Equivariant K-theory and wreath products

Weiqiang Wang^{∗}

Abstract

Given a finite group G and a G-space X, we show that a direct sum F_{G}(X) = ⨁_{n≥0} K_{G_{n}}(X^{n}) ⨂ ℂ admits a natural graded Hopf algebra structure, where G_{n} = G ∼ S_{n} is the wreath product of G and the symmetric group S_{n}. F_{G}(X) is shown to be isomorphic to a certain supersymmetric product in terms of K_{G}(X) ⨂ ℂ as a graded algebra. We prove that F_{G}(X) is a free λ-ring generated by K_{G}(X) ⨂ ℂ. We construct an infinite dimensional Heisenberg superalgebra acting irreducibly on F_{G}(X). As one of the several applications which we present, we compute the orbifold Euler characteristic e(X^{n}, G_{n}) in terms of e(X, G).

0 Introduction

Given a finite group G and a locally compact, Hausdorff, paracompact G-space X, the n-th direct product X^{n} admits a natural action of the wreath product G_{n} = G ∼ S_{n} which is a semi-direct product of the n-th direct product G^{n} of G and the symmetric group S_{n}. The main goal of the present paper is to study the equivariant topological K-theory K_{G_{n}}(X^{n}) for all n together. We will show that the direct sum

F_{G}(X) = ⨁_{n≥0} K_{G_{n}}(X^{n}) ⨂ ℂ

carries several wonderful structures.

More explicitly, we show that F_{G}(X) admits a natural Hopf algebra structure with a certain induction functor as multiplication and a certain restriction functor as comultiplication (cf. Theorem 2). When X is a point, K_{G_{n}}(X^{n}) is the Grothendieck ring R(G_{n}), we recover the standard Hopf algebra structure of ⨁_{n≥0} R(G_{n}) (cf. e.g. [M1, M2, Z]). A key lemma used here is a straightforward generalization to equivariant K-theory of a statement in the representation theory of finite groups concerning the restriction of an induced representation to a subgroup.

∗1991 Mathematics Subject Classification. Primary 19L47, 17B65.

We show that F_{G}(X) is a free λ-ring generated by K_{G}(X) ⨂ ℂ (cf. Proposition 4). We write down explicitly the Adams operations φ^{n}’s in F_{G}(X). Incidentally we also obtain an equivalent way of defining the Adams operations in K_{G}(X) ⨂ ℂ (not over ℤ) by means of the wreath products, generalizing a definition by Atiyah [A] in terms of the symmetric group in the ordinary (i.e. non-equivariant) K-theory setting. When X is a point we recover the λ-ring structure of ⨁_{n≥0} R(G_{n}) (cf. [M1]).

As a graded algebra F_{G}(X) has a simple description as a certain supersymmetric algebra in terms of K_{G}(X) ⨂ ℂ (cf. Theorem 3). The proof uses a theorem in [AS] and the structures of the centralizer group of an element in G_{n} and of the fixed point set of the action of a ∈ G_{n} on X^{n} which we work out in Sect. 1. In particular, this description indicates that F_{G}(X) has the size of a Fock space of a certain infinite-dimensional Heisenberg superalgebra which we will construct in terms of natural additive maps in K-theory (cf. Theorem 4).

Our results generalize Segal’s work [S2]. What Segal studied in [S2] is the space ⨁_{n≥0} K_{S_{n}}(X^{n}) ⨂ ℂ which is our special case when G is trivial and then G_{n} is the symmetric group S_{n}. The space ⨁_{n≥0} K_{S_{n}}(X^{n}) ⨂ ℂ was also considered earlier by Grojnowski [Gr]. In fact our present work grew out of an attempt to understand Segal’s insight [S2] and to fill in complete proofs of statements outlined in [S2]. Our main observation in this paper is that there is a natural way to add the group G into Segal’s scheme and this allows several interesting applications as discussed below. We present detailed proofs in our generalized (i.e. wreath products) setting. It is reasonable to expect that there is also a natural way to incorporate G into the remaining part of [S2].

By tensoring F_{G}(pt) with the group algebra of the Grothendieck ring R(G), we obtain the vertex algebra associated to the lattice R(G) [B, FLM]. When G is a Kleinian group, i.e. a finite subgroup of SU(2), this vertex algebra is closely related to the Frenkel-Kac-Segal vertex representation of an affine Kac-Moody Lie algebra, which can be viewed as a new form of McKay correspondence [Mc]. Detail along these lines will appear elsewhere.

An interesting case of our study is that X is the complex plane ℂ^{2} acted upon by a finite subgroup G of SU(2). Let ℂ^{2}/G^{}denote the corresponding ALE space, i.e. the minimal resolution of singularities of ℂ^{2}/G (cf. e.g. [N]). Via the McKay correspondence [Mc], we show that either K_{G_{n}}((ℂ^{2})^{n})⊗ ℂ or K_{S_{n}}((ℂ^{2}/G^{})^{n})⊗ ℂ has the same dimension as the homology group of the Hilbert scheme of n points on ℂ^{2}/G^{}[G]. It would be interesting to establish some direct isomorphisms among these equal-dimensional spaces. Our results indicate that the wreath product will play an important role in the study of the Hilbert scheme of n points on ℂ^{2}/G^{}. The fact that the dimension of \underline{K}_{S_{n}} (X^{n}) equals that of the homology group of the Hilbert scheme of n points on X holds for a more general surface X. Bezrukavnikov and Ginzburg [BG] have proposed a way to establish a direct isomorphism between these two spaces.

For a smooth manifold X acted upon by G, physicists [DHVW] introduced a notion of orbifold Euler characteristic e(X, G) in their study of string theory of orbifolds. We show that the orbifold Euler characteristic e(X^{n}, G_{n}) is uniquely determined by e(X, G) and n. In terms of a generating function, our formula reads (see Theorem 5):

∑_{n≥1} e(X^{n}, G_{n})q^{n} = \frac{1}{∏_{r=1}^{∞}(1 − q^{r})^{e(X,G)}} .  (1)

By putting G = 1 and thus e(X, G) = e(X), we recover a formula of Hirzebruch-Höfer [HH]. By using Eq. (1) and Göttsche’s formula [G], we show that X^{n}/G_{n} admits a resolution of singularities whose Euler characteristic coincides with the orbifold Euler characteristic e(X^{n}, G_{n}) assuming that X is a smooth quasi-projective surface and X/G has a resolution of singularities whose Euler characteristic is e(X, G).

The orbifold Euler characteristic is intimately related to the equivariant Ktheory (cf. [AS]).

From such a point of view, Eq. (1) can be derived as a consequence of Theorem 3. Content-Length: 27884 X-Lines: 705 Status: O

In this paper the language of equivariant K-theory is used. We should also mention the relevant construction of Heisenberg superalgebra in the homology groups of Hilbert scheme of ponts on a smooth quasi-projective surface, due to Nakajima (cf. [N] and references therein) and Grojnowski [Gr] independently. However the constructions and computations in terms of K-theory are simpler and work for more general spaces.

The plan of this paper is as follows. In Sect. 1 we give a presentation of the wreath product G_{n} and study its action on X^{n}. In Sect. 2 we construct a Hopf algebra structure on F_{G}(X). In Sect. 3 we give a description of F_{G}(X) as a graded algebra and study in detail the special case when X is the complex plane acted upon by a Kleinian group. In Sect. 4 we give a λ-ring structure on F_{G}(X). In Sect. 5 we construct the Heisenberg superalgebra which acts on F_{G}(X) irreducibly. In Sect. 6 we calculate the orbifold Euler characteristic e(X^{n}, G_{n}).

