Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
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Communicated by Robion C. Kirby, University of California, Berkeley, and approved February 24, 2006 (received for review September 27, 2005)
Abstract
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This technique in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence, simple unified proofs are obtained for formulas of Poincaré polynomials of toric hyperkähler varieties (recovering results of Bielawski–Dancer and Hausel–Sturmfels), Poincaré polynomials of Hilbert schemes of points and twisted Atiyah–Drinfeld–Hitchin–Manin (ADHM) spaces of instantons on ℂ^{2} (recovering results of Nakajima–Yoshioka), and Poincaré polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced.
Let 𝕂 be a field, which will be either the complex numbers ℂ or the finite field 𝔽_{q} in this work. Let G be a reductive algebraic group over 𝕂, 𝔤 its Lie algebra. Consider a representation ρ : G → GL(𝕍) of G on a 𝕂vector space 𝕍, inducing the Lie algebra representation ϱ : 𝔤 → 𝔤𝔩(𝕍). The representation ρ induces an action ρ : G → GL(𝕄) on 𝕄 = 𝕍 × 𝕍*. The vector space 𝕄 has a natural symplectic structure, defined by the natural pairing 〈v, w〉 = w(v), with v ∈ 𝕍 and w ∈ 𝕍*. With respect to this symplectic form a moment map of ρ is given at X ∈ 𝔤 by Let now ξ ∈ (𝔤*)^{G} be a central element, then the holomorphic symplectic quotient is defined by the affine geometric invariant theory (GIT) quotient which is the affine algebraic geometric version of the hyperkähler quotient construction of ref. 1. In particular our varieties, in addition to the holomorphic symplectic structure, will carry a natural hyperkähler metric, although the latter will not feature in what follows.
Our main proposition counts rational points on the varieties μ^{−1}(ξ) over the finite fields 𝔽_{q}, where q = p^{r} is a prime power. For convenience we will use the same letters 𝕍, G, 𝔤, 𝕄, ξ for the corresponding vector spaces, groups, Lie algebras, and matrices over the finite field 𝔽_{q}. We define the function a_{ϱ} : g → ℕ ⊂ ℂ at X ∈ 𝔤 as where we used the notation S for the number of elements in any set S. In particular a_{ϱ}(X) is always a power of q. For an element v ∈ V of any vector space we define the characteristic function δ_{v} : V → ℂ by δ_{v}(x) = 0 unless x = v when δ_{v}(v) = 1. We can now formulate our main proposition.
Proposition 1.
The number of solutions of the equation μ(v, w) = ξ over the finite field 𝔽_{q} equals To explain the last two terms in the proposition above, we need to define Fourier transforms (2) of functions f : 𝔤 → ℂ on the finite Lie algebra 𝔤, which here we think of as an abelian group with its additive structure. To define this fix Ψ : 𝔽_{q} → ℂ^{x} a nontrivial additive character, and then we define the Fourier transform ℱ(f) : 𝔤* → ℂ at a Y ∈ 𝔤*
Proof:
Using two basic properties of Fourier transform for X ∈ 𝔤 and for v ∈ V we get
Affine Toric Hyperkähler Varieties
We take G = 𝕋^{d} ≅ (ℂ^{×})^{d} a torus. A vector configuration A = (a_{1}, …, a_{n}) : ℤ^{n} → ℤ^{d} gives a representation ρ_{A} : 𝕋^{d} → 𝕋^{n} ⊂ GL(𝕍), where 𝕍 ≅ ℂ^{n} is an ndimensional vector space and 𝕋^{n} ⊂ GL(𝕍) is a fixed maximal torus. The corresponding map on the Lie algebras is ϱ_{A} : t^{d} → t^{n}. The holomorphic moment map of this action μ_{A} : 𝕍 × 𝕍* → (t^{d})* is given by Eq. 1 , which in this case takes the explicit form We take a generic ξ ∈ (t^{d})*. The affine toric hyperkähler variety is then defined as the affine geometric invariant theory quotient: ℳ(ξ, A) = μ_{A} ^{−1}(ξ)//𝕋^{d}. To use our main result, we need to determine a _{ϱ} (X). Note that the natural basis e_{1}, …, e_{n} ∈ (t^{n})* gives us a collection of hyperplanes H_{1}, …, H_{n} in t^{d}. Now for X ∈ t^{d} we have that a _{ϱ} (X) = q^{ca(X)} , where ca(X) is the number of hyperplanes that contain X. Finally, we take the intersection lattice L(A) of this hyperplane arrangement, i.e., the set of all subspaces of t^{d} that arise as the intersection of any collection of our hyperplanes, with partial ordering given by containment. The generic choice of ξ will ensure that ξ will not be trivial on any subspace in the lattice L(A). Thus, for any subspace V ∈ L(A), we have from Eq. 3 that Σ_{X∈V} Ψ (〈X, ξ〉) = 0.
