O ( C )]) = ( L − − n h G × G od m × g , D Ad pr G od m C, Ad ( pr G ) − pr g Ei . (4.2.3)We will simplify this by applying Lemma 2.3.6. Consider the decomposition G od m = G od + ⊕ G od − with G od + = { b ∈ G od m | b ⌊ m +12 ⌋ = · · · = b m − = 0 } G od − = { b ∈ G od m | b = · · · = b ⌊ m − ⌋ = 0 } . It follows from Lemma 4.2.5 below that there are functions ( h , h ) : G × g × G od + → (G od − ) ∨ × K such that (cid:10) Ad b C, Ad g − X (cid:11) = h h ( g, X, b + ) , b − i + h ( g, X, b + ) for all g ∈ G , X ∈ g and b =( b + , b − ) ∈ G od + ⊕ G od − . More explicitly h and h are given by h ( g, X, b + ) = (cid:10) Ad b + C, Ad g − X (cid:11) , h h ( g, X, b + ) , b − i = (cid:10) Ad b C − Ad b + C, Ad g − X (cid:11) . Applying Lemma 2.3.6 to this decomposition formula (4.2.3) becomes F ([ O ( C )]) = ( L − − n L dim G od − [ h − (0) , h ] . Now assume first m is even. In this case ⌊ m − ⌋ = ⌊ m − ⌋ and Lemma 4.2.5 implies h − (0) = { ( g, X, b + ) ∈ G × g × G od + | Ad g − X ∈ t , [ b + , Ad g − X ] = 0 } , h | h − (0) is independent of b + . For any λ ∈ P n the pullback i ∗ λ h − (0) → Z λ is the kernelof the vector-bundle endomorphism b + [ b + , Ad g − X ] on Z λ × G od + . The rank of the kernelis constant and equals m − ( N ( λ ) − n ) , thus we finally get i ∗ λ [ h − (0) , h ] = L m − ( N ( λ ) − n ) [Z λ , φ C ] . (4.2.4)Together with dim G od − = m (cid:0) n − n (cid:1) the theorem follows when m is even.If m is odd we have h − (0) = { ( g, X, b + ) ∈ G × g × G od + | Ad g − X ∈ t , [ b + , Ad g − X ] = [ b m − , Ad g − X ] } . If we decompose b m − = b l + b u into strictly lower and upper diagonal parts, then Lemma4.2.5 implies that h | h − (0) is affine linear in b u and independent of b u if and only if b l com-mutes with Ad g − X . As before we can now apply Lemma 2.3.6 and see that (4.2.4) also holdsfor m odd, which finishes the proof.We are left with proving Lemma 4.2.5, for which we need the explicit formula (2.2.6) forthe inverse of an element b = + zb + · · · + z m − b m − ∈ G m . Notice that for (cid:4) m +12 (cid:5) ≤ p ≤ m − , b p can appear at most once in each summand on the right hand side of (2.2.6). This isthe crucial observation in the proof of Lemma 4.2.5. Lemma 4.2.5. For X ∈ g , the function φ X : G od m → K b φ X ( b , b , . . . , b m − ) = h Ad b C, X i is affine linear in b ⌊ m +12 ⌋ , . . . , b m − . It is independent of those variables if and only if X ∈ t and b , b , . . . , b ⌊ m − ⌋ commute with X .In this case, if m is odd and we decompose b m − = b l + b u , where b l and b u are strictlylower and upper triangular respectively, then φ X is affine linear in b u and independent of b u if and only if b l commutes with X . Proof. It follows directly from the observation above that φ X depends linearly on b i for (cid:4) m +12 (cid:5) ≤ i ≤ m − .For b ∈ G od m , using the notation (2.2.6) we have h Ad b C, X i = tr m X i =1 i − X j =0 b j C i w i − j − X, (4.2.6)where we use the convention b = w = . We start by looking at the dependence of h Ad b C, X i when varying b m − . The terms in (4.2.6) containing b m − are given by tr( b m − C m X − C m b m − X ) = tr[ b m − , C m ] X. As C m ∈ t reg , the commutator [ b m − , C m ] can take any value in g od , thus tr[ b m − , C m ] X is independent of b m − if and only if X ∈ ( g od ) ⊥ = t .Assume from now on X ∈ t . We show now inductively that φ X is independent of b ⌊ m +12 ⌋ , . . . , b m − if and only if b , b , . . . , b ⌊ m − ⌋ all commute with X .22o do so, fix (cid:4) m +12 (cid:5) ≤ p ≤ m − and assume that b , . . . , b m − − p commute with X .Consider the element b ′ = + zb + · · · + z p − b p − + z p b p X + z p +1 b p +1 + · · · + z m − b m − ∈ G od m . The point now is that the b p -parts of the explicit formulas for h Ad b C, X i and Res tr( b ′ Cb ′− ) = Res tr( C ) = tr( C ) are very similar. Indeed, from (4.2.6) we see, that all the terms containing b p in h Ad b C, X i are contained in tr m X i = p +1 b p C i w i − p − X + m − X r = p m X i = r +1 b i − r − C i w r X. (4.2.7)To write a formula for Res tr( b ′ Cb ′− ) we write b ′− = + zw ′ + · · · + z m − w ′ m − . Thenwe can use a similar expression as (4.2.6) to conclude that all the terms containing b p in Res tr( b ′ Cb ′− ) are contained in tr m X i = p +1 b p XC i w ′ i − p − + m − X r = p m X i = r +1 b i − r − C i w ′ r . (4.2.8)Next we want to study the dependence of the difference (4.2.7) − (4.2.8) on b p . Noticefirst, that since [ b i , X ] = 0 for ≤ i ≤ m − − p also [ w i , X ] = 0 for ≤ i ≤ m − − p andfurthermore [ w m − p − , X ] = [ X, b m − p − ] . From (2.2.6) we also see w ′ i = w i for all ≤ i < p .Finally, we remark that w r X − w ′ r is independent of b p for p ≤ r ≤ m − and the termscontaining b p in w m − X − w ′ m − are given by b p [ b m − p − , X ] . Combining all this we see thatthe terms containing b p in (4.2.7) − (4.2.8) are just tr ( b p C m [ X, b m − p − ] + C m b p [ b m − p − , X ]) = tr[ b p , C m ][ X, b m − p − ] . Since C m ∈ t reg , the commutator [ b p , C m ] can take any value in g od as we vary b p ∈ g od .Hence in order for tr[ b p , C m ][ X, b m − p − ] to be constant, we need [ X, b m − p − ] ∈ t . Since X ∈ t this is only possible if [ X, b m − p − ] = 0 , which finishes the induction step.Finally, we consider the special case when m is odd. Take p = m − . Then by thesame argument as before, we obtain that all the terms in h Ad b C, X i which depend on b p are tr[ b p , C m ][ X, b p ] . Now using the decomposition b p = b l + b u we have tr[ b p , C m ][ X, b p ] = 2 tr( b u [ C m [ X, b l ]]) . Since the orthogonal complement of strictly upper triangular matrices are the upper trian-gular matrices we see that tr( b u [ C m [ X, b l ]]) is independent of b u if and only if [ X, b l ] = 0 . In this section we compute the motivic class of the generic open de Rham space [ M ∗ n, r ] ∈ M as defined in Definition 3.1.8. 23 heorem 4.3.1. The motivic class of [ M ∗ n, r ] ∈ M is given by L d r ( L − nd − X λ ∈P n ( − n ( d − l ( λ ) − ( l ( λ ) − u λ (cid:18) nλ (cid:19) d L ( r − ( N ( λ ) − n ) φ λ ( L ) d − , (4.3.2)where d r = ( n − n )( r + d ) − n − denotes the dimension of M ∗ n, r .We start by simplifying [ M ∗ n, r ] in the following standard way. Lemma 4.3.3. We have the following relation in M [ M ∗ n, r ] = ( L − µ − (0)][G] . (4.3.4) Proof. Define G m = Q di =1 G m i and T m = Q di =1 T m i . Taking a product in Lemma 2.2.9(c) weconsider the principal T m -bundle α : G m → Q di =1 O ( C i ) . Notice that α is G -equivariantwith respect the free G -action on G m given by diagonal left multiplication. By restriction weobtain a G -equivariant principal T m -bundle X → µ − (0) . Taking the (affine GIT-)quotientby G , we obtain a principal K × \ T m -bundle G \ X → M ∗ n, r . Also, X → G \ X is a principal G -bundle, as it is the restriction of G m → G \ G m . As the groups T m , G m and K × \ T m arespecial [Se, §4.3], all the principal bundles here are Zariski locally trivial and we get [ M ∗ n, r ] = [G \ X ][ K × \ T m ] = [ X ][G][ K × \ T m ] = [ µ − (0)][T m ]( L − m ] = ( L − µ − (0)][G] . By (4.3.4) it is enough to determine [ µ − (0)] = 0 ∗ [ O ( C ) ×· · ·×O ( C d )] , where we consider Spec K → g ∗ as a morphism. Using motivic convolution and, in particular, Proposition2.3.11 we have an equality of motivic Fourier transforms F " d Y i =1 O ( C i ) = F (cid:16) [ O ( C )] ∗ · · · ∗ [ O ( C d )] (cid:17) = d Y i =1 F ([ O ( C i )]) ∈ KExpVar g . (4.3.5)Notice that the last product is relative to g , hence we have by Theorem 4.2.1 for every λ ∈ P n i ∗ λ d Y i =1 F ([ O ( C i )]) = ( L − − nd L ( r ( n − n )+ dn + N ( λ )( r − d )) d Y i =1 h Z λ , φ C i i = ( L − − nd L ( r ( n − n )+ dn + N ( λ )( r − d )) " Z dλ , d X i =1 φ C i , with the notations r = P r i , Z dλ for the d -fold product Z λ × g · · · × g Z λ and P i φ C i for thefunction taking ( z , . . . , z d ) ∈ Z dλ to P i φ C i ( z i ) . By Fourier inversion (Proposition 2.3.9) wethus get [ µ − (0)] = 0 ∗ F d Y i =1 F ([ O ( C i )]) ! = ( L − − nd L ( r ( n − n )+ dn ) − n X λ ∈P n L N ( λ )( r − d ) " Z dλ , X i φ C i (4.3.6)24his leaves us with understanding h Z dλ , P i φ C i i as an element of KExpVar . We start bytaking a closer look at Z λ = { ( g, X ) ∈ G × g λ | Ad g − X ∈ t } . If we put t λ = t ∩ g λ we havean isomorphism Z λ ∼ −→ t λ × G ( g, X ) ( Ad g − X, g ) . (4.3.7)Next, we need to fix some notation to describe t λ combinatorially. To parametrize theeigenvalues of elements in t λ define for any e ∈ N the open subvariety A e ◦ ⊂ A e as thecomplement of ∪ i = j { x i = x j } .Furthermore we need some discrete data. A set partition of n is a partition I = ( I , I , . . . , I l ) of { , , . . . , n } , i.e., I i ∩ I j = ∅ for i = j and ∪ i I i = { , , . . . , n } . For λ = ( λ ≥ · · · ≥ λ l ) ∈P n we write P λ for the set of set partitions I = ( I , . . . , I l ) of n such that ( | I | , . . . , | I l | ) =( λ , . . . , λ l ) . Notice that I = ( I , I , . . . , I l ) is ordered and hence we have |P λ | = n ! Q li =1 λ i ! = (cid:18) nλ (cid:19) . Lemma 4.3.8. The morphism p : P λ × A l ◦ → t λ , ( I, α ) l X j =1 α j E I j , is a trivial covering of degree u λ , where E I j is defined as in (3.1.5). Proof. For e ∈ N let S e be the symmetric group on e elements. Then the lemma follows fromthe fact, that the subgroup Q j ≥ S m j ( λ ) of S l acts simply transitively on the fibers of p . Lemma 4.3.9. The following relation holds in KExpVar " Z dλ , d X i =1 φ C i = [G × L d − λ ] u λ X ( I ,...,I d ) ∈ ( P λ ) d " A l ◦ , d X i =1 (cid:10) C i , p I i (cid:11) , where p I i = p |{ I i }× A l ◦ : A l ◦ → t λ . Proof. For α ∈ A l ◦ the map p |P λ ×{ α } : P λ → t λ from Lemma 4.3.8 is injective and its imageare exactly the elements in t λ with eigenvalues given by α . Combining this with (4.3.7) wesee Z λ × g Z λ ∼ = ( t λ × G) × g ( t λ × G) ∼ = t λ × G × L λ × P λ . Applying this reasoning d − times and then using Lemma 4.3.8 we obtain a trivialcovering of degree u λ G × L d − λ × P dλ × A l ◦ → Z dλ . Keeping track of the isomorphisms gives the desired equality.25 roof of Theorem 4.3.1. Combining (4.3.4), (4.3.6) and Lemma 4.3.9 we are left with computingthe character sum h A l ◦ , P di =1 (cid:10) C i , p I i (cid:11)i for a fixed d -tuple ( I , . . . , I d ) of set partitions of n .For α ∈ A l ◦ we can write d X i =1 (cid:10) C i , p I i ( α ) (cid:11) = d X i =1 * C i , l X j =1 α j E I ij + = l X j =1 α j d X i =1 D C i , E I ij E . Now, for a fixed ≤ j ≤ l we have by definition | E I j | = · · · = | E I dj | . Thus by ourgenericity assumption (3.1.7) the numbers β j = P di =1 D C i , E I ij E satisfy the assumptions ofLemma 4.3.11 below, and we deduce " A l ◦ , d X i =1 (cid:10) C i , p I i (cid:11) = ( − l − ( l − L , (4.3.10)which proves Theorem 4.3.1. Lemma 4.3.11. Let β , . . . , β e ∈ C be such that P ej =1 β j = 0 and for J ⊂ { , , . . . , e } a propersubset, P j ∈ J β j = 0 . Then for the function h· , β i : A e ◦ → K , α e X j =1 α j β j , we have [ A e ◦ , h· , β i ] = ( − e − ( e − L ∈ KExpVar . Proof. We use induction on e . For e = 1 we have β = 0 and A ◦ = A , hence the statementis clear. For the induction step consider A e ◦ as a subvariety of A e − ◦ × A . As β e = 0 we have [ A e − ◦ × A , h· , β i ] = 0 , hence [ A e ◦ , h· , β i ] = − [ A e − ◦ × A \ A e ◦ , h· , β i ] . Now notice that the complement A e − ◦ × A \ A e ◦ has e − connected components, each ofwhich is isomorphic A e − ◦ , which implies the formula. Here, the notation will be the same as in Sections 3 and 4, especially Definition 3.1.8 andSection 4.1. In this section, we replace the motivic computations over an algebraically closedfield with arithmetic ones over a finite field F q . This allows us to study the number of F q -rational points of M ∗ µ , r any pair ( µ , r ) with s ≥ .First, we derive in Theorem 5.1.6 a closed formula for |M ∗ µ , r ( F q ) | using techniques similarto those of Section 4. Then, in Theorem 5.2.3 we give a second description of |M ∗ µ , r ( F q ) | interms of symmetric functions, which allows us to identify the E -polynomial of M ∗ µ , r with thepure part of the conjectural mixed Hodge polynomial of the corresponding character variety[HMW, Conjecture 1.2.2]. This gives strong numerical evidence for the purity Conjecture1.1.3, which was one of the main motivations of this paper.26 .1 de Rham spaces and finite fields Let F q be a finite field of characteristic coprime to n and q large enough (see Remark 3.1.9)so that there exists a regular generic tuple C = ( C , . . . , C d ) of formal types over F q for agiven pair ( µ , r ) with s ≥ . Then also M ∗ µ , r is defined over F q and we can prove a versionof Theorem 4.3.1 which also includes tame poles.For a variety X defined over F q we sometimes abbreviate | X ( F q ) | = | X | . We write g ′ ( F q ) ⊂ g ( F q ) for the matrices in g ( F q ) whose eigenvalues are in F q and for λ ∈ P n wewrite g ′ λ ( F q ) = g ′ ( F q ) ∩ g λ ( F q ) . The proof of Theorem 4.2.1 then implies that the Fouriertransform of the count function C : g ∗ ( F q ) → Z , Y 7→ | π − ( Y ) | associated to the coad-joint orbit π res : O ( C ) → g ∗ is supported on F λ ∈P n g ′ λ ( F q ) . Given such an X ∈ g ′ λ ( F q ) , the F q -version of formula (4.2.2) reads F ( C )( X ) = q n + ( m ( n − n )+( m − N ( λ ) )( q − n | L λ | X t ∈ t ∩O ( X )( F q ) Ψ( h C , t i ) , (5.1.1)where O ( X ) ⊂ g denotes the orbit of X under the adjoint action and F and Ψ are defined asin Section 2.3.We now spell out a similar formula for the Fourier transform of a tame pole. Let µ =( µ ≥ · · · ≥ µ l ) ∈ P n be a partition, C ∈ t ( F q ) ∩ g λ ( F q ) and C : g ( F q ) → Z the characteristicfunction of the adjoint orbit of C . Furthermore, for e ∈ N we write S e for the symmetricgroup on e elements and S µ = Q li =1 S µ i ⊂ S n . For any w ∈ S n we fix a maximal torus T w asin [HLRV1, Section 2.5.3] and define Q L λ T w = ( − n − rk q (T w ) | L λ | q ( N ( λ ) − n ) | T w | , where rk q denotes the F q -rank. This is a shorthand notation for the value at of the Greenfunction [DL, Theorem 7.1] and we can be even more explicit. If we write µ ( w ) ∈ P n for thepartition defined by the cycles of w ∈ S n , we have | T w | = l ( µ ( w )) Y i =1 ( q µ ( w ) i − . (5.1.2) Lemma 5.1.3. For λ ∈ P n and any X ∈ g ′ λ ( F q ) we have F (1 C ) ( X ) = q ( n − N ( µ )) µ ! X w ∈ S µ Q L λ T w X Y ∈ t w ∩O ( X )( F q ) Ψ( h C, Y i ) , (5.1.4)where t w ⊂ t denotes the locus fixed by w ∈ S µ . If C is regular, then F (1 C ) is supported on g ′ ( F q ) . Proof. The proof consists of writing out Equation (2.5.5) in [HLRV1] explicitly. In their nota-tion we have F (1 C ) = ǫ G ǫ L | W L | − X w ∈ W L q d L / R gt w (cid:0) F t w (1 T w C ) (cid:1) . L ∼ = Q li =1 GL µ i denotes the centralizer of C and W L ∼ = Q li =1 S µ i the normalizer ofthe standard maximal torus T in L . We have d L = dim G − dim L = n − N ( µ ) and since T issplit over F q also ǫ G ǫ L = ( − n ( − n = 1 . The formula then becomes F (1 C ) = q n − N ( µ )2 µ ! X w ∈ W L R gt w (cid:0) F t w (1 T w C ) (cid:1) . The T w -coadjoint orbit of C is just C and hence F t w (1 T w C )( Y ) = Ψ( h C, Y i ) for all Y ∈ t w .As X ∈ g ′ λ ( F q ) is semisimple [HLRV1, Equation (2.5.4)] gives R gt w (cid:0) F t w (1 T w C ) (cid:1) ( X ) = | L λ | − X { h ∈ G( F q ) | X ∈ Ad h t w } Q C G ( X ) h T w h − (cid:0) (cid:1) Ψ( h C, Ad h − X i ) . (5.1.5)By [DL, Theorem 7.1], for any h ∈ G( F q ) , Q C G ( X ) h T w h − (1) = Q L λ T w and thus R gt w (cid:0) F t w (1 T w C ) (cid:1) ( X ) = Q L λ T w | L λ | X { h ∈ G( F q ) | X ∈ Ad h t w } Ψ( h C, Ad h − X i ) = Q L λ T w X Y ∈ t w ∩O ( X )( F q ) Ψ( h C, Y i ) . Using [HLRV1, 2.5.3] we see that for every w ∈ W C G ( C ) there is a g ∈ C G ( C )( F q ) suchthat ( t w ∩ O ( X ))( F q ) = g ( t w ∩ O ( X )( F q )) g − and hence X Y ∈ t w ∩O ( X )( F q ) Ψ( h C, Y i ) = X Y ∈ t w ∩O ( X )( F q ) Ψ( (cid:10) C, gY g − (cid:11) ) = X Y ∈ t w ∩O ( X )( F q ) Ψ( h C, Y i ) . If C is regular we have W L = { } and for X ∈ g ( F q ) clearly { h ∈ G( F q ) | X s ∈ Ad h ( t ) } = ∅ unless X ∈ g ′ ( F q ) , which finishes the proof of the lemma.Recall that for λ ∈ P n we denote by P λ the set of ordered set partitions of λ . The naturalaction of S n on { , . . . , n } induces an action on P λ and for w ∈ S n we write P wλ ⊂ P λ for theset partitions invariant under w i.e. P wλ = { I = ( I , . . . , I l ) ∈ P λ | wI j = I j for all ≤ j ≤ l } . For two partitions λ, µ ∈ P n we then define ∆( λ, µ ) = 1 µ ! X w ∈ S µ Q L λ T w |P wλ | . Theorem 5.1.6. Let d = k + s and C = ( C , . . . , C k , C k +1 , . . . , C d ) generic regular formaltypes. If s ≥ we have |M ∗ µ , r ( F q ) | = (5.1.7) q d µ , r ( q − ns − X λ ∈P n ( − n ( s − l ( λ ) − ( l ( λ ) − u λ (cid:18) nλ (cid:19) s q ( r − ( N ( λ ) − n ) φ λ ( q ) s − k Y i =1 ∆( λ, µ i ) , where d µ , r denotes the dimension of M ∗ µ , r . 28 roof. We proceed similar as in the proof of Theorem 4.3.1. First, by a standard argument(see for example [HLRV1, Theorem 2.2.4]) we have |M ∗ µ , r ( F q ) | = ( q − | µ − (0) || G | , where µ is the moment map defined in (3.1.2). Using Fourier inversion and convolution overfinite fields we further have | µ − (0) | = q − n F k Y i =1 F (1 C i ) s Y j =1 F ( C k + j ) (0) = q − n X X ∈ g ( F q ) k Y i =1 F (1 C i )( X ) s Y j =1 F ( C k + j )( X )= q − n X λ ∈P n X X ∈ g ′ λ ( F q ) k Y i =1 F (1 C i )( X ) s Y j =1 F ( C k + j )( X ) . For the last equation we used that F ( C k + j ) is supported on F λ ∈P n g ′ λ ( F q ) for all ≤ j ≤ s and our assumption s ≥ . We now parametrize g ′ λ ( F q ) using the same notation as in Section4.3, i.e., we have a surjective map p : G( F q ) × P λ × A l ◦ ( F q ) → g ′ λ ( F q ) ( g, I, α ) g l X j =1 α j E I j g − , with each fiber of p having cardinality u λ (cid:0) nλ (cid:1) | L λ | . Since F (1 C i ) and F ( C k + j ) are class func-tions they depend only on α ∈ A l ◦ ( F q ) . In particular if X = p ( g, I, α ) we have for every w ∈ S n t w ∩ O ( X )( F q ) = p ( { Id } × P wλ × { α } ) , with P wλ = { I = ( I , . . . , I l ) ∈ P λ | wI j = I j for all ≤ j ≤ l } . As Lemma 4.3.11 clearly alsoholds over a finite field we finally deduce from (5.1.1) and (5.1.3) X X ∈ g ′ λ ( F q ) k Y i =1 F (1 C i )( X ) s Y j =1 F ( C k + j )( X )= q ( n ( d + r ) − nr − P N ( µ i ) ) + N ( λ )2 ( r − s ) | L λ | s − | G | ( q − ns u λ k Y i =1 µ i ! X w ∈ S µi Q L λ T w |P wλ | |P λ | s ( − l − ( l − q = q ( n ( d + r ) − nr − P N ( µ i ) ) + N ( λ )2 ( r − s )+1 | L λ | s − | G | ( q − ns u λ k Y i =1 ∆( λ, µ i ) (cid:18) nλ (cid:19) s ( − l − ( l − . Summing up over all λ ∈ P λ finishes the proof.Theorem 5.1.6 immediately implies that |M ∗ µ , r ( F q ) | is a rational function in q as we varythe finite field F q . Because it is the count of an algebraic variety we deduce that it is infact a polynomial in q , namely its weight polynomial by Theorem 2.3.4. By analyzing thecombinatorics of (5.1.7) we obtain the following Corollary 5.1.8. For every pair ( µ , r ) with d µ , r ≥ and s ≥ the counting polynomial for M ∗ µ , r is monic of degree d µ , r . In particular M ∗ µ , r is non-empty and connected, both over F q and C . 29 roof. The statement about the counting polynomial implies the second one, as the leadingcoefficient gives the number of components of M ∗ µ , r , see for example [HLRV2, Lemma 5.1.2].For n = 1 we have d µ , r = 0 and (5.1.6) reads indeed |M ∗ µ , r ( F q ) | = 1 . For n ≥ , d µ , r ≥ implies either r ≥ or r = s = 1 , k ≥ and there exists an ≤ i ≤ k with µ i = ( n ) . In bothcases it follows from Lemma 5.1.9 below, that there is only one summand in (5.1.7) of highestdegree, namely the one for λ = ( n ) . From the explicit formula for ∆(( n ) , µ ) in 5.1.9 we alsosee, that the coefficient of the highest q -power equals and the corollary follows. Lemma 5.1.9. For any λ, µ ∈ P n we have1. ∆(( n ) , µ ) = ( − n | G | q ( n − n ) φ µ ( q ) .2. deg ∆(( n ) , µ ) ≥ deg ∆( λ, µ ) , and equality holds if and only if λ = ( n ) or µ = ( n ) . Proof. The lemma boils down to the following identity. Let α ∈ N and write T α for thestandard maximal torus of GL α . Using (5.1.2) and [Mac, (2.14)] we have X w ∈ S α ( − α − rk q ( T αw ) | T αw | = ( − α α ! X ν ∈P α u ν Q i ν i Q i (1 − q ν i ) = ( − α α ! φ ( α ) ( q ) . Now every I ∈ P λ defines a refinement µ I of µ such that { w ∈ S µ | wI = I } = S µ I . Withthis we have ∆( λ, µ ) = 1 µ ! X w ∈ S µ Q L λ T w |P wλ | = 1 µ ! X I ∈P λ X w ∈ S µI Q L λ T w = | L λ | µ ! q ( N ( λ ) − n ) X I ∈P λ X w ∈ S µI ( − n − rk q (T w ) | T w | = ( − n | L λ | µ ! q ( N ( λ ) − n ) X I ∈P λ µ I ! φ µ I ( q ) . From this we get the formula for ∆(( n ) , µ ) as well as deg ∆( λ, µ ) = 12 ( N ( λ ) + n ) − min I ∈P λ 12 ( N ( µ I ) + n ) = 12 ( N ( λ ) − min I ∈P λ N ( µ I )) . The inequality deg ∆(( n ) , µ ) ≥ deg ∆( λ, µ ) is thus equivalent to n + min I ∈P λ N ( µ I )) ≥ N ( λ ) + N ( µ ) . (5.1.10)We can prove this inequality by induction on n as follows. Fix an I ∈ P λ and denote by µ I , λ and µ the smallest part of the respective partition. By lowering each of these smallestparts by one obtains partitions of n − and using the induction hypothesis the inequalitybecomes n + µ I ≥ λ + µ, which holds true by an inclusion-exclusion argument. The equality case follows from adirect inspection. 30 xample 5.1.11. Let us spell out explicitly formula (5.1.7) for n = 2 , . Recall that k denotesthe number of simple poles, s the number of higher order poles, d = k + s the total numberof poles, and r the sum of the Poincaré ranks of the poles. We will assume µ i = ( n ) for all ≤ i ≤ k , since the corresponding coadjoint orbits are points and do not contribute to thecount.For n = 2 , we have by assumption µ i = (1 ) for ≤ i ≤ k and thus |M ∗ µ , r ( F q ) | = q r + d − ( q r − ( q + 1) d − − d − ) q − . (5.1.12)For n = 3 , we write k = k + k with k = |{ i | µ i = (1 ) }| and k = |{ i | µ i = (2 , }| .Then |M ∗ µ , r ( F q ) | is given by q r +3 d − k − (cid:0) q r − ( q + q + 1) d − ( q + 1) s + k − − s + k q r − ( q + 1) s + k − ( q + 2) k + 3 d − s + k (cid:1) ( q − (5.1.13) Remark . If we take k = 0 , and thus s = d , formula (5.1.7) agrees with (4.3.2) if wereplace q by L . In fact by Theorem 2.3.4, (5.1.7) is a consequence of (4.3.2) provided thereexists a suitable spreading out of M ∗ n, r . We refer to [HLRV1, Appendix A], where a similarconstruction has been carried out in detail. Lemma 5.2.1. Let m ≥ and C any formal type of order m with C regular semisimple.Then we have an equality of Fourier transforms F ( C ) = F (1 C ) F (1 N ) m − , where N ⊂ g ( F q ) denotes the nilpotent cone and N its characteristic function on g ( F q ) ∼ = g ∨ ( F q ) . Proof. By [L2, Theorem 3.6], the Fourier transform F (1 N ) is given by F (1 N ) = U n = q dim O ( n ) St n , where St n denotes the Steinberg character and O ( n ) the regular nilpotent orbit, hence dim O ( n ) = n ( n − . By [L2, (2.2)], St n is supported on semisimple elements and its value on X ∈ g ′ λ ( F q ) isSt n ( X ) = ǫ G ǫ C G ( X ) | C G ( X ) | p = q P i λ i ( λ i − . Thus in total we obtain F (1 N )( X ) = q ( n − n + N ( λ )) . Comparing this with equation (5.1.1) and (5.1.4) we finally see F ( C ) = F (1 C ) F (1 N ) m − . λ ∈ P n we let H λ ( z, w ) = Y z a +2 − w l )( z a − w l +2 ) ∈ Q ( z, w ) , where the product is over the boxes in the Young diagram of λ and a and l are the arm lengthand the leg length of the given box. Then we define the Cauchy kernel Ω k ( z, w ) := X λ ∈P H λ ( z, w ) k Y i =1 ˜ H λ ( z , w ; x i ) ∈ Λ( x , . . . , x k ) ⊗ Z Q ( z, w ) , where we have the Macdonald polynomials of [GH] ˜ H λ ( z, w ; x ) = X µ ∈P n ˜ K λµ s µ ( x ) ∈ Λ ( x ) ⊗ Z Q ( z , w ) . Let now µ ∈ P kn , r = ( r , . . . , r s ) ∈ Z s> and r := r + · · · + r s be as in Definition 3.1.8. We let H µ ,r ( z, w ) =( − rn ( z − − w ) (cid:10) Log (Ω k + r ) , h µ ( x ) ⊗ · · · ⊗ h µ k ( x k ) ⊗ s (1 n ) ( x k +1 ) ⊗ · · · ⊗ s (1 n ) ( x k + s ) (cid:11) . Let now M µ , r B denote a generic wild character variety of type µ , r and g = 0 as studied in[HMW]. The main conjecture [HMW, Conjecture 1.2.2] in the form of [HMW, Lemma 5.2.1]is that the mixed Hodge polynomial of M µ , r B satisfies W H ( M µ , r B ; q, t ) = ( qt ) d µ, r / H ˜ µ ,r (cid:16) q / , − q − / t − (cid:17) , (5.2.2)where ˜ µ = ( µ , . . . , µ k , (1 n ) , . . . , (1 n )) ∈ P k + sn . The purity conjecture then is saying thatthe pure part of W H ( M µ , r B ; q, t ) should equal the Poincaré polynomial of the correspondingopen de Rham space.The following is our main Theorem 1.3.1. Theorem 5.2.3. In the notation of Theorem 5.1.6, we have E ( M ∗ µ , r ; q ) = q d µ , r / H ˜ µ ,r (0 , q / ) Proof. By Theorem 2.3.4 we have E ( M ∗ µ , r ; q ) = |M ∗ µ , r ( F q ) | and by Lemma 5.2.1 we can com-pute |M ∗ µ , r ( F q ) | = q − n | PGL n ( F q ) | F k Y i =1 F (1 C i ) s Y j =1 F (1 C k + j ) F (1 N ) r j (0) = q d µ , r / H ˜ µ ,r (0 , q / ) . (5.2.4) The last equation follows from combining [L1, Theorem 7.3.3] and [L1, Corollary 7.3.5] withthe observation in [L1, §6.10.3] that the symmetric function corresponding to a split semisim-ple adjoint orbit of type µ i is h µ i and in [L1, §6.10.4] the symmetric function correspondingto a regular nilpotent adjoint orbit is s (1 n ) . The result follows in the usual way, as in the proofof [HLRV2, Theorem 7.1.1]. Remark . The formula for |M ∗ µ , r ( F q ) | = E ( M ∗ µ , r ; q ) in (5.1.7) was obtained by computingthe same Fourier transform as in (5.2.4). Thus we can consider the result of Thm 5.1.6 as theexplicit form of Theorem 5.2.3. 