Monstrous Moonshine of higher weight

Chongying Dong^{1} and Geoffrey Mason^{2} Department of Mathematics, University of California, Santa Cruz, CA 95064

Abstract We determine the space of 1-point correlation functions associated with the Moonshine module: they are precisely those modular forms of non-negative integral weight which are holomorphic in the upper half plane, have a pole of order at most 1 at infinity, and whose Fourier expansion has constant 0. There are Monsterequivariant analogues in which one naturally associates to each element g in the Monster a modular form of fixed weight k, the case k = 0 corresponding to the original “Moonshine” of Conway and Norton.

1 Introduction

Suppose that V is a vertex operator algebra. One of the basic problems is that of determining the so-called n-point correlation functions associated to V. There is a recursive procedure whereby n-point functions determine n + 1-point functions (see [Z], for example), so that understanding 1-point functions become important. In this paper we will study the 1-point functions (on the torus) associated with the Moonshine module, which is of interest not only as an example of the general problem but because of connections with the Monster simple group 𝕄.

First we recall the definition of a 1-point function. Let the decomposition of V into homogeneous spaces be given by

V = ⨁_{n≥n_{0}} V_{n}.  (1.1)

Each v ∈ V is associated to a vertex operator

Y (v, z) = ∑_{n∈ℤ} v(n)z^{−n−1}  (1.2)

with v(n) ∈ EndV. If v is homogeneous of weight k, that is v ∈ V_{k}, we write wtv = k. The zero mode of v is defined for homogeneous v to be the component operator

o(v) = v(wtv − 1)  (1.3)

and one knows that o(v) induces an endomorphism of each homogeneous space. That is,

o(v) : V_{n} → V_{n}.  (1.4)

1Supported by NSF grant DMS-9700923 and a research grant from the Committee on Research, UC Santa Cruz.

2Supported by NSF grant DMS-9700909 and a research grant from the Committee on Research, UC Santa Cruz.

The 1-point function determined by v is then essentially the graded trace of o(v) on V. More precisely, if V has central charge c we define the 1-point function (on the torus) via

Z(v, q) = Z(v, τ) = tr|_{V} o(v)q^{L(0)−c/24} = q^{−c/24} ∑_{n≥n_{0}}(tr|_{V_{n}}o(v))q^{n}.  (1.5)

Here, L(0) is the usual degree operator and q may be taken either as an indeterminate or, less formally, to be e^{2πiτ} with τ in the upper half plane 𝔥. If g is an automorphism of V we define

Z(v, g, q) = Z(v, g, τ) = q^{−c/24} ∑_{n≥n_{0}}(tr|_{V_{n}}o(v)g)q^{n}.  (1.6)

These functions can be extended linearly to all v ∈ V by defining Z(v, g, q) = ∑_{i} Z(v_{i}, g, q) if v = ∑_{i} v_{i} is the decomposition of v into homogeneous components. In this way we obtain the space of 1-point functions associated to V, namely the functions Z(v, q) for v ∈ V.

In order to state our results efficiently we need some notation concerning modular forms. We denote F the ℂ-linear space spanned by those (meromorphic) modular forms f(τ) of level 1 and integral weight k ≥ 0 which satisfy

(i) f(τ) is holomorphic in 𝔥.

(ii) f(τ) has Fourier expansion of the form

f(τ) = ∑_{n=−1}^{∞} a_{n}q^{n}, a_{0} = 0.  (1.7)

Thus f(τ) has a pole of order at most 1 at infinity and constant 0. Let M be the space of holomorphic modular forms of level 1 and S the space of cusp-forms of level 1. Thus we have S = F ∩ M.

Among the elements of M are the Eisenstain series E_{k}(τ) for even k ≥ 4. We normalize them as in [DLM], namely

E_{k}(τ) = \frac{−B_{k}}{k!} + \frac{2}{(k − 1)!} ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}  (1.8)

with B_{k} the kth Bernoulli number defined by

\frac{t}{e^{t} − 1} = ∑_{k=0}^{∞} B_{k} \frac{t^{k}}{k!}.  (1.9)

If M_{k} is the space of forms f(τ) ∈ M of weight k then there is a differential operator ∂ : M_{k} → M_{k+2} defined via

∂ = ∂_{k} : f(τ) ↦ \frac{1}{2πi} \frac{d}{dτ} f(τ) + kE_{2}(τ)f(τ).  (1.10)

Here, E_{2}(τ) is again defined by (1.8), though E_{2} is not a modular form.

By a ∂-ideal we mean an ideal I in the commutative algebra M which also satisfies ∂(I) ⊂ I.

Theorem 1 Let V^{♮} be the Moonshine module. The space of 1-point functions associated to V^{♮} is precisely the linear space F defined above.

As we will explain in due course, it is a consequence of results in [Z] (see also [DLM]) that all 1-point functions associated to vectors v ∈ V^{♮} lie in F. The new result here is therefore an existence result: for each f(τ) ∈ F there is a v ∈ V^{♮} such that Z(v, τ) = f(τ).

Recall next that V^{♮} is a direct sum of irreducible highest weight modules M(c, k) for the Virasoro algebra V ir. Here, c = 24 and for k > 0, M(c, k) is the Verma module generated a highest weight vector v ∈ V_{k}^{♮}. Thus L(n)v = 0 for all n > 0 where L(n) are the usual generators for V ir, and L(0)v = kv.

The proof of Theorem 1 is facilitated by the next result.

Proposition 2 Let v ∈ V_{k}^{♮} be a highest weight vector of positive weight k. Then the following hold:

(a) Z(v, τ) is a cusp-form of weight k, possibly 0.

(b) The space of 1-point functions consisting of all Z(w, τ) for w in the highest weight module for V ir generated by v is the ∂-ideal generated by Z(v, τ).

While Proposition 2 actually holds for a wide class of vertex operator algebras, our final result is more closely tied to the structure of V^{♮}. It gives us a large set of highest weight vectors (for the Virasoro algebra) to which we can usefully apply the preceding proposition.

First recall that to each λ in the Leech Lattice Λ there is a corresponding element e^{λ} in the group algebra ℂ[Λ] and an element, also denoted e^{λ}, in the vertex operator algebra V_{Λ} associated to Λ. See [B1], [FLM] and Section 4 below for more details. The relation of V_{Λ} to V^{♮} shows that e^{λ} + e^{−λ} can be considered as an element of both vertex operator algebras.

Theorem 3 Let v(λ) = e^{λ} + e^{−λ} be as above and considered as an element of V^{♮}. Then v(λ) is a highest weight vector of weight k = \frac{⟨λ,λ⟩}{2} and if 0 ≠ λ ∈ 2Λ then

Z(v(λ), τ) = η(τ)^{12} \{(\frac{Θ_{1}(τ)}{2})^{⟨λ,λ⟩−12} + (\frac{Θ_{2}(τ)}{2})^{⟨λ,λ⟩−12} − (\frac{Θ_{3}(τ)}{2})^{⟨λ,λ⟩−12}\} . (1.11)

In (1.11), η(τ) is the Dedekind eta function and Θ_{1}, Θ_{2}, Θ_{3} are the usual Jacobi theta functions (see, for example [C], p. 69).