Now we can use Proposition 1. If we perform the sum we get a combinatorial expression where μ_{L(A)} is the Möbius function of the partially ordered set L(A), while ca(V) is the number of coatoms, i.e., hyperplanes containing V. Because the count above is polynomial in q and the mixed Hodge structure on ℳ(ξ, A) is pure, we get that for the Poincaré polynomial we need to take the opposite of the count polynomial, i.e., substitute q = 1/t ^{2} and multiply by t^{4(n−d)} . This way we get:
Theorem 1.
The Poincaré polynomial of the toric hyperkähler variety is given by One can prove by a simple deletion contraction argument (and it also follows from the second proof of proposition 6.3.26 of ref. 3) that for any matroid ℳ_{𝒜} and its dual ℳ_{ℬ} where is the hpolynomial of the dual matroid ℳ_{ℬ}. This way we recover a result of refs. 4 and 5; for a more recent arithmetic proof see (6):
Corollary 1.
The Poincaré polynomial of the toric hyperkähler variety is given by where B is a Gale dual vector configuration of A.
Hilbert Scheme of nPoints on ℂ^{2} and Atiyah–Drinfeld–Hitchin–Manin (ADHM) Spaces
Here G = GL(V), where V is an ndimensional 𝕂 vector space. We need three types of basic representations of G. The adjoint representation ρ_{ad} : GL(V) → GL(𝔤𝔩(V)), the defining representation ρ_{def} = Id : G → GL(V), and the trivial representations ρ_{triv} ^{k} = 1 : G → GL(𝕂^{k}). Fix k and n. Define 𝕍 = 𝔤𝔩(V) × V ⊗ 𝕂^{k}, 𝕄 = 𝕍 × 𝕍* and ρ : G → GL(𝕍) by ρ = ρ_{ad} × ρ_{def} ⊗ ρ_{triv} ^{k}. Then we take the central element ξ = Id_{V} ∈ 𝔤𝔩(V) and define the twisted ADHM space as where with A, B ∈ 𝔤𝔩(V), I ∈ Hom(𝕂^{k}, V) and J ∈ Hom(V, 𝕂^{k}).
The space ℳ(n, k) is empty when k = 0 (the trace of a commutator is always zero), diffeomorphic with the Hilbert scheme of npoints on ℂ^{2}, when k = 1, and is the twisted version of the ADHM space (7) of U(k) Yang–Mills instantons of charge n on ℝ^{4} (cf. ref. 8). By our main Proposition 1 the number of solutions over 𝕂 = 𝔽_{q} of the equation is the Fourier transform on 𝔤 of the function a _{ϱ}(X) = ker(ϱ(X)). First we determine a_{ϱ}(X) for X ∈ 𝔤 = 𝔤𝔩(V). By the definition of ϱ we have and so if a_{ϱad} (X) = ker(ϱ_{ad}(X)) and a_{ϱdef} = ker(ϱ_{def}), then we have Now Proposition 1 gives us We will perform the sum adjoint orbit by adjoint orbit. The adjoint orbits of 𝔤𝔩(n), according to their Jordan normal forms, fall into types, labeled by 𝒯(n), which stands for the set of all possible Jordan normal forms of elements in 𝔤𝔩(n). We denote by 𝒯 _{reg} (t) the types of the regular (i.e., nonsingular) adjoint orbits, while 𝒯_{nil}(s) = 𝒫(s) denotes the types of the nilpotent adjoint orbits, which are just given by partitions of s. First we do the k = 0 case where we know a priori, that the count should be 0, because the commutator of any two matrix is always tracefree and thus cannot equal ξ (for almost all q). Additionally, if we separate the nilpotent and regular parts of our adjoint orbits we get where C_{τ} and, respectively, ℭ_{τ} denotes the centralizer of an element X_{τ} of 𝔤 of type τ in the adjoint representation of G, respectively, 𝔤 on 𝔤. So if we define the generating series and then we have However, Φ_{nil} ^{0} is easy to calculate (9) thus we get Now the general case is easy to deal with: Thus, if we define the grand generating function by and then for the latter we get similarly to the argument for Eq. 4 in ref. 9 that For the grand generating function then we get Because the mixed Hodge structure is pure, and this count is polynomial, we get the compactly supported Poincaré polynomial. To get the ordinary Poincaré polynomial, we need to replace q = 1/t^{2} and multiply the nth term in Eq. 6 by t^{4kn} . This way we get Theorem 2.
