Quasifinite representations of classical Lie subalgebras of W_{1+∞}

Victor G. Kac^{1}, Weiqiang Wang^{2}^{∗} and Catherine H. Yan^{3}

1Department of Mathematics

MIT, Cambridge, MA 02139, USA

E-mail: kac@math.mit.edu

2Max-Planck Institut für Mathematik

Gottfried-Claren Str. 26, 53225 Bonn, Germany

E-mail: wqwang@mpim-bonn.mpg.de

3Courant Institute of Mathematical Sciences

251 Mercer Street, New York, NY 10012, USA

E-mail: yanhuaf@math1.cims.nyu.edu

Abstract

We show that there are precisely two, up to conjugation, anti-involutions σ_{±} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D̂^{±} of the Lie subalgebra of this algebra fixed by −σ_{±}, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra ℂ[u]/(u^{m+1}) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of D̂^{±} (including all the unitary ones) are obtained. We also realize a class of primitive representations of D̂^{±} in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V_{c} of D̂^{+} carries a vertex algebra structure and establish a relationship between V_{c} for c ∈ \frac{1}{2}ℤ and W-algebras.

∗On leave from Department of Mathematics, Yale University, USA

2 V. G. Kac, W. Wang and C. H. Yan

Contents

0 Introduction 2 1 Lie algebra 𝔤̂𝔩^{[m]} and its classical subalgebras 5

1.1 Lie algebra 𝔤̂𝔩^{[m]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Lie algebra b_{∞}^{[m]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Lie algebra c_{∞}^{[m]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Lie algebra d_{∞}^{[m]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Anti-involutions of D̂ preserving its principal gradation 14 3 Structure of parabolic subalgebras of D̂^{±} 17

3.1 D̂^{−} case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 D̂^{+} case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Characterization of quasifiniteness of HWM’s of D̂^{−} 22 5 Embedding of D̂^{−} into infinite rank classical Lie algebras 26 6 Realization of QHWM’s of D̂^{±} 31 7 Unitary QHWM’s of D̂^{±} 37 8 FFR’s of QHWM’s over D̂^{−} with c ∈ −ℕ/2 41

8.1 Dual pair (O(2l), D̂^{−}) . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.2 Dual pair (O(2l + 1), D̂^{−}) . . . . . . . . . . . . . . . . . . . . . . . 44

9 FFR’s of QHWM’s over D̂^{−} with c ∈ ℕ 46 10 FFR’s of QHWM’s over D̂^{−} with c ∈ ℕ − 1/2 50 11 FFR’s of QHWM’s over D̂^{+} with c ∈ ℕ 54 12 FFR’s of QHWM’s over D̂^{+} with c ∈ \frac{1}{2} + ℤ_{+} 58 13 FFR’s of QHWM’s over D̂^{+} with c ∈ −\frac{1}{2}ℕ 62

13.1 Case c ∈ −ℕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 13.2 Case c ∈ −ℕ + 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

14 Vertex algebra associated to D̂^{+} 65 14.1 Vertex algebra structure on the vacuum module of D̂^{+} . . . . . . . 65 14.2 Vertex algebra V_{c} for c ∈ \frac{1}{2}ℤ and W algebras . . . . . . . . . . . . . 69

15 Appendix 71

classical Lie subalgebras of W_{1+∞} 3

0 Introduction

The systematic study of quasifinite highest weight modules of the universal central extension D̂ of the Lie algebra of differential operators over the circle (described first in [KP]) was initiated by Kac and Radul in [KR1] and further studied in [Ma, FKRW, AFMO, KR2, W1] and many others. The Lie algebra D̂ is also known in the literature as W_{1+∞} [PRS] and as one of the universal W-algebras (cf.

[BS] [BEH^{3}] and references therein), and it has various connections with conformal field theory, the quantum Hall effect [CTZ] etc.

The difficulty in understanding representation theory of a Lie algebra of this sort is that although D̂ admits a natural “principal” ℤ-gradation and thus the associated triangular decomposition, each of the graded subspaces of D̂ is still infinite-dimensional in contrast to the more familiar cases such as the Virasoro algebra and Kac-Moody algebras. The study of the highest weight modules of D̂ with the finiteness requirement on the dimensions of their graded subspaces (which we will refer to as quasifiniteness condition throughout our paper) thus becomes a highly non-trivial problem. By analyzing for which parabolic subalgebras of D̂ the corresponding generalized Verma modules are quasifinite, Kac and Radul [KR1] were able to give an elegant characterization of quasifinite highest weight D̂-modules in terms of a certain generating function of highest weights. They constructed all these D̂-modules in terms of representations of the Lie algebra 𝔤̂𝔩^{[m]} which is the central extension of the Lie algebra 𝔤𝔩^{[m]} of infinite matrices with finitely many non-zero diagonals taking values in the truncated polynomial algebra R_{m} = ℂ[u]/(u^{m+1}). They also classified all such D̂-modules which are unitary.

It is well known that the Lie algebra 𝔤̂𝔩 [KP, DJKM], a special case of 𝔤̂𝔩^{[m]} with m = 0, has Lie subalgebras of B, C, D types as in the finite dimensional case [K]. So a natural question which arises here is: what are the Lie subalgebras of D̂ which correspond to the classical Lie subalgebras of 𝔤̂𝔩 of B, C, D types. It is the goal of this paper to give a complete answer to this question and to present the representation theory of these Lie subalgebras of D̂.

We show that there are two, up to conjugation, anti-involutions of D̂, denoted by σ_{±}, which preserve the principal ℤ-gradation. We thus obtain two different Lie subalgebras, denoted by D̂^{±}, fixed by −σ_{±}. The graded subspaces of D̂^{±} are still infinite-dimensional. We regard D̂^{+} to be more fundamental than D̂^{−} as we shall show that the vacuum module of D̂^{+} carries a canonical vertex algebra structure.

We first give a description of parabolic subalgebras of D̂^{±} in terms of the socalled characteristic polynomials {b_{k}(w)}_{k∈ℕ}. It turns out that the quasifiniteness of a generalized Verma module is completely determined by the non-vanishing of the first characteristic polynomial b_{1}(w). The corresponding condition in defining the generalized Verma module induced from a parabolic subalgebra P leads to a characterization of highest weights for which the corresponding D̂^{±}-module is quasifinite in terms of a generating function Δ(x). This is the content of our Theorem 4.1.

Next for each s ∈ ℂ we construct a Lie algebra homomorphism ϕ̂_{s}^{[m]} from D̂^{±} to 𝔤̂𝔩^{[m]}. While D̂^{±} is the central extension of a Lie algebra of differential operators with polynomial coefficients, it is important to consider a Lie algebra D̂^{O,±} which is an analytic completion of D̂^{±} and to extend the homomorphisms ϕ̂_{s}^{[m]} to D̂^{O,±}. It turns out that for s ∉ ℤ/2 the homomorphism ϕ̂_{s}^{[m]} from D̂^{O,±} to 𝔤̂𝔩^{[m]} is surjective, but for s ∈ ℤ/2, this is no longer true. The image for s ∈ ℤ/2 turns out to be various classical Lie subalgebras of 𝔤̂𝔩^{[m]}.

More generally we define a family of Lie algebra homomorphisms ϕ̂[m⃗] ; s⃗ from D̂^{±} to 𝔤^{[m⃗]} for a vector m⃗ = (m_{1}, …, m_{N}) ∈ ℤ_{+} and a vector s⃗ = (s_{1}, …, s_{N}) ∈ ℂ^{N} satisfying a certain condition (⋆±), where 𝔤^{[m⃗]} is a direct sum of Lie algebras b_{∞}^{[m_{i}]}, b̃_{∞}^{[m_{i}]}, c_{∞}^{[m_{i}]}, d_{∞}^{[m_{i}]}, and 𝔤̂𝔩^{[m_{i}]} satisfying a certain consistency condition. These homomorphisms ϕ̂[m⃗] ; s⃗ again extend to D̂^{O,±} and then become surjective. The principal ℤ-gradations on b_{∞}^{[m_{i}]}, c_{∞}^{[m_{i}]} and 𝔤̂𝔩^{[m_{i}]}, and the specialized ℤ-gradation on d_{∞}^{[m_{i}]} of type (2, 1, 1, ⋯) induce one on 𝔤^{[m⃗]}. The homomorphism ϕ̂[m⃗] ; s⃗ matches the induced ℤ-gradation on 𝔤^{[m⃗]} with the principal ℤ-gradation on D̂^{±}.

Quasifinite highest weight modules (QHWM’s) of 𝔤̂𝔩^{[m]} and its classical subalgebras are better understood (particularly when m = 0). Irreducible QHWM’s over 𝔤^{[m⃗]} can be regarded as modules over D̂^{±} via the homomorphisms ϕ̂[m⃗] ; s⃗. Our Theorem 6.1 asserts that they are irreducible over D̂^{±} as well. Our Theorem 6.2 asserts that all irreducible quasifinite highest weight modules over D̂^{±} can be realized in this way.

There is a natural anti-involution ω on D̂^{±} which is compatible with the standard Cartan involution on 𝔤̂𝔩 via the Lie algebra homomorphism ϕ̂_{s}. We show that unitary irreducible quasifinite highest weight modules over D̂^{±} with respect to ω are the pullbacks of those over 𝔤̂𝔩 via the homomorphisms ϕ̂_{s} for real vectors s⃗. We give a simple characterization for these unitary D̂^{±}-modules in terms of the generating function Δ(x) for the highest weights with respect to D̂^{±} and the central charge c. This is our Theorem 7.1. The proofs of Theorems 4.1, 6.1, 6.2 and 7.1 are similar to those in [KR1] for D̂.

The q-character formulas in accordance with the ℤ-gradation for unitary quasifinite highest weight modules of b_{∞}, c_{∞} and d_{∞} are worked out explicitly. This leads to the q-character formulas for the so-called positive primitive quasifinite D̂^{±}-modules, which include all unitary quasifinite D̂^{±}-modules.

In the second half of this paper we study primitive quasifinite representations of D̂^{±} in terms of free bosonic and fermionic fields. Recall that representations of b_{∞} or c_{∞} (resp. b̃_{∞} and d_{∞}) were realized in various Fock spaces [DJKM1, W2]. A number of duality results in various Fock spaces between finite dimensional irreducible representations of a finite dimensional Lie group or superalgebra and quasifinite representations of b_{∞} or c_{∞} (resp. b̃_{∞} or d_{∞}) were established in [W2]. Lie groups and Lie superalgebra appearing in these duality results are Sp(2l), P in(2l), Spin(2l + 1), O(2l), O(2l + 1) and 𝔬𝔰𝔭(1, 2l) respectively. Combining with our homomorphisms ϕ̂_{0} or ϕ̂_{1/2} (resp. ϕ̂_{−1/2}), we obtain free field realizations of D̂^{−} (resp. D̂^{+}) and corresponding dual pairs involving D̂^{−} (resp. D̂^{+}). We present explicit descriptions of the Fock space decompositions into isotypic subspaces with respect to the joint action of the corresponding dual pairs and calculate the corresponding exponents and their multiplicities (which characterize the highest weights for D̂^{±}) in each isotypic subspace. The explicit formulas for the highest weight vectors in isotypic subspaces were already given in [W2]. Free field realizations of D̂ were studied earlier in [FKRW, KR2] and a duality between the general linear Lie group and D̂ was established in these papers.

It is a remarkable new feature of dual pairs in our infinite dimensional setting that b_{∞} and c_{∞} (resp. b̃_{∞} and d_{∞}) involved in different dual pairs are uniformly replaced now by a single Lie algebra D̂^{−} (resp. D̂^{+}). The ranks of these finite dimensional Lie groups, while varying for different Fock spaces, correspond neatly to the central charges of the primitive D̂^{±}-modules appearing in these Fock spaces. Note that [W2] the Lie algebras of these Lie groups turn out to be the horizontal subalgebras of the twisted or untwisted affine algebras acting on the corresponding Fock spaces with level ±1 [F1] [KP] [FF].

We further show that the vacuum module M_{c} of D̂^{+} with central charge c and its irreducible quotient V_{c} admit a natural structure of a vertex algebra [B] [FLM]

[DL] [K2]. The module M_{c} is irreducible if and only if c ∉ \frac{1}{2}ℤ. For the central charge c = l, l + 1/2, −l or −l + 1/2, V_{c} is shown to be isomorphic to the vertex algebra of invariants of the various Fock spaces F^{⨂ l}, F^{⨂ l+ \frac{1}{2}} , F^{⨂ −l} or F^{⨂ −l+ \frac{1}{2}} (see the text for notations) with respect to the action of O(2l), O(2l + 1), Sp(2l) or 𝔬𝔰𝔭(1, 2l) respectively. Those quasifinite representations of D̂^{+} appearing from the Fock space decompositions are representations of the vertex algebra V_{c} for the corresponding c ∈ \frac{1}{2}ℤ.

It is well known that to any complex simple Lie algebra 𝔤 and c ∈ ℂ one canonically associates a vertex algebra with central charge c, called W-algebra and denoted by W𝔤, cf. [BS, FeF] and references therein. It is known [BS, F2] that the associated W-algebra WD_{l} with central charge l is isomorphic to the vertex algebra of the SO(2l)-invariants in the basic representation of the affine algebra 𝔰̂o(2l). We prove that WD_{l} with central charge l is a sum of the vertex algebra V_{l} and an irreducible representation of V_{l}. This provides us a new way of computing the q-character formula of WD_{l}.

In a similar fashion, we identify the vertex algebra of SO(2l+1)-invariants in the Fock space F^{⨂ l+ \frac{1}{2}} as a W-superalgebra WB(0, l) [Ito] with l bosonic generating fields of conformal weight 2, 4, …, 2l and a fermionic generating field of conformal weight l + 1/2. The even part of WB(0, l) is isomorphic to V_{l+1/2} while the odd part of WB(0, l) is an irreducible representation of V_{l+1/2}.

The approach of this paper can be modified to study the representation theory of another interesting Lie subalgebra of D̂ considered by Bloch [Bl]. This will be treated in a separate publication [W3].

The paper is organized as follows. In Section 1 we describe the infinite rank Lie algebras 𝔤̂𝔩^{[m]}, b_{∞}^{[m]}, b̃_{∞}^{[m]}, c_{∞}^{[m]}, d_{∞}^{[m]} and present the q-character formulas of unitary highest weight modules over them in the case m = 0 (see also Appendix). In Section 2 we review the Lie algebra D̂ and classify the anti-involutions of D̂ preserving the principal ℤ-gradation of D̂. In Section 3 we study the structure of parabolic subalgebras of D̂^{±}. In Section 4 we give a characterization for a D̂^{±}-module to be quasifinite. In Section 5 we study the connection between D̂^{±} and the infinite rank Lie algebras. In Section 6 we realize quasifinite D̂^{±}-modules in terms of modules over these infinite rank Lie algebras. In Section 7 we classify all unitary quasifinite D̂^{±}-modules. In Sections 8, 9 and 10, we realize a class of primitive D̂^{−}-modules with a half-integral central charge in some Fock spaces and establish some duality between this class of primitive D̂^{−}-modules with central charge c = −l (resp. −l − 1/2, l, l + 1/2) and finite dimensional irreducible modules of O(2l) (resp. O(2l + 1), Sp(2l) and P in(2l), 𝔬𝔰𝔭(1, 2l), Spin(2l + 1)). In Sections 11, 12 and 13, we realize in a similar way a class of primitive D̂^{+}-modules and establish various duality results for D̂^{+}. In Section 14, we study the representations of D̂^{±} from the viewpoint of vertex algebras.

Acknowledgement This paper is based on two preprints [KWY]. W. W. wishes to thank Max-Planck-Institut für Mathematik for its hospitality.

1 Lie algebra 𝔤̂𝔩^{[m]} and its classical subalgebras

1.1 Lie algebra 𝔤̂𝔩^{[m]}

Denote by R_{m} the quotient algebra ℂ[u]/(u^{m+1}) of the polynomial algebra ℂ[u] by the ideal generated by u^{m+1} (m ∈ ℤ_{+}). Denote by 1 the identity element of R_{m}. Denote by 𝔤𝔩_{f}^{[m]} the complex Lie algebra of all infinite matrices (a_{ij})_{i,j∈ℤ} with finitely many non-zero entries in R_{m}. Denote by 𝔤𝔩^{[m]} the Lie algebra of all matrices (a_{ij})_{i,j∈ℤ} with only finitely many nonzero diagonals with entries in R_{m}. Denote by E_{ij} the infinite matrix with 1 at (i, j) place and 0 elsewhere. Obviously 𝔤𝔩_{f}^{[m]} is a Lie subalgebra of 𝔤𝔩^{[m]}. There is a natural automorphism ν of 𝔤̂𝔩^{[m]} given by

ν(E_{i,j}) = E_{i+1,j+1}.  (1.1)

Let the weight of E_{ij} be j − i. This defines the principal ℤ-gradation 𝔤𝔩^{[m]} = ⨁_{j∈ℤ} 𝔤𝔩_{j}^{[m]}. Denote by 𝔤̂𝔩^{[m]} = 𝔤𝔩^{[m]} ⨁ R_{m} the central extension of 𝔤̂𝔩^{[m]} given by the following 2–cocycle with values in R_{m} [KP, DJKM]:

C(A, B) = Tr ([J, A]B)  (1.2)

where J = ∑_{j≤0} E_{ii}. The ℤ–gradation of Lie algebra 𝔤𝔩^{[m]} extends to 𝔤̂𝔩^{[m]} by putting the weight R_{m} to be 0. In particular, we have the triangular decomposition

𝔤̂𝔩^{[m]} = 𝔤̂𝔩_{+}^{[m]} ⨁ 𝔤̂𝔩_{0}^{[m]} ⨁ 𝔤̂𝔩_{−}^{[m]},

where

ĝl_{±} = ⨁_{j∈ℕ} 𝔤̂𝔩_{±j}^{[m]}, 𝔤̂𝔩_{0}^{[m]} = 𝔤𝔩_{0}^{[m]} ⨁ R_{m}.

Given Λ ∈ (𝔤̂𝔩_{0}^{[m]})^{∗}, we let

c_{j} = Λ(u^{j}),
aλ_{i}^{(j)} = Λ(u^{j}E_{ii}),
aH_{i}^{(j)} = u^{j}E_{ii} − u^{j}E_{i+1,i+1} + δ_{i,0}u^{j},
ah_{i}^{(j)} = Λ(^{a}H_{i}^{(j)}) = ^{a}λ_{i}^{(j)} −^{a}λ_{i+1}^{(j)} + δ_{i,0} c_{j},

where i ∈ ℤ, j = 0, …, m. The superscript a here denotes 𝔤̂𝔩^{[m]} which is of A type. Denote by L(𝔤̂𝔩^{[m]}; Λ) the highest weight 𝔤̂𝔩^{[m]}–module with highest weight Λ. The c_{j} are called the central charges and ^{a}λ_{i}^{(j)} are called the labels of L(𝔤̂𝔩^{[m]}; Λ).

In particular putting m = 0 we recover the well-known Lie algebras 𝔤𝔩 = 𝔤𝔩^{[0]}, 𝔤̂𝔩 = 𝔤̂𝔩^{[0]}. In this case, we drop the superscript [0]. Define ^{a}Λ_{j} ∈ 𝔤𝔩_{0}^{∗} (j ∈ ℤ) as follows:

aΛ_{j}(E_{ii}) = \{ \begin{matrix}1, & for 0 < i ≤ j \\ −1, & for j < i ≤ 0 \\ 0, & otherwise. \end{matrix}  (1.3)

Define ^{a}Λ̂_{0} ∈ 𝔤̂𝔩_{0}^{∗} by

aΛ̂_{0}(C) = 1, ^{a}Λ̂_{0}(E_{ii}) = 0 for all i ∈ ℤ

and extend Λ_{j} from 𝔤𝔩_{0}^{∗} to 𝔤̂𝔩_{0}^{∗} by letting Λ_{j}(C) = 0. Then

aΛ̂_{j} = ^{a}Λ_{j} +^{a}Λ̂_{0} (j ∈ ℤ)

are the fundamental weights, i.e. ^{a}Λ̂_{j}(^{a}H_{i}) = δ_{ij}.

Recall that the q-character (i.e. principally specialized character) formula for an integrable highest weight module L(λ) of a Kac-Moody algebra 𝔤 ([K], chap.

10) with respect to the principal gradation of 𝔤 is given by

ch_{q}L(λ) = ∏_{α^{∨}∈Δ_{+}^{∨}} (\frac{1 − q^{⟨λ+ρ,α^{∨}⟩}}{1 − q^{⟨ρ,α^{∨}⟩}} )^{mult α^{∨}}  (1.4)

where Δ_{+}^{∨} is the positive coroots of the Lie algebra 𝔤, and ρ satisfies ⟨ρ, α_{i}^{∨}⟩ = 1 for all simple coroots α_{i}^{∨}.

Since 𝔤̂𝔩 is a completed infinite rank affine algebra of Kac-Moody type, the above formula (1.4) applies. This gives the following explicit formula (cf. [FKRW])

ch_{q}L(𝔤̂𝔩, Λ) = \frac{∏_{1≤i<j≤c}(1 − q^{n_{i}−n_{j}+j−i})}{φ(q)^{c}}  (1.5)

where Λ = Λ̂_{n_{1}} +Λ̂_{n_{2}} +…+Λ̂_{n_{c}}, n_{1} ≥ n_{2} ≥ … ≥ n_{c}, c ∈ ℕ, and φ(q) = ∏_{j=1}^{∞}(1−q^{j}) is the Euler product.

1.2 Lie algebra b_{∞}^{[m]}

Now consider the vector space R_{m}[t, t^{−1}] and take its basis v_{i} = t^{i} (i ∈ ℤ) over R_{m}. The Lie algebra 𝔤𝔩^{[m]} acts on this vector space via the usual formula

E_{ij}v_{k} = δ_{j,k}v_{i}.  (1.6)

Let us consider the following ℂ-bilinear forms on this space:

B^{±}(u^{m}v_{i}, u^{n}v_{j}) = u^{m}(−u)^{n}(±1)^{i}δ_{i,−j}.