If Λ_{n} = \{λ ∈ Λ|⟨λ, λ⟩ = 2n\} then Λ_{2} = 0, so if 0 ≠ λ ∈ 2Λ then \frac{1}{2}⟨λ, λ⟩ = 4m with m ≥ 2. If m = 2 then Z(v(λ), τ) is a cusp form of level 1 and weight 8 by Proposition 2, and hence must be 0. Then (1.11) reduces to the identity Θ_{1}(τ)^{4} + Θ_{2}(τ)^{4} − Θ_{3}(τ)^{4} = 0, which is well-known in the theory of elliptic functions (loc. cit.). If m ≥ 3 then one can check that Z(v(λ), τ) ≠ 0 (e.g. by looking at the coefficient of q in the Fourier expansion), so Z(v(λ), τ) is a non-zero cusp form of level 1 and weight 4m = 12, 16, 20, · · · . One knows (see, for example [S]) that the cusp forms of level 1 and weights 12, 16, 20 are unique up to scalar (as are those of weight 18, 22 and 26) and given by Δ(τ), Δ(τ)E_{4}(τ), Δ(τ)E_{8}(τ) respectively, where Δ(τ) = η(τ)^{24} is the discriminant. Once we know that Δ(τ) can be realized as a 1-point function Z(v, τ) for some highest weight vector v, the fact that S = MΔ(τ) (loc. cit.) together with Proposition 2 then shows that every f(τ) ∈ S can be so realized. This in turn reduces the proof of Theorem 1 to dealing with forms which have a pole at infinity.

Our discussion so far has not taken into account the automorphisms g of V^{♮} (i.e., elements of the Monster). There are some general results, which follow from [DLM], which imply that if v ∈ V^{♮} is homogeneous of weight k with respect to a certain operator L[0], then Z(v, g, τ) is a modular form of weight k for each g ∈ 𝕄. Moreover the level is the same as that for the McKay-Thompson series Z(1, g, τ) described in [CN]. We describe the precise subgroup of SL(2, ℤ) which fixes Z(v, g, τ) in Theorem 6.1.

Group theorists may be disappointed to learn that if we fix v so that all Z(v, g, τ) are modular forms of weight k then in general the Fourier coefficients of the forms (for varying g) do not define characters, or even generalized characters. This is so even if Z(v, 1, τ) has integer coefficients. This does not mean, however, that these higher weight McKay-Thompson series are of no arithmetic interest. If we combine our results with some calculations of Harada and Lang [HL], for example, we find that for each of the weights k = 12, 16, 20 there is a unique vector v in the Moonshine module V^{♮} with the following properties:

(a) v is a highest weight vector for V ir which lies in V_{k}^{♮} and is Monster-invariant.

(b) The 1-point function Z(v, τ) = q + ⋯ is the unique normalized cusp form of level 1 and weight k.

Such a v may be obtained by averaging the vector v(λ) of Theorem 3 over the Monster (λ ∈ 2Λ_{m}, m = 3, 4 or 5). The unicity of such v makes them entirely analogous to the vacuum vector 1, and it is likely that the trace functions Z(v, g, τ) are of particular interest in these cases.

We can understand the representation-theoretic meaning of the functions Z(v, g, τ) as follows: since v is Monster-invariant then each g commutes with the zero mode o(v) and its semi-simple part o(v)_{s} with regard to its action on the homogeneous space V_{n}^{♮}. Thus if o(v)_{s} has distinct eigenvalues λ_{1}, ..., λ_{t} on V_{n}^{♮}, the corresponding eigenspaces V_{n,1}^{♮}, ..., V_{n,t}^{♮} are Monster-modules and the (n − 1)th Fourier coefficient of Z(v, g, τ) is equal to ∑_{i=1}^{t} λ_{i}tr_{V_{n,i}^{♮}}g.

We complete our discussion with two conjectures: (A) For each cusp form f(τ) ∈ S of weight k there is a (Monster-invariant) highest weight vector v ∈ V_{k}^{♮} with Z(v, τ) = f(τ); (B) If Z(v, τ) is a cusp form then so is Z(v, g, τ) for each Monster element g.

The paper is organized as follows: In Section 2 we review the required results from the theory of vertex operator algebra and prove Proposition 2. In Section 3 we reduce the proof of Theorem 1 to that of Theorem 3, which is proved by lengthy calculation in Section 4. In Section 5 we give an equivariant version of formula (1.11), that is, we calculate Z(v(λ), g, τ) for various (but not all!) elements g ∈ 𝕄, and in Section 6 we describe the invariance group of Z(v, g, τ) in SL(2, ℤ).

We thank Chris Cummins for useful comments on a prior version of this paper.

2 Proof of Proposition 2

We start by recalling some results from [Z] and [DLM]. If V is a vertex operator algebra as in (1.1) then there is a second VOA structure (V, Y []) defined on V with vertex operator Y [v, z]. The two VOAs are related by a change of variables and have the same vacuum vector 1 and central charge c. The conformal vectors are distinct, however, and we denote the standard Virasoro generators for the second VOA by L[n]. The relation between the L(n) and L[n] (cf. [Z]) shows that both Virasoro algebras have the same highest weight vectors v.

A most important identity for us is the following (cf. [Z] and [DLM], equation (5.8)): if w ∈ V then

Z(L[−2]w, τ) = ∂Z(w, τ) + ∑_{l=2}^{∞} E_{2l}(τ)Z(L[2l − 2]w, τ)  (2.1)

where we are using the notation of Section 1. We should emphasize that it is a consequence of the main results of [Z] and [DLM] that if v is homogeneous of weight k with respect to L[0], where we are taking V = V^{♮} to be the Moonshine module, then the trace function Z(v, τ) is indeed a meromorphic modular form of level 1 which lies in the space F defined in (1.7).

It is also shown in [Z] (cf. [DLM], equation (5.1)) that the following holds:

Z(L[−1]w, τ) = 0 for all w ∈ V.  (2.2)

We turn to the proof of Proposition 2, beginning with part (a), which is elementary. Namely, from the creation axiom

lim_{z→0} Y (v, z)1 = v

we get v(n)1 = 0 if n ≥ 0. So if v ∈ V_{k} with k > 0 then o(v)1 = 0, in which case we see that

Z(v, τ) = q^{−1} ∑_{n=2}^{∞} tr|_{V_{n}}o(v)q^{n}

is a modular form of level 1, holomorphic in 𝔥 with a zero of order at least 1 at ∞. So indeed Z(v, τ) is a cusp-form, as asserted in Proposition 2 (a).

We turn to the proof of (b) of Proposition 2, which is established by a systematic use of equations (2.1) and (2.2). Let v ∈ V_{k} be a highest weight vector. By a descendant of v we will mean a vector of the form L[n_{1}] ⋯ L[n_{t}]v with each n_{i} ≤ 0, or any linear combination of such vectors; we write v → w if w is a descendant of v.

Let I = ⟨Z(w, τ)|v → w⟩ be the linear span of the indicated forms and let J be the ∂-ideal generated by Z(v, τ). We must prove that I = J.