32e finish this section by observing that a combination of [L1, Theorem 6.10.1, Theorem7.4.1] implies that H ˜ µ ,r (0 , q / ) has non-negative coefficients, thus we have Corollary 5.2.6. The weight polynomial E ( M ∗ µ , r ; q ) has non-negative coefficients.This motivates the following Conjecture 5.2.7. The mixed Hodge structure on the cohomology of M ∗ µ , r is pure.If all poles in C are of order one this is proven in [HLRV1, Theorem 2.2.6] using thedescription of M ∗ µ , r as a quiver variety. Moreover if there is only one higher order pole,[Bo3, Theorem 9.11] also identifies M ∗ µ , r with a quiver variety, thus Conjecture 5.2.7 followsin this case too. In Section 6, we obtain a quiver like description of M ∗ µ , r for poles of anyorder, giving more evidence and possible strategy for Conjecture 5.2.7. The greater part of this section is devoted to a generalization of the theorem of Crawley-Boevey [CB, Theorem 1], which realizes an additive fusion product of coadjoint orbits for GL n ( K ) , where K is a fixed perfect ground field (the primary examples the reader shouldhave in mind are where K = C or K is a finite field), as a quiver variety associated to a star-shaped quiver. The generalization we wish to achieve is to replace the GL n ( K ) -coadjointorbits with those for the non-reductive group GL n (R m ) of Section 2.1 and to replace quiverswith “quivers with multiplicities” or “weighted quivers” which are defined in Section 6.1below. The main theorem is stated as Theorem 6.4.2. It states that an open de Rham spaceas defined in Section 3.1 may be realized as a variety associated to a weighted star-shapedquiver for an appropriate choice of multiplicities.The main novelty that enters when one introduces multiplicities is that the groups oneobtains are no longer reductive. As in the usual quiver case, one wants to define a varietyassociated to a quiver with multiplicities as an algebraic symplectic quotient. In the ap-proach we take here, we simply define this quotient as the spectrum of the appropriate ringof invariants. Of course, this makes good sense as an affine scheme, however, without thereductivity hypothesis, we are not guaranteed that this ring of invariants is a finitely gener-ated K -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 momentmap is a trivial principal bundle for the unipotent radical of the group (Proposition 6.1.14),the proof of which occupies Section 6.2. We are thus able to get around the issue of dividingby a non-reductive group.In Section 6.3, we explain how a certain symplectic quotient of a “leg”-shaped quiver (outof which one builds a star-shaped quiver) yields a coadjoint orbit for the group GL n (R m ) ,generalizing Boalch’s explanation of Crawley-Boevey’s theorem [Bo3, Lemma 9.10]. Thiswas done in the holomorphic category for “short legs” in [Y2, Lemma 3.7]; ours is an alge-braic version, along the lines described above.Quivers with multiplicities have been introduced in [Y2] for purposes quite similar toours (namely, the description of moduli spaces of irregular meromorphic connections), andindeed, the quivers of interest in that paper are a special case of those considered here [Y2,§6.2]. We would like to point out that the quotients referred to there are either taken in the33olomorphic category (when restricted to the stable locus, in the sense defined there) or sim-ply set-theoretic [Y2, Definition 3.3]. The results of Section 6.2, 6.3 show that these quotientsindeed make sense algebraically, which is of course essential to our results in Sections 4 and5. Finally, we gather together the results in Section 6.4, relating the varieties constructed inthis section to the open de Rham spaces of Section 3. Quivers with multiplicities are also dis-cussed by [GLS] in order to generalize certain classical results known for finite-dimensionalalgebras associated to symmetric Cartan matrices to the case where the Cartan matrix is onlysymmetrizable. It is well-known that given a simply laced affine Dynkin diagram, one canassociate a -dimensional quiver variety (depending on some parameters), which will in factbe a gravitational instanton (i.e., carry a complete hyperkähler metric). In a parallel general-ization, the construction above allows us to construct quiver varieties for non-simply lacedDynkin diagrams; this is explained in Section 6.4.2. In the next section, we will see that thesealso admit complete hyperkähler metrics (see Theorem 7.3.3 and Remark 7.3.7(i)). Here, we discuss a generalization of the definition of quiver representations to allow for mul-tiplicities at the vertices. This was first done in [Y2]. We follow the approach of [GLS, §1.4]and [Y3], using the greatest common divisor of neighbouring multiplicities in the definitionof a representation.Let K be a field. For a positive integer m ∈ Z ≥ , we will denote by R m the truncatedpolynomial ring R m := K [ z ] / ( z m ) , so that R = K . Observe that if m | ℓ , then there is a natural inclusion of K -algebras R m ֒ → R ℓ ,which makes R ℓ a free R m -module of rank ℓ/m .As in Section 2.1, we will use the groups GL n (R m ) , GL n (R m ) and their respective Liealgebras gl n (R m ) , gl n (R m ) . However, since the value of n will vary, we will not shortenthese to G or G m etc. We do adopt the identifications gl n (R m ) ∨ = z − m gl n (R m ) via the trace-residue pairing (2.1.5).As usual, a quiver Q = ( Q , Q , h, t ) is a finite directed graph, i.e., Q and Q are finitesets and one has head ant tail maps h , t : Q → Q . By a set m of multiplicities for Q we willmean an element m ∈ Z Q ≥ . The pair ( Q, m ) of a quiver and a set of multiplicities m can bereferred to as a quiver with multiplicities or a weighted quiver .A dimension vector is also an element n ∈ Z Q ≥ . The space of representations of ( Q, m ) fora given dimension vector n is defined as Rep( Q, m , n ) := M α ∈ Q Hom R α (cid:16) R ⊕ n t ( α ) m t ( α ) , R ⊕ n h ( α ) m h ( α ) (cid:17) where we have made the abbreviation R α := R ( m t ( α ) ,m h ( α ) ) . Here ( m t ( α ) , m h ( α ) ) denotes thegreatest common divisor of m t ( α ) and of m h ( α ) . In the case that m i = 1 for all i ∈ Q , this is34he usual definition of the space of representations Rep( Q, n ) for Q with dimension vector n . For a triple ( Q, m , n ) we define the group G Q, m , n = G m , n := Y i ∈ Q GL n i (R m i ) . We will denote its Lie algebra by g m , n . One has an action of G m , n on Rep( Q, m , n ) as in theusual case: for ( g i ) ∈ G m , n , ( ϕ a ) ∈ Rep( Q, m , n ) , ( g i ) · ( ϕ α ) = (cid:16) g h ( α ) ϕ a g − t ( α ) (cid:17) . The main difference between this and the case without multiplicities is that G m , n is notreductive if there is some i ∈ Q with m i ≥ .As usual, we denote by Q the doubled quiver: i.e., Q := Q ⊔ Q ′ , where Q ′ := { α ′ : α ∈ Q } and h ( α ′ ) = t ( α ) , t ( α ′ ) = h ( α ) for all α ∈ Q . Then Rep( Q, m , n ) = M α ∈ Q Hom R α (cid:16) R ⊕ n t ( α ) m t ( α ) , R ⊕ n h ( α ) m h ( α ) (cid:17) ⊕ Hom R α (cid:16) R ⊕ n h ( α ) m h ( α ) , R ⊕ n t ( α ) m t ( α ) (cid:17) (6.1.1) = T ∗ Rep( Q, m , n ) . Thus, we have the usual situation of a cotangent bundle, which has a canonical symplecticform, and an action on a group on the base, for which the induced action on the cotangentbundle is Hamiltonian. It is not difficult to write down the symplectic form explicitly, but aswe do not need it here, we omit it. An expression is given in Section A.5.We will, however, need to work with the moment map µ : Rep( Q, m , n ) → g ∨ m , n for the G m , n action; such is given by µ ( p, q ) = X α ∈ h − ( i ) p α q α − X β ∈ t − ( i ) q β p β i ∈ Q . (6.1.2)We refer the reader to Section A.4 for further explanation as to how the above expressionarises and should be interpreted. Definition 6.1.3. Let us fix an element γ ∈ g ∨ m , n whose G m , n -coadjoint orbit is a singleton.One defines the quiver scheme associated to Q ( m , n ) at γ as Q γ := Spec (cid:0) K [ µ − ( γ )] G m , n (cid:1) . (6.1.4) K [ µ − ( γ )] denotes the coordinate ring of the affine variety µ − ( γ ) , and K [ µ − ( γ )] G m , n itssubring of G m , n -invariants. Remark . By definition Q γ is a reduced affine scheme over K . However, as G m , n is notreductive, K [ µ − ( γ )] G m , n is not a priori a finitely generated K -algebra, therefore we cannotsay that Q γ is an affine variety without further justification.35n view of the preceding remark, we now describe a condition which is sufficient toconclude that Q γ is an affine variety. Then in Section 6.1.2 we will describe a class of quiverswith multiplicities and conditions on γ for which this condition is fulfilled.Observe that the group G m , n is a semi-direct product: there is a surjective morphism ofalgebraic groups G m , n → G n , where G n := Y i ∈ Q GL n i ( K ) is the usual group associated to the quiver Q and dimension vector n . We will call the kernel G m , n . This surjection splits, with GL n i ( K ) being the subgroup of “constant” group elementsin GL n i (R m i ) , and then taking the product of these inclusions (cf. Section 2.1). We may thuswrite G m , n = G n ⋉ G m , n . With this decomposition, we may similarly decompose the Lie algebra g m , n and its dualas direct sums: g m , n = g n ⊕ g m , n g ∨ m , n = g ∨ n ⊕ ( g m , n ) ∨ . (6.1.6)For γ ∈ g ∨ m , n , we will write γ res ∈ g ∨ n and γ irr ∈ ( g m , n ) ∨ for its components. Likewise, themoment map (6.1.2) will also have components µ res and µ irr corresponding to g ∨ n and ( g m , n ) ∨ ,respectively. Of course, each of these components is a moment map for the restriction of theaction to the respective subgroup. Lemma 6.1.7. Suppose that there is a G n -invariant closed subvariety M γ ⊆ µ − irr ( γ irr ) and a G m , n -equivariant isomorphism µ − irr ( γ irr ) ∼ −→ G m , n × M γ , where G m , n acts by left multiplication on the first factor (and trivially on M γ ); in other words, µ − irr ( γ irr ) is a trivial (left) principal G m , n -bundle over M γ . Then Q γ is an affine algebraicsymplectic variety, and hence we will often refer to it as a quiver variety . Remark . By “symplectic variety”, we mean that it is an algebraic symplectic manifoldalong its smooth locus. We are making no claims about the possible singular locus. Proof. We deal only with the finite generation of K [ µ − ( γ )] G m , n ; the symplectic structurearises as usual on the smooth locus. Since G m , n is normal in G m , n , it is not hard to see that µ res is constant on G m , n -orbits in Rep( Q, m , n ) . We have that µ − ( γ ) = µ − ( γ res ) ∩ µ − irr ( γ irr ) ∼ = µ − ( γ res ) ∩ (cid:0) G m , n × M γ (cid:1) . Using the fact just mentioned, we obtain an isomorphism µ − ( γ ) ∼ = G m , n × (cid:0) µ − ( γ res ) ∩ M γ (cid:1) . X := µ − ( γ res ) ∩ M γ for the second factor and note that this is an affine algebraic G n -variety. From this, K [ µ − ( γ )] G m , n = (cid:16)(cid:0) K [ G m , n ] ⊗ K K [ X ] (cid:1) G m , n (cid:17) G n = K [ X ] G n , and since G n is reductive, this is a finitely generated K -algebra. Remark . The idea in the proof comes from the procedure in the general theory of Hamil-tonian reduction known as “reduction in stages”. When taking a symplectic quotient by agroup with a semi-direct product decomposition, one obtains the same quotient by first re-ducing by the normal subgroup and then by the quotient group. See, for example, [MMOPR,§4.2]. We now describe a class of quivers with multiplicities and give conditions on the choice of γ ∈ g ∨ m , n for which the hypotheses of Lemma 6.1.7 will hold. Let Q be a star-shaped quiveras on [CB, p.340], with d legs and the i -th leg of length l i . Thus, the vertex set is Q = { } ∪ d [ i =1 { [ i, , . . . , [ i, l i ] } . (6.1.10)For notational convenience, for ≤ i ≤ d , we will often write [ i, for . Then for ≤ i ≤ d and ≤ j ≤ l i , the doubled quiver Q will have one arrow from [ i, j − to [ i, j ] and one from [ i, j ] to [ i, j − .For each ≤ i ≤ d , we fix m i ∈ Z > and choose the multiplicity vector m with m = 1 m [ i,j ] = m i ; (6.1.11)thus, the multiplicity is fixed on each leg (away from the central vertex ). The dimensionvector n will be n = n n [ i,j ] = n i,j (6.1.12)with n > n i, > . . . > n i,l i > for ≤ i ≤ d , so that the dimensions are decreasing as onemoves away from the central vertex on each leg.Observe that with this, G m , n = GL n ( K ) × d Y i =1 l i Y j =1 GL n i,j (R m i ); hence g m , n = gl n ( K ) ⊕ d M i =1 l i M j =1 gl n i,j (R m i ) g ∨ m , n = z − gl n ( K ) ⊕ d M i =1 l i M j =1 z − m i gl n i,j (R m i ) The same diagram also appears at [Y1, p.3] and at [HLRV1, p.348, Figure 1]. 37e choose γ ∈ g m , n of the form γ = ( γ I n , γ [ i,j ] I n i,j ) ≤ i ≤ d ≤ j ≤ l i with γ ∈ z − · K , γ [ i,j ] ∈ z − m i R m i . We further assume that for each ≤ i ≤ d and for ≤ j ≤ k ≤ l i , z m i (cid:16) γ [ i,j ] + · · · + γ [ i,k ] (cid:17) ∈ R × m i . (6.1.13) Proposition 6.1.14. With ( Q, m , n ) and γ as above, Q γ is an affine algebraic symplectic vari-ety.In the next subsection, we will show that the hypotheses of Proposition 6.1.14 implythose of Lemma 6.1.7, which then provides a proof of the Proposition. As just mentioned, we wish to prove the following. Proposition 6.2.1. Let ( Q, m , n ) and γ ∈ g ∨ m , n be as in Proposition 6.1.14. Then there is a G n -invariant closed subvariety M γ ⊆ µ − irr ( γ irr ) and a G m , n -equivariant isomorphism µ − irr ( γ irr ) ∼ −→ G m , n × M γ , (6.2.2)as in the hypothesis of Lemma 6.1.7. Simplified case: d = 1 . For notational simplicity, we will first assume that d = 1 , so thatwe have a quiver with a single leg. This allows us to drop the index i from the notation in(6.1.10), (6.1.11) and (6.1.12). Thus, the doubled quiver Q is the following ◦ p ( ( • q h h p ( ( • q h h · · · • p l − ( ( • q l − h h p l ( ( • q l h h , (6.2.3)with the vertices are labeled , . . . , l from left to right, multiplicity vector m = (1 , m, . . . , m ) and dimension vector n = ( n, n , . . . , n l ) with n > n > · · · > n l . To avoid having to writeout things separately for the vertex , we will often write n := n .Also, γ = (cid:0) γ I n , γ j I n j (cid:1) ≤ j ≤ l with γ ∈ z − · K , γ j ∈ z − m R m . The condition (6.1.13) on γ implies that z m (cid:16) γ j + · · · + γ k (cid:17) ∈ R × m (6.2.4)for ≤ j ≤ k ≤ l . 