Denote by \overline{b}_{∞}^{−[m]} (resp. \overline{b}_{∞}^{+[m]}) the Lie subalgebra of 𝔤𝔩^{[m]} which preserves the bilinear form B^{−} (resp. B^{+}), namely we have

\overline{b}_{∞}^{−[m]} = \{g ∈ 𝔤𝔩^{[m]} | B(a(u), v) + B(u, a(v)) = 0\}
= {(a_{ij}(u))_{i,j∈ℤ} ∈ 𝔤𝔩^{[m]} | a_{ij}(u) = (−1)^{i+j+1}a_{−j,−i}(−u)},
\overline{b}_{∞}^{+[m]} = \{g ∈ 𝔤𝔩^{[m]} | B(a(u), v) + B(u, a(v)) = 0\}
= {(a_{ij}(u))_{i,j∈ℤ} ∈ 𝔤𝔩^{[m]} | a_{ij}(u) = −a_{−j,−i}(−u)}.

Denote by b_{∞}^{[m]} = \overline{b}_{∞}^{−[m]} ⨁ R_{m} (resp. b̃_{∞}^{[m]} = \overline{b}_{∞}^{+[m]} ⨁ R_{m}) the central extension of \overline{b}_{∞}^{−[m]} (resp. \overline{b}_{∞}^{+[m]}) given by the 2-cocycle (1.2) restricted to \overline{b}_{∞}^{−[m]} (resp. \overline{b}_{∞}^{+[m]}). Then b_{∞}^{[m]} (resp. b̃_{∞}^{[m]}) inherits from 𝔤̂𝔩^{[m]} the principal ℤ-gradation and the triangular decomposition:

b_{∞}^{[m]} = ⨁_{j∈ℤ} b_{∞,j}^{[m]}, b_{∞}^{[m]} = b_{∞,+}^{[m]} ⨁ b_{∞,0}^{[m]} ⨁ b_{∞,−}^{[m]},
b̃_{∞}^{[m]} = ⨁_{j∈ℤ} b̃_{∞,j}^{[m]}, b̃_{∞}^{[m]} = b̃_{∞,+}^{[m]} ⨁ b̃_{∞,0}^{[m]} ⨁ b̃_{∞,−}^{[m]}

where

b_{∞,j}^{[m]} = b_{∞}^{[m]} ∩ 𝔤̂𝔩_{j}^{[m]}, b_{∞}_{±}^{[m]} = b_{∞}^{[m]} ∩ 𝔤𝔩_{±}^{[m]}, b_{∞,0}^{[m]} = b_{∞}^{[m]} ∩ 𝔤𝔩_{0}^{[m]},
b̃_{∞,j}^{[m]} = b̃_{∞}^{[m]} ∩ 𝔤̂𝔩_{j}^{[m]}, b̃_{∞,±}^{[m]} = b̃_{∞}^{[m]} ∩ 𝔤𝔩_{±}^{[m]} b̃_{∞0}^{[m]} = b̃_{∞}^{[m]} ∩ 𝔤𝔩_{0}^{[m]}.

Remark 1.1 The Lie algebra b̃_{∞}^{[m]} is isomorphic to b_{∞}^{[m]} by sending the elements u^{k}E_{ij} − (−u)^{k}E_{−j,−i} to u^{k}E_{ij} + (−1)^{i+j+1}(−u)^{k}E_{−j,−i}, i, j ∈ ℤ, k ∈ ℤ_{+}. Their Cartan subalgebras coincide.

Given Λ ∈ (b_{∞}^{[m]}_{0})^{∗}, we let

c_{j} = Λ(u^{j}),
bλ_{0}^{(j)} = Λ (2u^{j}E_{00}) (j odd),
bλ_{i}^{(j)} = Λ (u^{j}E_{ii} − (−u)^{j}E_{−i,−i}) ,
bH_{i}^{(j)} = u^{j}E_{ii} − u^{j}E_{i+1,i+1} + (−u)^{j}E_{−i−1,−i−1} − (−u)^{j}E_{−i,−i},
bH_{0}^{(j)} = 2 (u^{j}E_{−1,−1} − u^{j}E_{1,1}) + 2u^{j} (j even),
bH_{0}^{(j)} = (2u^{j}E_{0,0} − u^{j}E_{−1,−1} − u^{j}E_{1,1}) + u^{j} (j odd),
bh_{i}^{(j)} = Λ(^{b}H_{i}^{(j)}) = ^{b}λ_{i}^{(j)} −^{b}λ_{i+1}^{(j)},  (1.7)
bh_{0}^{(j)} = Λ(^{b}H_{0}^{(j)}) = −2 ^{b}λ_{1}^{(j)} + 2c_{j} (j even),
bh_{0}^{(j)} = Λ(^{b}H_{0}^{(j)}) = ^{b}λ_{0}^{(j)} −^{b}λ_{1}^{(j)} + c_{j} (j odd),

where i ∈ ℕ and j = 0, …, m. The superscript b here means B type. Denote by L(b_{∞}^{[m]}; Λ) (resp. L(b̃_{∞}^{[m]}; Λ)) the highest weight module over b_{∞}^{[m]} (resp. b̃_{∞}^{[m]}) with highest weight Λ. The c_{j} are called the central charges and ^{b}λ_{i}^{(j)} are called the labels of L(b_{∞}^{[m]}; Λ) or L(b̃_{∞}^{[m]}; Λ).

In particular when m = 0 we have the usual Lie subalgebras of 𝔤𝔩 and 𝔤̂𝔩, denoted by \overline{b}_{∞}^{−} and b_{∞} [K] (resp. \overline{b}_{∞}^{+} and b̃_{∞} [W2]). Denote by ^{b}Λ̂_{i} the i-th fundamental weight of b_{∞} (and b̃_{∞}), namely ^{b}Λ̂_{i}(^{b}h_{j}) = δ_{ij}.

The set of simple coroots of b_{∞} (and b̃_{∞}), denoted by Π^{∨}, can be described as follows:

\begin{matrix}Π^{∨} = \{α_{0}^{∨} = 2(E_{−1,−1} − E_{1,1}) + 2C, \\ α_{i}^{∨} = E_{i,i} + E_{−i−1,−i−1} − E_{i+1,i+1} − E_{−i,−i}, i ∈ ℕ\}. \end{matrix}

The set of roots is:

Δ = {±ϵ_{0}, ±ϵ_{i}, ±ϵ_{i} ± ϵ_{j}, i ≠ j, i, j ∈ ℕ}.

The set of positive coroots is:

Δ_{+}^{∨} = {α_{i}^{∨} + α_{i+1}^{∨} + … + α_{j}^{∨}, 1 ≤ i ≤ j}
∪{α_{0}^{∨} + 2α_{1}^{∨} + … + 2α_{i}^{∨} + α_{i+1}^{∨} + … + α_{j−1}^{∨}, 0 ≤ i < j}.

The set of simple roots is:

Π = {α_{0} = −ϵ_{1}, α_{i} = ϵ_{i} − ϵ_{i+1}, i ∈ ℕ}.

Here ϵ_{i} are viewed restricted to the restricted dual of the Cartan subalgebra of b_{∞}, so that ϵ_{i} = −ϵ_{−i}.

Given Λ = ^{b}Λ̂_{n_{1}} +^{b}Λ̂_{n_{2}} + … +^{b}Λ̂_{n_{k}} +^{b}h^{b}Λ̂_{0}, n_{1} ≥ n_{2} ≥ … ≥ n_{k} ≥ 1,^{b}h ∈ ℤ_{+}, the b_{∞}-module L(b_{∞}; Λ) has central charge c = k +^{b}h/2. Here and further we let bh = ^{b}h^{(0)} and ^{b}λ_{i} = ^{b}λ_{i}^{(0)}, see (1.7). Let

φ_{i}(q) = ∏_{j=1}^{i}(1 − q^{j}).

Further on we shall use the following notations: \overline{m} = 1 for m ∈ 2ℤ and \overline{m} = 0 for m ∈ 2ℤ + 1; for a real number x denote by [x] the integer no greater than and closest to x.

Proposition 1.1 The q-character formula of L(b_{∞}; Λ) (the same for L(b̃_{∞}; Λ)) corresponding to the principal gradation of b_{∞} is

ch_{q}L(b_{∞}; Λ) = \frac{∏_{1≤i<j≤k}(1 − q^{n_{i}−n_{j}+j−i})}{∏_{1≤i≤k} φ_{n_{i}+k−i}(q)} · \frac{φ(q^{2})\overline{2c+1} ∏_{j>0} φ_{2c−2j}(q)}{φ(q)^{[ \frac{2c+1}{2} ]}} ·

∏_{i=0}^{n_{1}−1} \frac{φ_{2c+i+n_{1}}(q)}{φ_{2c+n_{1}+i−^{b}λ_{i+1}}(q)} · ∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c+j+i−^{b}λ_{i+1}−^{b}λ_{j}}}{1 − q^{2c+j+i}} . (1.8)

Proof. Applying formula (1.4) to L(b_{∞}; Λ), we get

ch_{q}L(b_{∞}; Λ) = ∏_{1≤i<j} \frac{1 − q^{^{b}λ_{i}−^{b}λ_{j}+j−i}}{1 − q^{j−i}} · ∏_{0≤i<j} \frac{1 − q^{2c−^{b}λ_{i+1}−^{b}λ_{j}+j+i}}{1 − q^{i+j}} .  (1.9)

Denote by the first (resp. second) product on the right hand side of (1.9) to be A (resp. B). Consider the Young diagram Y_{Λ} corresponding to the partition (n_{1}, n_{2}, …, n_{k}), it is easy to see that (^{b}λ_{1},^{b}λ_{2}, …,^{b}λ_{n_{1}}) is its conjugate (cf. [M] for terminology). Also ^{b}λ_{i} = 0 for i > n_{1}. Thus

A = ∏_{1≤i<j} \frac{1 − q^{^{b}λ_{i}−^{b}λ_{j}+j−i}}{1 − q^{j−i}} = ∏_{j∈Y_{Λ}}(1 − q^{h_{j}})^{−1} = \frac{∏_{1≤i<j≤k}(1 − q^{n_{i}−n_{j}+j−i})}{∏_{1≤i≤k} φ_{n_{i}+k−i}(q)} (1.10)

where h_{j} are the hook numbers of the Young diagram Y_{Λ} (cf. [M]).

On the other hand, B can be computed as follows.

B = ∏_{0≤i<j} \frac{1 − q^{2c+j+i−^{b}λ_{i+1}−^{b}λ_{j}}}{1 − q^{i+j}}

= ∏_{0≤i<j} \frac{1 − q^{2c+j+i}}{1 − q^{i+j}} · \frac{1 − q^{2c+j+i−^{b}λ_{i+1}−^{b}λ_{j}}}{1 − q^{2c+j+i}}

= ∏_{0≤i<j} \frac{1 − q^{2c+j+i}}{1 − q^{i+j}} ∏_{0≤i<n_{1}<j} \frac{1 − q^{2c+j+i−^{b}λ_{i+1}−^{b}λ_{j}}}{1 − q^{2c+j+i}} ·

∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c+j+i−^{b}λ_{i+1}−^{b}λ_{j}}}{1 − q^{2c+j+i}} .

A little further manipulation shows that the first, second and third products of the right hand side of the equation above give rise to the second, third and fourth products of the right hand side of (1.8) respectively. □

1.3 Lie algebra c_{∞}^{[m]}

As before we take a basis v_{i} = t^{i} (i ∈ ℤ) of R_{m}[t, t^{−1}] over R_{m}. Consider on this space the following ℂ-bilinear form

C(u^{m}v_{i}, u^{n}v_{j}) = u^{m}(−u)^{n}(−1)^{i}δ_{i,1−j}.

Denote by \overline{c}_{∞}^{[m]} the Lie subalgebra of 𝔤𝔩^{[m]} which preserves this bilinear form:

\overline{c}_{∞}^{[m]} = \{g ∈ 𝔤𝔩^{[m]} | C(a(u), v) + C(u, a(v)) = 0\}
= {(a_{ij}(u))_{i,j∈ℤ} ∈ 𝔤𝔩^{[m]} | a_{ij}(u) = (−1)^{i+j+1}a_{1−j,1−i}(−u)}.

Denote by c_{∞}^{[m]} = \overline{c}_{∞}^{[m]} ⨁ R_{m} the central extension of \overline{c}_{∞}^{[m]} given by the 2-cocycle (1.2) restricted to \overline{c}_{∞}^{[m]}. Then c_{∞}^{[m]} inherits from 𝔤̂𝔩^{[m]} the natural ℤ-gradation and the triangular decomposition:

c_{∞}^{[m]} = ⨁_{j∈ℤ} c_{∞}^{[m]}_{j}, c_{∞}^{[m]} = c_{∞}^{[m]}_{+} ⨁ c_{∞}^{[m]}_{0} ⨁ c_{∞}^{[m]}_{−},

where c_{∞}^{[m]}_{j} = c_{∞}^{[m]} ∩ 𝔤̂𝔩_{j}^{[m]}, c_{∞}_{±}^{[m]} = c_{∞}^{[m]} ∩ 𝔤𝔩_{±}^{[m]} and c_{∞}^{[m]}_{0} = c_{∞}^{[m]} ∩ 𝔤𝔩_{0}^{[m]}.

Given Λ ∈ (c_{∞}_{0}^{[m]})^{∗}, we let

c_{j} = Λ(u^{j}),
cλ_{i}^{(j)} = Λ (u^{j}E_{ii} − (−u)^{j}E_{1−i,1−i}) ,
cH_{i}^{(j)} = u^{j}E_{ii} − u^{j}E_{i+1,i+1} + (−u)^{j}E_{−i,−i} − (−u)^{j}E_{1−i,1−i},
cH_{0}^{(j)} = (u^{j}E_{0,0} − u^{j}E_{1,1}) + u^{j} (j even),  (1.11)
ch_{i}^{(j)} = Λ(^{c}H_{i}^{(j)}) = ^{c}λ_{i}^{(j)} −^{c}λ_{i+1}^{(j)},
ch_{0}^{(j)} = Λ(^{c}H_{0}^{(j)}) = −^{c}λ_{1}^{(j)} + c_{j} (j even),

where i ∈ ℕ and j = 0, …, m. For later purpose, it is convenient to put^{c}h_{0}^{(j)} = c_{j} (j odd), j = 0, …, m. The superscript c here denotes c_{∞} which is of C type. Denote by L(c_{∞}^{[m]}; Λ) the highest weight c_{∞}^{[m]}–module with highest weight Λ. The c_{j} are called the central charges and ^{c}λ_{i}^{(j)} are called the labels of L(c_{∞}^{[m]}; Λ).

In particular, when m = 0 we have the usual \overline{c}_{∞} = \overline{c}_{∞}^{[0]}, c_{∞} = c_{∞}^{[0]} [K]. In this case, we drop the superscript [0]. Then we denote by ^{c}Λ̂_{i} the i-th fundamental weight of c_{∞}, namely ^{c}Λ̂_{i}(^{c}h_{j}) = δ_{ij}.

The set of simple coroots of c_{∞}, denoted by Π^{∨}, can be described as follows:

\begin{matrix}Π^{∨} = \{α_{0}^{∨} = (E_{0,0} − E_{1,1}) + C, \\ α_{i}^{∨} = E_{i,i} + E_{−i,−i} − E_{i+1,i+1} − E_{1−i,1−i}, i ∈ ℕ\}. \end{matrix}

The set of roots is:

Δ = {±ϵ_{i} ± ϵ_{j}, ±2ϵ_{i}, i ≠ j, i, j ∈ ℕ}.

The set of positive coroots is:

Δ_{+}^{∨} = {α_{i}^{∨} + α_{i+1}^{∨} + … + α_{j}^{∨}, 0 ≤ i ≤ j}
∪ {2α_{0}^{∨} + … + 2α_{i}^{∨} + α_{i+1}^{∨} + … + α_{j}^{∨}, 0 ≤ i < j}.

The set of simple roots is:

Π = {α_{0} = −2ϵ_{1}, α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, 2, …}.

Here ϵ_{i} are viewed restricted to 𝔥′, so that ϵ_{i} = −ϵ_{−i+1}.

Given Λ = ^{c}Λ̂_{n_{1}} +^{c}Λ̂_{n_{2}} + … +^{c}Λ̂_{n_{k}} +^{c}h^{c}Λ̂_{0}, n_{1} ≥ n_{2} ≥ … ≥ n_{k} ≥ 1,^{c}h ∈ ℤ_{+}, the c_{∞}-module L(c_{∞}; Λ) has central charge c = k +^{c}h.

Proposition 1.2 The q-character formula of L(c_{∞}; Λ) corresponding to the principal gradation of c_{∞} is

ch_{q}L(c_{∞}; Λ) = \frac{∏_{1≤i<j≤k}(1 − q^{n_{i}−n_{j}+j−i})}{∏_{1≤i≤k} φ_{n_{i}+k−i}(q)} · \frac{∏_{1≤j≤c−1} φ_{2j}(q)}{φ(q)^{c}} ·

∏_{i=0}^{n_{1}} \frac{φ_{2c+n_{1}+i}(q)}{φ_{2c+n_{1}+i−^{c}λ_{i}}(q)} · ∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c+i+j−^{c}λ_{i+1}−^{c}λ_{j}}}{1 − q^{2c+i+j}} . (1.12)

Proof. Applying formula (1.4) to L(c_{∞}; Λ), we get

ch_{q}L(c_{∞}; Λ) = ∏_{1≤i<j} \frac{1 − q^{^{c}λ_{i}−^{c}λ_{j}+j−i}}{1 − q^{j−i}} ·

∏_{0≤i<j} \frac{1 − q^{2c−^{c}λ_{i+1}−^{c}λ_{j+1}+j+i+2}}{1 − q^{i+j+2}} · ∏_{j>0} \frac{1 − q^{c−^{c}λ_{j}+j}}{1 − q^{j}}

= ∏_{1≤i<j} \frac{1 − q^{^{c}λ_{i}−^{c}λ_{j}+j−i}}{1 − q^{j−i}} · ∏_{0≤i<j} \frac{1 − q^{2c−^{c}λ_{i}−^{c}λ_{j}+j+i}}{1 − q^{i+j}} (1.13)

with the convention ^{c}λ_{0} = 0. The first product on the right hand side of (1.13) is already given by formula (1.10) with ^{b}λ_{i} substituted by ^{c}λ_{i}. Denote by C the second product on the right hand side of (1.13); it can be calculated in a similar way as calculating B in the proof of Proposition 1.1:

C = ∏_{0≤i<j} \frac{1 − q^{2c−^{c}λ_{i}−^{c}λ_{j}+i+j}}{1 − q^{i+j}}

= \frac{∏_{1≤j≤c−1} φ_{2j}(q)}{φ(q)^{c}} · ∏_{i=0}^{n_{1}} \frac{φ_{2c+n_{1}+i}(q)}{φ_{2c+n_{1}+i−^{c}λ_{i}}(q)} · ∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c+i+j−^{c}λ_{i+1}−^{c}λ_{j}}}{1 − q^{2c+i+j}} .
□

1.4 Lie algebra d_{∞}^{[m]}

As before we take a basis v_{i} = t^{i} (i ∈ ℤ) of R_{m}[t, t^{−1}] over R_{m}. Consider the following ℂ-bilinear form

D(u^{m}v_{i}, u^{n}v_{j}) = u^{m}(−u)^{n}δ_{i,1−j}, i, j ∈ ℤ.

Denote by \overline{d}_{∞}^{[m]} the Lie subalgebra of 𝔤𝔩^{[m]} which preserves this bilinear form:

\overline{d}_{∞}^{[m]} = \{g ∈ 𝔤𝔩^{[m]} | D(a(u), v) + D(u, a(v)) = 0\}
= {(a_{ij}(u))_{i,j∈ℤ} ∈ 𝔤𝔩^{[m]} | a_{ij}(u) = (−1)^{i+j+1}a_{1−j,1−i}(−u)}.

Denote by d_{∞}^{[m]} = \overline{d}_{∞}^{[m]} ⨁ R_{m} the central extension of \overline{d}_{∞}^{[m]} given by the 2-cocycle (1.2) restricted to \overline{d}_{∞}^{[m]}. Then d_{∞}^{[m]} inherits from 𝔤̂𝔩^{[m]} a natural ℤ-gradation and the triangular decomposition:

d_{∞}^{[m]} = ⨁_{j∈ℤ} d_{∞}^{[m]}_{j}
d_{∞}^{[m]} = d_{∞}^{[m]}_{+} ⨁ d_{∞}^{[m]}_{0} ⨁ d_{∞}^{[m]}_{−}

where d_{∞}^{[m]}_{j} = d_{∞}^{[m]} ∩ 𝔤̂𝔩_{j}^{[m]}, d_{∞}_{±}^{[m]} = d_{∞}^{[m]} ∩ 𝔤𝔩_{±}^{[m]} and d_{∞}^{[m]}_{0} = d_{∞}^{[m]} ∩ 𝔤𝔩_{0}^{[m]}.

Given Λ ∈ (d_{∞}_{0}^{[m]})^{∗}, we let

c_{j} = Λ(u^{j}),
dλ_{i}^{(j)} = Λ (u^{j}E_{ii} − (−u)^{j}E_{1−i,1−i}) ,
dH_{i}^{(j)} = u^{j}E_{ii} − u^{j}E_{i+1,i+1} + (−u)^{j}E_{−i,−i} − (−u)^{j}E_{1−i,1−i},
dH_{0}^{(j)} = ((−u)^{j}E_{0,0} + (−u)^{j}E_{−1,−1} − u^{j}E_{2,2} − u^{j}E_{1,1}) + 2u^{j}, (1.14)
dh_{i}^{(j)} = Λ(^{d}H_{i}^{(j)}) = ^{d}λ_{i}^{(j)} −^{d}λ_{i+1}^{(j)},
dh_{0}^{(j)} = Λ(^{d}H_{0}^{(j)}) = −^{d}λ_{1}^{(j)} −^{d}λ_{2}^{(j)} + 2c_{j},

where i ∈ ℕ and j = 0, …, m. The superscript d here denotes d_{∞} which is of D type. Denote by L(d_{∞}^{[m]}; Λ) the highest weight d_{∞}^{[m]}–module with highest weight Λ. The c_{j} are called the central charges and ^{d}λ_{i}^{(j)} are called the labels of L(d_{∞}^{[m]}; Λ).

In particular, when m = 0 we have the usual \overline{d}_{∞} = \overline{d}_{∞}^{[0]}, d_{∞} = d_{∞}^{[0]}, cf. [K1]. In this case, we drop the superscript [0]. Denote by Λ̂_{i}^{d} the i-th fundamental weight of d_{∞}, namely ^{d}Λ̂_{i}(^{d}h_{j}) = δ_{ij}.