First we show that I ⊂ J. We do this by proving by induction on wt[w] (the weight of w, homogeneous with respect to the second Virasoro algebra) that Z(w, τ) ∈ J. For this we may take w in the form w = L[n_{1}] ⋯ L[n_{t}]v with each n_{i} = −1 or −2. If n_{1} = −1 then Z(w, τ) = 0 by (2.2), so we may take n_{1} = −2. So w = L[−2]x where x = L[n_{2}] ⋯ L[n_{t}]v has weight equal to wt[w]-2.

By (2.1) we have

Z(w, τ) = ∂Z(x, τ) + ∑_{l=2}^{∞} E_{2l}(τ)Z(L[2l − 2]x, τ).  (2.3)

Since v → x and v → L[2k − 2]x, induction tells us that Z(x, τ) and Z(L[2l − 2]x, τ) both lie in J, whence so does of r.h.s. of (2.3) since J is a ∂-ideal. So indeed Z(w, τ) lies in J.

Next we show that I is also a ∂-ideal. Since Z(v, τ) is in I it follows from this that J ⊂ I and hence that I = J, as required.

Let r ≥ 1 with v → w and consider the vector x = L[−2]L[−1]^{2r}w. If 2l − 2 < 2r then L[2l − 2]L[−1]^{2r}w can be written as a linear combination of vectors of the shape L[−1]u for some u. Thus (2.2) tells us that Z(L[2l − 2]L[−1]^{2r}w, τ) = 0 if 2l − 2 < 2r. Now by (2.1) we get

Z(x, τ) = ∑_{l=r+1}^{∞} E_{2l}(τ)Z(L[2l − 2]L[−1]^{2r}w, τ).  (2.4)

Assuming that w is homogeneous with respect to the second Virasoro algebra, it follows in the same way that Z(L[2r]L[−1]^{2r}w, τ) is a non-zero multiple of Z(w, τ). If l > r + 1 then L[2l − 2]L[−1]^{2r}w has weight less than that of w, while if also v = w then L[2l − 2]L[−1]^{2r}v = 0. Thus (2.4) now reads

Z(x, τ) = αE_{2r+2}(τ)Z(w, τ) + ∑_{l=r+2}^{∞} E_{2l}(τ)Z(u_{l}, τ)  (2.5)

where v → u_{l}, wt[u_{l}]<wt[w] and α is a non-zero scalar. From (2.5) and what we have said it follows by induction on wt[w] that E_{2r+2}(τ)Z(w, τ) lies in I whenever r ≥ 1. Since the forms E_{2r+2}(τ) generate the space M of modular forms (in fact E_{4}(τ) and E_{6}(τ) suffice), it follows that I is an ideal in M. But then (2.1) shows that ∂Z(w, τ) lies in I whenever v → w, so I is a ∂-ideal. This completes the proof of Proposition 2 (b).

3 Trace functions with a pole

In this section we prove

Proposition 3.1 Let k be a non-negative integer. Then the trace function Z(L[−2]^{k}1, τ) is non-zero, and more precisely has a q-expansion of form ϵq^{−1} + ⋯ where (−1)^{k}ϵ > 0.

Set w = L[−2]^{k}1. Note that the truth of the proposition shows that Z(w, τ) is a form of level 1 and weight 2k which is non-zero with a pole at ∞. If we have two such trace functions of the same weight and the same residue at ∞ then they differ by a cusp-form. So together with Proposition 2 , this reduces the proof of Theorem 1 to showing that Δ(τ), say, can be realized as a trace function. As we have pointed out in Section 1, this is implicit in the statement of Theorem 3.

We turn to the proof of Proposition 3.1, using induction on k. The case k = 0 is obvious. Set x = L[−2]^{k−1}1, so that w = L[−2]x. By (2.1) and (1.10) we get

Z(w, τ) = q \frac{d}{dq} Z(x, τ) + ∑_{l=1}^{∞} E_{2l}(τ)Z(L[2l − 2]x, τ).  (3.1)

Now by another induction argument using the Virasoro relations, we easily find that if l ≥ 1 then there is an identity of the form

L[2l − 2]x = n_{l}L[−2]^{k−l}1  (3.2)

where n_{l} is positive and the right-side is interpreted as 0 if l > k.

From (1.8), the q-expansion of E_{2l}(τ) begins

−\frac{B_{2l}}{(2l)!} + ⋯

and it is easily seen from (1.9) that we have

(−1)^{l+1}B_{2l} > 0.  (3.3)

By induction we have Z(L[−2]^{r}1, τ) = ϵ(r)q^{−1} + ⋯ with (−1)^{r}ϵ(r) > 0 for 0 ≤ r < k. It follows that the coefficient of q^{−1} on the r.h.s. of (3.1) is equal to

−ϵ(k − 1) − ∑_{l=1}^{k} \frac{B_{2l}}{(2l)!}n_{l}ϵ(k − l)

= (−1)^{k} \{(−1)^{k−1}ϵ(k − 1) + ∑_{l=1}^{k}(−1)^{l+1} \frac{B_{2l}}{(2l)!}n_{l}(−1)^{k−l}ϵ(k − l)\} .

From what we have said, the sum of the terms in the braces is positive, so Proposition 3.1 is proved. □

4 Proof of Theorem 3

We have reduced the proof of Theorem 1 to that of Theorem 3, which we carry out in this section.

We first take over en bloc the notation of [FLM] with regard to the lattice VOA V_{Λ} and associated vertex operators, where Λ is the Leech lattice. In particular, 𝔥 = ℂ ⊗_{ℤ} Λ; _{𝔥̂_{ℤ}} is the corresponding Heisenberg algebra; M(1) is the associated irreducible induced module for _{ 𝔥̂_{ℤ}} such that the canonical central element of _{ 𝔥̂_{ℤ}} acts as 1; V_{Λ} = M(1) ⊗ ℂ[Λ];

Y (e^{α}, z) = E^{−}(−α, z)E^{+}(−α, z)e_{α}z^{α}

is the vertex operator associated to α ∈ Λ where

E^{±}(α, z) = exp (∑_{n∈ℕ} \frac{α(±n)}{±n}z^{∓n})

for α ∈ 𝔥 and e_{α} acts on ℂ[Λ] by

e_{α} : e^{β} ↦ ϵ(α, β)e^{α+β}

where ϵ(·, ·) is a bilinear 2-cocycle of Λ with values in {±1}; t is the automorphism of V_{Λ} of order 2 induced from the −1 isometry of Λ such that te^{α} = e^{−α}; t acts on M(1) by t(β_{1}(−n_{1}) ⋯ β_{k}(−n_{k})) = (−1)^{k}β_{1}(−n_{1}) ⋯ β_{k}(−n_{k}) for β_{i} ∈ 𝔥 and n_{i} > 0.