1 The wreath product and its action on X^{n}

Let G be a finite group. We denote by G_{∗} the set of conjugacy classes of G and R(G) the Grothendieck ring of G. R(G) ⨂_{ℤ} ℂ can be identified with the ring of class functions C(G) on G by taking the character of a representation. Denote by ζ_{c} the order of the centralizer of an element lying in the conjugacy class c in G. We define an inner product on C(G) as usual:

(χ|ψ) = \frac{1}{|G|} ∑_{g∈G} χ(g)\overline{ψ}(g), χ, ψ ∈ C(G).  (2)

Let G^{n} = G × … × G be the direct product of n copies of G. Denote by |G| the order of G, and by [g] the conjugacy class of g ∈ G. The symmetric group S_{n} acts on G^{n} by permuting the n factors: s(g_{1}, …, g_{n}) = (g_{s^{−1}(1)}, …, g_{s^{−1}(n)}). The wreath product G_{n} = G ∼ S_{n} is defined to be the semidirect product of G^{n} and S_{n}, namely the multiplication on G_{n} is given by (g, s)(h, t) = (g.s(h), st), where g, h ∈ G^{n}, s, t ∈ S_{n}. Note that G^{n} is a normal subgroup of G_{n} by identifying g ∈ G^{n} with (g, 1) ∈ G_{n}.

Given a = (g, s) ∈ G_{n} where g = (g_{1}, …, g_{n}). Write s ∈ S_{n} as a product of disjoint cycles: if z = (i_{1}, …, i_{r}) is one of them, the cycle-product g_{i_{r}}g_{i_{r−1}} … g_{i_{1}} of x corresponding to the cycle z is determined by g and z up to conjugacy. For each c ∈ G_{∗} and each integer r ≥ 1, let m_{r}(c) be the number of r-cycles in s whose cycle-product lies in c. Denote by ρ(c) the partition having m_{r}(c) parts equal to r (r ≥ 1) and ρ = (ρ(c))_{c∈G_{∗}} the corresponding partition-valued function on G_{∗}. Note that ||ρ|| := ∑_{c∈G_{∗}} |ρ(c)| = ∑_{c∈G_{∗},r≥1} rm_{r}(c) = n, where |ρ(c)| is the size of the partition ρ(c). Thus we have defined a map from G_{n} to P_{n}(G_{∗}), the set of partition-valued function ρ = (ρ(c))_{c∈G_{∗}} on G_{∗} such that ||ρ|| = n . The function ρ is called the type of a = (g, s) ∈ G_{n}. Denote P(G_{∗}) = ∑_{n≥0} P_{n}(G_{∗}).

Given a partition λ with m_{r} r-cycles (r ≥ 1), define z_{λ} = ∏_{r≥1} r^{m_{r}}m_{r}!. This is the order of the centralizer in S_{n} of an element of cycle-type λ. We shall denote by l(λ) = ∑_{r≥1} m_{r} the length of λ.

Given a partition-valued function ρ ∈ P(G_{∗}), we define l(ρ) = ∑_{c∈G_{∗}} l(ρ(c)) and

Z_{ρ} = ∏_{c∈G_{∗}} z_{ρ(c)}ζ_{c}^{l(ρ(c))}.

Denote by σ_{n}(c) the class function of G_{n} which takes value nζ_{c} at an n-cycle whose cycle-product lies in c ∈ G_{∗} and 0 otherwise. For ρ = {m_{r}(c)}_{c,r} ∈ P(G_{∗}), we define

σ^{ρ} = ∏_{c∈G_{∗},r≥1} σ_{r}(c)^{m_{r}(c)}.

We regard σ^{ρ} as the class function on G_{n} which takes value Z_{ρ} at elements of type ρ (where n = ||ρ||) and 0 elsewhere.

We formulate some well-known facts below (cf. e.g. [M2]) which will be needed later.

Proposition 1 Two elements in G_{n} are conjugate to each other if and only if they have the same type. The order of the centralizer in G_{n} of an element of type ρ is Z_{ρ}.

We want to calculate the centralizer Z_{G_{n}}(a) of a ∈ G_{n}. First we consider the typical case that a has one n-cycle. As the centralizers of conjugate elements are conjugate subgroups, we may assume that a is of the form a = ((g, 1, …, 1), τ) by Proposition 1, where τ = (12 … n). Denote by Z_{G}^{Δ}(g), or Z_{G}^{Δ_{n}}(g) when it is necessary to specify n, the following diagonal subgroup of G^{n} (and thus a subgroup of G_{n}):

Z_{G}^{Δ}(g) = {((h, …, h), 1) ∈ G^{n} | h ∈ Z_{G}(g)} .

The following lemma follows from a direct computation.

Lemma 1 The centralizer Z_{G_{n}}(a) of a in G_{n} is equal to the product Z_{G}^{Δ}(g) · ⟨a⟩, where ⟨a⟩ is the cyclic subgroup of G_{n} generated by a. Moreover, a^{n} ∈ Z_{G}^{Δ}(g) and |Z_{G_{n}}(a)| = n|Z_{G}(g)|.

Given a generic element a = (g, s) ∈ G_{n} of type ρ = (ρ(c))_{c∈G_{∗}}, where ρ(c) has m_{r}(c) r-cycles (r ≥ 1). By Proposition 1, we may assume (by taking a conjugation if necessary) that the m_{r}(c) r-cycles are of the form

g_{ur}(c) = ((g, 1, …, 1), (i_{u1}, …, i_{ur})), 1 ≤ u ≤ m_{r}(c), g ∈ c.

Denote g_{r}(c) = ((g, 1, …, 1), (12 … r)). Throughout the paper, ∏_{c,r} is understood as the product ∏_{c∈G_{∗},r≥1}.

Lemma 2 The centralizer Z_{G_{n}}(a) of a ∈ G_{n} is isomorphic to a direct product of the wreath products

∏_{c,r} (Z_{G_{r}}(g_{r}(c)) ∼ S_{m_{r}(c)}) .  (3)

Furthermore Z_{G_{r}}(g_{r}(c)) is isomorphic to Z_{G}^{Δ_{r}}(g) · ⟨g_{r}(c)⟩.

Proof. It follows from the first part of Lemma 1 that the centralizer Z_{G_{n}}(a) should contain a certain subgroup naturally isomorphic to (3). By the second part of Lemma 1 we can count that the order of (3) is equal to Z_{ρ}. The lemma now follows by comparing with the order of Z_{G_{n}}(a) given in Proposition 1. □

We will use ⋆ to denote the multiplication in C(G_{n}) which corresponds to the tensor product in R(G_{n}). We denote by \underline{n} the trivial representation of G_{n}, and \underline{1^{n}} the sign representation of G_{n} in which G^{n} acts trivially and S_{n} acts by ±1 depending on a permutation is even or odd. By abuse of notations, we also use the same symbols to denote the corresponding characters as well. The following lemma follows easily from the definitions.

Lemma 3 1) Given ρ, ρ̃ ∈ P_{n}(G_{∗}), σ^{ρ} ⋆ σ_{ρ} = δ_{ρ,_{ρ}}Z_{ρ}σ^{ρ}. In particular,

\underline{n} = ∑_{||ρ||=n} Z_{ρ}^{−1}σ^{ρ},  (4)
\underline{1^{n}} = ∑_{||ρ||=n}(−1)^{n−l(ρ)}Z_{ρ}^{−1}σ^{ρ}.  (5)

2) (σ^{ρ}|σ_{ρ}) = δ_{ρ,_{ρ}}Z_{ρ}. In other words, σ^{ρ} takes value Z_{ρ} at the elements in G_{n} of type ρ and 0 elsewhere.

For a G-space X, we define an action of G_{n} on X^{n} as follows: given a = (g_{1}, …, g_{n}), s), we let

a.(x_{1}, …, x_{n}) = (g_{1}x_{s^{−1}(1)}, …, g_{n}x_{s^{−1}(n)})  (6)

where x_{1}, …, x_{n} ∈ X. Next we want to determine the fixed point set (X^{n})^{a} for a ∈ G_{n}. Let us first calculate in the typical case a = ((g, 1, …, 1), τ) ∈ G_{n}. Note that the centralizer group Z_{G}(g) preserves the g-fixed point set X^{g}.

Lemma 4 The fixed point set is

(X^{n})^{a} = {(x, …, x) ∈ X^{n} | x = gx}

which can be naturally identified with X^{g}. The action of Z_{G_{n}}(a) on (X^{n})^{a} can be identified canonically with that of Z_{G}(g) on X^{g} together with the trivial action of the cyclic group ⟨a⟩ (cf. Lemma 1). Thus

(X^{n})^{a}/Z_{G_{n}}(a) ≈ X^{g}/Z_{G}(g).