38 xplicit description of the groups, Lie algebras and their duals. Let us now be explicitabout the groups that are involved. One has G m , n = GL n ( K ) × l Y i =1 GL n i (R m ) G m , n = l Y i =1 GL n i (R m ) , (6.2.5)with respective Lie algebras g m , n = gl n ( K ) ⊕ l Y i =1 gl n i (R m ) g m , n = l M i =1 gl n i (R m ) and duals g ∨ m , n = z − gl n ( K ) ⊕ l Y i =1 z − m gl n i (R m ) (cid:0) g m , n (cid:1) ∨ = l M i =1 z − m gl n i (R m ) (cid:14) z − gl n i (R m ) . One has direct sum decompositions as in (6.1.6) and the projections maps to each factorsimply omits the “irregular” part or the “residue” term, respectively, as in (2.1.6), (2.1.7). Asbefore, we write γ res and γ irr for the images of γ under the respective projections to g n andto g m , n . Explicit description of the space of representations Rep( Q, m , n ) and group actions. Ex-plicitly, the space Rep( Q, m , n ) is given by Hom K ( K n , R ⊕ n m ) ⊕ Hom K (R ⊕ n m , K n ) ⊕ l M i =2 (cid:0) Hom R m (R n i − m , R n i m ) ⊕ Hom R m (R n i m , R n i − m ) (cid:1) . We will write elements of Rep( Q, m , n ) as pairs ( p, q ) , where p = ( p , . . . , p l ) and q =( q , . . . , q l ) are each themselves s -tuples, where p ∈ Hom K ( K n , R ⊕ n m ) p i ∈ Hom R m (R n i − m , R n i m ) , ≤ i ≤ lq ∈ Hom K (R ⊕ n m , K n ) q i ∈ Hom R m (R n i m , R n i − m ) , ≤ i ≤ l. As explained in Remark A.4.3, we may realize the p i as elements p i ∈ M n i × n i − (R m ) , ≤ i ≤ l and dually, we will think of the elements q i as Laurent polynomials q i ∈ z − m M n i − × n i (R m ) , ≤ i ≤ l. In this notation, an element g = ( g , . . . , g l ) ∈ G m , n acts on ( p, q ) ∈ Rep( Q, m , n ) by g · ( p, q ) = (cid:0) g p ( g ) − , g p ( g ) − , . . . , g l p l ( g s − ) − ,g q ( g ) − , g q ( g ) − , . . . , g s − q l ( g l ) − (cid:1) . (6.2.6)Of course, here g ∈ GL n ( K ) and g i ∈ GL n i (R m ) .39ater we will need to be even more explicit, and so we will write p i = p i + zp i + · · · + z m − p im − q i = q im z m + · · · + q i z with p ij ∈ M n i × n i − ( K ) , q ij ∈ M n i − × n i ( K ) .When we will need to evaluate moment maps or group actions, we will need to regard,for example, the product p i q i as an element of gl n i (R m ) ∨ = z − m gl n i (R m ) . We do this bymultiplying p i and q i as matrices of Laurent polynomials and truncating the terms of degree ≥ . Explicitly, p i q i = p i q im z m + p i q im − + p i q im z m − + · · · + p i q i + p i q i + · · · + p im − q im z ∈ gl n i (R m ) ∨ . (6.2.7)Of course, products of the form q i p i or g i − q i ( g i ) − as in (6.2.6) are written similarly. Productssuch as g i p i ( g i − ) − in (6.2.6) will be considered as multiplication of the relevant matriceswith entries in R m .We will also write γ i = γ im z m + · · · + γ i z , ≤ i ≤ l with γ ij ∈ K . In particular, the assumption (6.2.4) implies that γ im ∈ K × , ≤ i ≤ l . Explicit description of the moment maps. The moment map µ : Rep( Q, m , n ) → g ∨ m , n hasthe explicit expression µ ( p, q ) = ( − q p , p q − q p , . . . , p s − q s − − q l p l , p l q l ) ∈ g ∨ m , n . (6.2.8)Composing with the projections arising from the direct sum decomposition (6.1.6), we getmoment maps µ res : Rep( Q, m , n ) → g ∨ n µ irr : Rep( Q, m , n ) → ( g m , n ) ∨ which are, in fact, the moment maps for the restricted G n - and G m , n -actions, respectively.We are primarily interested in the preimage µ − irr ( γ irr ) . The point ( p, q ) lies in this preim-age if and only if (in other words the moment map condition translates to) p i q i = q i +1 p i +1 + γ i I n i , ≤ i ≤ l − p l q l = γ l I n l . (6.2.9)We recall that ( g m , n ) ∨ has no component corresponding to the vertex . Thus if we writeout an explicit expression as in (6.2.7), we should ignore any contributions coming from theresidue terms, since we are considering these equations in ( g m , n ) ∨ . Definition of M γ and its G n -invariance. We now proceed with defining M γ appearing inLemma 6.1.7, where ( Q, m , n ) is as in Proposition 6.1.14 in the special case where d = 1 . Westart by defining M γ := µ − irr ( γ irr ) and define inductively a nested sequence of subvarieties of M γ by M i γ := (cid:8) ( p, q ) ∈ M i − γ : p i q im = γ im I n i + q i +1 m p i +10 (cid:9) M γ = M l γ := n ( p, q ) ∈ M l − γ : p l q lm = γ lm I n l o . (6.2.10)40 emark . On the left side of the defining equation for M i γ , the matrix p i q im is a prioria gl n i ( K ) -valued polynomial in z of degree ≤ m − , but the right hand side is, in fact, aconstant matrix. So the defining condition is equivalent to the further conditions p i q im = · · · = p im − q im = 0 ∈ gl n i ( K ) . (6.2.12)The fact that M γ is an affine variety is thus clear.Furthermore, from this description, it is easy to see that M γ is a G n -invariant subvari-ety of Rep( Q, m , n ) using (6.2.6) and the appropriate expressions in (6.2.7): the vanishingconditions (6.2.12) imposed by (6.2.10) are left unchanged. Remark . Equation (6.3.13) below implies that z m p i q i ∈ GL n i (R m ) for ≤ i ≤ l . Inparticular, the constant term, namely p i q im , must lie in GL n i ( K ) . This fact is essential in whatfollows.We now establish some properties of the M i γ with respect to the action of G m , n and itssubgroups. Let us consider each GL n i (R m ) , ≤ i ≤ l as a subgroup of G m , n via the obviousinclusion in (6.2.5) and will have use for the subgroups ( G m , n ) i := l Y j = i GL n j (R m ) ≤ G m , n , noting that we have a (decreasing) chain of inclusions ( G m , n ) l ≤ ( G m , n ) l − ≤ · · · ≤ ( G m , n ) ≤ ( G m , n ) = G m , n . Lemma 6.2.14. M i γ is invariant under ( G m , n ) i +1 . Proof. Let ( p, q ) ∈ M i γ , r = (1 , . . . , , r i +1 , . . . , r l ) ∈ ( G m , n ) i +1 . Then r · ( p, q ) = (cid:0) p , q , . . . , p i , q i , r i +1 p i +1 , q i +1 ( r i +1 ) − , r i +2 p i +2 ( r i +1 ) − , r i +1 q i +2 ( r i +2 ) − ,. . . , r l p l ( r l − ) − , r l − q l ( r l ) − (cid:1) . Observe that r · ( p, q ) ∈ M i − γ since the condition for this to hold depends only on the com-ponents p , q , . . . , p i , q i , which are unchanged under the action of r . We only need to checkthe condition for M i γ , which involves the terms q i +1 m , p i +10 . It suffices to see that ( r i +1 p i +1 ) = p i +10 (cid:0) q i +1 ( r i +1 ) − (cid:1) m = q i +1 m but this holds since the constant term of r i +1 and ( r i +1 ) − are both I n i +1 . Lemma 6.2.15. Suppose r i ∈ GL n i (R m ) , ( p, q ) ∈ M i γ . If r i · ( p, q ) ∈ M i γ then r i = I n i . Proof. Write r i = I n i + zr i + · · · + z m − r im − . Then r i · ( p, q ) = (cid:16) p , q , . . . , r i p i , q i ( r i ) − , p i +1 ( r i ) − , r i q i +1 , . . . , p l , q l (cid:17) . (6.2.16)The condition for r i · ( p, q ) ∈ M i γ is r i p i (cid:0) q i ( r i ) − (cid:1) m = γ im I n i + ( r i q i +1 ) m (cid:0) p i +1 ( r i ) − (cid:1) , (cid:0) q i ( r i ) − (cid:1) m = q im ( r i q i +1 ) m = q i +1 m (cid:0) p i +1 ( r i ) − (cid:1) = p i +10 , so we need r i p i q im = γ im I n i + q i +1 m p i +10 = p i q im . But by Remark 6.2.13, p i q im ∈ GL n i ( K ) ≤ GL n i (R m ) and hence this may be regarded as anequation in GL n i (R m ) and hence r i = I n i . Construction of the isomorphism (6.2.2) .Lemma 6.2.17. For ≤ i ≤ l , there exists a morphism ϕ i : M i − γ → GL n i (R m ) such that forall ( p, q ) ∈ M i − γ , ϕ i ( p, q ) · ( p, q ) ∈ M i γ . Proof. Let ( p, q ) ∈ M i − γ and r i ∈ GL n i (R m ) . Since r i ∈ ( G m , n ) i , r i · ( p, q ) ∈ M i − γ and so weneed only check the condition for M i γ . We recall the expression for r i · ( p, q ) from (6.2.16) andobserve that r i p i q im = p i q im + z ( p i + r i p i ) q im + z ( p i + r i p i + r i p i ) q im + · · · + z m − ( p im − + r i p im − + · · · + r im − p i ) q im . The condition we want is that all the non-constant (with respect to z ) terms vanish (Remark6.2.11). But now, by Remark 6.2.13, p i q im is invertible, and so starting with the coefficientof z above, we may solve for r i = − p i ( p i q im ) − so that this coefficient vanishes. It is thenclear that we may successively solve for r i , . . . , r im − algebraically as functions of p and q toeliminate the remaining non-constant terms. This produces ϕ i with the stated property. Corollary 6.2.18. (a) For r i ∈ GL n i (R m ) , x ∈ M i − γ , ϕ i ( r i · x ) = ϕ i ( x )( r i ) − .(b) For ¯ r ∈ ( G m , n ) i +1 , x ∈ M i − γ , ϕ i (¯ r · x ) = ϕ i ( x ) . Proof. (a) By definition, ϕ i ( x ) · x ∈ M i γ , so also ϕ i ( r i · x ) · ( r i · x ) ∈ M i γ , but the latter isequal to ϕ i ( r i · x ) · r i · ϕ i ( x ) − · (cid:0) ϕ i ( x ) · x (cid:1) . Since ϕ i ( x ) · x ∈ M i γ , by Lemma 6.2.15, ϕ i ( r i · x ) · r i · ϕ i ( x ) − = I n i .(b) We have ϕ i (¯ r · x ) · (¯ r · x ) ∈ M i γ but since ϕ i (¯ r · x ) ∈ GL n i (R m ) , ¯ r ∈ ( G m , n ) i +1 , ϕ i (¯ r · x )¯ r =¯ rϕ i (¯ r · x ) and hence ¯ r · (cid:0) ϕ i (¯ r · x ) · x (cid:1) ∈ M i γ , but then also ϕ i (¯ r · x ) · x ∈ M i γ by Lemma6.2.14. So again Lemma 6.2.15 yields ϕ i (¯ r · x ) = ϕ i ( x ) . Corollary 6.2.19. There is a ( G m , n ) i -equivariant isomorphism σ i : M i − γ → GL n i (R m ) × M i γ ,where the action of ( r i , ¯ r ) ∈ GL n i (R m ) × ( G m , n ) i +1 = ( G m , n ) i on ( s, y ) ∈ GL n i (R m ) × M i γ is ( r i , ¯ r ) · ( s, x ) = ( r i s, ¯ r · x ) . Proof. We define σ i : M i − γ → GL n i (R m ) × M i γ by σ i ( x ) = (cid:0) ϕ i ( x ) − , ϕ i ( x ) · x (cid:1) , which iswell-defined by Lemma 6.2.17. The inverse τ i : GL n i (R m ) × M i γ → M i − γ is simply given bythe action τ i ( s, y ) = s · y . This is well-defined since GL n i (R m ) ≤ ( G m , n ) i and M i γ ⊆ M i − γ and ( G m , n ) i acts on M i − γ by Lemma 6.2.14. It is clear that τ i ◦ σ i = M i − γ . Now, if ( s, y ) ∈ L n i (R m ) × M i γ , then ϕ i ( s · y ) = ϕ i ( y ) s − = s − since ϕ i ( y ) · y ∈ M i γ but already y ∈ M i γ , soone uses Lemma 6.2.15 to see that ϕ i ( y ) = I n i . Then σ i ◦ τ i ( s, y ) = σ i ( s · y ) = (cid:0) ϕ i ( s · y ) − , ϕ i ( s · y ) · ( s · y ) (cid:1) = (cid:0) s, s − · ( s · y ) (cid:1) = ( s, y ) . Using Corollary 6.2.18(a), it is easy to see that σ i is GL n i (R m ) -equivariant. Thus, it suf-fices to show that it is ( G m , n ) i +1 -equivariant. Let ¯ r ∈ ( G m , n ) i +1 , x ∈ M i − γ : σ i (¯ r · x ) = (cid:0) ϕ i (¯ r · x ) − , ϕ i (¯ r · x ) · (¯ r · x ) (cid:1) = (cid:0) ϕ i ( x ) − , ϕ i ( x ) · ¯ r · x (cid:1) = ¯ r · (cid:0) ϕ i ( x ) − , ϕ i ( x ) · x (cid:1) = ¯ r · σ i ( x ) . Conclusion in the case d = 1 . We can now put the σ i of Corollary 6.2.19 into a G m , n -equivariant isomorphism µ − irr ( γ irr ) = M γ ∼ −→ GL n (R m ) × M γ ∼ −→ GL n (R m ) × GL n (R m ) × M γ ∼ −→ · · · ∼ −→ GL n (R m ) × · · · × GL n l (R m ) × M l γ = G m , n × M γ . It was already noted in Remark 6.2.11 that M γ is G n -invariant. These are the hypotheses ofLemma 6.1.7, and so we may conclude in the case d = 1 . The general case of d legs. Now, consider the situation of Proposition 6.1.14, where thenumber of legs d ∈ Z > in the quiver Q is arbitrary. Since the multiplicity at the centralvertex is m = 1 , µ irr has no component at and thus µ − irr ( γ irr ) can be taken to be theproduct of the preimages on each of the legs. Likewise, G m , n also has no component at andthe action on µ − irr ( γ irr ) is a product action if we view the latter as a product in the way justmentioned. Thus, M γ can be constructed as the product of those subvarieties constructedon each leg. Invariance of M γ under G n is also easy to see: if we ignore the central vertex,then one may again use the product argument. Note that in the argument for each leg, wedid include the group GL n ( K ) at the central vertex, so M γ will also be invariant under theaction of this group. This completes the proof of Proposition 6.1.14. Here, we discuss the relationship between coadjoint orbits for the group GL n (R m ) for a fixed m ≥ and varieties associated to quivers with multiplicities, where the underlying quiver isa single leg. It may thus help the reader to refer back to the diagram (6.2.3). What will be trueis that coadjoint orbits of certain diagonal elements C ∈ t ∨ m in gl n (R m ) ∨ (2.2.1) can be realizedas reductions of the spaces Rep( Q, m , n ) that we considered in the case d = 1 in Section 6.2.To be able to make a precise statement, we will first need to explain the conditions on thecoadjoint orbits and set some notation.We will take C ∈ t ∨ m , and suppose that it is written in the form (2.2.2). We will make thefurther assumption that (2.2.3) holds. For such a C , we wish to describe a quiver Q , whichwill be a leg, as well as some data on it, from which we will recover O ( C ) . The quiver will bethe same as in (6.2.3), having l + 1 nodes and l arrows, and will have the same multiplicityvector m , with multiplicity at the vertex and all other vertices receiving multiplicity m . The dimension vector n will be defined by taking n l := λ l and n i := n i +1 + λ i , with ( λ , . . . , λ l ) given as in (2.2.2). 