For a t-stable subspace W of V_{Λ} we define W^{±} to be the eigenspaces of t with eigenvalues ±1. We start by considering the action of Y (e^{α} + e^{−α}, z) on V_{Λ}^{+}. Thus V_{Λ}^{+} is spanned by elements of the form

v ⊗ e^{β} + tv ⊗ e^{−β}  (4.1)

and we have

Y (v(α), z)(v ⊗ e^{β} + tv ⊗ e^{−β})
= z^{⟨α,β⟩}E^{−}(−α, z)E^{+}(−α, z)v ⊗ ϵ(α, β)e^{α+β}
+z^{−⟨α,β⟩}E^{−}(−α, z)E^{+}(−α, z)tv ⊗ ϵ(α, −β)e^{α−β}
+z^{−⟨α,β⟩}E^{−}(α, z)E^{+}(α, z)v ⊗ ϵ(−α, β)e^{−α+β}
+z^{⟨α,β⟩}E^{−}(α, z)E^{+}(α, z)tv ⊗ ϵ(−α, −β)e^{−α−β}  (4.2)

From this we see that non-zero contributions to the trace on V_{Λ}^{+} can arise only when α ∈ 2Λ, and more precisely when α = ±2β in (4.2).

For β ∈ Λ we set

V (β) = M(1) ⊗ (ℂe^{β} + ℂe^{−β})

which is t-stable. So the trace of o(v(λ)) on V_{Λ}^{+} is equal to the trace of o(v(λ)) on V (α)^{+} where λ = 2α ∈ Λ, which we now assume. Clearly v(λ) is a highest weight vector with weight \frac{⟨λ,λ⟩}{2}.

Note that ϵ(±λ, ±α) = 1. It follows from (4.2) that only expressions of the form

z^{−2⟨α,α⟩} (E^{−}(−2α, z)E^{+}(−2α, z)tv ⊗ e^{α} + E^{−}(2α, z)E^{+}(2α, z)v ⊗ e^{−α})  (4.3)

contribute to the trace. Thus we are essentially reduced to computing the trace of the degree zero operators of E^{−}(−2α, z)E^{+}(−2α, z) and E^{−}(2α, z)E^{+}(2α, z) on M(1).

Let A = ℂλ and 𝔥 = A⊥B be an orthogonal direct sum. Then M(1) = S(_{𝔥̂^{−}}) = S(Â^{−}) ⊗ S(B̂^{−}). Let x be a formal variable and define x^{N} ∈ (EndM(1))[x] such that x^{N}(α_{1}(−n_{1}) ⋯ α_{k}(−n_{k})) = x^{k}α_{1}(−n_{1}) ⋯ α_{k}(−n_{k}) for α_{i} ∈ 𝔥 and n_{i} > 0. Set

E^{−}(±λ, z)E^{+}(±λ, z) = ∑_{n∈ℤ} E^{±}(n)z^{−n}.

Lemma 4.1 We have

trE^{±}(0)q^{L(0)}x^{N}|_{S(Â^{−})} = \frac{exp (∑_{n>0} \frac{−⟨λ,λ⟩xq^{n}}{n(1−xq^{n})} )}{∏_{n>0}(1 − xq^{n})}.  (4.4)

Proof: Note that S(Â^{−}) has a basis

{λ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}}|k_{i} ≥ 0, n ≥ 1}.

In order to compute the trace it suffices to compute the coefficients of λ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}} in E(0)^{±}λ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}}. That is, we need to compute the projection

P_{k_{1},...,k_{n}} : E(0)^{±}λ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}} → ℂλ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}}.

Recall that

[λ(s), λ(t)] = s⟨λ, λ⟩δ_{s+t,0}

for s, t ∈ ℤ. Then

P_{k_{1},...,k_{n}}E(0)^{±}λ(−n)^{k_{n}} ⋯ λ(−1)^{k_{1}}

= ∑_{p_{i}≤k_{i}}(−1)^{p_{1}+···p_{n}} \frac{λ(−1)^{p_{1}}}{p_{1}!} ⋯ \frac{λ(−n)^{p_{n}}}{n^{p_{n}}p_{n}!} \frac{λ(1)^{p_{1}}}{p_{1}!} ⋯ \frac{λ(n)^{p_{n}}}{n^{p_{n}}p_{n}!} λ(−n)^{k_{1}} ⋯ λ(−1)^{k_{1}}

= ∑_{p_{i}≤k_{i}}(−1)^{p_{1}+···p_{n}} (∏_{i=1}^{n} \frac{⟨λ, λ⟩^{p_{i}}i^{p_{i}}k_{i}(k_{i} − 1) ⋯ (k_{i} − p_{i} + 1)}{(p_{i}!)^{2}i^{2p_{i}}}) λ(−n)^{k_{1}} ⋯ λ(−1)^{k_{1}}

= ∑_{p_{i}≤k_{i}} (∏_{i=1}^{n} \frac{\binom{k_{i}}{p_{i}}}{p_{i}!} (\frac{−⟨λ, λ⟩}{i})^{p_{i}} ) λ(−n)^{k_{1}} ⋯ λ(−1)^{k_{1}}.

Thus

trE^{±}(0)q^{L(0)}x^{N}|_{S(Â^{−})}

= ∑_{n≥1} ∑_{k_{i},p_{i}≥0} (∏_{i=1}^{n} \frac{\binom{k_{i}}{p_{i}}}{p_{i}!} (\frac{−⟨λ, λ⟩}{i})^{p_{i}} ) q^{k_{1}+2k_{2}+···+nk_{n}}x^{k_{1}+···k_{n}}

= ∏_{i≥1} ( ∑_{k_{i},p_{i}≥0} \frac{\binom{k_{i}}{p_{i}}}{p_{i}!} (\frac{−⟨λ, λ⟩}{i})^{p_{i}} q^{ik_{i}}x^{k_{i}} ) .

Note that if y is a formal variable and s is a nonnegative integer then

∑_{m≥s} \binom{m}{s}y^{m}

= y^{s} ∑_{m=0}^{∞} \binom{s + m}{s}y^{m}

= y^{s} ∑_{m=0}^{∞} \binom{s + m}{m}y^{m}

= \frac{y^{s}}{(1 − y)^{1+s}} .

Then for any p_{i} ≥ 0 we have

∑_{k_{i}≥0} \binom{k_{i}}{p_{i}}(xq^{i})^{k_{i}} = \frac{(xq^{i})^{p_{i}}}{(1 − xq^{i})^{1+p_{i}}} .

Hence

trE^{±}(0)q^{L(0)}x^{N}|_{S(Â^{−})}

= ∏_{i≥1} ∑_{p_{i}≥0} \frac{1}{(1 − xq^{i})} \frac{1}{p_{i}!} (\frac{−⟨λ, λ⟩xq^{i}}{i(1 − xq^{i})} )^{p_{i}}

= ∏_{n≥1} \frac{1}{(1 − xq^{n})} exp (\frac{−⟨λ, λ⟩xq^{n}}{n(1 − xq^{n})} )

= ∏_{n=1}^{∞} \frac{1}{(1 − xq^{n})} exp (∑_{n=1}^{∞} \frac{−⟨λ, λ⟩xq^{n}}{n(1 − xq^{n})} ) ,

as desired. □ Lemma 4.2 We have

trE^{±}(0)q^{L(0)}x^{N}|_{M(1)} = \frac{exp (∑_{n>0} \frac{−⟨λ,λ⟩xq^{n}}{n(1−xq^{n})} )}{∏_{n>0}(1 − xq^{n})^{24}}.  (4.5)

Proof: Since M(1) = S(Â^{−}) ⊗ S(B̂^{−}) and E^{±}(0) commute with β(n) for β ∈ B and n ∈ ℤ, we immediately have

trE^{±}(0)q^{L(0)}x^{N}|_{M(1)} = trE^{±}(0)q^{L(0)}x^{N}|_{S(Â^{−})}trq^{L(0)}x^{N}|_{S(B̂^{−})}

and also

trq^{L(0)}x^{N}|_{S(B̂^{−})} = \frac{1}{∏_{n>0}(1 − xq^{n})^{23}} .