Proof. Let (x_{1}, …, x_{n}) be in the fixed point set (X^{n})^{a}. By Eq. (6) we have

(x_{1}, x_{2}, x_{3}, …, x_{n}) = a.(x_{1}, x_{2}, x_{3}, …, x_{n})
= (gx_{n}, x_{1}, x_{2}, …, x_{n−1}).

So all x_{i}(i = 1, …, n) are equal to, say x, and gx = x. The remaining statements follow from Lemma 1. □

All Z_{G}(g) are conjugate and all X^{g} are homeomorphic to each other for different representatives g in a fixed conjugacy class c ∈ G_{∗}. Also the orbit space X^{g}/Z_{G}(g) can be identified with each other by conjugation for different representatives of g in c ∈ G_{∗}. We make a convention to denote Z_{G}(g) (resp. X^{g}, X^{g}/Z_{G}(g)) by Z_{G}(c) (resp. X^{c}, X^{c}/Z_{G}(c)) by abuse of notations when the choice of a representative g in c is immaterial.

Lemma 5 Retain the notations in Lemma 2. The fixed point set (X^{n})^{a} can be naturally identified with ∏_{c,r}(X^{c})^{m_{r}(c)}. Furthermore the orbit space (X^{n})^{a}/Z_{G_{n}}(a) can be naturally identified with

∏_{c,r} S^{m_{r}(c)} (X^{c}/Z_{G}(c))

where S^{m}(·) denotes the m-th symmetric product. Proof. The first part easily follows from Lemma 4. By Lemma 2 and Lemma 4, the action of Z_{G_{n}}(a) on (X^{n})^{a} can be naturally identified with that of

∏_{c,r} ((Z_{G}^{Δ_{r}}(g) · ⟨g_{r}(c)⟩) ∼ S_{m_{r}(c)})

on ∏_{c,r}(X^{c})^{m_{r}(c)} where S_{m_{r}(c)} acts by permutation and ⟨g_{r}(c)⟩ acts on X^{c} trivially. Thus the second part of the lemma follows. □

2 The Hopf algebra structure on F_{G}(X)

Given a compact Hausdorff G-space X, we recall [A1, S1] that K_{G}^{0}(X) is the Grothendieck group of G-vector bundles over X. One can define K_{G}^{1}(X) in terms of the K^{0} functor and a certain suspension operation, and one puts

K_{G}(X) = K_{G}^{0}(X) ⨁ K_{G}^{1}(X).

The tensor product of vector bundles gives rise to a multiplication on K_{G}(X) which is super (i.e. ℤ_{2}-graded) commutative. In this paper we will be only concerned about the free part K(X) ⊗ ℂ, which will be denoted by \underline{K}_{G}(X) subsequently. We denote by dim K_{G}^{i}(X)(i = 0, 1) the dimension of K_{G}^{i}(X) ⊗ ℂ.

If X is locally compact, Hausdorff and paracompact G-space, take the one-point compactification X^{+} with the extra point ∞ fixed by G. Then we define K_{G}^{0}(X) to be the kernel of the map

K_{G}^{0}(X^{+}) ⟶ K_{G}^{0}({∞})

induced by the inclusion map {∞} ↪ X^{+}. It is clear that this definition is equivalent to the earlier one when X is compact. We also define K_{G}^{1}(X) = K_{G}^{1}(X^{+}).

Note that K_{G}(pt) is isomorphic to the Grothendieck ring R(G) and \underline{K}_{G}(pt) is isomorphic to the ring C(G) of class functions on G. The bilinear map ⋆ induced from the tensor product

K_{G}(pt) ⨂ K_{G}(X) ⟶ K_{G}(X)

gives rise to a natural \underline{K}_{G}(pt)-module structure on \underline{K}_{G}(X). Thus \underline{K}_{G}(X) naturally decomposes into a direct sum over the set of conjugacy classes G_{∗} of G. The following theorem [AS] (also cf. [BC] ) gives a description of each direct summand.

Theorem 1 There is a natural ℤ_{2}-graded isomorphism

ϕ : \underline{K}_{G}(X) ⟶ ⨁_{[g]} \underline{K} (X^{g}/Z_{G}(g)) .

Given c ∈ G_{∗}, we denote by σ_{c} the class function which takes value ζ_{c} at an element of G lying in the conjugacy class c and 0 otherwise. Then an element in \underline{K}_{G}(X) can be written as of the form ∑_{c∈G_{∗}} ξ_{c}σ_{c}, where ξ_{c} ∈ \underline{K} (X^{g}/Z_{G}(g)) . More explicitly the isomorphism ϕ is defined as follows when X is compact: if E is a G-vector bundle over X its restriction to X^{g} is acted by g with base points fixed. Thus E|_{X^{g}} splits into a direct sum of subbundles E_{μ} consisting of eigenspaces of g fiberwise for each eigenvalue μ of g. Z_{G}(g) acts on X^{g} and one may check that ∑_{μ} μE_{μ} indeed lies in the Z_{G}(g)-invariant part of \underline{K}_{G}(X). Put

ϕ_{g}(E) = ∑_{μ} μE_{μ} ∈ \underline{K}^{0} (X^{g}/Z_{G}(g)) = \underline{K}^{0} (X^{g})^{Z_{G}(g)} .

The isomorphism ϕ on the K^{0} part is given by ϕ = ⨁_{[g]∈G_{∗}} ϕ_{g}. Then one easily extends the isomorphism ϕ to K^{1} as K_{G}^{1}(X) can be identified with the kernel of the map from K_{G}^{0}(X ×S^{1}) to K_{G}^{0}(X) given by the inclusion of a point in S^{1}. When X is a point the isomorphism ϕ becomes the map from a representation of G to its character. By some standard arguments using compact pairs [A1], the isomorphism remains valid for a locally compact, Hausdorff and paracompact G-space.

Lemma 6 Given a finite group G, a subgroup H of G, and a G-space X. There is a natural induction functor Ind = Ind_{H}^{G} : K_{H}(X) ⟶ K_{G}(X). In particular when X is a point, the functor Ind_{H}^{G} reduces to the familiar induction functor of representations.

Proof. Note that there is a G-equivariant isomorphism

G ×_{H} X ⟶^{Iso} G/H × X  (7)

by sending (g, x) ∈ G ×_{H} X to (gH, gx). We remark that although both sides of (7) remain well-defined for a H-spcase X without a G-action, the map Iso makes sense only for a G-space X. Denote by p : G ×_{H} X ⟶ X the composition of the projection G/H × X to X with the isomorphism (7). As is well known [S1], one has a natural isomorphism

K_{H}(X) ⟶ K_{G}(G ×_{H} X)

by sending an H-equivariant vector bundle V on X to the G-equivariant vector bundle G×_{H} V. The composition p∘π of the projection p : G×_{H} X ⟶ X with the bundle map π : G×_{H} V ⟶ G×_{H} X sends (g, v) to (g, π(v)). One easily check that this gives rise to a well-defined G-equivariant vector bundle on X, which induces the induction functor Ind_{H}^{G} : K_{H}(X) ⟶ K_{G}(X). □

We denote by Res_{H}^{G} (or Res_{H}, or even Res, if there is no ambiguity) the restriction functor from K_{G}(X) to K_{H}(X) by regarding a G-equivariant vector bundle as an H-equivariant vector bundle. Denote

F_{G}(X) = ⨁_{n≥0} \underline{K}_{G_{n}}(X^{n}), F_{G}^{q}(X) = ⨁_{n≥0}q^{n}\underline{K}_{G_{n}}(X^{n})

where q is a formal variable counting the graded structure of F_{G}(X). We introduce the notion of q-dimension:

dim_{q} F_{G}(X) = ∑_{n≥0} q^{n} dim K_{G_{n}}(X^{n}).