The set of simple coroots of d_{∞}, denoted by Π^{∨}, can be described as follows:

\begin{matrix}Π^{∨} = \{α_{0}^{∨} = (E_{0,0} + E_{−1,−1} − E_{2,2} − E_{1,1}) + 2C, \\ α_{i}^{∨} = E_{i,i} + E_{−i,−i} − E_{i+1,i+1} − E_{1−i,1−i}, i ∈ ℕ\}. \end{matrix}

The set of roots is:

Δ = {±ϵ_{i} ± ϵ_{j}, i ≠ j, i, j ∈ ℕ}.

The set of positive coroots is:

Δ_{+}^{∨} = {α_{i}^{∨} + α_{i+1}^{∨} + ⋯ + α_{j}^{∨} (0 ≤ i ≤ j), α_{0}^{∨} + α_{2}^{∨} + ⋯ + α_{j}^{∨} (j ≥ 2)}
∪ {α_{0}^{∨} + α_{1}^{∨} + 2α_{2}^{∨} + ⋯ + 2α_{i}^{∨} + α_{i+1}^{∨} + ⋯ + α_{j}^{∨}, 1 < i < j}.

The set of simple roots is:

Π = {α_{0} = −ϵ_{1} − ϵ_{2}, α_{i} = ϵ_{i} − ϵ_{i+1}, i ∈ ℕ}.

Here ϵ_{i} are viewed restricted to 𝔥′, so that ϵ_{i} = −ϵ_{−i+1}. The root vectors corresponding to ±α_{0} are e_{0} = E_{0,2}−E_{−1,1} and f_{0} = E_{2,0}−E_{1,−1}. Introduce ρ ∈ (d_{∞}_{0}^{[m]})^{∗} such that (ρ, α_{i}^{∨}) = 1, i ∈ ℤ_{+}.

Remark 1.2 The q-character formula (corresponding to the induced ℤ-gradation on d_{∞} from 𝔤̂𝔩) of a d_{∞}-module is given in the Appendix. Note that such a ℤgradation on d_{∞} is of type (2, 1, 1, …) rather than the principal ℤ-gradation of type (1, 1, …) as for 𝔤̂𝔩, b_{∞} and c_{∞} since f_{0} = E_{2,0} − E_{1,−1} has degree −2 instead of −1.

Example 1.1 Take a parabolic subalgebra of \overline{d}_{∞}

P_{0} = \{(a_{ij}) ∈ \overline{d}_{∞} | a_{ij} = 0 if i > 0 ≥ j\},

and let P̂_{0} = P_{0} ⨁ ℂC. The 2-cocycle (1.2) when restricted to P_{0} is trivial. Consider the so-called vacuum module M_{c}(d_{∞}), i.e. the induced d_{∞}-module from a 1-dimensional P_{0}-module on which P_{0} acts as zero and C as c ∈ ℂ. The irreducible quotient of M_{c}(d_{∞}) is L(d_{∞}; 2c ^{d}Λ̂_{0}). It follows by a standard argument (cf. [K1], Chapter 10) that

L(d_{∞}; 2c ^{d}Λ̂_{0}) = M_{c}(d_{∞})/⟨f_{0}^{2c+1}⟩ if c ∈ ℤ_{+}/2.

2 Anti-involutions of D̂ preserving its principal gradation

Let D_{as} be the associative algebra of regular differential operators on the circle. The elements

J_{k}^{l} = −t^{l+k}(∂_{t})^{l} (l ∈ ℤ_{+}, k ∈ ℤ)

form its basis, where ∂_{t} denotes \frac{d}{dt}. Another basis of D is

L_{k}^{l} = −t^{k}D^{l} (l ∈ ℤ_{+}, k ∈ ℤ)

where D = t∂_{t}. It is easy to see that J_{k}^{l} = −t^{k}[D]_{l}. Here and further we use the notation

[x]_{l} = x(x − 1) … (x − l + 1).  (2.15)

Let D denote the Lie algebra obtained from D_{as} by taking the usual bracket [a, b] = ab − ba. Denote by D̂ the central extension of D by a one-dimensional center with a generator C: D̂ = D + ℂC. The Lie algebra D̂ has the following commutation relations [KR1]

[t^{r}f(D), t^{s}g(D)] = t^{r+s} (f(D + s)g(D) − f(D)g(D + r))
+ Ψ (t^{r}f(D), t^{s}g(D)) C  (2.16)

where

Ψ (t^{r}f(D), t^{s}g(D)) = \{ \begin{matrix}∑_{−r≤j≤−1} f(j)g(j + r), & r = −s ≥ 0 \\ 0, & r + s ≠ 0. \end{matrix}  (2.17)

The 2-cocycle Ψ can be equivalently given in terms of another formula [KP]:

Ψ (f(t)(∂_{t})^{m}, g(t)(∂_{t})^{n}) = \frac{m!n!}{(m + n + 1)!}Res_{t=0}f^{(n+1)}(t)g^{(m)}(t)dt.  (2.18)

Let the weight of J_{k}^{l} be k and the weight of C be 0. This defines the principal ℤ-gradations of D_{as}, D and D̂:

D = ⨁_{j∈ℤ} D_{j}, D̂ = ⨁_{j∈ℤ} D̂_{j}

and so we have the triangular decomposition

D̂ = D̂_{+} ⨁ D̂_{0} ⨁ D̂_{−},

where

D̂_{±} = ⨁_{j∈±ℕ} D̂_{j}, D̂_{0} = D_{0} ⨁ ℂC.

An anti-involution σ of D_{as} is an involutive anti-automorphism of D, i.e. σ^{2} = I, σ(aX + bY) = aσ(X) + bσ(Y) and σ(XY) = σ(Y)σ(X), where a, b ∈ ℂ, X, Y ∈ D.

Proposition 2.1 Any anti-involution σ of D_{as} which preserves the principal ℤgradation is one of the following:

1) σ_{−,b}(t) = −t, σ_{−,b}(D) = −D + b;

2) σ_{+,b}(t) = t, σ_{+,b}(D) = −D + b, b ∈ ℂ.

Proof. Since σ preserves the principal ℤ-gradation, we can assume that σ(t) = tf(D), and σ(D) = g(D), where f and g are polynomials. Since σ is an antiinvolution we have

g(g(D)) = D,  (2.19)
f(g(D)) · (tf(D)) = t.  (2.20)

Equation (2.19) means that the polynomial g(w) satisfies g(g(w)) = w, which can only be possible when g(w) is linear in w. Hence g(w) = w or g(w) = −w + b for some b ∈ ℂ.

It follows from equation (2.20) that

t = f(g(D)) · (tf(D)) = tf(g(D + 1))f(D)

which implies f(w) = ±1. By applying the anti-involution σ to the equation [t, D] = −t, we have [σ(D), σ(t)] = −σ(t), which excludes the possibility g(w) = w.

An anti-involution σ of D_{as} is completely determined by σ(t) and σ(D). On the other hand, it is straightforward to check the σ_{±,b} listed in the proposition are indeed anti-involutions of D_{as}. This completes the proof of the proposition. □

It follows immediately that σ_{±,b}(∂_{t}) = ∓ (∂_{t} − (b + 1)t^{−1}) . Given s ∈ ℂ, denote by Θ_{s} the automorphism of D_{as} which sends t to t and D to D+s. Equivalently Θ_{s} is given by sending a ∈ D to t^{−s}at^{s}, the conjugate of a by t^{s}. Clearly Θ_{s} preserves the principal ℤ-gradation of D_{as}. We have

σ_{±,b} · Θ_{s} = σ_{±,b+s}, Θ_{−s} · σ_{±,b} = σ_{±,b+s}.  (2.21)

Denote by D^{±,b} the fixed Lie subalgebra of D by −σ_{±,b}, namely

D^{±,b} = {a ∈ D | σ_{±,b}(a) = −a}.

It inherits a ℤ-gradation from D since σ_{±,b} preserves the principal ℤ-gradation of D: D^{±,b} = ⨁_{j∈ℤ} D_{j}^{±,b}, where

D_{j}^{±,b} = { t^{j}f(D) | f(w) ∈ ℂ[w] and σ_{±,b}(t^{j}f(D)) = −t^{j}f(D)}.

Let us denote by ℂ[w]^{(1)} the set of all odd polynomials in ℂ[w], and by ℂ[w]^{(0)} the set of all even polynomials in ℂ[w]. As before we let \overline{k} = 0 if k is an odd integer and \overline{k} = 1 if k is even. The following lemma gives a complete description of D_{j}^{±,b}.

Lemma 2.1 We have

D_{j}^{−,b} = {t^{j}g (D + (j − b)/2) | g(w) ∈ ℂ[w]^{(j̄)}, j ∈ ℤ} ,
D_{j}^{+,b} = {t^{j}g (D + (j − b)/2) | g(w) ∈ ℂ[w]^{(1)}, j ∈ ℤ} .

Proof. Given t^{j}f(D) ∈ D_{j}^{−,b}, we have σ_{−,b}(t^{j}f(D)) = −t^{j}f(D), which means (−1)^{j}t^{j}f(−D − j + b) = −t^{j}f(D). Equivalently we have (−1)^{j}f(−w − j + b) = −f(w). Letting g(w) = f(w − \frac{j−b}{2}), we have g(−w) = (−1)^{j+1}g(w). □

The relation among D^{±,b} for different b ∈ ℂ is given by the following lemma which follows from Lemma 2.1.

Lemma 2.2 The Lie algebras D^{+,b} (resp. D^{−,b}) for different b ∈ ℂ are all isomorphic. More precisely, we have Θ_{s}(D^{±,b}) = D^{±,b−2s}.

Due to Lemma 2.2 we may choose among D^{−,b} a Lie algebra, say D^{−} = D^{−,0}. We see from Lemma 2.1 that D^{n} ∈ D^{−} for n ∈ 2ℤ + 1, and D^{n} ∉ D^{−} for n ∈ 2ℤ. Let (see notation (2.15))

T_{k}^{n,s} = −t^{k} ([D − s]_{n} + (−1)^{k+1}[−D − k − s]_{n}) (k ∈ ℤ, n ∈ ℤ_{+}, s ∈ ℂ). (2.22)

Clearly T_{k}^{n,s} ∈ D^{−}. It is also clear that T_{k}^{n,s} (k ∈ ℤ, n ∈ 2ℤ_{+} + 1) form a ℂ-basis of D^{−} for a fixed s ∈ ℂ. We denote T_{k}^{n,0} by T_{k}^{n}. A straightforward computation shows

T_{k}^{n,s} = − (t^{k+n+s}∂_{t}^{n}t^{−s} + (−1)^{k+n+1}t^{1−s}∂_{t}^{n}t^{k+n+s−1}) .  (2.23)

Although t^{m±s} (m ∈ ℤ, s ∈ ℂ) does not lie in D in general, the above expression (2.23) does lie in D^{−}.

We denote again by Ψ the restriction of the 2-cocycle Ψ to D^{−}, namely

\begin{matrix}Ψ (t^{r}f(D + r/2), t^{s}g(D + s/2)) \\ = & \{ \begin{matrix}∑_{−r≤j≤−1} f(j + r/2)g(j + r/2), & r = −s ≥ 0 \\ 0, & r + s ≠ 0 \end{matrix} \end{matrix}

where f(w) ∈ ℂ[w]^{(r̄)}, g(w) ∈ ℂ[w]^{(s̄)}. Denote by D̂^{−} the central extension of D^{−} by ℂC corresponding to the 2-cocycle Ψ. D̂^{−} is a Lie subalgebra of D̂ by definition.

The most convenient choice for the other family of Lie algebras is b = −1 (σ_{+,−1}(∂_{t}) = −∂_{t}). Denote D^{+} = D^{+,−1}. We shall see that there is a canonical structure of a vertex algebra on the vacuum module of the central extension of D^{+}. Let

W_{k}^{n,s} = −\frac{1}{2}t^{k} ([D + s]_{n} − [−D − k − 1 + s]_{n}) (k ∈ ℤ, n ∈ ℕ, s ∈ ℂ). (2.24)

An important linear basis of D^{+} is W_{k}^{n,s} (k ∈ ℤ, n ∈ 2ℤ_{+} + 1) for a fixed s ∈ ℂ. In particular we denote W_{k}^{n,0} by W_{k}^{n}. A straightforward computation shows that

W_{k}^{n} = −\frac{1}{2} (t^{k+n}∂_{t}^{n} + (−1)^{n+1}∂_{t}^{n}t^{k+n}) .  (2.25)

Note that the W_{k}^{1} (k ∈ ℤ) span a Virasoro algebra, namely we have

[W_{m}^{1}, W_{n}^{1}] = (m − n)W_{m+n}^{1} + δ_{m,−n} \frac{m^{3} − m}{12}C.

By abuse of notation we again denote by Ψ the restriction of the 2-cocycle Ψ to D^{+}:

\begin{matrix}Ψ (t^{r}f(D + (r + 1)/2), t^{s}g(D + (s + 1)/2)) \\ = & \{ \begin{matrix}∑_{−r≤j≤−1} f(j + (r + 1)/2)g(j + (r + 1)/2), & r = −s ≥ 0 \\ 0, & r + s ≠ 0, \end{matrix} \end{matrix}

where f(w), g(w) ∈ ℂ[w]^{(1)}. Denote by D̂^{+} the central extension of D^{+} by ℂC corresponding to the 2-cocycle Ψ. The Lie algebra D̂^{+} is a subalgebra of D̂ by definition.

3 Structure of parabolic subalgebras of D̂^{±}

3.1 D̂^{−} case

We define a parabolic subalgebra P of D̂^{−} as a subalgebra of the form P = ⊕_{j∈ℤ}P_{j}, where P_{j} = D̂_{j}^{−} if j ≥ 0, and P_{j} ≠ 0 for some j < 0.

For each positive integer k we have P_{−k} = {t^{−k}h(D−k/2) | h(w) ∈ I_{−k}}, where I_{−k} is some subspace of ℂ[w]^{(k̄)}. Given p(w) ∈ ℂ[w]^{(k̄)} and f(w) ∈ ℂ[w]^{(1)}, we have f(D), t^{−k}p(D − k/2) ∈ D̂^{−}. One calculates that

[f(D), t^{−k}p(D − k/2)] = t^{−k}(f(D − k) − f(D))p(D − k/2)
= t^{−k}g(D − k/2)p(D − k/2)  (3.26)

where g(w) = f(w −k/2)−f(w +k/2). As f ranges over all odd polynomials, g(w) ranges over all even polynomials. Thus equation (3.26) implies that if p(w) ∈ I_{−k} then ℂ[w]^{(0)}p(w) ⊂ I_{−k}. This means that I_{−k} is a submodule of ℂ[w]^{(k̄)} over ℂ[w]^{(0)}, where ℂ[w]^{(k̄)} is regarded as a module over ℂ[w]^{(0)} by multiplication. Clearly every non-zero submodule of ℂ[w]^{(k̄)} over ℂ[w]^{(0)} is a free rank 1 submodule generated by a monic polynomial. Denote by b_{k}(w) such a generator for I_{−k} if I_{−k} ≠ 0, and let b_{k}(w) = 0 if I_{−k} = 0. We call b_{k}(w) (k = 1, 2, …) the characteristic polynomials of P.

Lemma 3.1 Let {b_{k}, k ∈ ℕ} be the sequence of characteristic polynomials of a parabolic subalgebra P of the Lie algebra D̂^{−}. Then

1) b_{k}(w) ∈ ℂ[w]^{(k̄)};

2) b_{k}(w) divides wb_{k+1}(w − 1/2) and wb_{k+1}(w + 1/2) for all k ∈ ℕ;

3) b_{k+l}(w) divides wb_{k}(w − l/2)b_{l}(w + k/2) for all l, k ∈ ℕ;

4) P_{−k} ≠ 0 for all k ∈ ℕ.

In particular, all b_{k}(w) are non-zero.

Proof. Part 1) follows from the definition of the characteristic polynomials. From the commutation relation

\begin{matrix}[t, t^{−k−1}b_{k+1} (D − (k + 1)/2)] \\ = t^{−k} (b_{k+1} (D − (k + 1)/2) − b_{k+1} (D − (k − 1)/2)) \end{matrix}

we see that b_{k}(w − k/2) divides

b_{k+1} (w − (k + 1)/2) − b_{k+1} (w − (k − 1)/2) .  (3.27)

In particular b_{k+1} ≠ 0 implies b_{k} ≠ 0. From the commutation relation

[t, t^{−k−1} (D − (k + 1)/2)^{2} b_{k+1} (D − (k + 1)/2)]

= t^{−k} {{D − (k − 1)/2)^{2} (b_{k+1} (D − (k + 1)/2) − b_{k+1} (D − (k − 1)/2)}

−2 (D − k/2) b_{k+1} (D − (k + 1)/2) }

we see that b_{k} (w − k/2) divides

(w − (k − 1)/2)^{2} [b_{k+1} (w − (k + 1)/2) − b_{k+1} (w − (k − 1)/2)]
−2 (w − k/2) b_{k+1} (w − (k + 1)/2) .  (3.28)

From (3.27) and (3.28), we see that b_{k}(w) divides wb_{k+1}(w − 1/2). Noting that b_{k}(w) = ±b_{k}(−w) and b_{k+1}(w) = ±b_{k+1}(−w), b_{k}(w) also divides wb_{k+1}(w + 1/2). This proves part 2).

Part 3) can be similarly proved by calculating the following two commutators

[t^{−k}b_{k}(D − k/2), t^{−l}b_{l}(D − l/2)]

and

[t^{−k}(D − k/2)^{2}b_{k}(D − k/2), t^{−l}b_{l}(D − l/2)] .

In particular, b_{k} ≠ 0 and b_{l} ≠ 0 imply b_{k+l} ≠ 0. Part 4) follows from 2) and 3). □

Given a monic even polynomial b = b(w), denote by D̂_{0}^{−}(b) the subspace of D̂_{0}^{−} spanned by

f(D − 1/2)b(D − 1/2) − f(D + 1/2)b(D + 1/2) + f(−1/2)b(−1/2)C

where f(w) ∈ ℂ[w]^{(0)}. We have the following proposition.

Proposition 3.1 Let P be a parabolic subalgebra of D̂^{−} and let b = b_{1}(w) be its first characteristic polynomial. Then

[P, P] = D̂_{0}^{−}(b) ⨁ ⨁_{k≠0} P_{k}.

Proof. It follows from [D, t^{k}f(D)] = kt^{k}f(D) that P_{k} ⊂ [P, P] for k ≠ 0. Next

we claim [D_{1}^{−}, D_{k}^{−}] = D_{k+1}^{−} for k ∈ ℕ. Indeed, first consider the case when k is an even integer.

\begin{matrix}[t(D + 1/2)^{2l}, t^{k}(D + k/2)] \\ = & t^{k+1} ((D + k + 1/2)^{2l}(D + k/2) − (D + 1/2)^{2l}(D + k/2 + 1)) . \end{matrix} (3.29)

Note that the leading term of (D + k + 1/2)^{2l}(D + k/2) − (D + 1/2)^{2l}(D + k/2 + 1) is (2kl − 1)D^{2l}. It is clear that when l ranges over ℤ_{+}, the right hand side of (3.29) form a basis of D̂_{k+1}^{−}. The case when k is odd can be treated similarly.

Now from the fact that P_{k} = D_{k}^{−} for k ≥ 0, we have

\begin{matrix}[P_{k+1}, P_{−k−1}] ⊂ [[P_{1}, P_{k}], P_{−1−k}] \\ ⊂ & [[P_{k}, P_{−k−1}], P_{1}] + [[P_{−k−1}, P_{1}], P_{k}] ⊂ [P_{−1}, P_{1}] + [P_{−k}, P_{k}]. \end{matrix}

Hence by induction, [P, P]_{0} = [P_{1}, P_{−1}]. A direct computation shows that [P_{1}, P_{−1}] is exactly D̂_{0}^{−}(b). □

3.2 D̂^{+} case

Given a parabolic subalgebra P = ⨁_{j∈ℤ} P_{j} of D̂^{+}, we have:

P_{−k} = {t^{−k}h(D + (−k + 1)/2) | h(w) ∈ I_{−k}} (k ∈ ℕ),

where I_{−k} is some subspace of ℂ[w]^{(1)}. Given p(w), f(w) ∈ ℂ[w]^{(1)} we have: t^{−k}p(D + (−k + 1)/2), f(D + 1/2) ∈ D̂^{+}. One calculates that

\begin{matrix}[f (D + 1/2) , t^{−k}p (D + (1 − k)/2)] \\ = & t^{−k}g (D + (1 − k)/2) p (D + (1 − k)/2) , \end{matrix}  (3.30)

where g(w) = f(w −k/2)−f(w +k/2). As f ranges over all odd polynomials, g(w) ranges over all even polynomials. Thus (3.30) implies that if p(w) ∈ I_{−k} then p(w) multiplied by any even polynomial belongs to I_{−k}. Let b_{k}(w) (k ∈ ℕ) be the unique monic odd polynomial in I_{−k} of minimal degree when I_{−k} ≠ 0 and let b_{k}(w) = 0 when I_{−k} = 0. We call b_{k}(w) (k = 1, 2, …) the characteristic polynomials of P.

Lemma 3.2 Let {b_{k}, k ∈ ℕ} be the sequence of characteristic polynomials of a parabolic subalgebra P of the Lie algebra D̂^{+}. Then

1) b_{k}(w) divides w(w + (k + 1)/2)b_{k+1}(w + 1/2) for all k ∈ ℕ;

2) b_{k+l}(w) divides wb_{k}(w + l/2)b_{l}(w − k/2) for all l, k ∈ ℕ;

3) P_{−k} ≠ 0 for all k ∈ ℕ.