The lemma now follows from Lemma 4.1. □

Set f(q, x) = trE^{±}(0)q^{L(0)}x^{N}|_{M(1)}. Then one can easily see that

trE^{±}(0)q^{L(0)}|_{M(1)^{+}} = \frac{1}{2}(f(q, 1) + f(q, −1))

trE^{±}(0)q^{L(0)}|_{M(1)^{−}} = \frac{1}{2}(f(q, 1) − f(q, −1)).  (4.6)

Lemma 4.3 The contribution of V_{Λ}^{+} to Z(v(λ), τ) is

q^{\frac{1}{8} ⟨λ,λ⟩−1} ∏_{n=1}^{∞} \frac{(1 − q^{n})^{24}}{(1 − q^{2n})^{24}} ∏_{n=1}^{∞} \frac{(1 − q^{2n})^{2⟨λ,λ⟩}}{(1 − q^{n})^{⟨λ,λ⟩}} = \frac{η(2τ)^{2⟨λ,λ⟩−24}}{η(τ)^{⟨λ,λ⟩−24}} .  (4.7)

Proof: We have already seen that

tro(v(λ))q^{L(0)}|_{V_{Λ}^{+}} = tro(v(λ))q^{L(0)}|_{V (α)^{+}}.

Clearly, q^{L(0)}e^{±α} = q^{\frac{1}{8} ⟨λ,λ⟩}e^{±α}. From the proof of Lemma 4.1 we see that E^{±}(0) have the same eigenvectors and the corresponding eigenvalues are also the same. It follows from (4.3), (4.5) and (4.6) that

tro(v(λ))q^{L(0)}|_{V (α)^{+}}

= q^{\frac{1}{8} ⟨λ,λ⟩}(E^{±}(0)q^{L(0)}|_{M(1)^{+}} − trE^{±}(0)q^{L(0)}|_{M(1)^{−}})

= q^{\frac{1}{8} ⟨λ,λ⟩}f(q, −1)

= q^{\frac{1}{8} ⟨λ,λ⟩} \frac{exp (∑_{n>0} \frac{⟨λ,λ⟩q^{n}}{n(1+q^{n})})}{∏_{n>0}(1 + q^{n})^{24}}.

Next note that

∑_{n>0} \frac{q^{n}}{n(1 + q^{n})} = ∑_{n=1}^{∞} \frac{q^{n}}{n} ∑_{i=0}^{∞}(−1)^{i}q^{in}

= − ∑_{i=1}^{∞}(−1)^{i} ∑_{n=1}^{∞} \frac{q^{in}}{n}

= ∑_{n=1}^{∞}(−1)^{n} log(1 − q^{n}).

So tro(v(λ))q^{L(0)}|_{V (α)^{+}} may be written as

q^{\frac{1}{8} ⟨λ,λ⟩} ∏_{n=1}^{∞}(1 + q^{n})^{−24} ∏_{n=1}^{∞}(1 − q^{n})^{(−1)^{n}⟨λ,λ⟩}.  (4.8)

If now we incorporate the grade-shift of q^{−c/24} = q^{−1}, the lemma follows from (4.8).

□

At this point, recall [C] the Jacobi theta functions Θ_{i}, i = 1, 2, 3, considered as functions of τ i.e., with the “other” variable set equal to 0:

Θ_{1}(τ) = 2q^{\frac{1}{8}} ∏_{n=1}^{∞}(1 − q^{n})(1 + q^{n})^{2} = 2\frac{η(2τ)^{2}}{η(τ)}  (4.9)

Θ_{2}(τ) = ∏_{n=1}^{∞}(1 − q^{n})(1 − q^{n−1/2})^{2} = \frac{η(τ/2)^{2}}{η(τ)}  (4.10)

Θ_{3}(τ) = ∏_{n=1}^{∞}(1 − q^{n})(1 + q^{n−1/2})^{2} = \frac{η(τ)^{5}}{η(τ/2)^{2}η(2τ)^{2}} .  (4.11)

Combining (4.7) and (4.9) then yields Lemma 4.4 The contribution of V_{Λ}^{+} to Z(v(λ), τ) is equal to

η(τ)^{12} (\frac{Θ_{1}(τ)}{2})^{⟨λ,λ⟩−12} .

Now let V_{Λ}^{T} be the t-twisted V_{Λ}-module (cf. [FLM]). Then the moonshine module V^{♮} is the direct sum of V_{Λ}^{+} and (V_{Λ}^{T})^{+} where again + refer to the fixed points of the action t on V_{Λ}^{T} . The space V_{Λ}^{T} can be described as follows:

V_{Λ}^{T} = S(_{𝔥̂}[−1]^{−}) ⊗ T

where _{ 𝔥̂}[−1] = ∑_{n∈ℤ} 𝔥 ⊗ t^{n+1/2} ⊕ ℂc is the −1-twisted Heisenberg algebra, _{ 𝔥̂}[−1]^{−} = ∑_{n>0} 𝔥⊗t^{−n+1/2} and T is the 2^{12}-dimensional projective representation for Λ such that 2L acts on T trivially. The grading on V_{Λ}^{T} is the natural one together with an overall shift of q^{3/2}. Now t acts on T as multiplication by −1 and on S(_{𝔥̂}[−1]^{−}) by t(β_{1}(−n_{1}) ⋯ β_{k}(−n_{k}))= (−1)^{k}β_{1}(−n_{1}) ⋯ β_{k}(−n_{k}) for b_{i} ∈ 𝔥 and positive n_{i} ∈ \frac{1}{2} + ℤ. As before, for any t-stable subspace W of V_{Λ}^{T} , we denote by W^{±} the eigenspaces of t with eigenvalues ±1. Then (V_{Λ}^{T})^{+} is the tensor product of T and S(_{𝔥̂}[−1]^{−})^{−}.

The twisted vertex operator Y (e^{β}, z) for β ∈ Λ on V_{Λ}^{T} is defined to be

Y (e^{β}, z) = 2^{−⟨β,β⟩}E_{1/2}^{−}(−β, z)E_{1/2}^{+}(−β, z)e_{β}z^{−⟨β,β⟩/2}

where

E_{1/2}^{±}(h, z) = exp (∑_{n=0}^{∞} \frac{h(±(n + 1/2))}{±(n + 1/2)}z^{∓(n+1/2)})

for h ∈ 𝔥, and e_{β} acts on T. Because λ ∈ 2Λ then e_{λ} and e_{−λ} act trivially on T, and we see that

Y (v(λ), z) = 2^{−⟨λ,λ⟩}E_{1/2}^{−}(−λ, z)E_{1/2}^{+}(−λ, z)z^{−⟨β,β⟩/2} + 2^{−⟨λ,λ⟩}E_{1/2}^{−}(λ, z)E_{1/2}^{+}(λ, z)z^{−⟨β,β⟩/2}

on V_{Λ}^{T} . As before we set

E_{1/2}^{−}(±λ, z)E_{1/2}^{+}(±λ, z) = ∑_{n∈ℤ+1/2} E_{1/2}^{±}(n)z^{−n}.