43 emark . Observe that if C is a regular formal type (recall Definition 3.2.3), then for anycoordinate z , C z satisfies (2.2.3).From the data in (2.2.2), we set γ := c γ i := c i − c i − , ≤ i ≤ l. (6.3.2)Then (2.2.3) implies that (6.2.4) holds.The statement that we want is that the GL n (R m ) -coadjoint orbit of C as above is givenby a symplectic reduction of Rep( Q, m , n ) by a subgroup of G m , n . The subgroup in questionis that we obtain by leaving out the group GL n ( K ) corresponding to the vertex , namely, G m , n , := l Y i =1 GL n i (R m ) . We write g m , n , for its Lie algebra. The reason the vertex in (6.2.3) was drawn empty isbecause we want to consider only the symplectic quotient by G m , n , .Of course, G m , n is a normal subgroup of G m , n , with quotient G n , = l Y i =1 GL n i ( K ) which is precisely the group associated to the underlying quiver with dimension vector n ,ignoring the multiplicities, where again we are omitting the group GL n ( K ) correspondingto the vertex . Thus, we want to take a symplectic quotient by the semi-direct product G m , n , = G n , ⋉ G m , n . (6.3.3)From the inclusion g m , n , ⊆ g m , n we have a natural surjection of the duals g ∨ m , n → g ∨ m , n , , and the moment map for the G m , n , -action on Rep( Q, m , n ) is given by the composition µ : Rep( Q, m , n ) µ −→ g ∨ m , n → g ∨ m , n , . We will consider the element γ := ( γ I n , . . . , γ l I n l ) ∈ g ∨ m , n , , and define the symplectic quotient Rep( Q, m , n ) // γ G m , n , := Spec (cid:0) K [ µ − ( γ )] G m , n , (cid:1) . For less burdensome notation, we will often abbreviate the left hand side of the above to Rep //G m , n , . Observe that the assumption (2.2.3), the arguments of Section 6.2 and Lemma6.1.7 already show that Rep //G m , n , is an affine symplectic variety. We will write π : µ − ( γ ) → Rep //G m , n , for the quotient map; since this is defined as a GIT quotient, this is a categorical quotient.44 roposition 6.3.4. Suppose C ∈ z − m t (R m ) ⊆ gl n (R m ) ∨ is written in the form (2.2.2) andsatisfies (2.2.3) and that Q , m , and n are given as above. Then Rep( Q, m , n ) // γ G m , n , admitsa GL n (R m ) -action and there is a GL n (R m ) -equivariant isomorphism Rep( Q, m , n ) // γ G m , n , ∼ −→ O ( C ) . (6.3.5)The following is a slight generalization of [Bo3, Proposition D.1] which will be importantin the proof of the Proposition. Lemma 6.3.6. Let R be a commutative local ring and let m ≤ n ∈ Z > , p ∈ M m × n ( R ) , q ∈ M n × m ( R ) be such that pq ∈ GL m ( R ) . Then qp and (cid:20) n − m pq (cid:21) (6.3.7)are conjugate in GL n ( R ) . Proof. Observe that R n = ker p ⊕ im q . Given v ∈ R n , let w := ( pq ) − pv ∈ R m . Then p ( v − qw ) = 0 hence v = ( v − qw ) + qw ∈ ker p + im q , i.e., R n = ker p + im q . Furthermore, the sum is direct,for if v ∈ ker p ∩ im q , say v = qw with w ∈ R m and pv = pqw = 0 , then w = 0 and hence v = 0 .Furthermore, the assumption that pq is invertible also implies that ker p and im q are free R -modules. This can be seen via the Cauchy–Binet formula: for a subset I ⊆ { , . . . , n } ofsize m , one sets det I p to be the determinant of the m × m submatrix of p taking the columnswith indices in I ; one defines det I q the same way, using columns instead of rows; then theformula states that det pq = X I (det I p )(det I q ) , where the sum is over all subsets of size m . Since det pq ∈ R × and R is local, there mustbe some I with det I p ∈ R × (otherwise, all the terms in the sum would lie in the maximalideal and hence det pq could not be a unit). Hence there exists r ∈ GL m ( R ) for which thesubmatrix of rp corresponding to I is the identity matrix. That is, the matrix rp is in reducedrow echelon form and one can find a basis of ker p = ker rp as one does in a first-year linearalgebra class. A similar argument shows that the columns of q are linearly independent over R and hence already give a basis of im q .Now, we choose a basis of R n by taking the first n − m vectors as a basis of ker p andthe last m vectors as the columns of q . Then the fact that R n = ker p ⊕ im q implies that thematrix obtained in this way lies in GL n ( R ) . Writing qp with respect to this basis give thesecond matrix in (6.3.7).With this, the proof of Proposition 6.3.4 follows the idea of [Bo3, Lemma 9.10], whichexplains the proof of [CB, §3]. However, there one can rely on usual results of linear alge-bra over fields, while in our case working with the orbits of the unipotent groups involvedrequires a little care, which makes the arguments somewhat longer.45 roof of Proposition 6.3.4. For the reader’s convenience, we recall the explicit expressions forthe moment map (6.2.8) µ ( p, q ) = ( p q − q p , . . . , p s − q s − − q l p l , p l q l ) ∈ g ∨ m , n , = l M i =1 gl n i (R m ) ∨ (6.3.8)and the G m , n , -action: for h = ( h , . . . , h l ) ∈ G m , n , , one has h · ( p, q ) = ( h p , q h − , h p h − , h q h − , . . . , h l p l h − s − , h s − q l h − l ) . (6.3.9) GL n (R m ) -action on Rep //G m , n , . Of course Rep( Q, m , n ) admits a GL n (R m ) -action: for g ∈ GL n (R m ) , one has g · ( p, q ) = ( p g − , gq , p , q , . . . , p l , q l ) . (6.3.10)From (6.3.8), it is easy to check that µ − ( γ ) is invariant under this action. It is likewiseeasy to see that it commutes with the G m , n , -action (6.3.9). Thus, the action descends to thequotient Rep //G m , n , . Definition of the isomorphism Φ : Rep //G m , n , → O ( C ) . We begin by defining a mor-phism e Φ : µ − ( γ ) → O ( C ) by ( p, q ) q p + γ I n . (6.3.11)For this to define a GL n (R m ) -equivariant morphism Φ : Rep //G m , n , → O ( C ) , we needto verify three things: first, a priori, e Φ takes values only in gl n (R m ) ∨ , so we need to seethat it indeed takes values in O ( C ) ; second, we need to check that e Φ is G m , n , -invariant;finally, one wants to see that e Φ is GL n (R m ) -equivariant. The latter two statements are easy tocheck simply from their definitions: (6.3.9) for G m , n , -invariance and (6.3.10) for GL n (R m ) -equivariance. The first statement is a bit longer and so we will justify it in the next paragraph. e Φ takes values in O ( C ) . For ≤ i ≤ l , we define the diagonal matrix t i := γ i I λ i γ i,i +1 I λ i +1 . . . γ i,l − I λ l − γ i,l I λ l ∈ z − m gl n i (R m ) = gl n i (R m ) ∨ , where the γ i were defined in (6.3.2) and γ i,j := γ i + · · · + γ j , ≤ i < j ≤ l. Since (2.2.3) and hence (6.2.4) hold, z m t i ∈ GL n i (R m ) for ≤ i ≤ l . Also, we have (cid:20) λ i t i +1 (cid:21) + γ i I n i = t i , ≤ i ≤ l − t = C. (6.3.12)46e observe that if ( p, q ) ∈ µ − ( γ ) , then for ≤ i ≤ l , p i q i ∼ GL ni (R m ) t i , (6.3.13)where ∼ GL ni (R m ) means in the same GL n i (R m ) coadjoint orbit in gl n i (R m ) ∨ , and q i p i ∼ GL ni − (R m ) (cid:20) λ i − t i (cid:21) . (6.3.14)First, note that (6.3.14) follows from (6.3.13) and Lemma 6.3.6. Then (6.3.13) is easy to seeby decreasing induction on i . For i = l , (6.3.13) is the last component of the moment mapcondition (6.3.8), where we in fact have equality. Now, for the inductive step, one has p i q i = q i +1 p i +1 + γ i I n i ∼ (cid:20) λ i t i +1 (cid:21) + γ i I n i = t i , the first equality being the i th component of the moment map (6.3.8), the similarity (6.3.14)and the last equality following directly from the definition of the t i .Finally, to show that e Φ takes values in O ( C ) , we wish to show that for ( p, q ) ∈ µ − ( γ ) ,one has q p + γ I n ∈ O ( C ) . This now follows from (6.3.14) for i = 1 and (6.3.12). Definition of the inverse Ψ : O ( C ) → Rep //G m , n , . We start by defining a morphism Ψ ′ : GL n (R m ) → µ − ( γ ) . Of course, we can compose this with the projection π : µ − ( γ ) → Rep //G m , n , to obtain a map e Ψ : GL n (R m ) → Rep //G m , n , . Then, since the map η :GL n (R m ) → O ( C ) of Lemma 2.2.4(b) is a categorical quotient, in order to define Ψ : O ( C ) → Rep //G m , n , , it suffices to show that e Ψ is L λ,m -invariant, by Lemma 2.2.8(d).We first define a tuple ( p, q ) C by p iC := (cid:2) n i × λ i − I n i (cid:3) q iC := (cid:20) λ i − × n i t i (cid:21) , ≤ i ≤ l. (6.3.15)With (6.3.12) it is easy to check that ( p, q ) C ∈ µ − ( γ ) and e Φ( p, q ) C = C. (6.3.16)We now define Ψ ′ : GL n (R m ) → µ − ( γ ) using the action (6.3.10) g g · ( p, q ) C . We wish to show that the resulting e Ψ is L λ,m -invariant. Let f ∈ L λ,m . We may write f = diag ( f , . . . , f l ) with f i ∈ GL λ i (R m ) . Furthermore, for ≤ i ≤ l , we will set f i := diag ( f i , . . . , f l ) , noting that f = f . Then it is easy to check that for ≤ i ≤ l , p iC ( f i − ) − = ( f i ) − p iC f i − q iC = q iC f i . (6.3.17)47y an inductive argument using (6.3.17), it is straightforward to show that for g ∈ GL n (R m ) and f ∈ L λ,m as above, ( gf ) · ( p, q ) C = (cid:16) f , . . . , f l (cid:17) − · ( g · ( p, q ) C ) , where the right hand side is the action of G m , n , ; in other words, Ψ ′ ( gf ) and Ψ ′ ( f ) lie inthe same G m , n , -orbit. It follows that e Ψ is L λ,m -invariant, and hence induces Ψ : O ( C ) → Rep //G m , n , with Ψ ◦ η = e Ψ . Verification that Φ and Ψ are mutually inverse. We first check that Φ ◦ Ψ = O ( C ) . Now, Φ ◦ Ψ is the morphism induced via L λ,m -invariance from the map e Φ ◦ Ψ ′ : GL n (R m ) → O ( C ) ,which is, by (6.3.16), g g · ( p, q ) C Ad g C. But this is precisely η , as in Lemma 2.2.4(b), hence the induced map on the quotient must bethe identity.Furthermore, for ( p, q ) C , the similarity relations in (6.3.13) and (6.3.14) are, in fact, equal-ities. In particular, q C p C + γ I n = t = C and hence it is easy to check that Φ ◦ Ψ( A ) = A for all A ∈ O ( C ) .Finally, we show that Ψ ◦ Φ = Rep //G m , n , . Let ( p, q ) ∈ µ − ( γ ) . Then e Φ( p, q ) = q p + γ I n ; if this is Ad g C , for g ∈ GL n (R m ) , then Ψ ′ ◦ e Φ( p, q ) = g · ( p, q ) C . Thus, to show that Ψ ◦ Φ ◦ π ( p, q ) = π ( p, q ) , we wish to show that ( p, q ) and g · ( p, q ) C are inthe same G m , n , -orbit. This is again an (increasing) induction. Using (6.3.16), we have q p + γ I n = Ad g C = g ( q C p C + γ I n ) g − and hence we find q p = gq C p C g − (6.3.18)and multiplying by p on the left and by g on the right, we obtain p q p g = p gq C p C . (6.3.19)Let us now write p g =: (cid:2) d h (cid:3) , for some d ∈ M n × λ (R m ) and h ∈ gl n (R m ) . Substituting this into (6.3.19), and using theexplicit expressions for p C and q C (6.3.15), we get (cid:2) ( p q ) d ( p q ) h (cid:3) = (cid:2) n × λ h t (cid:3) z m p q is invertible and hence d = 0 . Therefore p g = (cid:2) h (cid:3) = h p C or p = h p C g − . (6.3.20)Since z m p q is invertible, p is of rank n and multiplication by g does not change this, so h must also be of rank n and hence h ∈ GL n (R m ) . Using this and substituting (6.3.20) into(6.3.18), we can conclude that q = gq C h − . Therefore, ( p, q ) = ( h p C g − , gq C h − , p , q , . . . , p l , q l )= ( h , , . . . , · ( p C g − , gq C , p h , h − q , p , q , . . . , p l , q l ) , thus, after relabeling p , q , ( p, q ) is in the same G m , n , -orbit as an element of the form ( p C g − , gq C , p , q , p , q , . . . , p l , q l ) .By induction, we may assume that ( p, q ) is of the form ( p, q ) = ( p C g − , gq C , p C , q C , . . . , p iC , q iC , p i +1 , q i +1 , . . . , p l , q l ) . Then the i th component of the moment map (6.3.8) gives q i +1 p i +1 = p iC q iC − γ i I n i = (cid:20) λ i t i +1 (cid:21) ; (in case i = 1 , we have the product ( p iC g − )( gq C ) = p C q C , and so this case yields the sameequation). Using the same argument as above, we write p i +1 as a block matrix with twoblocks and using the fact that z m p i +1 q i +1 is invertible, show that the square block is an in-vertible matrix h i +1 and the other block is zero. We then conclude that p i +1 = h i +1 p i +1 C andthen that q i +1 = q i +1 C h − i +1 . Hence ( p, q ) = ( p C g − , gq C , p C , q C , . . . , p iC , q iC , h i +1 p i +1 C , q i +1 C h − i +1 , . . . , p l , q l )= (1 , . . . , h i +1 , . . . , · ( p C g − , gq C , p C , q C , . . . , p i +1 C , q i +1 C , p i +2 C h i +1 , h − i +1 q i +2 C , . . . , p l , q l ) , and the induction hypothesis is satisfied for i + 1 . Continuing in this fashion, we see that ouroriginal ( p, q ) is in the G m , n , -orbit of g · ( p, q ) C and hence Ψ ◦ Φ ◦ π ( p, q ) = π ( p, q ) . Of course, the reason for the emphasis on the quiver with multiplicities described in Section6.1.2 is to relate the corresponding variety to an additive