Theorem 2.
The generating function of the Poincaré polynomials of the twisted ADHM spaces are given by This result appeared as corollary 3.10 in ref. 10.
Quiver Varieties of Nakajima
Here we recall the definition of the affine version of Nakajima’s quiver varieties (11). Let 𝒬 = (𝒱, ℰ) be a quiver, i.e., an oriented graph on a finite set 𝒱 = {1, …, n} with ℰ ⊂ 𝒱 × 𝒱 a finite set of oriented (perhaps multiple and loop) edges. To each vertex i of the graph, we associate two finite dimensional 𝕂 vector spaces V_{i} and W_{i} . We call (v _{1}, …, v _{n}, w _{1}, …, w _{n}) = (v, w) the dimension vector, where v _{i} = dim(V_{i}) and w _{i} = dim(W_{i}). To these data we associate the grand vector space the group and its Lie algebra and the natural representation with derivative The action is from both left and right on the first term and from the left on the second.
We now have G_{v} acting on 𝕄_{v,w} = 𝕍_{v,w} × 𝕍*_{v,w} preserving the symplectic form with moment map μ_{v,w} : 𝕍_{v,w} × 𝕍*_{v,w} → 𝔤*_{v} given by Eq. 1 . We take now ℰ_{v} = (Id_{V1}, …, Id_{Vn}) ∈ (𝔤*_{v})^{Gv}, and define the affine Nakajima quiver variety (11) as Here we determine the Betti numbers of ℳ(v, w) using our main Proposition 1, by calculating the Fourier transform of the function a _{ϱv,w} given in Eq. 2 .
First, we introduce, for a dimension vector w ∈ 𝒱^{ℕ}, the generating function where 𝒯_{nil}(s) is the set of types of nilpotent s × s matrices, where a type is given by a partition λ ∈ 𝒫(s) of s, X_{λ} denotes the typical s × s nilpotent matrix in 𝔤𝔩(s) in Jordan form of type λ, C_{λ} is the centralizer of X_{λ} under the adjoint action of GL(s) on 𝔤𝔩(s). We also introduce the generating function where 𝒯_{reg}(t) is the set of types τ, i.e., Jordan normal forms, of a regular t × t matrix X_{τ} in 𝔤𝔩(t), C_{τ} ⊂ GL(t) its centralizer under the adjoint action. Note also that for a regular element X ∈ 𝔤_{v}, a_{ϱv,w} (X) = a _{ϱv,0} (X) does not depend on w ∈ 𝒱^{ℕ}.
Now we introduce for w ∈ 𝒱^{ℕ} the grand generating function As in the previous section, our main Proposition 1 implies Finally, we note that when w = 0 we have ϱ_{v,0}(ξ*_{v}) = 0, where ξ*_{v} = (Id_{V1} , …, Id_{Vn}) ∈ 𝔤, thus by Eq. 1 〈μ_{v,0}(v, 0), ξ*_{v}〉 = 0. Because 〈ξ_{v}, ξ*_{v}〉 = Σ v _{i}, the equation μ _{v,0}(v, w) = ξ_{v} has no solutions (for almost all q). This way we get that Φ(0) = 1, and so Eq. 8 yields Φ_{reg} = 1/Φ_{nil}(0), giving the result Therefore, it is enough to understand Φ_{nil}(w), which reduces to a simple linear algebra problem of determining a _{ϱv,w} (X_{λ1}, …, X_{λn}). Putting together everything yields the following:
Theorem 3.