Proof. It follows from

\begin{matrix}[t(D + 1), t^{−k−1}b_{k+1}(D − k/2)] \\ = & t^{−k} ((D − k)b_{k+1}(D − k/2) − (D + 1)b_{k+1}(D − k/2 + 1)) \end{matrix}

that b_{k}(w + (−k + 1)/2) divides

(w − k)b_{k+1}(w − k/2) − (w + 1)b_{k+1}(w − k/2 + 1).  (3.31)

We see that b_{k}(w + (−k + 1)/2) divides

(w − k)(w − k/2)^{2}b_{k+1}(w − k/2) − (w + 1)(w − k/2 + 1)^{2}b_{k+1}(w − k/2 + 1) (3.32)

by computing [t(D + 1), t^{−k−1}(D − k/2)^{2}b_{k+1}(D − k/2)]. Thus b_{k}(w + (−k + 1)/2) divides (3.32) subtracted by (3.31) multiplied with (w − k/2)^{2} which is equal to (w + 1)(w + (−k + 1)/2)b_{k+1}(w − k/2 + 1). This proves 1). The above computation shows that b_{k+1}(w) ≠ 0 implies b_{k}(w) ≠ 0.

Part 2) can be similarly proved by computing the following two commutators

[t^{−k}b_{k}(D + (−k + 1)/2), t^{−l}b_{l}(D + (−l + 1)/2)],
[t^{−k}(D + (−k + 1)/2)^{2}b_{k}(D + (−k + 1)/2), t^{−l}b_{l}(D + (−l + 1)/2)].

Similarly it follows that b_{k}(w), b_{l}(w) ≠ 0 implies b_{k+l}(w) ≠ 0. Now part 3) follows from 1) and 2). □

Lemma 3.3 [D̂_{1}^{+}, D̂_{k}^{+}] = D̂_{k+1}^{+} (k > 1), D̂_{2}^{+} = [D̂_{1}^{+}, D̂_{1}^{+}] ⨁ ℂ t^{2}(D + 3/2).

Proof. First we have

\begin{matrix}[t(D + 1)^{l}, t^{k}(D + (k + 1)/2)^{m}] \\ = & t^{k+1} ((D + k + 1)^{l}(D + (k + 1)/2)^{m} − (D + 1)^{l}(D + (k + 3)/2)^{m}) . \end{matrix}

(3.33)

For odd positive integers l and m, t(D + 1)^{l}, t^{k}(D + (k + 1)/2)^{m} ∈ D̂^{+}. For m = 1, the leading term of the right hand side is (lk − 1)D^{l}. When k > 1, (lk − 1) ≠ 0 for l ∈ ℕ. Thus in the case k > 1 the right hand side of (3.33) span the whole D̂_{k+1}^{+} when l ranges over all odd positive integers. In the case k = 1, the right hand side of (3.33) when l ranges over all odd positive integers together with t^{2}(D + 1) span the whole D̂_{2}^{+}.

On the other hand, putting k = 1 in equation (3.33) we see that the right hand side always contains a factor (D + 1)(D + 2). So [D̂_{1}^{+}, D̂_{1}^{+}] does not contain t^{2}(D + 3/2). □

Let D̂_{0}^{+}(b_{1}, b_{2}) denote the subspace of D̂_{0}^{+} spanned by

{g(D − 1/2)b_{2}(D − 1/2) − g(D + 3/2)b_{2}(D + 3/2) + 2g(1/2)b_{2}(1/2)C,

f(D)b_{1}(D) − f(D + 1)b_{1}(D + 1), f, g ∈ ℂ[w]^{(1)}}.

Proposition 3.2 Let P be a parabolic subalgebra of D̂^{+} and let b_{i} = b_{i}(w) (i = 1, 2) be its first and second characteristic polynomials. Then

[P, P] = D̂_{0}^{+}(b_{1}, b_{2}) ⨁ ⨁_{k≠0} P_{k}.

Proof. It follows from [D, z^{k}f(D)] = kz^{k}f(D) and D ∈ D̂^{+} that P_{k} = [P, P]_{k} for k ≠ 0. Since P_{k} = D_{k}^{−} for k ≥ 0, it follows from Lemma 3.3 that for k > 1

\begin{matrix}[P_{k+1}, P_{−k−1}] ⊂ [[P_{1}, P_{k}], P_{−1−k}] \\ ⊂ & [[P_{k}, P_{−k−1}], P_{1}] + [[P_{−k−1}, P_{1}], P_{k}] ⊂ [P_{−1}, P_{1}] + [P_{−k}, P_{k}]. \end{matrix}

Hence it follows by induction that

[P, P]_{0} = [P_{1}, P_{−1}] + [P_{2}, P_{−2}].  (3.34)

A direct computation shows the right hand side of (3.34) is indeed D̂_{0}^{+}(b_{1}, b_{2}). □

Example 3.1 Let P = {W_{k}^{n} | n+k ≥ 0, k ∈ ℤ, n ∈ 2ℤ_{+} +1} and let P̂ = P ⨁ ℂC. By using (2.25) it is easy to see that P is closed under the Lie bracket and thus is a parabolic subalgebra of D^{+}. Geometrically, P consists of those differential operators in D̂^{+} which extend to the interior of the circle. It is clear from formula (2.18) that the 2-cocycle Ψ when restricted to P is trivial. Denote by M_{c} the generalized Verma module M(D̂^{+}, P, ξ_{0}) where ξ_{0} has labels Δ_{i} = 0 for all i and central charge c. Denote by V_{c} the irreducible quotient of the D̂^{+}-module M_{c}.

4 Characterization of quasifiniteness of HWM’s of D̂^{−}

Let 𝔤 = ⊕_{j∈ℤ}𝔤_{j} (possibly dim 𝔤_{j} = ∞) be a ℤ-graded Lie algebra over ℂ satisfying [𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j}, and let 𝔤_{+} = ⊕_{j>0}𝔤_{j}. A 𝔤-module V is called ℤ-graded if V = ⊕_{j∈ℤ}V_{j}, and 𝔤_{i}V_{j} ⊂ V_{j−i}. A graded 𝔤-module is called quasifinite if dim V_{j} < ∞ for all j.

Given Λ ∈ 𝔤_{0}^{∗}, a highest weight module (HWM) is a ℤ-graded 𝔤-module V (𝔤, Λ) = ⊕_{j∈ℤ_{+}}V_{j} generated by a highest weight vector v_{Λ} ∈ V_{0} which satisfies

hv_{Λ} = Λ(h)v_{Λ} (h ∈ 𝔤_{0}), 𝔤_{+}v_{Λ} = 0.

A non-zero vector v ∈ V (𝔤, Λ) is called singular if 𝔤_{+}v = 0.

A Verma module is defined as

M(𝔤; Λ) = U(𝔤) ⨂_{U(𝔤_{0}⊕𝔤_{+})} ℂ_{Λ}

where ℂ_{Λ} is the 1-dimensional (𝔤_{0} ⊕ 𝔤_{+})-module given by h ↦ Λ(h) if h ∈ 𝔤_{0} and 𝔤_{+} ↦ 0. Here and further U(𝔰) stands for the universal enveloping algebra of the Lie algebra 𝔰. Any highest weight module V (𝔤, Λ) is a quotient module of M(𝔤; Λ). The irreducible module L(𝔤; Λ) is the quotient of M(𝔤; Λ) by the maximal proper graded submodule.

Let P = ⊕_{j}P_{j} be a parabolic subalgebra of 𝔤, and let Λ ∈ 𝔤_{0}^{∗} be such that Λ|_{𝔤_{0}∩[P,P]} = 0. Then the (𝔤_{0} ⊕ 𝔤_{+})-module ℂ_{Λ} extends to a P-module by letting P_{j} act as 0 for j < 0, and we may construct the highest weight module

M(𝔤, P, Λ) = U(𝔤) ⨂_{U(P)} ℂ_{Λ}

which is usually referred to as the generalized Verma module. Clearly all the highest weight modules are graded.

In the following we consider 𝔤 = D̂^{±} and ξ ∈ (D̂^{±})_{0}^{∗}. Let b(w) be a monic even polynomial (resp. b_{1}(w), b_{2}(w) be two monic odd polynomials). Let ξ ∈ \binom{}{D_{0}^{−}}^{∗} be

such that ξ |_{_{D_{0}^{−}}(b)}= 0, (resp. ξ ∈ \binom{}{D_{0}^{+}}^{∗} be such that ξ |_{_{D_{0}^{+}}(b_{1},b_{2})}= 0). Consider a

parabolic subalgebra P of D̂^{−} (resp. D̂^{+}) whose first characteristic polynomial is b(w) (resp. whose first and second characteristic polynomials are b_{1}(w), b_{2}(w)). Denote by M(D̂^{−}; ξ, b) (resp. M(D̂^{+}; ξ, b_{1}, b_{2})) the generalized Verma module M(D̂^{−}, P, ξ) (resp. M(D̂^{+}, P, ξ)). The polynomial b (resp. b_{1}, b_{2}) does not determine P uniquely, but for our need, any corresponding parabolic P will do.

Proposition 4.1 The following conditions on ξ ∈ \binom{}{D_{0}^{±}}^{∗} are equivalent:

1) M(D̂^{−}; ξ, b) (resp. M(D̂^{+}; ξ, b_{1}, b_{2})) contains a singular vector in its first graded subspace; 2) L(D̂^{±}; ξ) is quasifinite;

3) L(D̂^{±}; ξ) is a quotient of a generalized Verma module M(D̂^{−}; ξ, b) (resp. M(D̂^{+}; ξ, b_{1}, b_{2})) for some monic even polynomial b (resp. some monic odd polynomials b_{1}, b_{2}).

Proof. We give the proof for the − case. The proof for the + case is essentially the same. 1) ⇒ 3). Denote by (t^{−1}b(D − 1/2))v_{ξ} the singular vector where b(w) is a monic even polynomial of minimal degree (note that t^{−1}b(D − 1/2) ∈ D̂_{−1}^{−}). Then it is easy to see that 3) holds for this particular monic even polynomial b. 3) ⇒ 2) and 2) ⇒ 1) follow by Lemma 3.1. □

Let L(ξ) be an irreducible QHWM over D̂^{±}. According to Proposition 4.1, we have (t^{−1}b(D − 1/2))v_{ξ} = 0 for some monic even polynomial b(w) in the − case, and (t^{−1}b(D))v_{ξ} = 0 for some monic odd polynomial b(w) in the + case. Such a monic polynomial of minimal degree is uniquely determined by ξ and is called the characteristic polynomial of L(ξ).

We shall characterize a weight ξ ∈ (D̂_{0}^{−})^{∗} (resp. ξ ∈ (D̂_{0}^{+})^{∗}) by its labels Δ_{n}^{−} = −ξ(D^{n}) (resp. Δ_{n}^{+} = −ξ((D + 1/2)^{n})), where n ∈ ℕ_{odd} = {1, 3, 5, …}, and the central charge c = ξ(C). Introduce the generating series

Δ_{ξ}^{±}(x) = ∑_{n∈ℕ_{odd}} \frac{x^{n}}{n!} Δ_{n}^{±}.

Sometimes we simply write Δ^{±}(x) instead of Δ_{ξ}^{±}(x), or even drop ± when no confusion may arise. Clearly we have

Δ^{−}(x) = −\frac{1}{2}ξ (e^{xD} − e^{−xD}) ,  (4.35)
Δ^{+}(x) = −\frac{1}{2}ξ (e^{x(D+1/2)} − e^{−x(D+1/2)}) .  (4.36)

A quasipolynomial is a finite linear combination of functions of the form p(x)e^{αx}, where p(x) is a polynomial and α ∈ ℂ. Quasipolynomials have the following wellknown simple characterization: a formal power series is a quasipolynomial if and only if it satisfies a non-trivial linear differential equation with constant coefficients. We have the following characterization of quasi-finiteness of an irreducible module L(ξ).

Theorem 4.1 A D̂^{±}-module L(D̂^{±}; ξ) is quasifinite if and only if

Δ^{±}(x) = \frac{F (x)}{2 sinh(x/2)}

where F (x) is an even quasipolynomial such that F (0) = 0.

Proof. We prove the + case first. It follows from Propositions 3.2 and 4.1 that L(D̂^{+}; ξ) is quasifinite if and only if there exist two monic odd polynomials b_{1}(w) and b_{2}(w) such that for all l ≥ 1 the following two equations hold:

ξ(D^{2l−1}b_{1}(D) − (D + 1)^{2l−1}b_{1}(D + 1)) = 0,  (4.37)

\begin{matrix}ξ ((D − 1/2)^{2l−1}b_{2}(D − 1/2) \\ −(D + 3/2)^{2l−1}b_{2}(D + 3/2) + 2(1/2)^{2l−1}b_{2}(1/2)C) = 0. \end{matrix}  (4.38)

Let

b_{1}(w) = ∑_{n=0}^{M} a_{n}w^{2n+1}, b_{2}(w) = ∑_{n=0}^{N} c_{n}w^{2n+1}.

Then we can rewrite equations (4.37) and (4.38) as

∑_{n=0}^{M} a_{n} ( ∑_{i∈ℕ_{odd}} \binom{2n + 2l}{i} (\frac{1}{2})^{2n+2l−i} Δ_{i}^{+}) = 0  (4.39)

∑_{n=0}^{M} c_{n} ( ∑_{i∈ℕ_{odd}} \binom{2n + 2l}{i}Δ_{i}^{+} + c (\frac{1}{2})^{2n+2l}) = 0.  (4.40)

Let F (x) = Δ^{+}(x) sinh \frac{x}{2} and G(x) = Δ^{+}(x) sinh x + c cosh \frac{x}{2}. It is straightforward to check that equations (4.39) and (4.40) can be equivalently reformulated as follows:

(∑_{n=0}^{M} a_{n}(\frac{d}{dx})^{2n+1})F (x) = 0

(∑_{n=0}^{N} c_{n}(\frac{d}{dx})^{2n+1})G(x) = 0.

Since G(x) = (2F (x) + c) cosh \frac{x}{2}, we see that L(D̂^{+}; ξ) is quasifinite if and only if F (x) is an even quasipolynomial.

In the − case, it follows from Propositions 3.1 and 4.1 that L(D̂^{−}; ξ) is quasifinite if and only if there exists a monic even polynomial b(w) = ∑_{n=0}^{M} β_{n}w^{2n} such that

ξ ((D − 1/2)^{2l}b(D − 1/2) − (D + 1/2)^{2l}b(D + 1/2) + (−1/2)^{2l}b(−1/2)C) = 0

for all l ∈ ℤ_{+}. As in the + case, one can show that this condition can be reformulated as

(∑_{n=0}^{M} β_{n} (d^{2}/dx^{2})^{n}) H(x) = 0.

where

H(x) = 2Δ^{−}(x) sinh(x/2) + c cosh(x/2), H(0) = c.  (4.41)

Thus L(ξ) is quasifinite if and only if H(x) is an even quasipolynomial such that H(0) = 0. By (4.41), letting F (x) = H(x) − c cosh(x/2) completes the proof. □

¿From the proof of Theorem 4.1 we obtain the following corollary.

Corollary 4.1 Let L(D̂^{+}; ξ) (resp. L(D̂^{−}; ξ)) be an irreducible quasifinite highest weight module over D̂^{+} (resp. D̂^{−}) with b(w) (resp. b_{1}(w)) as its first characteristic polynomial. Then

F (x) = Δ^{+}(x) sinh \frac{x}{2}, H(x) = 2Δ^{−}(x) sinh(x/2) + c cosh(x/2)

are even quasipolynomials. Let F^{(M)} + a_{M−1}F^{(M−1)} + ⋯ + a_{0} = 0 be the minimal order linear differential equation with constant coefficients satisfied by F (x) such that w^{M} +a_{M−1}w^{M−1}+…+a_{0} is an odd polynomial. Let H^{(N)}+β_{N−1}H^{(N−1)}+⋯+ β_{0} = 0 be the minimal order linear differential equation with constant coefficients satisfied by H(x) such that w^{N} + β_{N−1}w^{N−1} + … + β_{0} is an even polynomial. Then b(w) = w^{N} + β_{N−1}w^{N−1} + … + β_{0}, b_{1}(w) = w^{M} + a_{M−1}w^{M−1} + … + a_{0}.

Given a quasifinite irreducible highest weight D̂^{±}-module V with central charge c ∈ ℂ and with Δ(x) as in Theorem 4.1, write F (x) + c in the + case or F (x) + c cosh(x/2) in the − case as a finite sum of the form

∑_{i} p_{i}(x) cosh(e_{i}^{+}x) + ∑_{j} q_{j}(x) sinh(e_{j}^{−}x),  (4.42)

where p_{i}(x) (resp. q_{j}(x)) are non-zero even (resp. odd) polynomials and e_{i}^{+} (resp. e_{j}^{−}) are distinct complex numbers. Clearly

∑_{i} p_{i}(0) = c.  (4.43)

The expression (4.42) is unique up to a sign of e_{i}^{+} or a simultaneous change of signs of e_{j}^{−} and q_{j}(x). We call e_{i}^{+} (resp. e_{j}^{−}) the even type (resp. odd type) exponents of V with multiplicities p_{i}(x) (resp. q_{j}(x)). We denote by e^{+} the set of even type exponents e_{i}^{+} with multiplicity p_{i}(x) and by e^{−} the set of odd type exponents e_{j}^{−} with multiplicity q_{j}(x). Then the pair (e^{+}, e^{−}) determines V uniquely. We shall therefore denote this module by L(D̂^{±}; e^{+}, e^{−}). As we shall see, the following class of D̂^{±}-modules is especially important.

Definition 4.1 A quasifinite irreducible highest weight D̂^{±}-module V with central charge c ∈ ℂ is called primitive if the multiplicities of its exponents e_{i}^{+} are nonzero constants n_{i} ∈ ℂ and e^{−} = ∅. A primitive D̂^{−}-module V is called positive if n_{i} ∈ ℕ when e_{i} ≠ ±\frac{1}{2} and n_{i} ∈ \frac{1}{2}ℕ when e_{i} = ±\frac{1}{2}. A primitive D̂^{+}-module V is called positive if n_{i} ∈ ℕ when e_{i} ≠ 0 and −\frac{1}{2}n_{i_{0}} ≤ n_{i} ∈ \frac{1}{2}ℤ when e_{i} = 0, where i_{0} is the index such that e_{i_{0}} = 1. (In both ± cases the central charge c = ∑_{i} n_{i} ∈ \frac{1}{2}ℕ).

It is convenient to make the following convention.

Convention 4.1 For a primitive module V of D̂^{±}, we let e stand for the set of (even type) exponents with their multiplicities in e^{+} which are not equal to ±\frac{1}{2} (resp. 0) in the − (resp. +) case. The pair (e, c) determines uniquely the module V. We will denote this primitive module by L(D̂^{±}; e, c).

5 Embedding of D̂^{−} into infinite rank classical Lie algebras

Let O be the algebra of all holomorphic functions on ℂ with topology of uniform convergence on compact sets. Denote

O^{(1)} = {f ∈ O | f(w) = −f(−w)}
O^{(0)} = {f ∈ O | f(w) = f(−w)}.

We define a completion D^{O} of D consisting of all differential operators of the form t^{j}f(D) where f ∈ O and j ∈ ℤ. We similarly define a completion D^{O,−} (resp. D^{O,+}) of D^{−} (resp. D^{+}) consisting of all differential operators of the form t^{j}f(D) (resp. t^{j}f(D + j/2)) where f ∈ O (resp. O^{(j̄)}) and j ∈ ℤ.

Note that t^{j}f(D) acts on ℂ[t, t^{−1}] by

t^{j}f(D)t^{k} = f(k)t^{k+j}.

Formula (2.17) for the 2-cocycle Ψ on D (resp. D^{−}) extends to a 2-cocycle on D^{O} (resp. D^{O,−}). We denote the corresponding central extension by D̂^{O} = D^{O} ⨁ ℂC and D̂^{O,−} = D^{O,−} ⨁ ℂC. The commutation relations (2.16) extend as well.

The vector space R_{m}[t, t^{−1}]t^{s} (s ∈ ℂ) has a basis v_{i} = t^{−i+s} (i ∈ ℤ) over R_{m}. The Lie algebra 𝔤𝔩^{[m]} acts on this vector space by (1.6). The Lie algebras D and D^{O} also act on R_{m}[t, t^{−1}] naturally as differential operators. In this way we obtain a family of embeddings ϕ_{s}^{[m]} of the Lie algebra D (resp. D^{O}) to 𝔤𝔩^{[m]} defined by

ϕ_{s}^{[m]}(t^{k}f(D)) = ∑_{j∈ℤ} f(−j + s + u)E_{j−k,j} = ∑_{i=0}^{m} ∑_{j∈ℤ} \frac{f^{(i)}(−j + s)}{i!}u^{i}E_{j−k,j}

where f^{(i)} denotes the i-th derivative. When restricted to D^{−} and D^{O,−}, we have

ϕ_{s}^{[m]} (t^{k}f(D + k/2)) = ∑_{i=0}^{m} ∑_{j∈ℤ} \frac{f^{(i)}(−j + k/2 + s)}{i!}u^{i}E_{j−k,j} for f ∈ ℂ[w]^{(k̄)}. (5.44)

When restricted to D^{+} and D^{O,+}, we have

ϕ_{s}^{[m]} (t^{k}f(D + (k + 1)/2)) = ∑_{i=0}^{m} ∑_{j∈ℤ} \frac{f^{(i)}(−j + (k + 1)/2 + s)}{i!}u^{i}E_{j−k,j} (5.45)

for f ∈ O^{(1)}.

Remark 5.1 The principal ℤ-gradations on D^{±} and 𝔤𝔩^{[m]} are compatible under the homomorphisms ϕ_{s}^{[m]}.

Let

I_{s,k}^{[m],−} = {f ∈ O^{(k̄)} | f^{(i)}(n + k/2 + s) = 0 for all n ∈ ℤ, i = 0, 1, ..., m}
I_{s,k}^{[m],+} = {f ∈ O^{(1)} | f^{(i)}(n + (k + 1)/2 + s) = 0 for all n ∈ ℤ, i = 0, 1, ..., m}

and let

J_{s}^{[m],−} = ⨁_{k∈ℤ}{t^{k}f(D + k/2) | f ∈ I_{s,k}^{[m],−}},
J_{s}^{[m],+} = ⨁_{k∈ℤ}{t^{k}f(D + (k + 1)/2) | f ∈ I_{s,k}^{[m],+}}.

Now fix s⃗ = (s_{1}, s_{2}, …, s_{N}) ∈ ℂ^{N}, s_{i} − s_{j} ∉ ℤ if i ≠ j, s_{i} + s_{j} ∉ ℤ for all i, j. Also fix m⃗ = (m_{1}, m_{2}, …, m_{N}) ∈ ℤ_{+}^{N}. Let

𝔤𝔩^{[m⃗]} = ⨁_{i=1}^{N} 𝔤𝔩^{[m_{i}]}

and consider the homomorphism

ϕ[m⃗] ; s⃗ = ⨁_{i=1}^{N} ϕ_{s_{i}}^{[m_{i}],±} : D^{O,±} ⟶ 𝔤𝔩^{[m⃗]}.