Define a multiplication · on F_{G}(X) by a composition of the induction map and the Künneth isomorphism k:

\underline{K}_{G_{n}}(X^{n}) ⨂ \underline{K}_{G_{m}}(X^{m}) ⟶^{k} \underline{K}_{G_{n}×G_{m}}(X^{n+m}) \underrightarrow{Ind} \underline{K}_{G_{n+m}}(X^{n+m}).  (8)

We denote by 1 ∈ K_{G_{0}}(X^{0}) ≈ ℂ the unit.

On the other hand we can define a comultiplication Δ on F_{G}(X), given by a composition of the inverse of the Künneth isomorphism and the restriction from G_{n} to G_{k} × G_{n−k}:

\underline{K}_{G_{n}}(X^{n}) ⟶ ⨁_{m=0}^{n} \underline{K}_{G_{m}×G_{n−m}}(X^{n}) ⟶^{k^{−1}} ⨁_{m=0}^{n} \underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{n−m}}(X^{n−m}).

We define the counit ϵ : F_{G}(X) ⟶ ℂ by sending \underline{K}_{G_{n}}(X^{n}) (n > 0) to 0 and 1 ∈ K_{G_{0}}(X^{0}) ≈ ℂ to 1.

Theorem 2 (F_{G}(X), ·, Δ, 1, ϵ) is a graded Hopf algebra.

To prove Theorem 2, we will need some preparation. Given two subgroups H and L of a finite group Γ and a Γ-space Y, and let V be an H-equivariant vector bundle on Y. We denote the action of H on V by ρ. Choose a set of representatives S of the double coset H\Γ/L. H_{s} = sHs^{−1} ∩ L is a subgroup of L for s ∈ S. We denote by V_{s} the H_{s}-equivariant vector bundle on Y which is the same as V as a vector bundle and has the conjugated action

ρ^{s}(x) = ρ(s^{−1}xs), x ∈ H_{s}.  (9)

Lemma 7 Res_{L}Ind_{H}^{Γ}V is isomorphic to the direct sum of the L-equivariant vector bundles Ind_{H_{s}}^{L}V_{s} for all s ∈ H\Γ/L.

One easily shows that one can extend V in Lemma 7 to the whole K_{H}(Y). In the case Y = pt, an H-equivariant vector bundle is just an H-module, and the induction and restriction functors become the more familiar ones in representation theory. In such a case, the above lemma is standard (cf. e.g. [Ser], Proposition 2.2). In view of our construction of the induction functor and the restriction functor, the proof of Lemma 7 is essentially the same as in the case X = pt which we refer to [Ser] for a proof.

Proof of Theorem 2. We will show below that the comultiplication Δ is an algebra homomorphism. The other Hopf algebra axioms are easy to check.

We apply Lemma 7 to the case Y = X^{N}, L = G_{m} × G_{n}, H = G_{l} × G_{r}, and Γ = G_{N}, where l + r = m + n = N. In this case the double coset H\Γ/L is isomorphic to (S_{l} × S_{r})\S_{N}/(S_{m} × S_{n}) since G_{N} = G^{N} · S_{N} and G_{l} × G_{r} = G^{N} · (S_{l} × S_{r}). Furthermore (S_{l} × S_{r})\S_{N}/(S_{m} × S_{n}) is parametrized by the 2 × 2 matrices

[ \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ]  (10)

satisfying

a_{ij} ∈ ℤ_{+}, i, j = 1, 2,
a_{11} + a_{12} = m, a_{21} + a_{22} = n,  (11)
a_{11} + a_{21} = l, a_{12} + a_{22} = r.

We denote by M the set of all the 2 × 2 matrices of the form (10) satisfying the conditions (11).

Given E ∈ \underline{K}_{G_{l}}(X^{l}), F ∈ \underline{K}_{G_{r}}(X^{r}), we calculate by using Lemma 7 as follows:

Res_{(m,n)}Ind_{(l,r)}^{N}(E ⊠ F) = ⨁_{A∈M} Ind_{A}^{(m,n)}(1_{a_{11}} ⊗ T^{(a_{12},a_{21})} ⊗ 1_{a_{22}})(Res_{A′}(E ⊠ F))
= ⨁_{A∈M} Ind_{(a_{11},a_{12})}^{m}F_{1} ⊠ Ind_{(a_{21},a_{22})}^{n}F_{2}.  (12)

Here the superscript or subscript (a, b) is a short-hand notations for G_{a} × G_{b}. 1_{a} stands for the identity operator from \underline{K}_{G_{a}}(X^{a}) to itself, and Ind_{(a,b)}^{a+b} for the induction functor from \underline{K}_{G_{a}×G_{b}}(X^{a+b}) to \underline{K}_{G_{a+b}}(X^{a+b}). T^{(a,b)} denotes the canonical functor from \underline{K}_{G_{a}}(X^{a})⊗\underline{K}_{G_{b}}(X^{b}) to \underline{K}_{G_{b}}(X^{b})⊗\underline{K}_{G_{a}}(X^{a}) by switching the factors with an appropriate sign coming from the ℤ_{2} grading of K-theory. Given A ∈ M of the form (10), the A in the expressions Res_{A}, Ind_{A} etc stands for G_{A} ≡ G_{a_{11}} × G_{a_{12}} × G_{a_{21}} × G_{a_{22}} while A′ for G_{A′} ≡ G_{a_{11}} × G_{a_{21}} × G_{a_{12}} × G_{a_{22}}. We wrote (1 ⊗ T^{(a_{12},a_{21})} ⊗ 1)(Res_{A′}(E ⊠ F)) as F_{1} ⊠ F_{2} instead of a direct sum of the form F_{1} ⊠F_{2} in order to simplify notations, with F_{i} (i = 1, 2) the corresponding elements in K_{G_{a_{i1}}×G_{a_{i2}}}(X^{a_{11}+a_{12}}).

Now it is straightforward to check that the statement that Δ is an algebra homomorphism is just a reformulation of the identity obtained by summing Eq. (12) over all possible (m, n) with m + n = N. □

3 A description of F_{G}(X) as a graded algebra

In this section, we give an explicit description of F_{G}(X) as a graded algebra which in particular tells us the dimension of \underline{K}_{G_{n}}(X^{n}).

Theorem 3 As a (ℤ_{+} × ℤ_{2})-graded algebra F_{G}^{q}(X) is isomorphic to the supersymmetric algebra S (⨁_{n≥1} q^{n}\underline{K}_{G}(X)). In particular,

dim_{q} F_{G}(X) = \frac{∏_{r≥1}(1 + q^{r})^{dim K_{G}^{1}(X)}}{∏_{r≥1}(1 − q^{r})^{dim K_{G}^{0}(X)}} .

The supersymmetric algebra here is equal to the tensor product of the symmetric algebra S (⨁_{n≥1} q^{n}\underline{K}_{G}^{0}(X)) and the exterior algebra Λ (⨁_{n≥1} q^{n}\underline{K}_{G}^{1}(X)).

Proof. Take a ∈ G_{n} of type ρ = {m_{r}(c)}_{c,r} as in Sect. 1. By Lemma 2 and Lemma 5 and the Künneth formula, we have

K((X^{n})^{a}/Z_{G_{n}}(a)) ≈ ⨂_{c∈G_{∗},r≥1} ((K(X^{c})^{Z_{G}(c)})^{⨂ m_{r}(c)})^{S_{mr(c)}}
≈ ⨂_{c∈G_{∗},r≥1} S^{m_{r}(c)}(K(X^{c}/Z_{G}(c))).  (13)

Thus if we take a summation of Eq. (13) over all conjugacy classes of G_{n} and all over n ≥ 0, we obtain:

F_{G}^{q}(X) ≈ ⨁_{n≥0} ⨁_{\{m_{r}(c)\}_{c,r}∈P_{n}(G_{∗})} q^{n} ⨂_{c,r} S^{m_{r}(c)}(\underline{K}(X^{c}/Z_{G}(c)))  by Theorem 1,
= ⨁_{\{m_{r}(c)\}_{c,r}∈P(G_{∗})} ⨂_{c,r} S^{m_{r}(c)} (q^{r}\underline{K}(X^{c}/Z_{G}(c)))

= ⨁_{\{m_{r}\}_{r}} ⨂_{r≥1} S^{m_{r}} ( ⨁_{c∈G_{∗}} q^{r}\underline{K}(X^{c}/Z_{G}(c))) by letting m_{r} = ∑_{c∈G_{∗}} m_{r}(c),
= ⨁_{\{m_{r}\}_{r}} ⨂_{r≥1} S^{m_{r}}(q^{r}\underline{K}_{G}(X))  by Theorem 1,

= S (⨁_{r≥1} q^{r}\underline{K}_{G}(X)) .