Let 𝒬 = (𝒱, ℰ) be a quiver, with 𝒱 = {1, …, n} and ℰ ⊂ 𝒱 × 𝒱, with possibly multiple edges and loops. Fix a dimension vector w ∈ ℕ^{𝒱}. The Poincaré polynomials P_{t}(ℳ(v , w)) of the corresponding Nakajima quiver varieties are given by the generating function where d(v, w) = 2Σ_{(i,j)∈ℰ} v _{i} v _{j} + 2Σ_{i∈𝒱} v _{i}(w _{i} − v _{i}) is the dimension of ℳ(v, w), and T^{v} = ∏_{i∈𝒱} T_{i} ^{vi}. 𝒫(s) stands for the set of partitions ^{¶}
The notation for partitions is that of ref. 12.
of s ∈ ℕ. For two partitions λ = (λ_{1} , …, λ_{l}) ∈ 𝒫(s) and μ = (μ_{1}, …, μ_{m}) ∈ 𝒫(s) we define n(λ, μ) = Σ_{i,j} min(λ_{i}, μ_{j}), and if we write λ = (1^{m1(λ)}, 2^{m2(λ)}, …) ∈ 𝒫(s), then we can define l(λ) = Σ m_{i}(λ) = l the number of parts in λ. With this notation n(λ^{i}, (1^{wi})) = w _{i}l(λ^{i}) in the above formula.Remark:
This single formula encompasses a surprising amount of combinatorics and representation theory. When v = (1, …, 1), the Nakajima quiver variety is a toric hyperkähler variety, thus Eq. 9 gives a new formula for its Poincaré polynomial, which was given in Corollary 1. If additionally w = (1, 0, …, 0) then ℳ(v, w) is the toric quiver variety of ref. 5. Therefore, its Poincaré polynomial, which is the reliability polynomial ^{‖}
Incidentally, the reliability polynomial measures the probability of the graph remaining connected when each edge has the same probability of failure, a concept heavily used in the study of reliability of computer networks (13).
of the graph underlying the quiver (5) also can be read off from the above formula 9.When the quiver is just a single loop on one vertex, our formula 9 reproduces Theorem 2. When the quiver is of type A_{n} Nakajima (14) showed, that the Poincaré polynomials of the quiver variety are related to the combinatorics of Youngtableaux, while in the general ADE case, Lusztig (15) conjectured a formula for the Poincaré polynomial, in terms of formulae arising in the representation theory of quantum groups. When the quiver is starshaped recent work (T.H., unpublished work and T.H., E. Letellier, and F. Rodriguez–Villegas, unpublished work) calculates these Poincaré polynomials using the character theory of reductive Lie algebras over finite fields (2) and arrives at formulas determined by the Hall–Littlewood symmetric functions (12), which arose as the pure part of Macdonald symmetric polynomials (12). In the case when the quiver has no loops, Nakajima (16) gives a combinatorial algorithm for all Betti numbers of quiver varieties, motivated by the representation theory of quantum loop algebras. Finally, through the paper (17) of CrawleyBoevey and Van den Bergh, Poincaré polynomials of quiver varieties are related to the number of absolutely indecomposable representations of quivers in the work of Kac (18), which were eventually completely determined by Hua (19).
In particular, formula 9, when combined with results in refs. 19 and 11 and the Weyl–Kac character formula in the representation theory of Kac–Moody algebras (20), yields a simple proof of conjecture 1 of Kac (18). Consequently, formula 9 can be viewed as a qdeformation of the Weyl–Kac character formula (20).
A detailed study of the above generating function 9, its relationship to the wide variety of examples mentioned above, and details of the proofs of the results of this work will appear elsewhere.
Acknowledgments
I thank Ed Swartz for ref. 3, Fernando RodriguezVillegas for ref. 9, Balázs Szendröi for ref. 10, and Hiraku Nakajima for suggestions regarding Eq. 9 . This work was supported by a Royal Society University Research Fellowship, National Science Foundation Grant DMS0305505, an Alfred P. Sloan Research Fellowship, and a Summer Research Assignment of the University of Texas at Austin.
Footnotes
 ^{†}Email: hausel{at}maths.ox.ac.uk

Author contributions: T.H. performed research and wrote the paper.

↵ ‖ Incidentally, the reliability polynomial measures the probability of the graph remaining connected when each edge has the same probability of failure, a concept heavily used in the study of reliability of computer networks (13).

Conflict of interest statement: No conflicts declared.
 Abbreviations:
 ADHM,
 Atiyah–Drinfeld–Hitchin–Manin.
Abbreviation
 © 2006 by The National Academy of Sciences of the USA
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