Then the contribution of (V_{Λ}^{T})^{+} to Z(v(λ), τ) is equal to

q^{−1}2^{12−⟨λ,λ⟩}tr(E_{1/2}^{+}(0) + E_{1/2}^{−}(0))q^{L(0)}|_{S(_{𝔥̂}[−1]^{−})^{−}}.

For a formal variable x we define the operator x^{N} ∈ (EndS(_{𝔥̂}[−1]^{−}))[x] as before, so that x^{N}(β_{1}(−n_{1}) ⋯ β_{k}(−n_{k})) = x^{k}β_{1}(−n_{1}) ⋯ β_{k}(−n_{k}) for b_{i} ∈ 𝔥 and positive n_{i} ∈ \frac{1}{2} + ℤ. Set

g(q, x) = q^{3/2} exp (∑_{n=0}^{∞} \frac{−⟨λ, λ⟩xq^{n+1/2}}{(n + 1/2)(1 − xq^{n+1/2})}) ∏_{n>0}(1 − xq^{n−1/2})^{−24}.

By a proof not essentially different to that of Lemmas 4.1 and 4.2 we find the following:

Lemma 4.5 The traces trE_{1/2}^{+}(0)q^{L(0)}x^{N}|_{S(_{𝔥̂}[−1]^{−})} and trE_{1/2}^{−}(0)q^{L(0)}x^{N}|_{S(_{𝔥̂}[−1]^{−})} are the

same and equal to g(q, x).

One can easily see that

tr(E_{1/2}^{+}(0) + E_{1/2}^{−}(0))q^{L(0)}|_{S(_{𝔥̂}[−1]^{−})^{−}} = g(q, 1) − g(q, −1).

Next,

∑_{n=0}^{∞} \frac{xq^{n+1/2}}{(n + 1/2)(1 − xq^{n+1/2})}

= ∑_{n=0}^{∞} \frac{xq^{n+1/2}}{(n + 1/2)} ∑_{i=0}^{∞} x^{i}q^{i(n+1/2)}

= ∑_{i=1}^{∞} x^{i} ∑_{n=0}^{∞} \frac{q^{i(n+1/2)}}{(n + 1/2)}

= − ∑_{i=1}^{∞} x^{i} log (\frac{1 − q^{\frac{i}{2}}}{1 + q^{\frac{i}{2}}} ) .

Then the contribution of (V_{Λ}^{T})^{+} to Z(v(λ), τ) is equal to

2^{−⟨λ,λ⟩+12}q^{1/2} ∏_{n=1}^{∞}(1 − q^{n−1/2})^{−24} ∏_{i=1}^{∞} (\frac{1 − q^{\frac{i}{2}}}{1 + q^{\frac{i}{2}}} )^{⟨λ,λ⟩}

−2^{−⟨λ,λ⟩+12}q^{1/2} ∏_{n=1}^{∞}(1 + q^{n−1/2})^{−24} ∏_{i=1}^{∞} (\frac{1 − q^{\frac{i}{2}}}{1 + q^{\frac{i}{2}}} )^{(−1)^{i}⟨λ,λ⟩}

= 2^{−⟨λ,λ⟩+12}q^{1/2} ∏_{n=1}^{∞} \frac{(1 − q^{n})^{24}}{(1 − q^{n/2})^{24}} ∏_{i=1}^{∞} \frac{(1 − q^{\frac{i}{2}} )^{2⟨λ,λ⟩}}{(1 − q^{i})^{⟨λ,λ⟩}}

−2^{−⟨λ,λ⟩+12}q^{1/2} ∏_{n=1}^{∞} \frac{(1 − q^{n/2})^{24}(1 − q^{2n})^{24}}{(1 − q^{n})^{48}} ∏_{i=1}^{∞} \frac{(1 − q^{i})^{5⟨λ,λ⟩}}{(1 − q^{2i})^{2⟨λ,λ⟩}(1 − q^{i/2})^{2⟨λ,λ⟩}}

= η(τ)^{12} \{(\frac{Θ_{2}(τ)}{2})^{⟨λ,λ⟩−12} − (\frac{Θ_{3}(τ)}{2})^{⟨λ,λ⟩−12}\} .

Thus we have proved

Lemma 4.6 The contribution of (V_{Λ}^{T})^{+} to Z(v(λ), τ) is equal to

η(τ)^{12} \{(\frac{Θ_{2}(τ)}{2})^{⟨λ,λ⟩−12} − (\frac{Θ_{3}(τ)}{2})^{⟨λ,λ⟩−12}\} .

Theorem 3 is an immediate consequence of Lemmas 4.4 and 4.6.

5 A generalization of Theorem 3

In this section we generalize Theorem 3 by computing explicitly the trace function Z(v(λ), g, τ) for certain automorphism g of the Moonshine module. As before, Λ is the Leech lattice. To describe the result we first recall some facts about Aut(V^{♮}), that is to say, the Monster simple group 𝕄.

The centralizer of an involution in 𝕄 (of type 2B) is a quotient of a group Ĉ, partially described by the following short exact sequence:

1 → Q → Ĉ → Aut(Λ) → 1

where Q ≅ 2_{+}^{1+24} is an extra-special group of type + and order 2^{25}. For more information on this and other facts we use below, see [G] or [FLM]. The group Ĉ acts on both S(_{𝔥̂^{−}}) and S(𝔥[−1]^{−}) through the natural action of Aut(Λ), i.e., with kernel Q. It acts on ℂ[Λ] with kernel the center Z(Q) of Q, and on T with kernel a subgroup of Z(Ĉ) of order 2 distinct from Z(Q). Then the quotient C of Ĉ by the third subgroup of Z(Ĉ) of order 2 acts faithfully on V^{♮}.

Let us fix 0 ≠ λ ∈ 2Λ, and let H < Ĉ be the subgroup defined as follows:

1 → Q → H → (Aut Λ)_{λ} → 1

where (Aut Λ)_{λ} is the subgroup of Aut Λ which fixes λ. We will compute Z(v(λ), h, τ) for h ∈ H. The action of h on V_{Λ} is described by a pair (ξ, a) where ξ ∈ \frac{1}{2}Λ/Λ and a ∈ (Aut Λ)_{λ}; a acts in the natural manner, and ξ acts via

ξ : v ⊗ e^{β} ↦ e^{2πi⟨ξ,β⟩}v ⊗ e^{β}.