The statement concerning dim_{q} F_{G}(X) is an immediate consequence. □

Remark 1 1) Theorem 3 in the case when G is the trivial group (and so G_{n} = S_{n}) is due to Segal [S2].

2) If G acts on X freely, so does G^{n} on X^{n}. Then we have the isomorphism K(X/G) ≈ K_{G}(X). Note that G^{n} is a normal subgroup of the wreath product G_{n} and the quotient G_{n}/G^{n} is isomorphic to S_{n}. By Proposition 2.1 in Segal [S1] we see that

K_{G_{n}}(X^{n}) ≈ K_{G_{n}/G^{n}}(X^{n}/G^{n}) = K_{S_{n}}((X/G)^{n}).  (14)

Therefore Theorem 3 follows from the special case G = 1 of Theorem 3 (applying to X/G) and Eq. (14).

3) When X is a point, F_{G}(pt) = ⨁_{n≥0}R(G_{n}), and σ^{ρ}, ρ ∈ P(G_{∗}) form a linear basis for F_{G}(pt) (cf. [M2]). In particular,

dim_{q} F_{G}(pt) = ∏_{r≥1}(1 − q^{r})^{−|G_{∗}|}.

4) One may reformulate Theorem 3 in terms of the de-localized equivariant cohomology [BBM, BC] via the Chern character.

We consider a special case in detail. Let X be the complex plane ℂ^{2} acted upon by a Kleinian group G, i.e. a finite subgroup of SU(2). It is well known that G is one of the members in the following list: the cyclic, binary dihedral, tetrahedral, octahedral and icosahedral groups. Via the McKay correspondence [Mc], there is a one-to-one correspondence between the finite subgroups of SU(2) and the Dynkin diagrams of simply-laced types A_{n}, D_{n}, E_{6}, E_{7} and E_{8}. Let us denote by 𝔤 the simple Lie algebra corresponding to G.

From the exact correspondence we know that the rank of 𝔤 is |G_{∗}|−1. Content-Length: 34187 X-Lines: 919 Status: O

The quotient ℂ^{2}/G has an isolated simple singularity at 0. There exists a minimal resolution ℂ^{2}/G^{}of ℂ^{2}/G well known as ALE-spaces (cf. e.g. [N]). It is known that the second homology group H_{2}(ℂ^{2}/G^{}, ℤ) is isomorphic to the root lattice of 𝔤, cf. e.g. [N]. In particular

dim K \binom{}{ℂ^{2}/G} = dim H \binom{}{ℂ^{2}/G} = |G_{∗}|.  (15)

So if we apply Theorem 3 to the case ℂ^{2}/G^{}with a trivial group action, we have

∑_{n≥0} dim K_{S_{n}} (ℂ^{2}/G^{}^{n}) q^{n} = ∏_{r≥1}(1 − q^{r})^{−|G_{∗}|}.

On the other hand, by the Thom isomorphism [S1] we have

K_{G_{n}}(X^{n}) ≈ K_{G_{n}}(pt) = R(G_{n}).

It follows from the part 3) of Remark 1 and Eq. (15) that

∑_{n≥0} dim K_{G_{n}} (ℂ^{2n}) q^{n} = ∏_{r≥1}(1 − q^{r})^{−|G_{∗}|}.

We can obtain another numerical coincidence from a somewhat different point of view as follows. For a general quasi-projective smooth surface Y, a well-known result of Fogarty says that the Hilbert scheme of n points on Y, denoted by Y^{[n]}, is a smooth 2n dimensional manifold. The Betti numbers of Y^{[n]} were computed by Göttsche [G]. In particular, Göttsche’s formula yields the dimension of the homology group of Y^{[n]}:

∑_{n≥0} dim H(Y^{[n]})q^{n} = ∏_{r≥1}(1 − q^{r})^{− dim H(Y)}.  (16)

We apply Eq. (16) to the case Y = ℂ^{2}/G^{}. It follows from Eq. (15) that

∑_{n≥0} dim H(ℂ^{2}/G^{}^{[n]})q^{n} = ∏_{r≥1}(1 − q^{r})^{−|G_{∗}|}.

Therefore we have proved the following.

Proposition 2 The spaces K_{G_{n}} (ℂ^{2n}) ⨂ ℂ, K_{S_{n}}(ℂ^{2}/G^{}^{n}) ⨂ ℂ and H(ℂ^{2}/G^{}^{[n]}) have the same dimension.

It is an interesting and non-trivial problem to establish direct isomorphisms among these equal-dimensional spaces.

Remark 2 The fact that the dimension of K_{S_{n}} (X^{n}) equals that of the homology group of the Hilbert scheme of n points of X for a more gerenal surface X holds by the same argument above.

Proposition 2 can be generalized as follows. Assume that X is a quasi-projective surface acted upon by G and there exists a smooth resolution of singularities X/G^{} of X/G such that the dimension of K_{G}^{i}(X) (i = 0, 1) equals that of the even (resp. odd) dimensional homology group of X/G^{}. The assumptions here may not be so artificial as it seems to be if one reinterprets the dimension of K_{G}^{0}(X) in terms of the orbifold Euler number (say in the case when K_{G}^{1}(X) vanishes), cf. Sect. 6. Then we conclude that the dimension of K_{G_{n}} (X^{n}) is equal to that of the homology group of the Hilbert scheme of n points on X/G^{}.

We believe that this is just a first indication of intriguing relations between the Hilbert scheme of n points on X/G^{}and the wreath product G_{n}. At the moment we can formulate some explicit conjectures but are unable to prove it in general. We hope to elaborate on this in another occasion. When G is trivial, it reduces to well-known relations between the Hilbert scheme of n points and the symmetric group S_{n} (cf. [N]).

4 The λ-ring structure on F_{G}(X)

Let us denote by c_{n} the conjugacy class in G_{n} which has the type of an n-cycle and whose cycle product lies in the conjugacy class c ∈ G_{∗}. We consider the following diagram of K-theory maps:

\underline{K}_{G}(X) ⟶^{⊠n} \underline{K}_{G_{n}}(X^{n}) ⟶^{ϕ_{n}} ⨁_{[a]∈(G_{n})_{∗}} \underline{K} ((X^{n})^{a}/Z_{G_{n}}(a))

ι^{⇄^{pr}}  ⨁_{c∈G_{∗}} \underline{K} ((X^{n})^{c_{n}}/Z_{G_{n}}(c_{n})) ⟶^{ϑ} ⨁_{c∈G_{∗}} \underline{K} (X^{c}/Z_{G}(c))
⟵^{ϕ} \underline{K}_{G}(X).

Given a G-equivariant vector bundle V, we define a G_{n}-action on the n-th outer tensor product V^{⊠n} by letting

((g_{1}, …, g_{n}), s).v_{1} ⊗ … ⊗ v_{n} = g_{1}v_{s^{−1}(1)} ⊗ … ⊗ g_{n}v_{s^{−1}(n)}  (17)

where g_{1}, …, g_{n} ∈ G, s ∈ S_{n}. Clearly V^{⊠n} endowed with such a G_{n} action is a G_{n}- equivariant vector bundle over X^{n}. Sending V to V^{⊠n} gives rise to the K-theory map ⊠n. ϕ_{n} is the isomorphism in Theorem 1 when applying to the case X^{n} with the action of G_{n}. pr is the projection to the direct sum over the conjugacy classes of G_{n} which are of the type of an n-cycle while ι denotes the inclusion map. ϑ denotes the natural identification given by Lemma 5. Finally the last map ϕ is the isomorphism given in Theorem 1.

We shall now define a λ-ring structure on F_{G}(X). It suffices to define the Adams operations ψ^{n} on F_{G}(X). We also introduce several other K-theory operations which will be needed later.