fusion product of coadjoint orbitsand hence open de Rham spaces as in Proposition 3.1.10. Suppose we are given a d -tuple m := ( m i ) di =1 of positive integers and coadjoint orbits O ( C i ) , ≤ i ≤ d , for some diagonalelements C i ∈ gl n ( R m i ) ∨ , which we will take to be written in the form (2.2.2). We use this todefine the data for a quiver with multiplicities:49. For ≤ i ≤ d , the integer l i is defined as in (2.2.2) and the quiver Q as in (6.1.10) witharrows defined immediately thereafter.2. The tuple m of multiplicities can then be chosen as in (6.1.11).3. We define the dimension vector n as follows. We set n := n . Again from (2.2.2),for each ≤ i ≤ d , one gets a series of positive integers λ [ i, , . . . , λ [ i,l i − , and we set n [ i,k +1] := n [ i,k ] − λ [ i,k ] . This defines ( n [ i, , . . . , n [ i,l i ] ) for ≤ i ≤ d and we use these todefine the remaining entries of n . We have now defined Q ( m , n ) , and hence also G m , n , g m , n , etc.4. Finally, we wish to define an element γ ∈ g ∨ m , n . Once again from (2.2.2) and (6.3.2), foreach ≤ i ≤ d , we obtain elements γ [ i, , . . . , γ [ i,l i ] ∈ z − m i R m i ; we take γ [ i,j ] I n [ i,j ] ∈ z − m i gl n [ i,j ] ( R m i ) to be the component of γ at all vertices except the central vertex . There, we take γ z − I n ∈ gl n ( K ) ∨ , where γ := d X i =1 res z =0 γ [ i, . (6.4.1)5. Furthermore, we will assume that for ≤ i ≤ d , (2.2.3) is satisfied. Theorem 6.4.2. With Q ( m , n ) and γ ∈ g m , n chosen as above, one has an isomorphism of theassociated quiver variety with the additive fusion product of coadjoint orbits Q γ ∼ = d Y i =1 O ( C i ) ! (cid:30)(cid:30) GL n ( K ) . (6.4.3)In particular, if C , . . . , C d are regular generic we have Q γ ∼ = M ∗ µ , r . Furthermore, in termsof the quiver data, the dimension of Q γ is given by the formula (cf. (3.1.11)) dim Q γ = 2 d X i =1 m i l X k =1 n [ i,k ] ( n [ i,k − − n [ i,k ] ) − n + 1 ! . (6.4.4) Proof. Taking each leg one at a time, Proposition 6.3.4 takes ( p, q ) ∈ µ − ( γ ) and gives us a d -tuple ( A , . . . , A d ) with A i ∈ O ( C i ) , ≤ i ≤ d ; more explicitly (6.3.11), one has A i = q [ i, p [ i, + γ [ i, I n . The moment map condition for the quiver at the vertex is π res − d X i =1 q [ i, p [ i, ! = γ I n That for the additive fusion product, that is the right hand side of (6.4.3), is π res d X i =1 A i ! = π res d X i =1 q [ i, p [ i, + γ [ i, I n ! = 0 50o by definition (6.4.1), it is clear that one moment map condition is satisfied if and only ifthe other is. Finally, we remark the the remaining group action is the the diagonal actionof GL n ( K ) , with g ∈ GL n ( K ) acting on q [ i, as gq [ i, and on p [ i, by p [ i, g − ; this clearlytranslates into conjugation on the A i .For the dimension formula, we can use the expression (6.4.3) and compute the total di-mension by summing those of the coadjoint orbits O ( C i ) . To obtain these in terms of thequiver data, we use the expression (6.3.5). For the i th leg, the dimension of the space of rep-resentations for the arrows (going in opposite directions) joining the ( k − th and k th nodesis m i n [ i,k ] n [ i,k − ; the dimension of the group at the node [ i, k ] is m i n i,k ] . Summing over thenodes on the i th leg and accounting for the preimage of central elements in the dual of theLie algebra, we get dim O ( C i ) = 2 m i l i X k =1 n [ i,k ] n [ i,k − − m i l i X k =1 n i,k ] ! = 2 m i l i X k =1 n [ i,k ] ( n [ i,k − − n [ i,k ] ) . The term − n − 1) = − n − of course comes from the quotient by P GL n = GL n /Z . Let Q = ( Q , Q , h, t ) be a quiver, n ∈ Z Q > a dimension vector and m ∈ Z Q > the multiplic-ity vector. As explained in [Y3] this data is equivalent with the following symmetrizablegeneralized Cartan matrix C = ( c ij ) i,j ∈ Q by c ii = 2 and for i = jc i,j := − n i gcd( n i , n j ) a i,j where a i,j := |{ a ∈ Q | h ( a ) = i, t ( a ) = j or h ( a ) = j, t ( a ) = i }| . This we can record by a not necessarily simply-laced Dynkin diagram.Let now γ ∈ g ∨ m , n . Then the quiver variety with multiplicity Q γ has dimension given byformula (6.4.4). Thus Q γ is a surface if and only if d X i =1 m i l X k =1 n [ i,k ] ( n [ i,k − − n [ i,k ] ) = n . For instance, in the example of F ( ) below, one has m = ( m , m [1 , , m [1 , , m [1 , , m [2 , ) = (1 , , , , n = ( n , n [1 , , n [1 , , n [1 , , n [2 , ) = (4 , , , , and the condition is readily verified.Below we will list the star-shaped non-simply laced affine Dynkin diagrams, which cor-respond to open de Rham spaces in Theorem 6.4.2 of dimension . The simply-laced star-shaped ones ˜ D , ˜ E , ˜ E , ˜ E correspond to open de Rham spaces with logarithmic singulari-ties. 51 xample: A ( ) 11 21 21 ⇐ The diagram part depicts the non-simply laced Dynkin diagram. The integers writtenbelow each node depict the dimension vector, while the ones above the node give the multi-plicity vector. This corresponds to the open de Rham space M ∗ (1 , ) , (1) of type ((1 , ) , (1)) .By [Bo3, Theorem 9.11] M ∗ (1 , ) , (1) is isomorphic with an A ALE space. In particular themixed Hodge structure is pure on H ∗ ( M ∗ (1 , ) , (1) ) and W H ( M ∗ (1 , ) , (1) ; q, t ) = 1 + 3 qt , which is compatible with |M ∗ (1 , ) , (1) ( F q ) | = q + 3 q from (5.1.12) with d = 3 , r = 1 . Example: C ( ) 21 12 21 ⇒ ⇐ This corresponds to the open de Rham space M ∗ , (1 , of irregular type (1 , . This is theonly example from the list of star-shaped non-simply-laced Dynkin diagrams which is notisomorphic with a Nakajima quiver variety. Using the explicit equation for M ∗ , (1 , in [Bie,(3.1)] we can deduce that it is homotopic with a wedge of spheres by [ST, Theorem 3.1]. Inorder to match the virtual weight polynomial computation W H c ( M ∗ , (1 , , q, − 1) = q + 2 q from (5.1.12) with d = 2 , r = 2 , we must have that the mixed Hodge structure is pure and W H ( M ∗ , (1 , , q, t ) = 1 + 2 qt . This case is special in that Boalch’s [Bo3] identification of M ∗ , (1 , with a quiver varietydoes not apply, as we have two irregular poles. In fact there is no ALE space which isisomorphic with M ∗ , (1 , as the intersection form on H c ( M ∗ , (1 , ) is divisible by , whilethe intersection form of the A ALE space is not divisible by . Example: D ( ) 11 12 31 ⇚ Here the corresponding open de Rham space is M ∗ , (2) . By [Bo3, Theorem 9.11] it isisomorphic to an ˜ A ALE space, thus in particular W H ( M ∗ , (2) ; q, t ) = 1 + 2 qt . Because after a hyperkähler rotation the manifold becomes a resolution of ( C × × C ) / Z where Z acts bythe inverse (see Example 7.3.6 and [Hit1, Da]). Thus H c ( M ∗ , (1 , , Q ) has a basis represented by the two disjointexceptional divisors. xample: A ( ) 12 41 Here the corresponding open de Rham space is M ∗ , (3) . By [Bo3, Theorem 9.11] it isisomorphic to an A ALE space, thus in particular W H ( M ∗ , (3) ; q, t ) = 1 + qt . Example: G ( ) 31 32 13 ⇛ Here the corresponding open de Rham space is M ∗ , (2) . By [Bo3, Theorem 9.11] it isisomorphic to an A ALE space, thus in particular W H ( M ∗ , (2) ; q, t ) = 1 + 2 qt . Example: F ( ) 21 22 23 14 12 ⇒ Here the corresponding open de Rham space is M ∗ (2 ) , (1) . By [Bo3, Theorem 9.11] it isisomorphic to an D ALE space, thus in particular W H ( M ∗ (2 ) , (1) ; q, t ) = 1 + 4 qt . Example: E ( ) 11 12 13 22 21 ⇐ Here the corresponding open de Rham space is M ∗ (1 ) , (1) . By [Bo3, Theorem 9.11] it isisomorphic to an D ALE space, thus in particular W H ( M ∗ (1 ) , (1) ; q, t ) = 1 + 4 qt . In all these cases the mixed Hodge polynomial is compatible with the weight polynomialcomputed from (5.1.12) in the rank cases and (5.1.13) in the rank cases. The only exampleof rank is F (1)4 where one can compute the weight polynomial directly from (5.1.7).53 Hyperkähler considerations Our purpose in this section is to prove Theorem 7.3.3, which says that some of the open deRham spaces M ∗ ( C ) that we have been discussing, namely those for which all the formaltypes are of order ≤ , admit canonical complete hyperkähler metrics. To first give a roughexplanation as to how these arise, let us recall that such an M ∗ ( C ) is an additive fusion prod-uct of coadjoint orbits in g ∨ and those for the group G (using the notation of Section 2.1).Each coajdoint orbit of the latter type may be realized as an algebraic symplectic quotientof T ∗ G by left multiplication by a maximal torus (Lemma 7.3.1). It is well known that T ∗ G admits a hyperkähler metric, by an infinite-dimensional hyperkähler quotient via Nahm’sequations [Kr1]. Furthermore, if K ≤ G is a maximal compact subgroup then K × K acts byleft and right multiplication (7.2.1) and these actions admit hyperkähler moment maps [DS].For coadjoint orbits in g ∨ , hyperkähler metrics were first constructed by [Kr3] for regularsemisimple orbits and for general semisimple orbits in [Biq] and [Ko]. The coadjoint G -action can be restricted to K and it can be shown (Lemma 7.2.4), in a manner similar to thatfor T ∗ G , that this action also admits a hyperkähler moment map.Now, if a hyper-hamiltonian action of a compact group extends to a holomorphic actionof its complexification, then the hyperkähler reduction can be understood as a holomorphicsymplectic reduction [HKLR, §3(D)]. We wish to go in the opposite direction: M ∗ ( C ) isgiven as an algebraic symplectic quotient; we wish to show that it, in fact, arises as a hy-perkähler quotient. A special case of a version of the Kempf–Ness theorem due to Mayrand[May] gives sufficient conditions for algebraic symplectic quotients of the type we have beenconsidering to be upgraded to hyperkähler quotients. Let us begin by incorporating Mayrand’s statement into the following, which will give usthe criterion we will apply later to obtain the theorem. Proposition 7.1.1. Suppose ( M, g, I , J , K ) is a hyperkähler manifold. We suppose that ( M, I ) is a (smooth) complex affine variety and refer to M as such with the complex structure I inmind. Suppose G is a complex reductive group with an algebraic action on M for which therestriction to its maximal compact K admits a hyperkähler moment map µ I , µ J , µ K : M → k ∨ . As usual, we will write µ R := µ I µ C := µ J + iµ K for the real and complex components of the moment map; we will assume that µ C : M → g ∨ = k ∨ ⊕ i k ∨ is algebraic. Let λ ∈ ( g ∨ ) G be such that G acts freely on the affine variety µ − C ( λ ) ; thus, the algebraic symplectic quotient M// λ G := Spec C [ µ − C ( λ )] G is smooth. Then if there exists a K -invariant, proper global Kähler potential for ω I | µ − C ( λ ) ,which is bounded below, then there exists λ R ∈ k ∨ and, for the complex structures inducedfrom I , a natural biholomorphism M/// ( λ R ,λ ) K ∼ = M// λ G. roof. The Kähler potential produces a moment map ˜ µ R for the K -action with respect to ω I [May, Proposition 4.1]. This must differ from the given µ R by a constant, i.e., there exists λ R ∈ k ∨ such that µ R = ˜ µ R + λ R . Thus, ˜ µ − R (0) = µ − R ( λ R ) . Then [May, Proposition 4.2] givesa homeomorphism at the last step of the sequence M/// ( λ R ,λ ) K = (cid:0) µ − R ( λ R ) ∩ µ − C (0) (cid:1) /K = (cid:0) ˜ µ − R (0) ∩ µ − C (0) (cid:1) /K ∼ = M// λ G. Furthermore, by the freeness of the action, there is a single orbit type stratum and hence thehomeomorphism is, in fact, a biholomorphism, again from [May, Proposition 4.2]. Here we show that the two kinds of factors that appear in the relevant additive fusion prod-uct each admit hyper-hamiltonian group actions. Let G be a complex reductive group with Lie algebra g and let K ≤ G be a maximal compactsubgroup. We recall that for T ∗ G = G × g ∨ , there is an algebraic hamiltonian action of G × G given by ( g, h ) · ( a, X ) = ( gah − , Ad h X ) , (7.2.1)for which the moment map is ( a, X ) ( Ad a X, − X ) . (7.2.2) Proposition 7.2.3. T ∗ G admits a hyperkähler metric for which the restriction of the action(7.2.1) to K × K admits a hyperkähler moment map. Furthermore, the complex part of thismoment map is given by (7.2.2), and for the natural complex (Kähler) structure there existsa ( K × K ) -invariant, proper and bounded below global Kähler potential. Proof. As mentioned, the hyperkähler structure is due to [Kr1, Proposition 1]. The existenceof the hyperkähler moment map is [DS, §3 Lemma 2]. The expression for the complex partof the moment map is obtained by comparing [DS, Equations (4), (5)] and the expressionsin the statement of [DS, §3 Lemma 2]. Finally, the existence of the global Kähler potential is[May, Proposition 4.6]. Let G , g , K be as above and let O be a semisimple coadjoint orbit in g ∨ . Of course, G actsalgebraically on O via the coadjoint action and the moment map is simply the inclusion O ֒ → g ∨ . Exactly the same statement as in Proposition 7.2.3 holds for O . Proposition 7.2.4. O admits a complete hyperkähler metric for which the restriction of thecoadjoint action to K admits a hyperkähler moment map. Furthermore, the complex part ofthis moment map is given by the inclusion O ֒ → g ∨ , and for the natural complex (Kähler)structure there exists a K -invariant, proper and bounded below global Kähler potential.55 roof. As mentioned earlier, existence of the hyperkähler metrics can be found at [Kr3, The-orem 1.1] for regular semisimple orbits and at [Biq, Théorème 1] and [Ko, Theorem 1.1] forgeneral semisimple orbits. The fact that the conjugation action of K admits a hyperkählermoment map can be proved in the same way as [DS, Lemma 2]. The existence of the Kählerpotential with the indicated properties uses the same argument as that of [May, Proposition4.6]. Further details can be found in Appendix B. Remark . Let S := T ∩ K be a maximal torus in the maximal compact group K ≤ G and let s be its Lie algebra. Then, as in the references [Kr3, Biq, Ko], once the coadjoint orbitis fixed, the family of such hyperkähler metrics is parametrized by an element τ ∈ s (seeAppendix B). We first give a lemma describing coadjoint orbits for G as algebraic symplectic reductionsof T ∗ G . Lemma 7.3.1. Consider the diagonal element (2.2.1) C := C z + C z ∈ t ∨ with C regular (i.e., having distinct eigenvalues). Then the G coadjoint orbit O ( C ) ⊆ g ∨ isisomorphic to the algebraic symplectic quotient T ∗ G// C T, for the T -action t · ( a, X ) = ( ta, X ) . Remark . This is a special case of [Bo1, Lemma 2.3(2), see also Lemma 2.4]. Proof. Recall that the moment map for this action is µ : T ∗ G → t ∨ ( a, X ) π t ( Ad a X ) . The map µ − ( C ) → O ( C ) ( a, X ) Ad a − C z + Xz is readily seen to be T -invariant, so descends to a morphism T ∗ G// C T → O ( C ) .Now, as in Lemma 2.2.4(b), O ( C ) may be realized as the geometric quotient G /T . Ifwe write an element of G in the form g ( I + zH ) for some g ∈ G , H ∈ g , then we define G → T ∗ G// C T by g ( I + zH ) [ g − , Ad g ( C + [ H, C ])] , with the square brackets indicating the class mod T . It is straightforward to check that thisis well-defined and that it descends to O ( C ) → T ∗ G// C T to give the inverse. Theorem 7.3.3. Consider a generic open de Rham space M ∗ ( C ) for which the orders of allthe formal types C i are ≤ . Then M ∗ ( C ) admits a canonical complete hyperkähler metric.56 emark . By “canonical”, we mean that the space of such metrics is parametrized by afinite-dimensional space with the choice coming from each factor of a semisimple coadjointorbit in g ∨ (see Remark 7.2.5). Proof. As in Section 3.1, we label the coadjoint orbits so that O ( C i ) ⊆ g ∨ for ≤ i ≤ k and O ( C i ) is a coadjoint orbit for G for k + 1 ≤ i ≤ d ; we will abbreviate O ( C i ) to O i in thefollowing. By Proposition 3.1.10 and Lemma 7.3.1, we have M ∗ ( C ) ∼ = d Y i =1 O ( C i ) (cid:30)(cid:30) G = k Y i =1 O i × d Y i = k +1 O i ! (cid:30)(cid:30) G ∼ = k Y i =1 O i × d Y i = k +1 T ∗ G// C i T ! (cid:30)(cid:30) G ∼ = k Y i =1 O i × m Y i =1 T ∗ G ! (cid:30)(cid:30) ( C , T m × G, (7.3.5) where C denotes the tuple ( C k +11 , . . . , C d ) of residue terms of the formal types. Each factorof T acts via the left action on the corresponding T ∗ G factor as in (7.2.1) and the G factor actsdiagonally: by the coadjoint action on O i , for ≤ i ≤ k and with the right action in (7.2.1)for the factors indexed by k + 1 ≤ i ≤ d . We have thus expressed M ∗ ( C ) as an algebraicsymplectic quotient and we now wish to apply Proposition 7.1.1. But now the hypothesesare verified for each factor in Propositions 7.2.3 and 7.2.4, and can therefore easily be verifiedfor the product. Example 7.3.6. Consider the case of a rank open de Rham space M ∗ ( C ) with two poleseach of order . Proposition 3.1.10 tells us that this is a smooth affine algebraic surface, andindeed by performing the reduction in (7.3.5), first taking the quotient of T ∗ G × T ∗ G by G , one can see that it is a quotient of the form T ∗ G//T (for appropriate values of the mo-ment map), which is precisely the reduction carried out at [Da, pp.88–89]. Hence M ∗ ( C ) is isometric to the deformation of the D singularity, the hyperkähler metric on which waspreviously constructed via twistor methods in [Hit1, §7], and proved to be ALF in [CK, §5.3].It is also worth noting that this space can also be described via a slice construction [Bie, (3.1)]which, although an algebraic operation, also yields the metric [Bie, §4]. Furthermore, we ex-pect the metrics on higher dimensional open de Rham spaces to exhibit “higher dimensionalALF behaviour”. Remark . (i) It is true more generally, and not much harder to prove, that with no re-striction on the order of the formal types, the spaces M ∗ ( C ) always admit completehyperkähler metrics. However, there is some degree of choice in the metric, and sowe cannot say that a canonical metric exists. In general, a coadjoint orbit for G m maybe realized as an algebraic symplectic quotient of T ∗ G × O ( C ′ ) ([Bo1, Lemma 2.3(2),Lemma 2.4], cf. Remark 7.3.2), where O ( C ′ ) is a coadjoint orbit for the unipotent group G m . As such, O ( C ′ ) is an even-dimensional complex affine space, hence, upon somechoice of coordinates, admits a flat hyperkähler metric. One can show that the coordi-nates can be chosen so that S acts on pairs of coordinates with opposite weights, andhence with a hyperkähler moment map. Of course, an appropriate norm would givethe required Kähler potential and we could conclude as before. However, the possiblechoices of coordinates differ by arbitrary complex analytic automorphisms of the affinespace, which form an infinite-dimensional group; thus, the metric is not canonical inthe sense of Remark 7.3.4. This observation was already mentioned at [Bo2, p.3]. T ∗ G × O ( C ′ ) are (isomorphic to) what are known as “extended orbits” andthese can be arranged into a moduli space by taking an additive fusion product (this isreferred to as the “extended” moduli space in [Bo1, Definition 2.6] [HY, Definition 2.4].This moduli space will admit an action of T l , where l is the number of irregular poles,and M ∗ ( C ) can thus be realized as a hyperkähler quotient of the extended modulispace from this action. Indeed, in the last expression in (7.3.5), if we first take thequotient by G , then we obtain the extended moduli space. A Addendum to Section 6 A.1 Notation and duals Here, we use the notation of Section 6.1. In particular, R c := K [ z ] / ( z c ) . As in Section 2.1, weidentify the K -dual R ∨ , K c with z − c R c , where the pairing is h z − c f, g i := res z =0 z − c f g. Suppose a = ce , R a = K [ x ] / ( x a ) ; we have an inclusion R c ֒ → R a z x e . Similarly, the R c -dual R ∨ ,R c a is identified with x − e R a , where we take residues with respect to x . A.2 Representations of doubled quivers and duals For an arbitrary quiver Q , the space of representations is a sum over the arrows α ∈ Q q .Thus, to simplify notation, we will assume that we have a quiver with a single arrow, i.e., Q = { α } . Suppose the multiplicity and dimension vectors are given by m = ( m t ( α ) , m h ( α ) ) = ( a, b ) n = ( n t ( α ) , n h ( α ) ) = ( r, s ) , respectively. Let c := gcd( a, b ) . Then V := Rep( Q, m , n ) = Hom R c (cid:0) R ⊕ ra , R ⊕ sb (cid:1) . When one doubles the quiver, the arrow α ′ in the opposite direction yields the summand e V := Hom R c (cid:0) R ⊕ sb , R ⊕ ra (cid:1) . We will write Rep := Rep( Q, m , n ) = V ⊕ e V . Lemma A.2.1. One has e V = V ∨ .The pairing between p ∈ V , q ∈ e V is given as follows. Observe that pq ∈ End R c (cid:0) R ⊕ sb (cid:1) qp ∈ End R c (cid:0) R ⊕ ra (cid:1) . Thus, tr pq = tr qp ∈ R c . Then the pairing is h p, q i := Res z =0 z − c tr pq = Res z =0 z − c tr qp. .3 The symplectic form Since Rep is of the form V ⊕ V ∨ , it carries a canonical symplectic form ω . For x := ( p, q ) ∈ Rep , T x Rep = V ⊕ V ∨ , and for v = ( v , v ) , w = ( w , w ) ∈ T x Rep , ω is given by ω ( v, w ) := h v , w i − h w , v i . (A.3.1) A.4 Group actions and moment maps Explicitly, the actions of GL r ( R a ) and GL s ( R b ) on Rep are given by g · ( p, q ) = ( pg − , gq ) h · ( p, q ) = ( hp, qh − ) . The moment map µ : Rep → gl r ( R a ) ∨ ⊕ gl s ( R b ) ∨ is given by µ ( p, q ) = ( − qp, pq ) . (A.4.1)These expressions should be interpreted in the following way. One has − qp ∈ End R c ( R ⊕ ra ) ,but we want to view this as an element of gl r ( R a ) ∨ , K . Observe End R c (cid:0) R ⊕ ra (cid:1) = End R c (cid:0) R ⊕ rc ⊗ R c R a (cid:1) = gl r ( R c ) ⊗ R c End R c ( R a ) = gl r ( R c ) ⊗ R c R ∨ ,R c a ⊗ R c R a = gl r ( R c ) ⊗ R c x − e R a ⊗ R c R a → gl r ( R c ) ⊗ R c x − e R a = x − e gl r ( R a ) ∼ −→ z − a gl r ( R a ) = gl r ( R a ) ∨ , K . The only map that is not a canonical isomorphism here is x − e R a ⊗ R c R a → x − e R a , whicharises simply from multiplication in R a , x − e being a formal placeholder. Of course, one hasa similar interpretation of pq as an element of gl s ( R b ) ∨ , K . Lemma A.4.2. µ as defined above is a moment map.This is proved in the usual way. Proof. Let ξ ∈ gl r ( R a ) , w = ( w , w ) ∈ T x Rep . We compute v ξ ( x ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 exp( ǫξ ) · ( p, q ) = ( − pξ, ξq ) . Thus, ω ( v ξ , w ) = ω (cid:0) ( − pξ, ξq ) , ( w , w ) (cid:1) = − ( h pξ, w i + h w , ξq i ) . Now, dµ ξ ( w ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 µ ξ ( p + ǫw , q + ǫw ) = − ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 h ( q + ǫw )( p + ǫw ) , ξ i = −h qw + w p, ξ i . Of course, one has to interpret the expressions in the way described above.59 emark A.4.3 . There are two special cases which occur in our computations in Section 6. Thefirst is where a = b = c = m . Then V = Hom R m (cid:0) R ⊕ rm , R ⊕ sm (cid:1) = Hom K (cid:0) K ⊕ r , K ⊕ s (cid:1) ⊗ K R m , and similarly for V ∨ . Thus, we may think of p and q as matrices with entries in R m . In theinterpretation of the moment map value, one has − qp ∈ End R m (R ⊕ rm ) = gl r (R m ) ⊗ R m x − R m ∼ −→ x − m gl r (R m ) = gl r (R m ) ∨ , K . Thus, we are simply multiplying q and p (i.e., matrices with coefficients in R m ) and thenformally multiplying by x − m , to view it as an element of the dual space gl r (R m ) ∨ , K .The second case is when a = c = 1 , b = m . Then V = Hom K (cid:0) K ⊕ r , R ⊕ sm (cid:1) = Hom K (cid:0) K ⊕ r , K ⊕ s (cid:1) ⊗ K R m , and again we may think of p and q as matrices with R m coefficients. In this case, pq ∈ End K (R ⊕ sm ) = gl s ( K ) ⊗ K x − m R m ⊗ K R m → x − m gl s (R m ) = gl s (R m ) ∨ , K with the map coming from multiplication in R m . One notes that this case is asymmetric inthat − qp ∈ End K ( K ⊕ r ) = gl R ( K ) but there is no problem of interpretation here.In both cases, we think of p and q as matrices over R m and when we want to view theproduct as an element of the dual of a Lie algebra, we multiply formally by x − m . Equiva-lently, we maybe multiply q first to view it as a matrix with entries in x − m R m (so a Laurentpolynomial), and then multiplying by p as matrices over the ring of Laurent polynomials; wethen truncate the non-negative powers of x (recall that since we are using the trace-residuepairing, such terms are superfluous). A.5 General formulae for the symplectic form and moment map Now, for an arbitrary quiver Q , multiplicity vector m and dimension vector n , as in Section6.1, one can write down the symplectic form and moment map explicitly as follows. We willuse the notation of (6.1.1): we will also denote an element Rep := Rep( Q, m , n ) by x = ( p, q ) ,where p = ( p α ) α ∈ Q , q = ( q α ) α ∈ Q are tuples indexed by Q , with p α ∈ V α := Hom R α (cid:16) R ⊕ n t ( α ) m t ( α ) , R ⊕ n h ( α ) m h ( α ) (cid:17) q α V ∨ α = ∈ Hom R α (cid:16) R ⊕ n h ( α ) m h ( α ) , R ⊕ n t ( α ) m t ( α ) (cid:17) . For the symplectic form, we simply sum the symplectic form in (A.3.1): a tangent vectorto Rep at x is given by an element of ⊕ α V α ⊕ V ∨ α . Let v , w ∈ T x Rep , so that v = ( v α , v α ) , w = ( w α , w α ) . Then the symplectic form is given by ω ( v, w ) = X α ∈ Q h v α , w α i − h w α , v α i . As the Lie algebra g m , n is indexed by Q rather than Q , we cannot simply sum over Q ,but at a given vertex i ∈ Q , we sum the expressions in the components of (A.4.1) over allarrows with i as an endpoint, according to whether it is the tail or the head. The explicitexpression is already given in (6.1.2). 