Proposition 5.1 Given s⃗ and m⃗ as above, we have the following exact sequence of Lie algebras:

0 ⟶ J [m⃗],± ; s⃗ ⟶ D^{O,±} ⟶^{ϕ[m⃗] ; s⃗} 𝔤𝔩^{[m⃗]} ⟶ 0

where J [m⃗],± ; s⃗ = ∩_{i=1}^{N}J_{s_{i}}^{[m_{i}],±}.

Proof. We will prove the proposition only in the − case. The proof in the + case is parallel. For the sake of simplicity of notations, we prove it in the case N = 1: m⃗ = m ∈ ℤ_{+} and s⃗ = s ∈ ℂ (s ∉ ℤ/2 by the assumption on s⃗). The general case is similar.

It is clear from the definition of J_{s}^{[m]} that ker ϕ_{s}^{[m]} = J_{s}^{[m]}. To show the surjectivity of ϕ_{s}^{[m]}, it suffices to find a preimage of

g = ∑_{i=0}^{m} ∑_{j∈ℤ} \frac{p_{ij}}{i!} u^{j}E_{j−k,j} ∈ 𝔤𝔩^{[m]} (p_{ij} ∈ ℂ)

for a fixed k ∈ ℤ. We need to quote the following well-known theorem: for every discrete sequence of points in ℂ and a non-negative integer m there exists p(w) ∈ O having prescribed values of its first m derivatives at these points.

Since s ∉ ℤ/2, the two sequences \{−j + \frac{k}{2} +s\}_{j∈ℤ} and \{j − \frac{k}{2} −s\}_{j∈ℤ} are disjoint. Thus there exists p(w) ∈ O such that

p^{(i)}(−j + k/2 + s) = p_{ij}, p^{(i)}(j − k/2 − s) = (−1)^{k+i+1}p_{ij}.

Now let

f(w) = \frac{p(w) − (−1)^{k}p(−w)}{2} ∈ O^{(k̄)}.

Then t^{k}f(D + k/2) is the preimage of g via ϕ_{s}^{[m]}. □

Now we want to extend the homomorphism ϕ_{s}^{[m]} to a homomorphism between the central extensions of the corresponding Lie algebras. Introduce the following functions:

η_{j}(x; μ) = \frac{1}{2j!}x^{j}(e^{μx} + (−1)^{j}e^{−μx}) (j ∈ ℤ_{+}, μ ∈ ℂ).

The functions η_{j}(x; μ) satisfy:

η_{j}(−x; μ) = η_{j}(x; μ), η_{j}(x; −μ) = (−1)^{j}η_{j}(x; μ), η_{0}(x; μ) = cosh(μx). (5.46)

Note that, being an even quasipolynomial, F (x) in Theorem 4.1 is a finite linear combination of the functions η_{j}(x; μ).

Proposition 5.2 1) The ℂ-linear map ϕ̂_{s}^{[m]} : D̂^{+} ⟶ 𝔤̂𝔩^{[m]} defined by

ϕ_{s}^{[m]}(C) = 1  (5.47)
ϕ̂_{s}^{[m]}|_{D_{j}^{+}} = ϕ_{s}^{[m]}|_{D_{j}^{+}} if j ≠ 0
ϕ̂_{s}^{[m]} (e^{x(D+1/2)} − e^{−x(D+1/2)}) = ϕ_{s}^{[m]} (e^{x(D+1/2)} − e^{−x(D+1/2)})
−\frac{cosh sx − 1}{sinh \frac{x}{2}}1 − ∑_{j=1}^{m} \frac{u^{j}η_{j}(x; s)}{2 sinh \frac{x}{2}} (5.48)

is a homomorphism of Lie algebras over ℂ.

2) The ℂ-linear map ϕ̂_{s}^{[m]} : D̂^{−} ⟶ 𝔤̂𝔩^{[m]} defined by

ϕ̂_{s}^{[m]}(C) = 1 ∈ R_{m}, ϕ̂_{s}^{[m]} |_{D_{j}^{−}}= ϕ_{s}^{[m]} |_{D_{j}^{−}} (j ≠ 0),  (5.49)

ϕ̂_{s}^{[m]} (e^{xD} − e^{−xD}) = ϕ_{s}^{[m]} (e^{xD} − e^{−xD}) − \frac{cosh(s − 1/2)x − cosh(x/2)}{sinh(x/2)}1
− ∑_{j=1}^{m} \frac{u^{j}η_{j}(x; s − 1/2)}{sinh(x/2)}  (5.50)

is a homomorphism of Lie algebras over ℂ.

Proof. We will prove part 1). The proof of part 2) is similar. Part 1) follows directly from the computation of the difference

ϕ̂_{s}^{[m]} (e^{x(D+1/2)} − e^{−x(D+1/2)})

by using the following lemma (also cf. Proposition 4.4, [KR1]). □

Lemma 5.1 The ℂ-linear map ϕ̂_{s}^{[m]} : D̂ ⟶ 𝔤̂𝔩^{[m]} defined by

ϕ_{s}^{[m]}(C) = 1
ϕ̂_{s}^{[m]}|_{D_{j}} = ϕ_{s}^{[m]}|_{D_{j}} if j ≠ 0
ϕ̂_{s}^{[m]} (e^{x(D+1/2)}) = ϕ_{s}^{[m]} (e^{x(D+1/2)})
−\frac{e^{(s+1/2)x} − e^{x/2}}{e^{x} − 1} 1 − ∑_{i=1}^{m} \frac{x^{i}e^{(s+1/2)x}}{e^{x} − 1} u^{i}/i!.  (5.51)

is a homomorphism of Lie algebras over ℂ.

Proof. Introduce two formal variables α, β and let x = α + β. It suffices to check that for a given k ∈ ℕ

[ϕ̂_{s}^{[m]}(t^{k}e^{α(D+1/2)}), ϕ̂_{s}^{[m]}(t^{−k}e^{β(D+1/2)})] = ϕ̂_{s}^{[m]} [(t^{k}e^{α(D+1/2)}), (t^{−k}e^{β(D+1/2)})] . (5.52)

By a straightforward computation using (5.45) we obtain

\begin{matrix}[ϕ̂_{s}^{[m]}(t^{k}e^{α(D+1/2)}), ϕ̂_{s}^{[m]}(t^{−k}e^{β(D+1/2)})] \\ = & [ϕ_{s}^{[m]}(t^{k}e^{α(D+1/2)}), ϕ_{s}^{[m]}(t^{−k}e^{β(D+1/2)})] \\ = & (e^{−αk} − e^{βk}) (∑_{j∈ℤ} e^{x(−j+s+1/2)}E_{j,j} + e^{xu}e^{x(s+1/2)} \frac{1}{1−e^{x}} ) \\ = & (e^{−αk} − e^{βk}) (ϕ_{s}^{[m]} (e^{x(D+1/2)}) + \frac{e^{x(s+1/2)}}{1−e^{x}} 1 + ∑_{j+1}^{m} \frac{x^{i}e^{x(s+1/2)}}{1−e^{x}} u^{i}/j!) . \end{matrix} (5.53)

On the other hand, using (2.16) we have

[t^{k}e^{α(D+1/2)}, t^{−k}e^{β(D+1/2)}] = (e^{−αk} − e^{βk}) (e^{x(D+1/2)} + \frac{e^{x/2}}{1 − e^{x}} C) . (5.54)

By applying ϕ̂_{s}^{[m]} to (5.54) and then comparing with (5.53) with the help of (5.51), we obtain (5.52). □

The homomorphism ϕ_{s}^{[m]} : D^{O,−} ⟶ 𝔤𝔩^{[m]} is defined for any s ∈ ℂ. However for s ∈ ℤ/2, it is no longer surjective. The case s = 0 is described by the following proposition.

Proposition 5.3 We have the following exact sequence of Lie algebras:

0 ⟶ J_{0}^{[m],−} ⟶ D^{O,−} ⟶^{ϕ_{0}^{[m]}} \overline{b}_{∞}^{−[m]} ⟶ 0

0 ⟶ J_{0}^{[m],+} ⟶ D^{O,+} ⟶^{ϕ_{0}^{[m]}} \overline{d}_{∞}^{[m]} ⟶ 0.

Proof. By the definition of ϕ_{0}^{[m]}, it is easy to see that the image of ϕ_{0}^{[m]} lies in \overline{b}_{∞}^{−[m]} (resp. \overline{d}_{∞}^{[m]}). The proof of the rest of the proposition is similar to that of Proposition 5.1. □

Similarly for s = \frac{1}{2} we have the following proposition.

Proposition 5.4 We have the following exact sequence of Lie algebras:

0 ⟶ J_{1/2}^{[m],−} ⟶ D^{O,−} ⟶^{ϕ_{1/2}^{[m]}} \overline{c}_{∞}^{[m]} ⟶ 0

0 ⟶ J_{−1/2}^{[m],+} ⟶ D^{O,+} ⟶^{ϕ_{−1/2}^{[m]}} \overline{b}_{∞}^{+[m]} ⟶ 0.

Remark 5.2 For s ∈ ℤ, the image of D̂^{−} (resp. D̂^{+}) under the homomorphism ϕ_{s}^{[m]} is ν^{s}(\overline{b}_{∞}^{−[m]}) (resp. ν^{s}(\overline{d}_{∞}^{[m]})) (recall that ν is defined in (1.1) ). For s ∈ \frac{1}{2} + ℤ, the image of D̂^{−} (resp. D̂^{+}) under ϕ_{s}^{[m]} is ν^{s−1/2}(\overline{c}_{∞}^{[m]}) (resp. ν^{s+1/2}(\overline{b}_{∞}^{+[m]})). Hence we will only need to consider s = 0, \frac{1}{2} in the − case and s = 0, −\frac{1}{2} in the + case whenever s ∈ ℤ/2 throughout the paper. Note that the principal ℤ-gradation of D̂^{+} is compatible with the gradation of type (2, 1, 1, …) on \overline{d}_{∞} via the homomorphism ϕ_{0}.

Definition 5.1 We say that the vector s⃗ = (s_{1}, s_{2}, …, s_{N}) ∈ ℂ^{N} satisfies the (⋆−) (resp. (⋆+)) condition if s_{i} ∈ ℤ implies s_{i} = 0, s_{i} ∈ \frac{1}{2} + ℤ implies s_{i} = \frac{1}{2} (resp. −\frac{1}{2}), and s_{i} ≠ ±s_{j} mod ℤ for i ≠ j.

Given m⃗ = (m_{1}, …, m_{N}) ∈ ℤ_{+}^{N} and s⃗ = (s_{1}, s_{2}, …, s_{N}) ∈ ℂ^{N} satisfying the (⋆−) (resp. (⋆+)) condition, combining Propositions 5.1, 5.3 and 5.4, we obtain a homomorphism of Lie algebras over ℂ:

ϕ̂[m⃗] ; s⃗ = ⨁_{i=1}^{N} ϕ̂_{s_{i}}^{[m_{i}]} : D̂^{±} ⟶ 𝔤^{[m⃗]} := ⨁_{i=1}^{N} 𝔤^{[m_{i}]},  (5.55)

where the following consistency condition is always assumed throughout the paper: in the − case,

𝔤^{[m_{i}]} = \{ \begin{matrix}b_{∞}^{[m_{i}]}, & if s_{i} = 0 \\ c_{∞}^{[m_{i}]}, & if s_{i} = \frac{1}{2} \\ 𝔤̂𝔩^{[m_{i}]}, & if s_{i} ∉ ℤ/2, \end{matrix}

while in the + case,

𝔤^{[m_{i}]} = \{ \begin{matrix}d_{∞}^{[m_{i}]}, & if s_{i} = 0 \\ b̃_{∞}^{[m_{i}]}, & if s_{i} = −\frac{1}{2} \\ 𝔤̂𝔩^{[m_{i}]}, & if s_{i} ∉ ℤ/2. \end{matrix}

Furthermore, we can prove the following proposition in the same way as Proposition 5.1.

Proposition 5.5 The homomorphism ϕ̂[m⃗] ; s⃗ extends to a surjective homomorphism of Lie algebras over ℂ which is denoted again by ϕ̂[m⃗] ; s⃗:

ϕ̂[m⃗] ; s⃗ = ⨁_{i=1}^{N} ϕ̂_{s_{i}}^{[m_{i}]} : D̂^{O,±} ⟶ 𝔤^{[m⃗]}.

6 Realization of QHWM’s of D̂^{±}

Let 𝔤^{[m]} stand for 𝔤̂𝔩^{[m]}, or one of its classical Lie subalgebras. The proof of the following simple proposition is standard.

Proposition 6.1 The 𝔤^{[m]}-module L(𝔤^{[m]}; Λ) is quasifinite if and only if all but finitely many of the ^{∗}h_{k}^{(j)} are zero, where ∗ represents a, b, c or d depending on whether 𝔤^{[m]} is 𝔤̂𝔩^{[m]}, b_{∞}^{[m]}, b̃_{∞}^{[m]}, c_{∞}^{[m]} or d_{∞}^{[m]}.

Take a quasifinite Λ(i) ∈ (𝔤_{0}^{[m_{i}]})^{∗} for each i = 1, …, N, and let L (𝔤^{[m_{i}]}; Λ(i)) be the corresponding irreducible 𝔤^{[m_{i}]}-module. Then the outer tensor product

L (𝔤^{[m⃗]}; Λ⃗) ≡ ⨂_{i=1}^{N} L (𝔤^{[m_{i}]}; Λ(i))

is an irreducible 𝔤^{[m⃗]}-module. The module L (𝔤^{[m⃗]}; Λ⃗) can be regarded as a D̂^{±}-

module via the homomorphism ϕ̂[m⃗] ; s⃗ given by (5.55), which we shall denote by

L[m⃗],± ; s⃗(Λ⃗).

We will need a technical lemma whose proof is analogous to that of Proposition 4.3 in [KR1].

Lemma 6.1 Let V be a quasifinite D̂^{±}-module. Then the action of D̂^{±} on V naturally extends to the action of D̂_{k}^{O,±} on V for any k ≠ 0.

Theorem 6.1 Let V be a quasifinite 𝔤^{[m⃗]}-module, which can be regarded as a quasifinite D̂^{±}-module via the homomorphism ϕ̂[m⃗] ; s⃗. Then any D̂^{±}-submodule of V is also a 𝔤^{[m⃗]}-submodule. In particular, the D̂^{±}-modules L[m⃗],± ; s⃗(Λ⃗) are irreducible if s⃗ = (s_{1}, s_{2}, …, s_{N}) satisfies the (⋆±) condition.

Proof. Take any D̂^{±}-submodule W of V. By the Lemma 6.1 we can extend the action of D̂^{±} to D̂_{j}^{O,±} (j ≠ 0). Then by Proposition 5.5, we see that the subspace W is preserved by the action of the graded subspace 𝔤_{j}^{[m⃗]} (j ≠ 0) of 𝔤^{[m⃗]}. Since 𝔤^{[m⃗]} coincides with its derived algebra, W is preserved by the action of the whole 𝔤^{[m⃗]}.

□

We will show that in fact all the quasifinite D̂^{±}-modules can be realized as some L[m⃗],± ; s⃗(Λ⃗). But first let us calculate the generating function Δ_{m,s,Λ}^{±}(x) of highest weight for L[m⃗],± ; s⃗(Λ⃗) in some typical cases.

Proposition 6.2 Consider the embedding ϕ̂_{s}^{[m]} : D̂^{−} ⟶ 𝔤̂𝔩^{[m]} with s ∉ ℤ/2. The 𝔤̂𝔩^{[m]}-module L(𝔤̂𝔩^{[m]}; Λ) regarded as a D̂^{−}-module is isomorphic to L(D̂^{−}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents s − i − 1/2 (i ∈ ℤ) with multiplicities

∑_{0≤j≤m, j even}^{a}h_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{a}h_{i}^{(j)}x^{j}/j!  (6.56)

respectively (see Section 1.1 for notations; the exponents with zero multiplicities are dropped).

Proof. By Theorem 6.1 and Proposition 6.1 the D^{−}-module L[m⃗],− ; s⃗(λ⃗) is an irreducible quasifinite highest weight module. By formula (5.49) we see the central charge c = c_{0}. By applying Λ to (5.50) and using formulas (4.36) and (5.44) we obtain:

2Δ_{m,s,Λ}^{−}(x) = − ∑_{j=0}^{m} ∑_{i∈ℤ}(^{a}λ_{i}^{(j)}/j!)x^{j} (e^{(s−i)x} − (−1)^{j}e^{(i−s)x})

+c_{0} \frac{cosh(s − 1/2)x − cosh(x/2)}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; s − 1/2)}{sinh(x/2)}

= − ∑_{j=0}^{m} ∑_{i∈ℤ}(^{a}λ_{i}^{(j)}/j!)\frac{(η_{j}(x; s − i + 1/2) − η_{j}(x; s − i − 1/2))}{sinh(x/2)}

+c_{0} \frac{cosh(s − 1/2)x − cosh(x/2)}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; s − 1/2)}{sinh(x/2)}

=^{(1)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} ^{a}h_{i}^{(j)}η_{j}(x; s − i − 1/2)}{sinh(x/2)} − \frac{c_{0} cosh(x/2)}{sinh(x/2)}.

The identity (1) above is obtained by shifting the index i to i+1 in the first half of the first summation of the left hand side of (1). Now the proposition follows from the definition of exponents and their multiplicities. □

Proposition 6.3 Consider the embedding ϕ̂_{0}^{[m]} : D̂^{−} ⟶ b_{∞}^{[m]}. The b_{∞}^{[m]}-module L(b_{∞}^{[m]}; Λ) regarded as a D̂^{−}-module is isomorphic to L(D̂^{−}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents −i − 1/2 (i ∈ ℤ_{+}) with multiplicities

∑_{0≤j≤m, j even}^{b}h̃_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{b}h̃_{i}^{(j)}x^{j}/j!  (6.57)

respectively, where ^{b}h̃_{i}^{(j)} = ^{b}h_{i}^{(j)} (i > 0) and ^{b}h̃_{0}^{(j)} = \frac{1}{2} ^{b}h_{0}^{(j)} (see Section 1.2 for notations).

Proof. We will only need to calculate Δ_{m,s,Λ}^{−}(x). The rest of the statement is clear, cf. the proof of Proposition 6.2. We have:

2Δ_{m,s,Λ}^{−}(x) =^{(1)} − ∑_{j=0}^{m} ∑_{i∈ℤ}(Λ(u^{j}E_{ii})/j!)\frac{(η_{j}(x; −i + 1/2) − η_{j}(x; −i − 1/2))}{sinh(x/2)}

+ ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; −1/2)}{sinh(x/2)}

=^{(2)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} Λ(u^{j}E_{ii} − u^{j}E_{i+1,i+1})η_{j}(x; −i − 1/2)}{sinh(x/2)}

+ ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; −1/2)}{sinh(x/2)}

=^{(3)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ_{+}} ^{b}h̃_{i}^{(j)}η_{j}(x; −i − 1/2)}{sinh(x/2)} − \frac{c_{0} cosh(x/2)}{sinh(x/2)}.

The identity (1) is obtained from a similar identity in the proof of Proposition 6.2 by putting s = 0. The identity (2) is obtained by shifting the index i to i + 1 in the first half of the first summation of the left hand side of (2). The identity (3) is obtained by splitting the summation into two, ∑_{i∈ℤ} = ∑_{i∈ℤ_{+}} + ∑_{i∈−ℕ}, changing the index i to −i − 1 in the second summation, and then using formulas (1.7) and (5.46). □

Proposition 6.4 Consider the embedding ϕ̂_{1/2}^{[m]} : D̂^{−} ⟶ c_{∞}^{[m]}. The c_{∞}^{[m]}-module L(c_{∞}^{[m]}; Λ) regarded as a D̂^{−}-module is isomorphic to L(D̂^{−}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents s − i − 1/2 (i ∈ ℤ_{+}) with multiplicities

∑_{0≤j≤m, j even}^{c}h_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{c}h_{i}^{(j)}x^{j}/j!  (6.58)

respectively (see Section 1.3 for more notations).

Proof. It suffices to calculate Δ_{m,s,Λ}^{−}(x):

2Δ_{m,s,Λ}^{−}(x) =^{(1)} − ∑_{j=0}^{m} ∑_{i∈ℤ}(Λ(u^{j}E_{ii})/j!)\frac{(η_{j}(x; −i + 1) − η_{j}(x; −i))}{sinh(x/2)}

+c_{0} \frac{1 − cosh(x/2)}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; 0)}{sinh(x/2)}

=^{(2)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} Λ(u^{j}E_{ii} − u^{j}E_{i+1,i+1})η_{j}(x; −i)}{sinh(x/2)}

+c_{0} \frac{1 − cosh(x/2)}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; 0)}{sinh(x/2)}

=^{(3)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ_{+}} ^{c}h_{i}^{(j)}η_{j}(x; −i)}{sinh(x/2)} − \frac{c_{0} cosh(x/2)}{sinh(x/2)}.

The identity (1) is obtained from a similar identity in the proof of Proposition 6.2 by putting s = \frac{1}{2}. The identity (2) is obtained by shifting the index i to i + 1 in the first half of the first summation of the left hand side of (2). The identity (3) is obtained by splitting the summation into two, ∑_{i∈ℤ} = ∑_{i∈ℤ_{+}} + ∑_{i∈−ℕ}, changing the index i to −i in the second summation, and then using formulas(1.11) and (5.46). □

Proposition 6.5 Consider the embedding ϕ̂_{s}^{[m]} : D̂^{+} ⟶ 𝔤̂𝔩^{[m]} with s ∉ ℤ/2. The 𝔤̂𝔩^{[m]}-module L(𝔤̂𝔩^{[m]}; Λ) regarded as a D̂^{+}-module is isomorphic to L(D̂^{+}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents s − i (i ∈ ℤ) with multiplicities

∑_{0≤j≤m, j even}^{a}h_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{a}h_{i}^{(j)}x^{j}/j!  (6.59)

respectively (see Section 1.1 for notations).