Definition 1 We define the following composition maps:

ψ^{n} := nϕ^{−1} ∘ ϑ ∘ pr ∘ ϕ_{n} ∘ ⊠n : \underline{K}_{G}(X) ⟶ \underline{K}_{G}(X),
φ^{n} := nϕ_{n}^{−1} ∘ ι ∘ pr ∘ ϕ_{n} ∘ ⊠n : \underline{K}_{G}(X) ⟶ \underline{K}_{G_{n}}(X^{n}),
ch_{n} := nϕ^{−1} ∘ ϑ ∘ pr ∘ ϕ_{n} : \underline{K}_{G_{n}}(X^{n}) ⟶ \underline{K}_{G}(X),
ω_{n} := ϕ_{n}^{−1} ∘ ι ∘ ϑ^{−1} ∘ ϕ : \underline{K}_{G}(X) ⟶ \underline{K}_{G_{n}}(X^{n}).

We list some properties of these K-theory maps whose proof is straightforward.

Proposition 3 The following identities hold:

ch_{n} ∘ ω_{n} = n Id, ω_{n} ∘ ψ^{n} = φ^{n}, ch_{n} ∘ φ^{n} = nψ^{n}.

Recall that σ_{n}(c) is the class function of G_{n} which takes value nζ_{c} at an n-cycle whose cycle-product lies in c ∈ G_{∗} and 0 otherwise.

Lemma 8 1) φ^{n}(V) = ∑_{c∈G_{∗}} ζ_{c}^{−1}V^{⊠n} ⋆ σ_{n}(c).

2) Both ψ^{n} and φ^{n} are additive K-theory maps.

Proof. Note that the order of the centralizer of an element lying in the conjugacy class c_{n} is equal to nζ_{c}. Part 1) now follows from the definition of φ^{n} and Lemma 3.

Given V, W two G-equivariant vector bundles, we will check that

φ^{n}([V] − [W]) = φ^{n}([V]) − φ^{n}([W])

where [V] denotes the corresponding element of V in K_{G}(X). (In general we use V itself to denote the corresponding element in K_{G}(X) by abuse of notation). The proof that φ^{n}([V] + [W]) = φ^{n}([V]) + φ^{n}([W]) is the same.

Recall that we can define the equivariant K-theory in terms of complexes of G-equivariant vector bundles [S1]. Denote by U the graded vector bundle with U_{0} = V and U_{1} = W. Then

U^{⊠n} = ∑_{j=0}^{n}(−1)^{j}Ind_{G_{n−j}×G_{j}}^{G_{n}}[V^{⊠(n−j)} ⊠ W^{⊠j}] ∈ K_{G_{n}}(X^{n}).

Here V^{⊠(n−j)} endows the standard G_{n−j}-action given by substituting n with n − j in Eq. (17), and G_{j} acts on W^{⊠j} by the tensor product of the standard G_{j}-action with the sign representation of G_{j}.

For 0 < j < n, we may write

Ind_{G_{n−j}×G_{j}}^{G_{n}}[V^{⊠(n−j)} ⊠ W^{⊠j}] = ∑_{ρ∉{c_{n},c∈G_{∗}}^{ρ∈(Gn)∗}} a_{ρ} ⋆ σ^{ρ}

for some a_{ρ} ∈ \underline{K}_{G_{n}}(X^{n}). It follows that φ^{n}([V^{⊠(n−j)} ⊠ W^{⊠j}]) = 0 for 0 < j < n by part 1) and Lemma 3. Since the sign of a j-cycle is (−1)^{j−1}, we have

φ^{n}([V] − [W]) = φ^{n}([V]) + (−1)^{n}(−1)^{n−1}φ^{n}([W])
= φ^{n}([V]) − φ^{n}([W]).

It follows from the definition of φ^{n} that the additivity for φ^{n} is equivalent to the additivity for pr ∘ ϕ_{n} ∘ ⊠n. As the composition of ϕ^{−1}, ϑ and pr ∘ ϕ_{n} ∘ ⊠n, ψ^{n} is also additive since ϕ and ϑ are linear isomorphisms. □

ψ^{n}’s are the Adams operations on \underline{K}_{G}(X), giving rise to the λ-ring structure on \underline{K}_{G}(X). Theorem 3 ensures us that F_{G}(X) as a λ-ring is free and generated by \underline{K}_{G}(X).

Proposition 4 F_{G}(X) is a free λ-ring generated by K_{G}(X) ⨂ ℂ, with φ^{n}’s as the Adams operations.

Remark 3 If X is a point, then K_{G_{n}}(pt) = R(G_{n}) and F_{G}(pt) = ⨁_{n≥0} R(G_{n}). Our result reduces to the fact that F_{G}(pt) is a free λ-ring generated by G_{∗} [M1]. In the case when G = 1, the proposition was due to Segal [S2].

Denote by F̂_{G}^{q}(X) the completion of F_{G}^{q}(X) which allows formal infinite sums. Given V ∈ \underline{K}_{G}(X), we introduce H(V, q), E(V, q) ∈ F̂_{G}^{q}(X) as follows:

H(V, q) = ⨁_{n≥0} q^{n}V^{⊠n},
E(V, q) = ⨁_{n≥0} q^{n}V^{⊠n} ⋆ \underline{1^{n}}.

Proposition 5 One can express H(V, q) and E(V, q) in terms of φ^{r}(V) as follows:

H(V, q) = exp (∑_{r>0} \frac{1}{r}φ^{r}(V)q^{r}) ,  (18)
E(V, −q) = exp (− ∑_{r>0} \frac{1}{r}φ^{r}(V)q^{r}) .

Proof. We shall prove Eq. (18) by using Eq. (4). The formula for E(V, −q) can be similarly obtained by using Eq. (5).

H(V, q) = ⨁_{n≥0} q^{n}V^{⊠n} ⋆ \underline{n}

= ⨁_{n≥0} q^{n}V^{⊠n} ⋆ ( ∑_{||ρ||=n} Z_{ρ}^{−1}σ^{ρ})
= ⨁_{n≥0} ∑_{||ρ||=n} (Z_{ρ}^{−1}q^{n}V^{⊠n} ⋆ σ^{ρ})

=^{(⋆)} ∏_{c∈G_{∗},r≥1} \frac{1}{m_{r}(c)!} (\frac{1}{r}q^{r}ζ_{c}^{−1}V^{⊠r} ⋆ σ_{r}(c))^{m_{r}(c)}

= exp (∑_{r≥1} \frac{1}{r}q^{r} ∑_{c∈G_{∗}} ζ_{c}^{−1}V^{⊠r} ⋆ σ_{r}(c))  by Lemma 8,

= exp (∑_{r≥1} \frac{1}{r}q^{r}φ^{r}(V)) .

Here the equation (⋆) is understood by means of the multiplication in F_{G}(X) given by the composition (8). □

Corollary 1 The λ-operations λ^{n} on the λ ring F_{G}(X) sends V ∈ \underline{K}_{G}(X) to V^{⊠n} ⋆ \underline{1^{n}}.

Combining with the additivity of φ^{r}, we have the following.

Corollary 2 The following equations hold for V, W ∈ \underline{K}_{G}(X):

H(−V, q) = E(V, −q)
H(V ⨁ W, q) = H(V, q)H(W, q).

5 F_{G}(X) and a Heisenberg superalgebra

We see from Theorem 3 that F_{G}(X) has the same size of the tensor product of the Fock space of an infinite-dimensional Heisenberg algebra of rank dim K_{G}^{0}(X) and that of an infinite-dimensional Clifford algebra of rank dim K_{G}^{1}(X). It is our next step to actually construct such an Heisenberg/Clifford algebra. We will simply refer to as Heisenberg superalgebra from now on.