60 Details of proof of Proposition 7.2.4 Here, we will be a bit more explicit about the proof of Proposition 7.2.4. The statements thatneed to be proved are: the existence of a hyperkähler moment map on the coajdoint orbit O and its complex part is simply the inclusion map into the dual of the complex Lie algebra,which is Lemma B.3 below; and the existence of a Kähler potential with the appropriateproperties, which is Lemma B.6.To proceed, we will need to fix notation, and so we will adopt that of [Biq, §3, L’espace desmodules ]. As such, G will now denote a compact Lie group, which is of course the maximalcompact subgroup of its complexification G C , whereas in Section 7.2 it denoted the com-plexification and K a maximal compact subgroup; we hope this will cause the reader noconfusion. Of course, g will denote the Lie algebra of G and g C its complexification, and h , i will denote a Ad-invariant inner product on g .Let us recall how the coadjoint orbit O is identified with a moduli space of solutionsto Nahm’s equations. Let S ≤ G be a maximal torus (which is, of course, compact) withLie algebra s , and respective complexifications S C and s C . As usual, using the invariantinner product, we identify g ∨ = g and g ∨ C = g C . Viewing O as a subset of g C , as it is asemisimple orbit, its intersection with s C is a singleton; we write this element as τ + iτ with τ , τ ∈ s . One chooses a third element τ ∈ s so that we have a triple τ = ( τ , τ , τ ) ∈ s ,so that for an appropriate ς > , we can make sense of the space Ω ∇ ; ς as described at [Biq,§3, p.265]. We will write an element ∇ + a ∈ A ς as a quadruple T = ( T , T , T , T ) ofsmooth maps T i : ( −∞ , → g satisfying an asymptotic condition depending on τ : one has ∇ + a = d + T ds + P ei =1 T i dθ i . We consider such T which are solutions to Nahm’s equations: dT i ds + [ T , T i ] + [ T j , T k ] = 0 (B.1)for cyclic permutations ( i j k ) of (1 2 3) . The quotient of the space of such solutions by thegroup G ς , also defined at [Biq, §3, p.265], will be referred to as the moduli space M = M ( τ ) of solutions to Nahm’s equations. We will often write [ T ] for the G ς -orbit of a solution T .The isomorphism M ∼ −→ O is given by [Biq, Corollaire 4.5] (see also the definition beforeEquations (IIIa) and (IIIb)) as [ T ] T (0) + iT (0) . (B.2) Lemma B.3. The map µ : M → g ⊕ given by [ T ] ( T (0) , T (0) , T (0)) (B.4)yields a hyperkähler moment map for the (co)adjoint G -action. Furthermore, the complexpart of the moment map coincides with the inclusion of O in the dual of the complex Liealgebra. Proof. As mentioned in Section 7.2.2, the proof mirrors that of [DS, Lemma 2]. Consider thegroup G + ς := (cid:8) g : R − → G : ( ∇ g ) g − ∈ Ω ∇ ; ς (cid:9) which has Lie algebra Lie ( G + ς ) = (cid:8) u : R − → g : ∇ u ∈ Ω ∇ ; ς (cid:9) . G ς is a normal subgroup of G + ς with quotient G and the quotient map G + ς → G is simplyevaluation at s = 0 , with of course a parallel statement for the Lie algebras. Furthermore, G + ς acts on the space of solutions to Nahm’s equations inducing the adjoint action of G on O , as is easily seen via the map (B.2).A tangent vector to M at [ T ] is represented by a quadruple w = ( w , w , w , w ) ∈ Ω ∇ ,ς satisfying dw i ds = − [ T , w i ] − [ w , T i ] − [ T j , w k ] − [ w j , T k ] (B.5)for cyclic permutations ( i j k ) of (1 2 3) (cf. [DS, Equations (6)-(9)]; note that there is a slightdifference in the complex structures there and in [Biq] accounting for the sign differences inthe equations). These equations are obtained simply by linearizing Nahm’s equations (B.1).Let ξ ∈ g and choose a lift u ( s ) ∈ Lie ( G + ς ) , so that u (0) = ξ . A representative for thetangent vector v ξ ([ T ]) to M at [ T ] generated by the infinitesimal action of ξ is given by v ξ ([ T ]) ↔ (cid:18) [ u, T ] − duds , [ u, T ] , [ u, T ] , [ u, T ] (cid:19) . We evaluate using (B.5) ω I ( v ξ , w ) = Z −∞ − (cid:28) [ u, T ] − duds , w (cid:29) + h [ u, T ] , w i − h [ u, T ] , w i + h [ u, T ] , w i ds = Z −∞ h u, − [ T , w ] − [ w , T ] − [ T , w ] − [ w , T ] i + (cid:28) duds , w (cid:29) ds = Z −∞ (cid:28) u, dw ds (cid:29) + (cid:28) duds , w (cid:29) ds = Z −∞ dds h u, w i ds = h ξ, w (0) i , since u → as s → −∞ .On the other hand, pairing ξ with the moment map µ I gives the function µ ξ I : M → R [ T ] 7→ h ξ, T (0) i . Hence dµ ξ I ( w ) = h ξ, w (0) i = ω I ( v ξ , w ) and this is exactly the moment map condition. The same computation can be repeated forthe complex structures J and K .The statement about the complex part of the moment map is obvious from the expres-sions (B.2) and (B.4). Lemma B.6. The semisimple coajdoint orbit O admits a global G -invariant Kähler potential(for the complex structure I ) which is proper and bounded below.As mentioned, the proof here is adapted from that of [May, Lemma 4.5]. Proof. A global Kähler potential ϕ I : M → R for the Kähler form ω I is given by (see [HKLR,§3(E)], cf. [DS, p.64]) [ T ] Z −∞ h T , T i + h T , T i ds. 62t is then sufficient to show that the ϕ I is G -invariant, proper and bounded below. G -invariance follows from that of the bilinear form h , i and lower-boundedness is obvious,so properness is essentially all that needs to be proved.Let ξ = ( ξ , ξ , ξ ) ∈ g ⊕ . Then the existence and uniqueness theorem for systems ofordinary differential equations gives a unique solution T ξ = ( T ξ ≡ , T ξ , T ξ , T ξ ) to thereduced Nahm’s equations dT i dt + [ T j , T k ] = 0 (B.7)(this is just (B.1) with T = 0 ) with T ξi (0) = ξ i for i = 1 , , . This allows us to define afunction e ϕ I : g ⊕ → R by ξ Z −∞ h T ξ , T ξ i + h T ξ , T ξ i ds, (B.8)which is (at the very least) continuous, again by the assertions of the existence and unique-ness theorem on the dependence on the initial conditions.Let us explain how this is related to ϕ I . Consider the map ev C : g ⊕ → g C ξ ξ + iξ . As O is a semi-simple (co)adjoint orbit, O is closed in g C and hence so is ev − C ( O ) . We maythen identify O with the subset of ξ ∈ ev − C (0) for which T ξ is gauge equivalent to an elementof A ς . We observe that this will be closed, as we are imposing an asymptotic condition on T ξ . As the definition of ϕ I is independent of the gauge equivalence class, one sees that underthe identification of O with the described subset of ev − C ( O ) , ϕ I = e ϕ I | O . It now suffices to show that there is a closed subset V ⊆ g ⊕ containing O (using the iden-tification above) for which e ϕ I | V is proper, since the restriction of a proper map to a closedsubset is still proper.For this, we will take V := (cid:8) r · ξ : ξ ∈ ev − C ( O ) , r ∈ R ≥ (cid:9) , which is closed in g ⊕ . Let S ⊆ g ⊕ be the unit sphere (here, we may take the invariant innerproduct on g in each factor); this is compact. Then S ∩ V is a compact subset of g ⊕ , andhence e ϕ I has a minimum value m > . It is non-zero, for if r · ξ ∈ V is such that e ϕ I ( rξ ) = 0 then, assuming that O 6 = { } (which we may of course do), then one finds ev C ( rξ ) = 0 , andhence r = 0 ; but this would contradict rξ ∈ S .By uniqueness of the solutions T ξ to (B.7), for r ∈ R > , it is easy to see that T rξ ( s ) = rT ξ ( rs ) . From this, we obtain for any ξ ∈ g ⊕ , e ϕ I ( rξ ) = 12 r Z −∞ h T ξ ( rs ) , T ξ ( rs ) i + h T ξ ( rs ) , T ξ ( rs ) i ds = 12 r Z −∞ h T ξ ( t ) , T ξ ( t ) i + h T ξ ( t ) , T ξ ( t ) i dt = r e ϕ I ( ξ ) . (B.9)63ow, given ξ ∈ V , one has ξ k ξ k ∈ S ∩ V and so (B.9) gives e ϕ I ( ξ ) = k ξ k e ϕ I (cid:18) ξ k ξ k (cid:19) ≥ m k ξ k or k ξ k ≤ e ϕ I ( ξ ) m . From this, one finds that the preimage of a bounded set in R under e ϕ I | V is bounded in V . Bycontinuity, the preimage of a closed set is closed, and so e ϕ I | V is proper. Remark B.10 . Since O is semisimple, it is a closed subset in the Euclidean space g ∨ C . As such, anorm function on g ∨ C would restrict to a bounded below, proper function on O . Indeed, look-ing at the form of e ϕ I (B.8), there is some similarity with the norm function on O . However, itis not clear (and indeed, probably not true) that the inclusion O ֒ → g ∨ C is a metric embeddingfor some hyperkähler metric on g ∨ C . C Additional proofs Here we include proofs of a couple results in the text, in the case the reader finds the detailsgiven there insufficient. C.1 Proof of Lemma 4.2.5 From (2.2.7), one has φ X ( b ) = * C + m − X i =1 [ b i , C i +1 ] + m − X i =1 m − i − X j =1 [ b i , C i + j +1 ] w j , X + . (C.1)We note that the pairing h A, B i := tr AB is the usual one on g = gl n , satisfying the properties h AB, C i = h A, BC i h [ A, B ] , C i = h A, [ B, C ] i . These properties will often be used without mention in what follows.We wish to show that if φ X is independent of b ⌊ m +12 ⌋ , . . . , b m − , then X ∈ t and [ X, b ] = · · · = [ X, b ⌊ m − ⌋ ] = 0 . We go by induction, say on p , looking at the terms in φ X involving b m − p . For p = 1 , the only term involving b m − is h [ b m − , C m ] , X i = h b m − , [ C m , X ] i and this vanishes if φ X is independent of b m − , so if this happens [ C m , X ] ∈ ( g od ) ⊥ = t ; butas C m ∈ t , also [ C m , X ] ∈ g od , so [ C m , X ] = 0 . As C m ∈ t reg , it follows that X ∈ t . Conversely,if X ∈ t , then [ C m , X ] = 0 and φ X is independent of b m − .Now, by induction, we assume that X ∈ t and that [ X, b ] = · · · = [ X, b p − ] = 0 . Weshow that if φ X is independent of b m − p , then [ X, b p − ] = 0 . We look at the terms of (C.1) thatinvolve b m − p ; from (2.2.6), we must also include terms w j with j ≥ m − p . Changing indices,we may write this as * [ b m − p , C m − p +1 ] + m X k = m − p +2 [ b m − p , C k ] w k + p − m − + k − X j = m − p [ b k − j − , C k ] w j , X + . (C.2)64bserve that since C m − p +1 ∈ t , [ b m − p , C m − p +1 ] ∈ g od and as X ∈ t , the first term abovevanishes, so m X k = m − p +2 * [ b m − p , C k ] w k + p − m − + k − X j = m − p [ b k − j − , C k ] w j , X + . (C.3)Let us fix k ∈ [ m − p + 2 , m − and consider the terms for the corresponding value of k in the sum above. Since k − j − ≤ p − if k ≤ m − , we can rewrite * k − X j = m − p [ b k − j − , C k ] w j , X + = * k − X j = m − p [ w j , b k − j − ] , XC k + (C.4)By definition of the w j , (by multiplying out the inverse relation and looking at the coefficientof z k − ), one has w k − + k − X j =1 w j b k − j − + b k − = 0 b k − + k − X j =1 b k − j − w j + w k − = 0 . (C.5)Using this, (C.4) becomes * m − p − X j =1 [ b k − j − , w j ] , XC k + , but now the only term involving b m − p is h [ b m − p , w k + p − m − ] , XC k i . Therefore the terms of(C.3) dependent on b m − p are h [ b m − p , C k ] w k + p − m − , X i + h [ b m − p , w k + p − m − ] , XC k i and using the fact that [ X, w k + p − m − ] = [ X, C k ] = 0 , we see that this vanishes.Therefore the dependence on b m − p comes only in the terms which also involve C m ,namely * [ b m − p , C m ] w p − + m − X j = m − p [ b m − j − , C m ] w j , X + . We separate out the first term in the sum (since [ X, b p − ] = 0 is what we wish to prove) andthen simplify as above, where we used (C.5), to get h [ b m − p , C m ] w p − + [ b p − , C m ] w m − p , X i + * m − p X j =1 [ b m − j − , w j ] , XC m + . Now, the terms involving b m − p are (and again, we are replacing w m − p with − b m − p , as this isthe only dependence on b m − p ) h [ b m − p , C m ] w p − + [ C m , b p − ] b m − p , X i + h [ b m − p , w p − ] + [ b m − p , b p − ] , XC m i . p ≤ ⌊ m ⌋ .) Now, since w p − + b p − is a function of the b i with i ≤ p − (2.2.6), the induction hypothesis gives [ X, w p − + b p − ] = 0 , and using this, one simplifies the above to h [ b m − p , C m ] , [ w p − , X ] i = h [ b m − p , C m ] , [ X, b p − ] i = h b m − p , [ C m , [ X, b p − ]] i . Arguing as above, if this is independent of b m − p , then [ C m , [ X, b p − ]] ∈ t and as C m ∈ t reg ,we have [ X, b p − ] ∈ t ; but as X ∈ t and b p − ∈ g od , we get [ X, b p − ] = 0 . Again, conversely, if [ X, b p − ] = 0 , we can conclude that φ X is independent of b m − p . C.2 Proof of (5.1.10) We write λ = ( λ , . . . , λ ℓ ) , µ = ( µ , . . . , µ m ) , I = ( I , . . . , I ℓ ) , M = { , . . . , µ } , M i = { P i − p =1 µ p + 1 , . . . , P ip =1 µ p } for ≤ i ≤ m , so that { , . . . , n } = ` mi =1 M i . Then µ I is ob-tained by writing the numbers | M i ∩ I j | , ≤ i ≤ m, ≤ j ≤ ℓ in (some) non-increasing order. Hence N ( µ I ) = m X i =1 ℓ X k =1 | M i ∩ I k | . (C.1)Observe that if λ j = λ j +1 and I is as above, if we take I ′ := ( I , . . . , I j +1 , I j , . . . , I ℓ ) , then N ( µ I ) = N ( µ I ′ ) as the terms in (C.1) are simply permuted.Now, let I ∈ P λ be such that N ( µ I ) is minimal. Suppose n ∈ I j . By the observationabove, we may assume that λ j +1 < λ j (for otherwise we could interchange I j and I j +1 andthe resulting I ′ would still give a minimal value), so that ˜ λ := ( λ , . . . , λ j − , λ j − , λ j +1 , . . . , λ ℓ ) is still a partition (of n − ). We therefore wish to show that N ( µ I ) ≥ N ( λ ) + N ( µ ) − n , and we will do this by induction.Let ˆ µ := ( µ , . . . , µ m − , µ m − , and let ˆ M i be defined as above; so that ˆ M i = M i for ≤ i ≤ m − , ˆ M m = M m \ { n } . We have N ( λ ) = N (˜ λ ) + 2 λ j − N ( µ ) = N (ˆ µ ) + 2 µ m − . By induction, we know min J ∈P ˜ λ N (ˆ µ J ) ≥ N (˜ λ ) + N (ˆ µ ) − ( n − . Therefore it suffices to show that min J ∈P ˜ λ N (ˆ µ J ) ≤ N ( µ I ) + 2( n − λ j − µ m ) + 1 . (C.2)66onsider J = ( I , . . . , I j − , I j \ { n } , I j +1 , . . . , I ℓ ) ∈ P ˜ λ . Then using (C.1) and noting that if i = m or k = j , then | ˆ M i ∩ J k | = | M i ∩ I k | , so N (ˆ µ J ) − N ( µ I ) = m X i =1 ℓ X k =1 | ˆ M i ∩ J k | − m X i =1 ℓ X k =1 | M i ∩ I k | = | ˆ M m ∩ J j | − | M m ∩ I j | = ( | M m ∩ I j | − − | M m ∩ I j | = − | M m ∩ I j | + 1= 2( | M m ∪ I j | − λ j − µ m ) + 1 ≤ n − λ j − µ m ) + 1 and taking the minimum on the left side of (C.2), we see that the inequality holds. 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