Proof. By Theorem 6.1 and Proposition 6.1 L[m⃗],+ ; s⃗(λ⃗) is an irreducible quasifinite highest weight D̂^{+}-module. By formula (5.47) we see the central charge c = c_{0}. By applying Λ to (5.48) and using formulas (4.36) and (5.45) we obtain

2Δ_{m,s,Λ}^{+}(x) = − ∑_{j=0}^{m} ∑_{i∈ℤ}(^{a}λ_{i}^{(j)}/j!)x^{j} (e^{(s−i+1/2)x} − (−1)^{j}e^{(i−s−1/2)x})

+c_{0} \frac{cosh(sx) − 1}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; s)}{sinh(x/2)}

= − ∑_{j=0}^{m} ∑_{i∈ℤ}(^{a}λ_{i}^{(j)}/j!)\frac{(η_{j}(x; s − i + 1) − η_{j}(x; s − i))}{sinh(x/2)}

+c_{0} \frac{cosh(sx) − 1}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; s)}{sinh(x/2)}

=^{(1)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} ^{a}h_{i}^{(j)}η_{j}(x; s − i)}{sinh(x/2)} − \frac{c_{0}}{sinh(x/2)}.

The identity (1) above is obtained by shifting the index i to i + 1 in the first half of the first summation of the left hand side of (1). This completes the proof of this proposition. □

Proposition 6.6 Consider the embedding ϕ̂_{0}^{[m]} : D̂^{+} ⟶ d_{∞}^{[m]}. The d_{∞}^{[m]}-module L(d_{∞}^{[m]}; Λ) regarded as a D̂^{+}-module is isomorphic to L(D̂^{−}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents −i (i ∈ ℤ_{+}) with multiplicities

∑_{0≤j≤m, j even}^{d}h̃_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{d}h̃_{i}^{(j)}x^{j}/j!  (6.60)

respectively, where ^{d}h̃_{i}^{(j)} = ^{d}h_{i}^{(j)} (i > 0) and ^{d}h̃_{0}^{(j)} = \frac{1}{2} (^{d}h_{0}^{(j)} −^{d}h_{1}^{(j)}) (see Section 1.4 for notations).

Proof. We will only need to calculate Δ_{m,s,Λ}^{+}(x). The rest of the statement is clear, cf. the proof of Proposition 6.2.

2Δ_{m,s,Λ}^{+}(x) =^{(1)} − ∑_{j=0}^{m} ∑_{i∈ℤ}(Λ(u^{j}E_{ii})/j!)\frac{(η_{j}(x; −i + 1) − η_{j}(x; −i))}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; 0)}{sinh(x/2)}

=^{(2)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} Λ(u^{j}E_{ii} − u^{j}E_{i+1,i+1})η_{j}(x; −i)}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; 0)}{sinh(x/2)}

=^{(3)} \frac{∑_{j=0}^{m} (∑_{i∈ℕ} ^{d}h_{i}^{(j)}η_{j}(x; −i) + \frac{1}{2}(h_{0}^{(j)} − h_{1}^{(j)})η_{j}(x; 0))}{sinh(x/2)} − \frac{c_{0}}{sinh(x/2)}.

The identity (1) is obtained from a similar identity in the proof of Proposition 6.2 by putting s = 0. The identity (2) is obtained by shifting the index i to i + 1 in the first half of the first summation of the left hand side of (2). The identity (3) is obtained by splitting the summation into two, ∑_{i∈ℤ} = ∑_{i∈ℤ_{+}} + ∑_{i∈−ℕ}, changing the index i to −i in the second summation, and then using formulas (1.14) and (5.46). □

Proposition 6.7 Consider the embedding ϕ̂_{−1/2}^{[m]} : D̂^{+} ⟶ b̃_{∞}^{[m]}. The b̃_{∞}^{[m]}-module L(b̃_{∞}^{[m]}; Λ) regarded as a D̂^{+}-module is isomorphic to L(D̂^{−}; e^{+}, e^{−}) where e^{+} and e^{−} consist of the exponents −i − 1/2 (i ∈ ℤ_{+}) with multiplicities

∑_{0≤j≤m, j even}^{b}h̃_{i}^{(j)}x^{j}/j! and ∑_{0≤j≤m, j odd}^{b}h̃_{i}^{(j)}x^{j}/j!  (6.61)

respectively, where ^{b}h̃_{i}^{(j)} = ^{b}h_{i}^{(j)} (i > 0) and ^{b}h̃_{0}^{(j)} = \frac{1}{2} ^{b}h_{0}^{(j)} (see Section 1.2 for notations).

Proof. Again it suffices to calculate Δ_{m,s,Λ}^{+}(x).

2Δ_{m,s,Λ}^{+}(x) =^{(1)} − ∑_{j=0}^{m} ∑_{i∈ℤ}(Λ(u^{j}E_{ii})/j!)\frac{(η_{j}(x; −i + 1/2) − η_{j}(x; −i − 1/2))}{sinh(x/2)}

+c_{0} \frac{cosh(x/2) − 1}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; −1/2)}{sinh(x/2)}

=^{(2)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ} Λ(u^{j}E_{ii} − u^{j}E_{i+1,i+1})η_{j}(x; −i − 1/2)}{sinh(x/2)}

+c_{0} \frac{cosh(x/2) − 1}{sinh(x/2)} + ∑_{j=1}^{m} \frac{c_{j}η_{j}(x; −1/2)}{sinh(x/2)}

=^{(3)} \frac{∑_{j=0}^{m} ∑_{i∈ℤ_{+}} ^{b}h̃_{i}^{(j)}η_{j}(x; −i − 1/2)}{sinh(x/2)} − \frac{c_{0}}{sinh(x/2)}.

The identity (1) is obtained from a similar identity in the proof of Proposition 6.2 by putting s = −\frac{1}{2}. The identity (2) is obtained by shifting the index i to i + 1 in the first half of the first summation of the left hand side of (2). The identity (3) is obtained by splitting the summation into two, ∑_{i∈ℤ} = ∑_{i∈ℤ_{+}} + ∑_{i∈−ℕ}, changing the index i to −i − 1 in the second summation, and then using formulas(1.7) and (5.46). □

Take an irreducible quasifinite highest weight D̂^{−}-module V with central charge c and

Δ(x) = \frac{F (x)}{2 sinh(x/2)}

where F (x) is an even quasipolynomial such that F (0) = 0 (see Theorem 4.1). We may write

F (x) + c cosh(x/2) = ∑_{s∈ℂ} ∑_{j=0}^{m_{s}} a_{s,j}η_{j}(x; s − 1/2)  (6.62)

where a_{s,j} ∈ ℂ and a_{s,j} ≠ 0 for only finitely many s ∈ ℂ. We can fix the ambiguities in expressing F (x) in the form (6.62) caused by the symmetries (5.46) by the following rules in choosing the parameter s: when s ∈ ℤ, we require s ≤ 0; when s ∈ \frac{1}{2} + ℤ, we require s ≤ \frac{1}{2}; when s ∉ \frac{1}{2}ℤ, we require that Im s > 0 if Im s ≠ 0, or s − [s] < \frac{1}{2} if s ∈ ℝ, where Im s denotes the imaginary part of s and [s] the closest integer to s which is not larger than s.

Decompose the set {s ∈ ℂ | a_{s,j} ≠ 0 for some j} into a disjoint union of equivalence classes under the equivalence condition: s ∼ s′ if and only if s = ±s′ ( mod ℤ). Pick a representative s in an equivalence class S such that s = 0 if the equivalence class lies in ℤ and s = \frac{1}{2} if the equivalence class lies in \frac{1}{2} + ℤ. Let S = {s, s − k_{1}, s − k_{2}, …} be such an equivalence class and let m = max_{s∈S} m_{s}. Put k_{0} = 0. It is easy to see by the rules in picking the parameter s that if s = 0 or \frac{1}{2}, then k_{1}, k_{2}, ⋯ ∈ ℕ.

We associate to S the 𝔤^{[m]}-module L_{s}^{[m]}(λ_{s}) in one of the following three ways: First, if s ∉ ℤ/2, let ^{a}h_{k_{r}}^{(j)} = a_{s+k_{r},j} (j = 0, …, m_{s}, r = 0, 1, 2, …). We associate

to S a 𝔤̂𝔩^{[m]}-module L_{s}^{[m]}(λ_{S}) with central charges and labels

c_{j} = ∑_{k_{r}} ^{a}h_{k_{r}}^{(j)}, ^{a}λ_{i}^{(j)} = ∑_{k_{r}≥i}^{a}h̃_{k_{r}}^{(j)}

where ^{a}h̃_{k}^{(j)} = ^{a}h_{k}^{(j)} − c_{j}δ_{k,0}.

Secondly, if s = 0, let ^{b}h̃_{k_{r}}^{(j)} = a_{k_{r},j}(j = 0, …, m_{0}, r ∈ ℤ_{+}). We associate to S a b_{∞}^{[m_{0}]}-module L_{0}^{[m_{0}]}(λ_{S}) with central charges and labels

c_{j} = ∑_{k_{r}} ^{b}h̃_{k_{r}}^{(j)} (j even), c_{j} = 0 (j odd)
bλ_{0}^{(j)} = ∑_{k_{r}≥0} h_{k_{r}}^{(j)} (j odd), ^{b}λ_{i}^{(j)} = ∑_{k_{r}≥i} h_{k_{r}}^{(j)}

where i ∈ ℕ, j = 0, …, m_{0}.

Thirdly, if s = \frac{1}{2}, let ^{c}h_{k_{r}}^{(j)} = a_{1/2+k_{r},j}(j = 0, …, m_{1/2}, r ∈ ℤ_{+}). We associate to S the c_{∞}^{[m_{1/2}]}-module L[m_{1/2}]_{\frac{1}{2}}(λ_{S}) with central charges and labels

c_{j} = ∑_{k_{r}} ^{c}h_{k_{r}}^{(j)} (j even), c_{j} = 0 (j odd), ^{c}λ_{i}^{(j)} = ∑_{k_{r}≥i}^{c}h_{k_{r}}^{(j)}

where i ∈ ℕ, j = 0, …, m_{1/2}.

Denote by {s_{1}, s_{2}, …, s_{N}} a set of representatives of equivalence classes in the set {s ∈ ℂ | a_{s,j} ≠ 0 for some j}. By Theorem 6.1, the D̂^{−}-module L[m⃗],− ; s⃗(λ⃗) is irreducible for s⃗ = (s_{1}, s_{2}, …, s_{N}) satisfying the (⋆−) condition. Then we have

Δ_{m⃗,s⃗,λ⃗}^{−}(x) = ∑_{i} Δ_{m_{i},s_{i},λ(i)}^{−}(x), c = ∑_{i} c_{0}(i).

Summarizing the above, together with Theorem 6.1, Propositions 6.2, 6.3 and 6.4, we have proved the following theorem for the − case. Similarly one can prove the + case with the help of Theorem 6.1, Propositions 6.5, 6.6 and 6.7.

Theorem 6.2 Let V be an irreducible quasifinite highest weight D̂^{±}-module with central charge c and

Δ^{±}(x) = \frac{F (x)}{2 sinh(x/2)}

for some even quasipolynomial F (x) which is written in the form (6.62). Then V is isomorphic to the tensor product of the modules L_{s}^{[m],±}(Λ_{S}) for different equivalence classes S.

Remark 6.1 A different choice of representative s ∉ \frac{1}{2}ℤ in an equivalence class S has the effect of shifting 𝔤̂𝔩^{[m]} via the automorphism ν^{i} for some i. It is not difficult to see that any irreducible quasifinite highest weight module L(D̂^{±}; ξ) can be obtained as above in an essentially unique way up to the shift. Note that we have always put c_{j} = 0 (j odd) when s = 0 or \frac{1}{2} in the − case (resp. −\frac{1}{2} in the + case). We could have defined b_{∞}^{[m]}, b̃_{∞}^{[m]}, c_{∞}^{[m]} and d_{∞}^{[m]} to be the central extensions of \overline{b}_{∞}^{−[m]}, \overline{b}_{∞}^{+[m]}, \overline{c}_{∞}^{[m]} and \overline{d}_{∞}^{[m]} by R′_{m} = ⨁_{j=1}^{[m/2]} ℂu^{2j} ⊂ R_{m}. We have made our choice for the convenience of notations.

7 Unitary QHWM’s of D̂^{±}

Consider the following anti-linear anti-involution ω of the Lie algebra D̂:

ω(t^{k}f(D)) = t^{−k}\overline{f}(D − k), ω(C) = C,

where \overline{f} denotes the complex conjugate. Note that D̂^{±} is ω-invariant:

ω(t^{k}f(D + k/2)) = t^{−k}\overline{f}(D − k/2)
ω(t^{k}f(D + (k + 1)/2)) = t^{−k}\overline{f}(D + (−k + 1)/2).

The goal of this section is to classify all unitary quasifinite highest weight modules over D̂^{±} with respect to the anti-involution ω.

Remark 7.1 1) The anti-involution ω on D is compatible with the standard anti-involution ω′ of 𝔤𝔩 (given by the complex conjugate transpose of a matrix) under the homomorphism ϕ_{s} = ϕ_{s}^{[0]} : D^{O,−} ⟶ 𝔤𝔩:

ω′(ϕ_{s}(t^{k}f(D))) = ϕs̄(ω(t^{k}f(D))),

where s ∈ ℂ and s̄ denotes its complex conjugate.

2) With respect to the anti-involution ω′ of 𝔤̂𝔩 defined by ω′(A) = ^{t}\overline{A} and ω′(c) = \overline{c}, a highest weight 𝔤̂𝔩-module with highest weight Λ and central charge c is unitary if and only if all ^{a}h_{i} are non-negative integers and c = ∑_{i} ^{a}h_{i}. This follows from the unitarity of finite dimensional modules over 𝔤l_{n}.

Similarly, a highest weight module of b_{∞} or b̃_{∞} with highest weight Λ and central charge c is unitary with respect to ω′ if and only if the numbers bh_{i} (i ∈ ℤ_{+}) are non-negative integers and c = \frac{1}{2}^{b}h_{0} + ∑_{i≥1} ^{b}h_{i}.

A highest weight c_{∞}-module with highest weight Λ and central charge c is unitary if and only if the numbers ^{c}h_{i} (i ∈ ℤ_{+}) are non-negative integers and c = ∑_{i≥0} ^{c}h_{i}.

A highest weight d_{∞}-module with highest weight Λ and central charge c is unitary if and only if the numbers ^{d}h_{i} (i ∈ ℤ_{+}) are non-negative integers and c = (^{d}h_{0} +^{d}h_{1})/2 + ∑_{i≥2} ^{d}h_{i}.

Remark 7.2 A highest weight d_{∞}-module with highest weight Λ and central charge c is unitary if and only if the numbers ^{d}h_{i} ∈ ℤ_{+} (i ∈ ℕ), c ∈ \frac{1}{2}ℤ_{+} and c ≥ dh_{1}/2 + ∑_{i≥2} ^{d}h_{i}.

Lemma 7.1 1) Let V be a unitary quasifinite highest weight module over D̂^{−}, and let b(w) be its characteristic polynomial. Then b(w) has only real roots and any non-zero root of b(w) is simple; 0 is a double root of b(w) if it is a root.

2) Let V be a unitary quasifinite highest weight module over D̂^{+}, and let b(w) be its characteristic polynomial. Then b(w) has only real simple roots.

Proof. We will prove part 1). The proof of part 2) is similar.

Let v_{Λ} be a highest weight vector of V. Since b(w) is an even polynomial, we may assume that deg b(w) = 2n, and b(w) = g(w^{2}) for some polynomial g(w). Then the first graded subspace V_{1} of V has a basis

{t^{−1}(D − 1/2)^{2l}v_{Λ}, 0 ≤ l < n}.

Consider the action of S = −\frac{1}{12}(4D^{3} + 4Δ_{3} + 1) ∈ D̂^{−} on End V_{1}. It is straightforward to check that

S^{j} (t^{−1}v_{Λ}) = (t^{−1}(D − 1/2)^{2j}) v_{Λ}.

It follows that g(S)(t^{−1}v_{Λ}) = 0 and {S^{j}(t^{−1}v_{Λ}), 0 ≤ j < n} is a basis of V_{1}. We conclude from the above that g(w) is the characteristic polynomial of the operator S on V_{1}. Since the operator S is self-adjoint, all the roots of g(w) are real.

Similarly, let b_{2}(w) be the polynomial of minimal degree such that

t^{−2}b_{2}(D − 1)v_{Λ} = 0.

Since b_{2}(w) is an odd polynomial, we may assume b_{2}(w) = wg_{2}(w^{2}) for some polynomial g_{2}(w) of degree k. Consider the action of T = −\frac{1}{6}(D^{3} + Δ_{3} + 2) ∈ D̂^{−} on the second graded subspace V_{2} of V. One can check that

T^{j} ((t^{−2}(D − 1)) v_{Λ}) = (t^{−2}(D − 1)^{2j+1}) v_{Λ}.

It follows that g_{2}(T)((t^{−2}(D − 1))v_{Λ}) = 0 and {T^{j}(t^{−2}(D − 1))v_{Λ}), 0 ≤ j < n} is a basis of V_{2}.

We then conclude that g_{2}(w) is the characteristic polynomial of the operator T on V_{2}. Since the operator T is self-adjoint, all the roots of g_{2}(w) are real. By Lemma 3.1, b(w) divides wb_{2}(w − 1/2). Given a root a of b(w), it follows that

a(a − 1/2)g_{2} ((a − 1/2)^{2}) = 0.

So either a is equal to 0, \frac{1}{2}, or (a − 1/2)^{2} is real. But we know already that a^{2} is real as well since a^{2} is a root of g(w). Thus a is real.

Now let r be a root of g(w) of multiplicity m, then we may write g(w) = h(w)(w − r)^{m} for some polynomial h(w). Denote v = (S − r)^{m−1}h(S)(t^{−1}v_{λ}). It is a non-zero vector in V_{1}. However, on the other hand,

(v, v) = (c(S)(t^{−1}v_{λ}), (S − r)^{2m−2}c(S)(t^{−1}v_{λ})) = 0 if m ≥ 2.

Hence the unitarity condition forces m = 1. Consequently, any non-zero root of b(w) = g(w^{2}) has multiplicity 1 and 0 is a double root if it is a root. □

Lemma 7.2 If L[m⃗],± ; s⃗(Λ⃗) is a unitary quasifinite irreducible highest weight module over D̂^{±}, then m = 0.

Proof. By Theorem 6.2, Corollary 4.1 and Lemma 7.1, we see that a_{s,j} = 0 (j ≥ 1) in (6.62) in the − case. The argument for the + case is the same. □

Theorem 7.1 1) An irreducible quasifinite highest weight D̂^{−}-module with central charge c is unitary if and only if it is positive primitive with real exponents e_{i}, or equivalently if and only if

Δ(x) = \frac{∑_{i} n_{i}(cosh(e_{i}x) − cosh(x/2))}{2 sinh(x/2)}  (7.63)

where e_{i} are distinct real numbers different from ±\frac{1}{2}, n_{i} ∈ ℕ, c ∈ \frac{1}{2}ℤ_{+} and c ≥ ∑_{i} n_{i}.

2) A quasifinite irreducible highest weight D̂^{+}-module with central charge c is unitary if and only if it is positive primitive with real exponents e_{i} or equivalently if and only if

Δ(x) = \frac{∑_{i} n_{i}(cosh(e_{i}x) − 1)}{2 sinh(x/2)}

where e_{i} are distinct non-zero real numbers, n_{i} ∈ ℕ, c ∈ \frac{1}{2}ℤ_{+} and c ≥ ∑_{i≠i_{0}} n_{i} + n_{i_{0}}/2. Here i_{0} is the index such that e_{i_{0}} = 1.

3) Any unitary quasifinite highest weight D̂^{±}-module can be obtained by taking a tensor product of N unitary irreducible highest weight modules over 𝔤^{[0]} and restricting to D̂^{±} via ϕ̂[0] ; s⃗, where s⃗ = (s_{1}, s_{2}, …, s_{N}) (s_{i} ∈ ℝ) satisfies the (⋆±) condition.

Proof. The “only if” part of 1) was implied by the Lemma 7.2. The part 3) follows from Remark 7.1. Now the “if” part of 1) follows from part 3) since by Theorem 6.2 we have realized all irreducible unitary quasifinite highest weight D̂^{−}-modules with Δ(x) of the form (7.63). The proof of part 2) is similar to part 1) (cf. Remark 7.2). □

By Theorem 6.2, a positive primitive D̂^{±}-module is the pullback of a tensor product of N unitary irreducible highest weight modules V_{i} over 𝔤^{[0]} via a homomorphism ϕ̂[0] ; s⃗, where s⃗ = (s_{1}, s_{2}, …, s_{N}) (s_{i} ∈ ℂ) satisfies the (⋆±) condition. Since the ℤ-gradations of D̂^{±} and 𝔤^{[0]} are compatible under the homomorphism ϕ̂[0] ; s⃗, the q-character formula of a positive primitive module of D̂^{±} is given by the product of the q-character formulas of V_{i}, which are in turn given by formulas (1.5), (1.8), and (1.12) in − case (resp. (15.115) in + case).

Remark 7.3 One can introduce a tensor category O_{b,±} consisting of all positive primitive modules of D̂^{±} with non-negative half-integral central charge as was done for W_{1+∞} in [W1] (also cf. [W2]) equipped with the usual tensor product. One can show that O_{b,±} is a semisimple tensor category. A tensor product of two irreducible modules in such a category is decomposed into an (infinite) sum of irreducibles with finite multiplicities. Certain reciprocity laws can be established between these multiplicities and some branching coefficients in finite dimensional Lie groups as a formal consequence of various duality results we will establish in the next sections.

8 FFR’s of QHWM’s over D̂^{−} with c ∈ −ℕ/2

8.1 Dual pair (O(2l), D̂^{−})

In this subsection we study the free field realizations (FFR’s) of certain primitive D̂^{−}-modules with negative integral central charges in some bosonic Fock spaces and establish certain duality results. These duality results are intimately related to duality results obtained in [W2]. We refer the reader to [W2] for more detail. In the case of D̂ similar results were obtained in [KR2].