The dual of \underline{K}_{G}(X), denote by \underline{K}_{G}(X)^{∗}, is naturally ℤ_{2}-graded as identified with \underline{K}_{G}^{0}(X)^{∗} ⨁ \underline{K}_{G}^{1}(X)^{∗}. Denote by ⟨·, ·⟩ the pairing between \underline{K}_{G}(X)^{∗} and \underline{K}_{G}(X). For any n, m ≥ 1 and η ∈ \underline{K}_{G}(X)^{∗}, we define an additive map

a_{−m}(η) : \underline{K}_{G_{n}}(X^{n}) ⟶ \underline{K}_{G_{n−m}}(X^{n−m})  (19)

as the composition

\underline{K}_{G_{n}}(X^{n}) \underrightarrow{Res} \underline{K}_{G_{m}×G_{n−m}}(X^{n}) ⟶^{k^{−1}} \underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{n−m}}(X^{n−m})
⟶^{ch_{m}⊗1} \underline{K}_{G}(X) ⨂ \underline{K}_{G_{n−m}}(X^{n−m}) ⟶^{η⊗1} \underline{K}_{G_{n−m}}(X^{n−m}).

On the other hand, we define for any m ≥ 1 and V ∈ \underline{K}_{G}(X) an additive map

a_{m}(V) : \underline{K}_{G_{n−m}}(X^{n−m}) ⟶ \underline{K}_{G_{n}}(X^{n})  (20)

as the composition

\underline{K}_{G_{n−m}}(X^{n−m}) ⟶^{ω_{m}(V)⊠·} \underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{n−m}}(X^{n−m})
⟶^{k}  \underline{K}_{G_{m}×G_{n−m}}(X^{n}) \underrightarrow{Ind} \underline{K}_{G_{n}}(X^{n}).

Let H be the linear span of the operators a_{−m}(η), a_{m}(V), m ≥ 1, η ∈ \underline{K}_{G}(X)^{∗}, V ∈ \underline{K}_{G}(X). Clearly H admits a natural ℤ_{2}-gradation induced from that on \underline{K}_{G}(X) and \underline{K}_{G}(X)^{∗}. Below we shall use [·, ·] to denote the supercommutator as well. It is understood that [a, b] is the anti-commutator ab + ba when a, b ∈ H are both odd elements according to the ℤ_{2}-gradation.

Theorem 4 When acting on F_{G}(X), H satisfy the Heisenberg superalgebra commutation relations, namely for m, l ≥ 1, η, η′ ∈ \underline{K}_{G}(X)^{∗}, V, W ∈ \underline{K}_{G}(X),

[a_{−m}(η), a_{l}(V)] = lδ_{m,l}⟨η, V ⟩,  (21)
[a_{m}(W), a_{l}(V)] = 0,  (22)
[a_{−m}(η), a_{−l}(η′)] = 0.  (23)

Furthermore F_{G}(X) is an irreducible representation of the Heisenberg superalgebra.

Proof. We may assume that V, η are homogeneous, say of degree v and e where v, e ∈ \{0, 1\}, according to the ℤ_{2}-grading of \underline{K}_{G}(X) and its dual. We keep using the notations in the proof of Theorem 2.

Given E ∈ \underline{K}_{G_{r}}(X^{r}), we first observe by the definitions (19) and (20) that a_{−m}(η)a_{l}(V)E (resp. (−1)^{ve}a_{l}(V)a_{−m}(η)E) is given by the composition from the top to the bottom along the left (resp. right) side of the diagram below:

\underline{K}_{G_{r}}(X^{r})
↓ω_{l}(V) ⊠ ·
\underline{K}_{G_{l}}(X^{l}) ⨂ \underline{K}_{G_{r}}(X^{r})
Ind↙  ↘1 ⊗ Res
\underline{K}_{G_{N}} (X^{N})  \underline{K}_{G_{l}}(X^{l}) ⨂ \underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{r−m}}(X^{r−m})
∥  ↓ T^{(l,m)} ⊗ 1
\underline{K}_{G_{N}} (X^{N})  \underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{l}}(X^{l}) ⨂ \underline{K}_{G_{r−m}}(X^{r−m})
{Res \atop ↘}  P ↙Ind ⊗ 1
\underline{K}_{G_{m}}(X^{m}) ⨂ \underline{K}_{G_{n}}(X^{n})
↓ch_{m} ⊗ 1
\underline{K}_{G}(X) ⨂ \underline{K}_{G_{n}}(X^{n})
↓η ⊗ 1
\underline{K}_{G_{n}}(X^{n})

Here and below it is understood that when a negative integer appears in indices the corresponding term is 0. To simplify notations, we put Res (resp. Ind) instead of the composition k^{−1} ∘ Res (resp. Ind ∘ k) in the above diagram and below.

We denote by M′ the set of all the 2 × 2 matrices of the form (10) satisfying (11) except the following two matrices

[ \begin{matrix}0 & m \\ l & r − m \end{matrix} ] , [ \begin{matrix}m & 0 \\ l − m & r \end{matrix} ] .

As in the proof of Theorem 2, we apply Lemma 7 to the case Y = X^{N}, Γ = G_{N}, H = G_{l} × G_{r}, and L = G_{m} × G_{n}, where l + r = m + n = N.

Res_{(m,n)}Ind_{(l,r)}^{N}(ω_{l}(V) ⊠ E)
= ⨁_{A∈M} Ind_{A}^{(m,n)}(1_{a_{11}} ⊗ T^{(a_{12},a_{21})} ⊗ 1_{a_{22}})(Res_{A′}(ω_{l}(V) ⊠ E))
= ⨁_{A∈M} Ind_{(a_{11},a_{12})}^{m}F_{1} ⊠ Ind_{(a_{21},a_{22})}^{n}F_{2}
= ⨁_{A∈M′} Ind_{(a_{11},a_{12})}^{m}F_{1} ⊠ Ind_{(a_{21},a_{22})}^{n}F_{2}
⨁ F_{1} ⊠ Ind_{(l,r−m)}^{n}F_{2} ⨁ F_{1} ⊠ Ind_{(l−m,r)}^{n}F_{2}
= ⨁_{A∈M′} Ind_{(a_{11},a_{12})}^{m}F_{1} ⊠ Ind_{(a_{21},a_{22})}^{n}F_{2}
⨁(1_{m} ⊗ Ind_{(l,r−m)}^{n})(T^{(l,m)} ⊗ 1_{l−m})(1_{l} ⊗ Res_{(m,r−m)})(ω_{l}(V) ⊠ E) (24)
⨁(1_{m} ⊗ Ind_{(l−m,r)}^{n}))(Res_{(m,l−m)} ⊗ 1_{r})(ω_{l}(V) ⊠ E).

We get 0 when applying the map ch_{m} ⊗ 1 to Ind_{(a_{11},a_{12})}^{m}F_{1} ⊠ Ind_{(a_{21},a_{22})}^{n}F_{2} for A ∈ M′ in (24) by Lemma 3. When applying (η ⊗ 1) ∘ (ch_{m} ⊗ 1) to the second term of the r.h.s. of (24), we obtain (−1)^{ve}a_{l}(V)a_{−m}(η)E. When applying ch_{m} ⊗ 1 to the third term of the r.h.s. of (24), we get 0 if m ≠ l by Lemma 3. In the case m = l, the third term of the r.h.s. of (24) is simply ω_{l}(V) ⊠ E. When applying (η ⊗ 1) ∘ (ch_{m} ⊗ 1) to it, we get l⟨η, V ⟩E by Proposition 3. Putting all these pieces together, we have proved Eq. (21).

We may assume that W is homogeneous of degree w ∈ {0, 1} according to the ℤ_{2}-grading of \underline{K}_{G}(X). Eq. (22) is a consequence of the transitivity of the induction functor:

Ind_{G_{m}×G_{l+r}}^{G_{m+l+r}} (ω_{m}(W) ⊠ Ind_{G_{l}×G_{r}}^{G_{l+r}}(ω_{l}(V) ⊠ E))
= Ind_{G_{m}×G_{l+r}}^{G_{m+l+r}} (Ind_{G_{m}×G_{l}×G_{r}}^{G_{m}×G_{l+r}}(ω_{m}(W) ⊠ ω_{l}(V) ⊠ E))
= Ind_{G_{m}×G_{l}×G_{r}}^{G_{m+l+r}}(ω_{m}(W) ⊠ ω_{l}(V) ⊠ E)
= (−1)^{vw}Ind_{G_{l}×G_{m+r}}^{G_{m+l+r}} (Ind_{G_{l}×G_{m}×G_{r}}^{G_{l}×G_{m+r}}(ω_{l}(V) ⊠ ω_{m}(W) ⊠ E))
= Ind_{G_{l}×G_{m+r}}^{G_{m+l+r}} (ω_{l}(V) ⊠ Ind_{G_{m}×G_{r}}^{G_{m+r}}(ω_{m}(W) ⊠ E)) .