We let −a denote the element ta ∈ Aut Λ, and define a modified theta-function as follows:

θ_{ξ,−a}(τ) = ∑_{γ∈Λ,aγ=−γ} e^{2πi⟨ξ,γ⟩}q^{\frac{1}{2} ⟨γ,γ⟩}.  (5.1)

(5.1) is a modification of the theta-series of the sublattice of Λ fixed by −a, and as such is a modular form of weight equal to one half the dimension of the −a fixed sublattice. Finally, let η_{a}(τ) and η_{−a}(τ) by the “usual” eta-products associated to a and −a (with regard to their action on Λ) (cf. [CN], [M]). We will establish

Theorem 5.1 Let 0 ≠ λ = 2α, α ∈ Λ, and let h ∈ H be associated to (ξ, a) as above. Then we have

Z(v(λ), h, τ) = e^{2πi⟨ξ,α⟩} \frac{θ_{ξ,a}(τ)}{η_{−a}(τ)} (\frac{Θ_{1}(τ)}{2})^{⟨λ,λ⟩}

+tr_{T} (h) \{\frac{η_{a}(τ)}{η_{a}(τ/2)} (\frac{Θ_{2}(τ)}{2})^{⟨λ,λ⟩} − \frac{η_{−a}(τ)}{η_{−a}(τ/2)} (\frac{Θ_{3}(τ)}{2})^{⟨λ,λ⟩}\} .  (5.2)

Note that η_{−a}(τ) is a form of the same weight as θ_{ξ,a}(τ) (loc.cit.), so that (5.2) is indeed a form of the same weight as Z(v(λ), τ), as expected. The proof of Theorem 5.1 is a modification of that of Theorem 3.

We begin with the appropriate modification of (4.2), concerning the action of Y (v(λ), z)h on V_{Λ}^{+}. We have, setting h = h(ξ, a),

Y (v(λ), z)h(v ⊗ e^{β} + tv ⊗ e^{−β})
= Y (v(λ), z)h(ξ, 1)(a(v) ⊗ e^{a(β)} + ta(v) ⊗ e^{−a(β)})
= e^{2πi⟨ξ,a(β)⟩}Y (v(λ), z)(a(v) ⊗ e^{a(β)} + ta(v) ⊗ e^{−a(β)})
= e^{2πi⟨ξ,a(β)⟩} {z^{⟨λ,a(β)⟩}E^{−}(−λ, z)E^{+}(−λ, z)a(v) ⊗ e^{λ+a(β)}
+z^{−⟨λ,a(β)⟩}E^{−}(−λ, z)E^{+}(−λ, z)ta(v) ⊗ e^{λ−a(β)}
+z^{−⟨λ,a(β)⟩}E^{−}(λ, z)E^{+}(λ, z)a(v) ⊗ e^{−λ+a(β)}
+z^{⟨λ,a(β)⟩}E^{−}(λ, z)E^{+}(λ, z)ta(v) ⊗ e^{−λ−a(β)}} .  (5.3)

Lemma 5.2 We may take λ − a(β) = β in (5.3). This holds if, and only if, α − β = δ for some δ ∈ Λ satisfying −a(δ) = δ.

Proof: We see from (5.3) that contributions to the trace of o(v(λ))h on V_{Λ}^{+} potentially only arise when λ + α(β) = ±β or λ − a(β) = ±β. If λ = ±(a(β) − β) then λ is both a commutator (i.e., lies in [a, Λ]) and a fixed-point of a (by hypothesis). This leads to the contradiction that λ = 0, so in fact λ + a(β) = −β or λ − a(β) = β. Since β and −β are essentially interchangeable in (5.3), we may assume that indeed

λ − a(β) = β.  (5.4)

Applying a to (5.4) yields λ − a^{2}(β) = a(β) = λ − β, so that a^{2}(β) = β. This may be written as (a + 1)(a − 1)β = 0. Set

a(β) − β = 2δ.  (5.5)

Hence a(δ) + δ = 0, that is, 2δ lies in the sublattice of Λ fixed by −a. Moreover (5.4) and (5.5) yield λ − 2β = 2δ, so remembering that λ = 2α we get

α − β = δ.  (5.6)

On the other hand, if (5.6) holds, application of a yields

α − a(β) = −δ  (5.7)

and (5.6), (5.7) imply that λ − a(β) = β. □

From the Lemma and (5.3) we see that only expressions of the form

e^{2πi⟨ξ,α−δ⟩}z^{−⟨λ,α−δ⟩} (E^{−}(−λ, z)E^{+}(−λ, z)ta(v) ⊗ e^{α−δ} + E^{−}(λ, z)E^{+}(λ, z)a(v) ⊗ e^{−α+δ}) contribute to the trace, where δ ranges over the −a fixed sublattice of Λ.

We now follow the analysis of Section 4 which follows (4.3). Since a fixes λ then the contribution from S(Â^{−}) is identical to that of (4.4). As for S(B̂^{−}), the operators E^{±}(0) are trivial, and we need to calculate

trq^{L(0)}ax^{N}|_{S(B̂^{−})}.  (5.8)

If x = 1 this is precisely η_{a}(τ)/η(τ), by definition^{3}. If x = −1 then ax^{N} is just the action of ta, and (5.8) is then \frac{η_{−a}(τ)η(2τ)}{η(τ)}.

Combining (4.4) and the above, we obtain the analogue of Lemma 4.2, namely:

Lemma 5.3 We have for x = ±1,

trE^{±}(0)aq^{L(0)}x^{N}|_{M(1)} = exp (∑_{n>0} \frac{−⟨λ, λ⟩xq^{n}}{n(1 − xq^{n})} ) η_{xa}(τ)^{−1}.  (5.9)

Now use this, Lemma 5.2, and the proof of Lemma 4.3 to see that the contribution of V_{Λ}^{+} to Z(v(λ), h, τ) is equal to

∑_{δ∈Λ,−a(δ)=δ} e^{2πi⟨ξ,α−δ⟩}q^{\frac{1}{2} ⟨α−δ,α−δ⟩} exp (∑_{n>0} \frac{⟨λ, λ⟩q^{n}}{n(1 + q^{n})}) η_{−a}(τ)^{−1}.  (5.10)

Note that ⟨α, δ⟩ = 0. Then (5.10) is equal to

e^{2πi⟨ξ,α⟩}θ_{ξ,−a}(τ) (\frac{Θ_{1}(τ)}{2})^{⟨λ,λ⟩} η_{−a}(τ)^{−1}

which is the first summand of (5.2).

The other two summands of (5.2) arise from the contribution of (V_{Λ}^{T})^{+} to the trace. The proofs are as before, and are easier than the part just completed as there is no thetafunction to deal with. We leave details to the reader. This completes our discussion of Theorem 5.1.

6 The invariance group of Z(v, g, τ)

We will determine the subgroup of Γ = SL(2, ℤ) which leaves Z(v, g, τ) invariant. More precisely, if v is homogeneous of weight k with respect to L[0], so that Z(v, g, τ) is modular of weight k by [DLM], we will describe in Theorem 6.1 below the action of Γ_{0}(n) on Z(v, g, τ), where n is the order of g.

The case where v = 1 is the vacuum (and k = 0) is covered by results in [CN] and [B2]. Precisely, one knows that there is a character ϵ_{g} of Γ_{0}(n) such that

Z|γ(1, g, τ) := Z(1, g, γτ) = ϵ_{g}(γ)Z(1, g, τ)  (6.1)

3 This takes into account the corresponding grade-shift.

for γ ∈ Γ_{0}(n). Moreover ker ϵ_{g} = Γ_{0}(N) where N = nh, and h divides gcd(n, 24).