Let us take a pair of bosonic ghost fields

γ^{+}(z) = ∑_{n∈ \frac{1}{2} +ℤ} γ_{n}^{+}z^{−n− \frac{1}{2}} , γ^{−}(z) = ∑_{n∈ \frac{1}{2} +ℤ} γ_{n}^{−}z^{−n− \frac{1}{2}}

with the following commutation relations

[γ_{m}^{+}, γ_{n}^{−}] = δ_{m+n,0}.

We define the Fock space F^{⨂ −1} of the fields γ^{+}(z) and γ^{−}(z), generated by the vacuum |0⟩, satisfying

γ_{n}^{+}|0⟩ = γ_{n}^{−}|0⟩ = 0, n ∈ \frac{1}{2} + ℤ_{+}.

Now we take l pairs of bosonic ghost fields γ^{+,p}(z), γ^{−,p}(z) (p = 1, …, l) and consider the corresponding Fock space F^{⨂ −l}.

It is well known and straightforward to verify that

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1}w^{−j} = − ∑_{p=1}^{l} : γ^{+,p}(z)γ^{−,p}(w) :  (8.64)

defines an action of 𝔤̂𝔩 on F^{⨂ −l} with central charge −l.

It is known [FF] that the Fourier components of the following generating functions

e_{∗∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗∗}^{pq}(n)z^{−n−1} =: γ^{+,p}(z)γ^{+,q}(−z) : (p ≠ q)  (8.65)
e^{pq}(z) ≡ ∑_{i,j∈ℤ} e^{pq}(n)z^{−n−1} =: γ^{−,p}(z)γ^{−,q}(−z) : (p ≠ q)  (8.66)
e_{∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗}^{pq}(n)z^{−n−1} =: γ^{+,p}(z)γ^{−,q}(z) : (p, q = 1, …, l) (8.67)

span an affine algebra 𝔤𝔩^{(2)}(2l) of type A_{2l−1}^{(2)} of central charge −1. The horizontal subalgebra of the affine algebra 𝔤𝔩^{(2)}(2l) spanned by e_{∗∗}^{pq}(0), e_{∗}^{pq}(0), e^{pq}(0) (p, q = 1, …, l) is isomorphic to the Lie algebra 𝔰𝔬(2l). The Borel subalgebra of 𝔰𝔬(2l) is taken to be the one spanned by e_{∗}^{pq}(p ≤ q), e_{∗∗}^{pq}, p, q = 1, …, l and the Cartan subalgebra is spanned by e_{∗}^{pp}, p = 1, …, l. The action of 𝔰𝔬(2l) can be lifted to an action of SO(2l) on F^{⨂ −l} and then extends to O(2l) naturally. For example the operator which commutes with ψ^{±,k}(z) (k = 1, …, l − 1) and sends ψ^{±,l}(z) to ψ^{∓,l}(z) lies in ∈ O(2l) − SO(2l). Put

\begin{matrix}∑_{i,j∈ℤ}(E_{ij} − (−1)^{i+j}E_{1−j,1−i})z^{i−1}w^{−j} \\ = & ∑_{k=1}^{l} (γ^{+,k}(z)γ^{−,k}(w) − γ^{+,k}(−w)γ^{−,k}(−z)) . \end{matrix}  (8.68)

The operators E_{ij} −(−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) span c_{∞} with central charge −l. It is known [W2] that the actions of O(2l) and c_{∞} commute with each other and form a dual pair on F^{⨂ −l} in the sense of Howe [H1, H2].

O(2l) is a semi-direct product of SO(2l) by ℤ_{2}. If λ is a representation of SO(2l) of highest weight (m_{1}, m_{2}, …, m_{l}) (m_{l} ≠ 0), then the induced representation of (m_{1}, m_{2}, …, m_{l}) to O(2l) is irreducible and its restriction to SO(2l) is a sum of (m_{1}, m_{2}, …, m_{l}) and (m_{1}, m_{2}, …, −m_{l}). We denote this irreducible representation λ of O(2l) by (m_{1}, m_{2}, …, \overline{m}_{l}), where m_{l} is chosen to be greater than 0. If m_{l} = 0, the representation λ = (m_{1}, m_{2}, …, m_{l−1}, 0) extends to two different representations of O(2l), denoted by λ and λ ⨂ det, where det is the 1-dimensional non-trivial representation of O(2l). We denote

Σ(D) = \{(m_{1}, m_{2}, …, \overline{m}_{l}) | m_{1} ≥ m_{2} ≥ … ≥ m_{l} > 0, m_{i} ∈ ℤ;
(m_{1}, m_{2}, …, m_{l−1}, 0) ⨂ det,
(m_{1}, m_{2}, …, m_{l−1}, 0) | m_{1} ≥ m_{2} ≥ … ≥ m_{l−1} ≥ 0, m_{i} ∈ ℤ} .

Define a map Λ^{𝔡𝔠} from Σ(D) to c_{∞}_{0}^{∗} by sending λ = (m_{1}, ⋯, \overline{m}_{l}) (m_{l} > 0) to

Λ^{𝔡𝔠}(λ) = (−l − m_{1})^{c}Λ̂_{0} + ∑_{k=1}^{l}(m_{k} − m_{k+1})^{c}Λ̂_{k} (m_{l+1} = 0),

sending (m_{1}, ⋯, m_{j}, 0, ⋯, 0) (j < l) to

Λ^{𝔡𝔠}(λ) = (−l − m_{1})^{c}Λ̂_{0} + ∑_{k=1}^{j}(m_{k} − m_{k+1})^{c}Λ̂_{k},

and sending (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ⨂ det to

Λ^{𝔡𝔠}(λ) = (−l − m_{1})^{c}Λ̂_{0} + ∑_{k=1}^{j−1}(m_{k} − m_{k+1})^{c}Λ̂_{k} + (m_{j} − 1)^{c}Λ̂_{j} +^{c}Λ̂_{2l−j},

if m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0.

The following theorem was proved in [W2].

Theorem 8.1 We have the following (O(2l), c_{∞})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(D)} V (O(2l); λ) ⊗ L (c_{∞}; Λ^{𝔡𝔠}(λ), −l)  (8.69)

where V (O(2l); λ) is the irreducible O(2l)-module parametrized by λ ∈ Σ(D) and L (c_{∞}; Λ^{𝔡𝔠}(λ), −l) is the irreducible c_{∞}-module of highest weight Λ^{𝔡𝔠}(λ) and central charge −l.

Note that D̂^{−} acts on F^{⨂ −l} via the composition of the homomorphism ϕ̂_{1/2} and the action of c_{∞} given by (8.68). Introduce the following generating function

T_{1/2}^{n}(z) = ∑_{k∈ℤ} T_{k}^{n,1/2}z^{−k−n−1} (n ∈ 2ℤ_{+} + 1)

for the basis T_{k}^{n,1/2} defined in (2.22) of D̂^{−}. It follows from Proposition 5.2 that

c = −l  (8.70)
ϕ̂_{1/2} (e^{xD} − e^{−xD}) = ϕ_{1/2} (e^{xD} − e^{−xD}) − l tanh(x/4).  (8.71)

It follows that

ϕ̂_{1/2}(T_{0}^{n,1/2}) = ϕ_{1/2}(T_{0}^{n,1/2}) − α_{n}l  (8.72)

for some constant α_{n} determined by (8.71) depending on n only.

Lemma 8.1 One can recast the action of D̂^{−} in terms of T_{1/2}^{n}(z) as

T_{1/2}^{n}(z) = − ∑_{k=1}^{l} (: γ^{+,k}(z)∂_{z}^{n}γ^{−,k}(z) : + : ∂_{z}^{n}γ^{+,k}(−z)γ^{−,k}(−z) :) − α_{n}lz^{−n−1}.

Proof. We have the following calculations.

T_{1/2}^{n}(z) = ∑_{k∈ℤ} T_{k}^{n,1/2}z^{−k−n−1}
=^{(1)} − ∑_{k∈ℤ}([−j]_{n} + (−1)^{k+1}[j − k − 1]_{n})E_{j−k,j}z^{−k−n−1} − α_{n}lz^{−n−1}
=^{(2)} − ∑_{k,j∈ℤ}[−j]_{n} (E_{j−k,j} + (−1)^{k+1}E_{1−j,k−j+1}) z^{−k−n−1} − α_{n}lz^{−n−1}

=^{(3)} − ∑_{k=1}^{l} (: γ^{+,k}(z)∂_{z}^{n}γ^{−,k}(z) : + : ∂_{z}^{n}γ^{+,k}(−z)γ^{−,k}(−z) :) − α_{n}lz^{−n−1},

where equation (1) is given by the homomorphism ϕ̂_{1/2} (cf. (5.44) and (8.72)), (2) is obtained by change of variables in the second summation (replacing j − k − 1 with j), and (3) follows by taking the n-th derivative of (8.68) with respect to w and then putting w = z. □

For λ = (m_{1}, ⋯, \overline{m}_{l}) ∈ Σ(D) (m_{l} > 0) we let e(λ) be the set of the exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, l) and the exponent 0 with multiplicity −l − m_{1}, where m_{l+1} = 0; for λ = (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ∈ Σ(D) (j < l), we let e(λ) be the set of the exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, j) and the exponent 0 with multiplicity −l − m_{1}, where m_{j+1} = 0; for (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ⨂ det ∈ Σ(D) (j < l) we let e(λ) be the set of the exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, j − 1), the exponent j with multiplicity m_{j} −1, the exponent 2l−j with multiplicity 1 and the exponent 0 with multiplicity −l − m_{1}.

Now we can state a duality theorem between O(2l) and D̂^{−} (recall Convention 4.1).

Theorem 8.2 O(2l) and D̂^{−} form a dual pair on F^{⨂ −l}. More explicitly we have the following (O(2l), D̂^{−})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(D)} I_{λ} ≡ ⨁_{λ∈Σ(D)} V (O(2l); λ) ⊗ L (D̂^{−}; e(λ), −l) .  (8.73)

Proof. Since the actions of O(2l) commutes with c_{∞}, the action of O(2l) commutes with the action of D̂^{−} given in Lemma 8.1 by Proposition 5.4. So the decomposition of the Fock space into isotypic subspaces with respect to the dual pair (Sp(2l), c_{∞}) can be regarded as decomposition with respect to the dual pair (Sp(2l), D̂^{−}) as well. By Theorems 8.1 and 6.1 each isotypic subspace is irreducible under the joint action of Sp(2l) and D̂^{−}. By Proposition 6.4 and Theorem 8.1 the highest weight of the representation I_{λ} with respect to D̂^{−} is given as follow.

Δ_{m,s,Λ}(x) = \frac{(−l − m_{1}) + ∑_{k=1}^{j}(m_{k} − m_{k+1}) cosh(kx)}{2 sinh(x/2)} − \frac{−l cosh(x/2)}{2 sinh(x/2)}

for λ = (m_{1}, ⋯, \overline{m}_{l}) ∈ Σ(D) (m_{l} > 0).

The cases λ = (m_{1}, ⋯, m_{j}, 0, ⋯, 0) (j < l) and (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ⨂ det can be treated similarly. □

8.2 Dual pair (O(2l + 1), D̂^{−})

In this subsection we will realize certain primitive D̂^{−}-modules with negative halfodd-integral central charges in some Fock spaces and establish some duality theorems. We need a bosonic field χ(z) = ∑_{n∈ \frac{1}{2} +ℤ} χ_{n}z^{−n− \frac{1}{2}} which satisfies the following commutation relations:

[χ_{m}, χ_{n}] = (−1)^{m+ \frac{1}{2}} δ_{m,−n}, m, n ∈ \frac{1}{2} + ℤ.

Let F^{⨂ −l− \frac{1}{2}} be the tensor product of the Fock space of l pairs of bosonic ghost fields γ^{±,k}(z) (k = 1, …, l) and the Fock space F^{⨂ − \frac{1}{2}} of χ(z).

It is known [FF] that the Fourier components of the generating functions (8.65), (8.66), (8.67) and the following generating functions

ζ(z) ≡ ∑_{n∈ℤ} ζ(n)z^{−n−1} = : χ(z)χ(−z) :
e^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n−1} = : γ^{−,p}(z)χ(−z) :
e_{∗}^{p}(z) ≡ ∑_{n∈ℤ} e_{∗}^{p}(n)z^{−n−1} = : γ^{+,p}(z)χ(z) :

span an affine algebra A_{2l}^{(2)} of central charge −1 when acting on F^{⨂ − \frac{1}{2}} . Its horizontal subalgebra is isomorphic to 𝔰𝔬(2l+1). We take the Borel subalgebra 𝔟(𝔰𝔬(2l+1)) to be the one spanned by e_{∗}^{pq}(0)(p ≤ q), e_{∗}^{p}(0), e_{∗∗}^{pq}(0), p, q = 1, …, l. The Cartan subalgebra 𝔥(𝔰𝔬(2l + 1)) is spanned by e_{∗}^{pp}(0), p = 1, …, l.

The action of 𝔰𝔬(2l + 1) can be lifted to SO(2l + 1) on F^{⨂ −l− \frac{1}{2}} and then extends naturally to O(2l + 1). For example the operator which commutes with Ψ^{±,k}(z), k = 1, …, l and sends χ(z) to −χ(z) lies in O(2l + 1) − SO(2l + 1). Let

\begin{matrix}∑_{i,j∈ℤ}(E_{ij} − (−1)^{i+j}E_{1−j,1−i})z^{i−1}w^{−j} \\ = & ∑_{k=1}^{l} (: γ^{+,k}(z)γ^{−,k}(w) : + : γ^{+,k}(−w)γ^{−,k}(−z) :) + : χ(z)χ(−w) : . \end{matrix} (8.74)

The operators E_{ij} −(−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) span c_{∞} with central charge −l−1/2. It is shown [W2] that O(2l + 1) and c_{∞} form a dual pair on F^{⨂ −l− \frac{1}{2}} .

An irreducible modules of SO(2l + 1) is parametrized by its highest weight λ = (m_{1}, …, m_{l}), m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ. It is well known that O(2l + 1) is isomorphic to the direct product SO(2l + 1) × ℤ_{2} by sending the minus identity matrix to −1 ∈ ℤ_{2} = {±1}. Denote by det the non-trivial one-dimensional representation of O(2l + 1). An irreducible representation λ of SO(2l + 1) extends to two different irreducible representations of O(2l+1) by tensoring with the trivial representation and non-trivial representation of ℤ_{2}, denoted by λ and λ ⨂ det. All irreducible representations of O(2l + 1) can be obtained this way. Then we can parametrize irreducible representations of O(2l + 1) by Σ(B) consisting of highest weights (m_{1}, …, m_{l}) and (m_{1}, …, m_{l}) ⨂ det.

Define a map Λ^{𝔟𝔠} from Σ(B) to c_{∞}_{0}^{∗} by sending λ = (m_{1}, m_{2}, …, m_{l}) to

Λ^{𝔟𝔠}(λ) = (−l − m_{1} − 1/2)^{c}Λ̂_{0} + ∑_{k=1}^{j}(m_{k} − m_{k+1})^{c}Λ̂_{k}

and sending λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det to

Λ^{𝔟𝔠}(λ) = (−l − m_{1} − 1/2)^{c}Λ̂_{0} + ∑_{k=1}^{j−1}(m_{k} − m_{k+1})^{c}Λ̂_{k} + (m_{j} − 1)^{c}Λ̂_{j} +^{c}Λ̂_{2l−j+1},

where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0. The following theorem is proved in [W2].

Theorem 8.3 We have the following (O(2l + 1), c_{∞})-module decomposition:

F^{⨂ −l− \frac{1}{2}} = ⨁_{λ∈Σ(B)} V (O(2l + 1); λ) ⊗ L (c_{∞}; Λ^{𝔟𝔠}(λ), −l − 1/2)

where V (O(2l+1); λ) is the irreducible O(2l+1)-module parametrized by λ ∈ Σ(B) and L (c_{∞}; Λ^{𝔟𝔠}(λ), −l − 1/2) is the irreducible highest weight c_{∞}-module of highest weight Λ^{𝔟𝔠}(λ) and central charge −l − 1/2.

Note that D̂^{−} now acts on F^{⨂ −l− \frac{1}{2}} via the composition of the homomorphism ϕ̂_{1/2} and the action of c_{∞} given by formula (8.74). The following lemma is analogous to Lemma 8.1.

Lemma 8.2 We can recast the action of D̂^{−} in terms of generating function T_{1/2}^{n}(z) = ∑_{k∈ℤ} T_{k}^{n,1/2}z^{−k−n−1} as

T_{1/2}^{n}(z) = − ∑_{k=1}^{l} (: γ^{+,k}(z)∂_{z}^{n}γ^{−,k}(z) : + : ∂_{z}^{n}γ^{+,k}(−z)γ^{−,k}(−z) :)
− : χ(z)∂_{z}^{n}χ(−z) : −α_{n}(l + 1/2)z^{−k−n−1}, n ∈ 2Z_{+} + 1.

For λ = (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ∈ Σ(B), we let e(λ) be the set of the exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, j), where m_{j+1} = 0, and the exponent 0 with multiplicity −l −m_{1} −1/2; for λ = (m_{1}, ⋯, m_{j}, 0, ⋯, 0) ⨂ det ∈ Σ(B) we let e(λ) be the set of the exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, j − 1), the exponent j with multiplicity m_{j} − 1, the exponent 2l − j + 1 with multiplicity 1, and the exponent 0 with multiplicity −l − m_{1} − 1/2.

Now we have the following duality theorem on the joint action on F^{⨂ −l− \frac{1}{2}} of the dual pair (O(2l + 1), D̂^{−}). The proof is similar to that of Theorem 8.2 which is now based on Theorem 8.3 and Proposition 6.4.

Theorem 8.4 We have the following (O(2l + 1), D̂^{−})-module decomposition:

F^{⨂ −l− \frac{1}{2}} = ⨁_{λ∈Σ(B)} V (O(2l + 1); λ) ⊗ L (D̂^{−}; e(λ), −l − 1/2) .

9 FFR’s of QHWM’s over D̂^{−} with c ∈ ℕ

In this section we will realize certain primitive D̂^{−}-modules with positive integral central charges in some fermionic Fock spaces. Similar results for D̂ were obtained in [FKRW] (see also [KR2]). Let us take a pair of fermionic fields

ψ^{+}(z) = ∑_{n∈\underline{ℤ}} ψ_{n}^{+}z^{−n− \frac{1}{2} +ϵ}, ψ^{−}(z) = ∑_{n∈\underline{ℤ}} ψ_{n}^{−}z^{−n− \frac{1}{2} +ϵ}, \underline{ℤ} = \frac{1}{2} + ℤ or ℤ

with the following anti-commutation relations

[ψ_{m}^{+}, ψ_{n}^{−}]_{+} = δ_{m+n,0}
[ψ_{m}^{±}, ψ_{n}^{±}]_{+} = 0.

We take the convention here and below that ϵ = 0 if \underline{ℤ} = \frac{1}{2} + ℤ; and ϵ = \frac{1}{2} if \underline{ℤ} = ℤ. Denote by F the Fock space of the fields ψ^{−}(z) and ψ^{+}(z), generated by the vacuum |0⟩, satisfying

ψ_{n}^{+}|0⟩ = ψ_{n}^{−}|0⟩ = 0 (n ∈ \frac{1}{2} + ℤ_{+}), when \underline{ℤ} = \frac{1}{2} + ℤ,
ψ_{n}^{+}|0⟩ = ψ_{n+1}^{−}|0⟩ = 0 (n ∈ ℤ_{+}), when \underline{ℤ} = ℤ.

Now we take l pairs of fermionic fields, ψ^{±,p}(z) (p = 1, …, l) and consider the corresponding Fock space F^{⨂ l}.

Introduce the following “twisted” generating functions

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1+2ϵ}w^{−j} = ∑_{k=1}^{l} : ψ^{+,k}(z)ψ^{−,k}(w) :,  (9.75)
e^{pq}(z) ≡ ∑_{n∈ℤ} e^{pq}(n)z^{−n−1+2ϵ} =: ψ^{−,p}(z)ψ^{−,q}(−z) :,  (9.76)
e_{∗∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{+,q}(−z) :,  (9.77)
e_{∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{−,q}(z) : +δ_{p,q}ϵz^{−1}, (9.78)

where p, q = 1, …, l. It is known [FF] that the Fourier components of these generating functions span a representation of the twisted affine algebra 𝔤l^{(2)}(2l) of type A_{2l−1}^{(2)} with central charge 1.

I. Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (Sp(2l), D̂^{−})

The horizontal subalgebra of 𝔤l^{(2)}(2l) spanned by the operators e^{pq}(0), e_{∗}^{pq}(0), e_{∗∗}^{pq}(0), (p, q = 1, ⋯, l) is isomorphic to Lie algebra 𝔰𝔭(2l). We identify the Borel subalgebra 𝔟(𝔰𝔭(2l)) with the one generated by e_{∗}^{pq}(0) (p ≤ q), e_{∗∗}^{pq}(0) (p, q = 1, …, l) and the Cartan subalgebra with the one generated by e_{∗}^{pp}(0) (p = 1, …, l). Let

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − (−1)^{i+j}E_{1−j,1−i})z^{i−1}w^{−j} \\ = & ∑_{k=1}^{l} (: ψ^{+,k}(z)ψ^{−,k}(w) : + : ψ^{+,k}(−w)ψ^{−,k}(−z) :) . \end{matrix}  (9.79)

The operators E_{i,j} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) span c_{∞} with central charge l. The action of 𝔰𝔭(2l) on F^{⨂ l} can be integrated to Sp(2l). It is known [W2] that the actions of Sp(2l) and c_{∞} commute with each other on F^{⨂ l} and they form a dual pair.

Finite dimensional irreducible modules of Sp(2l) are parametrized by the highest weights in

Σ(C) = {λ = (m_{1}, …, m_{l}), m_{1} ≥ … ≥ m_{l}, m_{i} ∈ ℤ_{+}}.

We define a map Λ^{𝔠𝔠} from Σ(C) to c_{∞}_{0}^{∗} by sending (m_{1}, …, m_{l}) to

Λ^{𝔠𝔠}(λ) = (l − j)^{c}Λ̂_{0} + ∑_{k=1}^{j}^{c}Λ̂_{m_{k}},

where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0. We quote the following theorem from [W2].

Theorem 9.1 We have the following (Sp(2l), c_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (c_{∞}; Λ^{𝔠𝔠}(λ), l)  (9.80)

where V (Sp(2l); λ) is the irreducible Sp(2l)-module parametrized by λ ∈ Σ(C), and L (c_{∞}; Λ^{𝔠𝔠}(λ), l) is the irreducible highest weight c_{∞}-module of highest weight Λ^{𝔠𝔠}(λ) and central charge l.