Similarly Eq. (23) is a consequence of the transitivity of the restriction functor. The irreducibility of F_{G}(X) as a representation of H follows immediately from the q-dimension formula for F_{G}(X) given in Theorem 3. □

In the special case G = 1, the Heisenberg superalgebra was constructed in [S2].

Remark 4 One may consider the enlarged space

V_{G} := F_{G}(pt) ⨂ ℂ[R(G)]

where ℂ[R(G)] is the group algebra of the lattice R(G). V_{G} admits a natural vertex algebra structure [B, FLM]. In particular, when G is a Kleinian group, namely a finite subgroup of SU(2), V_{G} is closely related to the Frenkel-Kac-Segal vertex representation of an affine Lie algebra. In this way, we are able to obtain a new link between the Kleinian groups and the affine Lie algebras widely known as “McKay correspondence”. Connections among symmetric functions, F_{G}(pt), V_{G}, and vertex operators will be developed elsewhere.

More generally, one may consider

V_{G}(X) := F_{G}(X) ⨂ ℂ[K_{G}(X)]  (25)

when K_{G}(X) is torsion-free, where ℂ[K_{G}(X)] is the group algebra of the lattice ℂ[K_{G}(X)] (if K_{G}(X) is not torsion-free we replace K_{G}(X) in (25) by the free part of K_{G}(X) over ℤ).

6 The orbifold Euler characteristic e(X^{n}, G_{n})

In the study of string theory on orbifolds, physicists [DHVW] came up with a notion of orbifold Euler characteristic defined as follows:

e(X, G) = \frac{1}{|G|} ∑_{g_{1}g_{2}=g_{2}g_{1}} e(X^{g_{1},g_{2}}),

where X is a smooth G-manifold. X^{g_{1},g_{2}} denotes the common fixed point set of g_{1} and g_{2} and e(·) the usual Euler characteristic. One easily shows [HH] that the orbifold Euler characteristic can be equivalently defined as

e(X, G) = ∑_{[g]∈G_{∗}} e(X^{g}/Z_{G}(g)).  (26)

Denote by X^{(n)} the n-th symmetric product of X. Recall that Macdonald’s formula [M] relates e(X^{(n)}) to e(X) as follows:

∑_{n=0} e(X^{(n)})q^{n} = (1 − q)^{−e(X)}.  (27)

The following theorem relates the orbifold Euler characteristic e(X^{n}, G_{n}) to e(X, G).

Theorem 5 ∑_{n≥0} e(X^{n}, G_{n})q^{n} = ∏_{r=1}^{∞}(1 − q^{r})^{−e(X,G)}.

Proof. By the definition of the orbifold Euler characteristic, Lemmas 2 and Lemma 5, we have

∑_{n≥0} e(X^{n}, G_{n})q^{n} = ∑_{n≥0} ∑_{[a]∈(G_{n})_{∗}} e ((X^{n})^{a}/Z_{G_{n}}(a)) q^{n}  by Eq. (26),
=^{(A)} ∑_{n≥0} _{∑_{r} rm_{r}(c)=n}^{∑} ∏_{c∈G_{∗}} e ((X^{c}/Z_{G}(c))^{m_{r}(c)}) q^{n}

= ∏_{c∈G_{∗}} ∏_{r≥1} ( ∑_{m_{r}(c)≥0} e ((X^{c}/Z_{G}(c))^{m_{r}(c)}) q^{m_{r}(c)})
= ∏_{c∈G_{∗}} ∏_{r≥1}(1 − q^{r})^{−e(X^{c}/Z_{G}(c))} by applying (27) to X^{c}/Z_{G}(c),
= \frac{1}{∏_{r=0}^{∞}(1 − q^{r})^{e(X,G)}}  by Eq. (26).

Here Eq. (A) follows from Lemma 2 and Lemma 5. □

In the case when G is trivial, we recover a formula given in [HH].

Remark 5 According to Atiyah and Segal [AS] the orbifold Euler characteristic can be calculated in terms of equivariant K-theory:

e(X, G) = dim K_{G}^{0}(X) − dim K_{G}^{1}(X).  (28)

Theorem 5 follows from Theorem 3 by applying Eq. (28) to K_{G_{n}}(X^{n}).

One is interested [DHVW, HH] in finding a resolution of singularities

X/G^{}⟶ X/G

with the property

e(X, G) = e(X/G^{}).

We assume that X is a smooth quasi-projective surface with such a property in the following discussions. Denote by X^{[n]} the Hilbert scheme of n points on X. According to Göttsche [G], the Euler characteristic of X^{[n]} is given by:

∑_{n≥0} e(X^{[n]})q^{n} = ∏_{r=0}^{∞}(1 − q^{r})^{−e(X)}.  (29)

We note that X^{n}/G_{n} is naturally identified with (X/G)^{n}/S_{n}. The following commutative diagram

X/G^{}^{[n]} → \binom{}{X/G}^{n} /S_{n}
↓  ↓
X^{n}/G_{n} ≡ (X/G)^{n}/S_{n}

implies that the Hilbert scheme X/G^{}^{[n]} is a resolution of singularity of X^{n}/G_{n}. By applying Eq. (29) to X/G^{}and comparing with Theorem 5, we have the following corollary.

Corollary 3 1) Let X be a smooth quasi-projective surface and assume that there exists a smooth resolution X/G^{}of singularities of the orbifold X/G such that e(X/G^{}) = e(X, G). Then there exists a resolution of singularities

of X^{n}/G_{n} given by X/G^{}^{[n]} satisfying

e(X^{n}, G_{n}) = e (X/G^{}^{[n]}) .

2) Let G be a finite subgroup of SU(2) and ℂ^{2}/G^{}be the minimal resolution of ℂ^{2}/G. Then the Hilbert scheme of n points on ℂ^{2}/G^{}is a resolution of singularities of ℂ^{2n}/G_{n} whose Euler characteristic is equal to the orbifold Euler characteristic e(ℂ^{2n}, G_{n}).

We note that the assumption of the existence of the resolution X/G^{}of singularities of X/G above is necessary as this is the special case of X^{n}/G_{n} for n = 1. Part 2) of the corollary is a special however important case of Part 1). The fact that the Euler characteristic of the minimal resolution of ℂ^{2}/G is equal to e(ℂ^{2}, G) (which is |G_{∗}|) is well known (cf. [HH]).

Remark 6 Let us consider a special case in the part 2) of Corollary 3 by putting G = ℤ_{2}. The wreath product G_{n} in this case is exactly the Weyl group of B_{n} or C_{n}. One can easily show that the Hilbert scheme of n points on ℂ^{2n}/G_{n}^{}is a semismall resolution of singularities of ℂ^{2n}/G_{n} (with respect to a natural stratification). A “Hilbert scheme” associated to a reductive group of type B_{n} (or C_{n}) was introduced abstractly in [BG] and it is expected to be also a semismall resolution of singularities of ℂ^{2n}/G_{n} for G = ℤ_{2}. It is interesting to understand how this abstract “Hilbert scheme” is related to the Hilbert scheme of n points on ℂ^{2n}/G_{n}^{}.

Acknowledgement. The starting point of this work is the insight of Segal [S2]. I am grateful to Igor Frenkel who first aroused my interest in [S2] and to Mikio Furuta who taught me much about K-theory for inspiring and fruitful discussions. I also thank Mark de Cataldo for helpful conversations. It is a pleasure to acknowledge the warm hospitality of the Max-Planck Institut für Mathematik at Bonn.

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22 Weiqiang Wang

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Department of Mathematics, Yale University, New Haven, CT 06520, USA E-mail: wqwang@math.yale.edu Max-Planck Institut für Mathematik, 53225 Bonn, Germany