To describe our generalization of this result, we need to recall some further results. Let A_{𝕄}(⟨g⟩) = N_{𝕄}(⟨g⟩)/C_{𝕄}(⟨g⟩) be the automizer of ⟨g⟩ in the Monster 𝕄. Then A_{𝕄}(⟨g⟩) is the group of automorphisms of ⟨g⟩ induced by conjugation in 𝕄. As such, A_{𝕄}(⟨g⟩) has a canonical embedding

i_{g} : A_{𝕄}(⟨g⟩) → U_{n}  (6.2)

in which U_{n} is the group of units of ℤ/nℤ, and t ∈ N_{𝕄}(⟨g⟩) satisfying tgt^{−1} = g^{d} maps to d under i_{g}. From the character table of 𝕄 [Cal], we see that the following is true: [U_{n} : imi_{g}] ≤ 2, with equality if, and only if, g is not conjugate to g^{−1} in 𝕄. In this case, U_{n} = imi_{g} × {±1}.

Since Γ_{0}(n)/Γ_{1}(n) is naturally isomorphic to U_{n}, we may define a subgroup Γ_{g}(n) of Γ_{0}(n) via the following diagram (rows being short exact)

\begin{aligned}1 & → & Γ_{1}(n) & → & Γ_{0}(n) & → & U_{n} & → & 1 \\ & & ∥ & & ↑ & & ↑ \\ 1 & → & Γ_{1}(n) & → & Γ_{g}(n) & → & i_{g}(A_{𝕄}(⟨g⟩)) & → & 1 \end{aligned}  (6.3)

In (6.3), γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(n) maps to d ∈ U_{n}. From what we have said, we have

[Γ_{0}(n) : Γ_{g}(n)] ≤ 2, and Γ_{0}(n) = Γ_{g}(n) × {±1} if we have equality.

Let χ range over the irreducible, complex characters of the normalizer N_{𝕄}(⟨g⟩) of ⟨g⟩ in 𝕄. We will be particularly interested in those χ satisfying C_{𝕄}(⟨g⟩) ⊂ ker χ. Such χ are 1-dimensional, and induce characters

χ : A_{𝕄}(⟨g⟩) → ℂ^{∗}.  (6.4)

Using the lower row of (6.3), we can pull-back χ to a character of Γ_{g}(n), also denoted by χ. If [Γ_{0}(n) : Γ_{g}(n)] = 2 then Γ_{0}(n) = Γ_{g}(n) × ⟨−I⟩ (where I is the 2 × 2 identity matrix) and we then define a character χ_{k} (k ∈ ℤ) of Γ_{0}(n) so that its restriction to Γ_{g}(n) is the earlier χ, and its value on −I is (−1)^{k}. So in all cases we have defined characters χ_{k} of Γ_{0}(n), with the convention that χ_{k} = χ if Γ_{g}(n) = Γ_{0}(n).

We decompose V^{♮} into homogeneous subspaces V_{[k]}^{♮} with respect to the L[0]-operator. This commutes with the action of the Monster 𝕄, and we let V_{[k],χ}^{♮} be the χ-isotypic subspaces of V_{[k]}^{♮} considered as a N_{𝕄}(⟨g⟩)-module. We can now state our result:

Theorem 6.1 Fix g ∈ 𝕄, and let the notation be as above. Suppose that v ∈ V_{[k],χ}^{♮} for some simple character χ of N_{𝕄}(⟨g⟩). Then the following hold:

(a) If C_{𝕄}(⟨g⟩) ⊄ ker χ then Z(v, g, τ) = 0.

(b) If C_{𝕄}(⟨g⟩) ⊂ ker χ then

Z|_{k}γ(v, g, τ) = ϵ_{g}(γ)\overline{χ_{k}(γ)}Z(v, g, τ)  (6.5)

for γ ∈ Γ_{0}(n).

Proof: We first prove (a). Since χ is a simple character of N_{𝕄}(⟨g⟩) and C_{𝕄}(⟨g⟩) is normal in N_{𝕄}(⟨g⟩), the assumption C_{𝕄}(⟨g⟩) ⊄ ker χ means that C_{𝕄}(⟨g⟩) does not leave v invariant if 0 ≠ v ∈ V_{[k],χ}^{♮}. Then v can be written as a linear combination v = ∑_{i} v_{i} with each v_{i} ∈ V_{[k],χ}^{♮} and t_{i}v_{i} = λ_{i}v_{i} for each i, some t_{i} ∈ C_{𝕄}(⟨g⟩), and 1 ≠ λ_{i} ∈ ℂ^{∗}.

We may thus assume that v = v_{i}, with tv = λv for some t ∈ C_{𝕄}(⟨g⟩) and some 1 ≠ λ ∈ ℂ^{∗}. But then

Z(v, g, τ) = q^{−1} ∑_{n}(tr|_{V_{n}^{♮}}o(v)g)q^{n}

= q^{−1} ∑_{n}(tr|_{V_{n}^{♮}}to(v)gt^{−1})q^{n}

= q^{−1} ∑_{n}(tr|_{V_{n}^{♮}}o(tv)g)q^{n}

= Z(tv, g, τ) = λZ(v, g, τ).

Since λ ≠ 1, we get Z(v, g, τ) = 0, as required.

To prove (b) we need some results from [DLM], which we assume the reader is familiar

with. In particular, since g has order n then a matrix γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(n) maps the

(1, g) conformal block to the (1, g^{d}) conformal block. Since the trace functions Z(v, g, τ), Z(v, g^{d}, τ) span these conformal blocks, there is a scalar η_{g}(γ), independent of v, such that

Z|γ(v, g, τ) = η_{g}(γ)Z(v, g^{d}, τ).  (6.6)

Here, if v ∈ V_{[k]}^{♮} then

Z|γ(v, g, τ) = (cτ + d)^{−k}Z(v, g, γτ).  (6.7)

Taking v = 1, k = 0 in (6.6)-(6.7) and comparing with (6.1) then yields η_{g}(γ) = ϵ_{g}(γ), that is

Z|γ(v, g, τ) = ϵ_{g}(γ)Z(v, g^{d}, τ).  (6.8)

Suppose that d ∈ i_{g}(A_{𝕄}(⟨g⟩), that is γ ∈ Γ_{g}(n). Then g^{d} = tgt^{−1} for some t ∈ N_{𝕄}(⟨g⟩), and we calculate as before:

Z(v, g^{d}, τ) = Z(v, tgt^{−1}, τ) = Z(t^{−1}v, g, τ) = χ(t^{−1})Z(v, g, τ).

Then (6.8) reads

Z|γ(v, g, τ) = ϵ_{g}(γ)χ(t^{−1})Z(v, g, τ).  (6.9)

By our conventions, χ(t^{−1}) = \overline{χ(γ)}, so (6.9) is what we require.

Now assume that γ ∉ Γ_{g}(n). From our earlier remarks, it suffices to take γ = −I. In this case γ ∈ Γ_{0}(N), so ϵ_{g}(γ) = 1, and (6.7) reads

Z|_{k}γ(v, g, τ) = (−1)^{k}Z(v, g, τ),

which is what (6.5) says in this case. The proof of theorem is now complete. □

Remark 6.2 By Theorem 2 of [DM], each χ occurs in V^{♮}, that is, given χ as above, there is k such that V_{[k],χ} ≠ 0.

References

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