Note that D̂^{−} acts on F^{⨂ l} via the composition of the homomorphism ϕ̂_{1/2} and the action of c_{∞} given by formula (9.79). Similarly as in Lemma 8.1 we can rewrite the action of D̂^{−} in terms of generating function T_{1/2}^{n}(z) as

T_{1/2}^{n}(z) = − ∑_{k=1}^{l} (: ψ^{+,k}(z)∂_{z}^{n}ψ^{−,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(−z)ψ^{−,k}(−z) :)
+α_{n}lz^{−k−n−1}.  (9.81)

Given λ = (m_{1}, …, m_{l}) ∈ Σ(C), where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0, let e(λ) be the set of exponents m_{k} (k = 1, …, j) with multiplicity 1 and the exponent 0 with multiplicity l − j.

Theorem 9.2 Sp(2l) and D̂^{−} form a dual pair on F^{⨂ l}. Furthermore we have the following (Sp(2l), D̂^{−})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(C)} I_{λ} ≡ ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (D̂^{−}; e(λ), l) .

Proof. Since the actions of Sp(2l) commutes with c_{∞}, the action of Sp(2l) commutes with the action of D̂^{−} given by formula (9.81) by Proposition 5.4. So the decomposition of the Fock space into isotypic subspaces with respect to the dual pair (Sp(2l), c_{∞}) can be regarded as decomposition with respect to the dual pair (Sp(2l), D̂^{−}) as well. By Theorem 6.1 each isotypic subspace is irreducible under the joint action of Sp(2l) and D̂^{−}. By Proposition 6.4 and Theorem 9.1 the highest weight of the representation I_{λ} with respect to D̂^{−} is given as follows.

Δ_{m,s,Λ}(x) = \frac{(l − j) + ∑_{k=1}^{j} cosh(m_{k}x)}{2 sinh(x/2)} − \frac{l cosh(x/2)}{2 sinh(x/2)}.
□

II. Case \underline{ℤ} = ℤ: dual pair (Pin(2l), D̂^{−})

In this case the horizontal subalgebra of 𝔤l^{(2)}(2l) spanned by the operators e^{pq}, e_{∗}^{pq}, e_{∗∗}^{pq}, (p, q = 1, ⋯, l) is isomorphic to Lie algebra 𝔰𝔬(2l). In particular, the operators e_{∗}^{pq} (p, q = 1, ⋯ l) form a subalgebra 𝔤𝔩(l) in the horizontal 𝔰𝔬(2l). We identify the Borel subalgebra 𝔟(𝔰𝔬(2l)) with the one spanned by e_{∗}^{pq} (p ≤ q), e_{∗∗}^{pq}, p, q = 1, ⋯, l. The action of 𝔰𝔬(2l) can be lifted to P in(2l) on F^{⨂ l}. Recall that P in(2l) (resp. Spin(2l)) is the double covering group of O(2l) (resp. SO(2l)).

Denote 1_{l} = (1, 1, …, 1) ∈ ℤ^{l} and ¯1_{l} = (1, 1, …, 1, −1) ∈ ℤ^{l}. An irreducible representation of Spin(2l) which does not factor to SO(2l) is an irreducible representation of 𝔰𝔬(2l) given by its highest weight

λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l})  (9.82)

or

λ = \frac{1}{2} ¯1_{l} + (m_{1}, m_{2}, …, −m_{l})  (9.83)

where m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ. A representation of P in(2l) induced from λ of Spin(2l) of the form (9.82) is decomposed into a sum of two irreducible Spin(2l)-modules of highest weights (9.82) and (9.83). We use λ = \frac{1}{2}|1_{l}| + (m_{1}, m_{2}, …, \overline{m}_{l}), m_{l} ≥ 0 to denote this irreducible representation of P in(2l). Denote

Σ(P in) = \{\frac{1}{2}|1_{l}| + (m_{1}, m_{2}, …, \overline{m}_{l}), m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ\}. (9.84)

Let

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − (−1)^{i+j}E_{−j,−i})z^{i}w^{−j} \\ = & ∑_{k=1}^{l} (: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(−w)ψ^{−,k}(−z) :) . \end{matrix}  (9.85)

It is known [W2] that the operators E_{i,j} − (−1)^{i+j}E_{−j,−i}(i, j ∈ ℤ) span b_{∞} with central charge l and P in(2l) and b_{∞} form a dual pair on F^{⨂ l}. We define a map Λ^{𝔡𝔟} from Σ(P in) to b_{∞}_{0}^{∗} by sending λ = (m_{1}, …, \overline{m}_{l}) to

Λ^{𝔡𝔟}(λ) = (2l − 2j)^{b}Λ̂_{0} + ∑_{k=1}^{j}^{b}Λ̂_{m_{k}}

if m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0.

We need to quote the following theorem from [W2].

Theorem 9.3 We have the following (P in(2l), b_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(P in)} V (P in(2l); λ) ⊗ L (b_{∞}; Λ^{𝔡𝔟}(λ), l)

where V (P in(2l); λ) is the irreducible P in(2l)-module parametrized by λ ∈ Σ(P in), and L (b_{∞}; Λ^{𝔡𝔟}(λ), l) is the irreducible highest weight b_{∞}-module of highest weight Λ^{𝔡𝔟}(λ) and central charge l.

Note that D̂^{−} acts on F^{⨂ l} via the composition of the homomorphism ϕ̂_{0} and the action of b_{∞} given by (9.85). We can rewrite the action of D̂^{−} in terms of generating function T^{n}(z) (n ∈ 2ℤ_{+} + 1) as

T^{n}(z) = − ∑_{k=1}^{l} (: ψ^{+,k}(z)∂_{z}^{n}ψ^{−,k}(z) : − : ∂_{z}^{n}ψ^{+,k}(−z)ψ^{−,k}(−z) :) .

Given λ = (m_{1}, …, \overline{m}_{l}) ∈ Σ(P in), where

m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0,

let e(λ) be the set of exponents m_{k}+1/2 (k = 1, …, j) with multiplicity 1. The following theorem can be proved in the same way as Theorem 9.2, based on Theorem 9.3 and Proposition 6.3.

Theorem 9.4 P in(2l) and D̂^{−} form a dual pair on F^{⨂ l}. More explicitly we have the following (P in(2l), D̂^{−})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(P in)} V (P in(2l); λ) ⊗ L (D̂^{−}; e(λ), l) .

10 FFR’s of QHWM’s over D̂^{−} with c ∈ ℕ − 1/2

In this section we will realize certain primitive D̂^{−}-modules with positive halfintegral central charges in some Fock spaces.

I. Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (𝔬𝔰𝔭(1, 2l), D̂^{−})

We need a bosonic field χ(z) = ∑_{n∈ \frac{1}{2} +ℤ} χ_{n}z^{−n− \frac{1}{2}} which satisfies the following commutation relations:

[χ_{m}, χ_{n}] = (−1)^{m+ \frac{1}{2}} δ_{m,−n}, m, n ∈ \frac{1}{2} + ℤ.

Denote by F^{⨂ − \frac{1}{2}} the Fock space of χ(z) generated by a vacuum vector which is annihilated by χ_{n}, n ∈ \frac{1}{2} + ℤ_{+}. Let F^{⨂ l− \frac{1}{2}} be the tensor product of the Fock space of l pairs of fermionic fields ψ^{±,k}(z) (k = 1, …, l) and the Fock space F^{⨂ − \frac{1}{2}} of χ(z).

It is known [FF] that the Fourier components of the following “twisted” generating functions together with (9.76), (9.77) and (9.78)

ζ(z) ≡ ∑_{n∈ℤ} ζ(n)z^{−n−1} =: χ(z)χ(−z) :,
e^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n−1} =: ψ^{−,p}(z)χ(−z) :,
e_{∗}^{p}(z) ≡ ∑_{n∈ℤ} e_{∗}^{p}(n)z^{−n−1} =: ψ^{+,p}(z)χ(z) :,

generate a representation of the affine superalgebra 𝔤𝔩^{(2)}(1, 2l) of type A^{(2)}(0, 2l−1)

[K1] with central charge 1. Denote

e^{p} ≡ e^{p}(0), e_{∗}^{p} ≡ e_{∗}^{p}(0), e^{pq} ≡ e^{pq}(0), e_{∗}^{pq} ≡ e_{∗}^{pq}(0), e_{∗∗}^{pq} ≡ e_{∗∗}^{pq}(0), p, q = 1, ⋯, l.

The horizontal subalgebra in 𝔤𝔩^{(2)}(1, 2l) spanned by the operators e^{p}, e_{∗}^{p}, e^{pq}, e_{∗}^{pq}, e_{∗∗}^{pq}, (p, q = 1, ⋯, l) is isomorphic to Lie superalgebra 𝔬𝔰𝔭(1, 2l). We identify the Borel subalgebra 𝔟(𝔬𝔰𝔭(1, 2l)) with the one generated by e_{∗∗}^{pq}, e_{∗}^{pq} (p ≤ q), p, q = 1, ⋯, l. Let

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − (−1)^{i+j}E_{1−j,1−i})z^{i−1}w^{−j} \\ = & ∑_{k=1}^{l} (: ψ^{+,k}(z)ψ^{−,k}(w) : + : ψ^{+,k}(−w)ψ^{−,k}(−z) :) + : χ(z)χ(−w) : \end{matrix}(10.86)

It is shown in [W2] that E_{i,j} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) span c_{∞} with central charge l − \frac{1}{2} and (𝔬𝔰𝔭(1, 2l), c_{∞}) form a dual pair on F^{⨂ l− \frac{1}{2}} .

Finite dimensional irreducible representations of 𝔬𝔰𝔭(1, 2l) are parametrized by the highest weights [K1]

Σ(Osp) = {(m_{1}, m_{2}, …, m_{l}) | m_{1} ≥ m_{2} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ}.

Define a map Λ^{𝔬𝔰𝔭𝔠} from Σ(Osp) to c_{∞}_{0}^{∗} by sending λ = (m_{1}, …, m_{l}) to

Λ^{𝔬𝔰𝔭𝔠}(λ) = (l − j − 1/2)^{c}Λ̂_{0} + ∑_{k=1}^{j}^{c}Λ̂_{m_{k}},

if m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0. We quote the following theorem from [W2].

Theorem 10.1 We have the following (𝔬𝔰𝔭(1, 2l), c_{∞})-module decomposition:

F^{⨂ l− \frac{1}{2}} = ⨁_{λ∈Σ(Osp)} V (𝔬𝔰𝔭(1, 2l); λ) ⊗ L (c_{∞}; Λ^{𝔬𝔰𝔭𝔠}(λ), l − 1/2)

where V (𝔬𝔰𝔭(1, 2l); λ) is the irreducible module of 𝔬𝔰𝔭(1, 2l) parametrized by λ ∈ Σ(Osp), and L (c_{∞}; Λ^{𝔬𝔰𝔭𝔠}(λ), l − 1/2) is the irreducible highest weight c_{∞}-module of highest weight Λ^{𝔬𝔰𝔭𝔠}(λ) and central charge l − 1/2.

Note that D̂^{−} acts on F^{⨂ l} via the the composition of homomorphism ϕ̂_{1/2} and the action of c_{∞} given by (10.86). We can rewrite the action of D̂^{−} in terms of generating function T_{1/2}^{n}(z) (n ∈ 2ℤ_{+} + 1) as

T_{1/2}^{n}(z) = − ∑_{k=1}^{l} (: ψ^{+,k}(z)∂_{z}^{n}ψ^{−,k}(z) : − : ∂_{z}^{n}ψ^{+,k}(−z)ψ^{−,k}(−z) :)
− : χ(z)∂_{z}^{n}χ(−z) : .

Given λ = (m_{1}, m_{2}, …, m_{l}) ∈ Σ(Osp), where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0, let e(λ) be the set of exponents m_{k} (k = 1, …, j) with multiplicity 1 and the exponent 0 with multiplicity l − j − \frac{1}{2}. The following duality theorem can be proved in a similar way as Theorem 9.2, based on Theorem 10.1 and Proposition 6.4.

Theorem 10.2 𝔬𝔰𝔭(1, 2l) and D̂^{−} form a dual pair on F^{⨂ l− \frac{1}{2}} . Furthermore we have the following (𝔬𝔰𝔭(1, 2l), D̂^{−})-module decomposition:

F^{⨂ l− \frac{1}{2}} = ⨁_{λ∈Σ(Osp)} V (𝔬𝔰𝔭(1, 2l); λ) ⊗ L (D̂^{−}; e(λ), l − 1/2) .

II. Case \underline{ℤ} = ℤ: dual pair (Spin(2l + 1), D̂^{−})

Recall that a fermionic field φ(z) = ∑_{n∈ℤ} φ_{n}z^{−n} which satisfy the following commutation relations:

[φ_{m}, φ_{n}]_{+} = (−1)^{m}δ_{m,−n}, m, n ∈ ℤ.

In this case the Fock space F^{⨂ l+ \frac{1}{2}} is the tensor product of the Fock space of l pairs of fermionic fields ψ^{±,k}(z), k = 1, …, l and the Fock space F^{⨂ \frac{1}{2}} of φ(z) generated by a vacuum vector which is annihilated by φ_{m}, m ∈ ℕ.

The Fourier components of the following generating functions together with (9.76), (9.77) and (9.78)

η(z) ≡ ∑_{n∈ℤ} η(n)z^{−n} =: φ(z)φ(−z) :,
e^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n} =: ψ^{−,p}(z)φ(−z) :,
e_{∗}^{p}(z) ≡ ∑_{n∈ℤ} e_{∗}^{p}(n)z^{−n} =: ψ^{+,p}(z)φ(z) :

span an affine algebra of type A_{2l}^{(2)} on F^{⨂ l+ \frac{1}{2}} . The horizontal subalgebra of A_{2l}^{(2)} is isomorphic to 𝔰𝔬(2l + 1). The action of 𝔰𝔬(2l + 1) can be lifted to an action of Spin(2l + 1) on F^{⨂ l+ \frac{1}{2}} . It is well known that an irreducible representation of Spin(2l + 1) which does not factor to SO(2l + 1) is an irreducible representation of 𝔰𝔬(2l + 1) parametrized by its highest weight

λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}), m_{1} ≥ … ≥ m_{l} ≥ 0.  (10.87)

Denote

Σ(P B) = \{\frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}) | m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ\} .

Let

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − (−1)^{i+j}E_{−j,−i})z^{i}w^{−j} \\ = & ∑_{k=1}^{l} (: ψ^{+,k}(z)ψ^{−,k}(w) : + : ψ^{+,k}(−w)ψ^{−,k}(−z) :) + : φ(z)φ(−w) : \end{matrix}(10.88)

The Fock space F^{⨂ l+ \frac{1}{2}} splits into a sum of two subspaces F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} , where F_{e}^{⨂ l+ \frac{1}{2}} consists all even vectors while F_{o}^{⨂ l+ \frac{1}{2}} consists all odd vectors according to the ℤ_{2} gradation on the vector superspace F^{⨂ l+ \frac{1}{2}} . The action of 𝔰𝔬(2l + 1) can be lifted to Spin(2l) on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} . Spin(2l) and b_{∞} form a dual pair on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} [W2].

Note that D̂^{−} acts on F^{⨂ l+ \frac{1}{2}} via the composition of the homomorphism ϕ̂_{0} and the action of b_{∞} given by (10.88). We can rewrite the action of D̂^{−} in terms of generating function T^{n}(z) (n ∈ 2ℤ_{+} + 1) as

T^{n}(z) = − ∑_{k=1}^{l} (: ψ^{+,k}(z)∂_{z}^{n}ψ^{−,k}(z) : +(−1)^{n} : ∂_{z}^{n}ψ^{+,k}(−z)ψ^{−,k}(−z) :)
−(−1)^{n} : φ(z)∂_{z}^{n}φ(−z) : .

Given λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}) ∈ Σ(P B), where

m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0,

let V (Spin(2l+1); λ) be the irreducible Spin(2l+1)-module parametrized by λ and let e(λ) be the set of exponents m_{k} + 1/2 (k = 1, …, j) with multiplicity 1 (as in Section 9). The following duality theorem on the commuting actions of P in(2l) and D̂^{−} follows from a corresponding duality theorem of a dual pair (Spin(2l + 1), b_{∞}) in [W2] and similar argument as in Theorem 9.2.

Theorem 10.3 We have the following (Spin(2l + 1), D̂^{−})-module decomposition:

F_{e}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (D̂^{−}; e(λ), l + 1/2)

F_{o}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (D̂^{−}; e(λ), l + 1/2) .

11 FFR’s of QHWM’s over D̂^{+} with c ∈ ℕ

Let us take a pair of fermionic fields

ψ^{±}(z) = ∑_{n∈\underline{ℤ}} ψ_{n}^{±}z^{−n− \frac{1}{2} +ϵ}, \underline{ℤ} = \frac{1}{2} + ℤ or ℤ.

In the case \underline{ℤ} = \frac{1}{2} + ℤ the anti-commutation relations among ψ_{n}^{±} is equivalent to the following operator product expansions (OPE)

ψ^{+}(z)ψ^{−}(w) ∼ \frac{1}{z − w}, ψ^{+}(z)ψ^{+}(w) ∼ 0, ψ^{−}(z)ψ^{−}(w) ∼ 0.

Take l pairs of fermionic fields, ψ^{±,p}(z) (p = 1, …, l) and consider the corresponding Fock space F^{⨂ l}. Introduce the following generating functions

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1+2ϵ}w^{−j} = ∑_{p=1}^{l} : ψ^{+,p}(z)ψ^{−,p}(w) :  (11.89)

e^{pq}(z) ≡ ∑_{n∈ℤ} e^{pq}(n)z^{−n−1+2ϵ} =: ψ^{−,p}(z)ψ^{−,q}(z) : (p ≠ q) (11.90)
e_{∗∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{+,q}(z) : (p ≠ q) (11.91)
e_{∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{−,q}(z) : +δ_{p,q}ϵ  (11.92)

where p, q = 1, …, l, and the normal ordering :: means that the operators annihilating |0⟩ are moved to the right and multiplied by −1.

It is well known [F1, KP] that the operators e^{pq}(n), e_{∗}^{pq}(n), e_{∗∗}^{pq}(n), p, q = 1, …, l, n ∈ ℤ form a representation of the affine algebra ŝo(2l) of level 1. The operators e^{pq}(0), e_{∗}^{pq}(0), e_{∗∗}^{pq}(0) (p, q = 1, ⋯, l) form the horizonal subalgebra 𝔰𝔬(2l) in 𝔰̂𝔬(2l). We identify the Borel subalgebra 𝔟(𝔰𝔬(2l)) with the one generated by e_{∗∗}^{pq} (p ≠ q), e_{∗}^{pq} (p ≤ q), p, q = 1, ⋯, l.

¿From now on we need to treat the two cases \underline{ℤ} = \frac{1}{2} + ℤ or ℤ separately. First consider the case \underline{ℤ} = \frac{1}{2} + ℤ. It follows from (11.89) that

\begin{matrix}∑_{i,j∈ℤ}(E_{ij} − E_{1−j,1−i})z^{i−1}w^{−j} = \\ ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :). \end{matrix}  (11.93)

One can show that E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) span d_{∞}. The action of the horizontal subalgebra 𝔰𝔬(2l) can be integrated into an action of SO(2l) and extended to an action of O(2l) naturally. It is known [W2] that the action of d_{∞} commutes with the action of O(2l) on F^{⨂ l} and moreover d_{∞} and O(2l) form a dual pair^{1} by the same argument as in finite dimensional dual pairs case [H1, H2].

We define a map Λ^{𝔡𝔡} : Σ(D) ⟶ d_{∞}_{0}^{∗} by sending λ = (m_{1}, ⋯, \overline{m}_{l}) (m_{l} > 0) to

Λ^{𝔡𝔡}(λ) = (l − i)^{d}Λ̂_{0} + (l − i)^{d}Λ̂_{1} + ∑_{k=1}^{i}^{d}Λ̂_{m_{k}},

sending (m_{1}, ⋯, m_{j}, 0, …, 0) (j < l) to

Λ^{𝔡𝔡}(λ) = (2l − i − j)^{d}Λ̂_{0} + (j − i)^{d}Λ̂_{1} + ∑_{k=1}^{i}^{d}Λ̂_{m_{k}},

and sending (m_{1}, …, m_{j}, 0, …, 0) ⨂ det (j < l) to

Λ^{𝔡𝔡}(λ) = (j − i)^{d}Λ̂_{0} + (2l − i − j)^{d}Λ̂_{1} + ∑_{k=1}^{i}^{d}Λ̂_{m_{k}},

if m_{1} ≥ … m_{i} > m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0.

The following theorem was proved in [W2].

1This fact was also known to V. Kac and C. Yan.

Theorem 11.1 We have the following (O(2l), d_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(D)} V (O(2l); λ) ⊗ L (d_{∞}; Λ^{𝔡𝔡}(λ), l)

where V (O(2l); λ) is the irreducible O(2l)-module parametrized by λ ∈ Σ(D) and L (d_{∞}; Λ^{𝔡𝔡}(λ), l) is the irreducible highest weight d_{∞}-module of highest weight Λ^{𝔡𝔡}(λ) and central charge l.

We can obtain the action of D̂^{+} on F^{⨂ l} by composing the action of d_{∞} and the homomorphism ϕ̂_{0} given by the formula (5.45). Introduce the generating function

W^{n}(z) = ∑_{k∈ℤ} W_{k}^{n}z^{−k−n−1} (n ∈ 2ℤ_{+} + 1).  (11.94)

Lemma 11.1 On F^{⨂ l} we have

W^{n}(z) = \frac{1}{2} ∑_{k=1}^{l}(: ∂_{z}^{n}ψ^{−,k}(z)ψ^{+,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(z)ψ^{−,k}(z) :).  (11.95)

Proof. We calculate W^{n}(z) as follows.

W^{n}(z) = −\frac{1}{2} ∑_{k,j∈ℤ}([−j]_{n} − [j − k − 1]_{n})E_{j−k,j}z^{−k−n−1}  (11.96)
= −\frac{1}{2} ∑_{k,j∈ℤ}[−j]_{n}(E_{j−k,j} − E_{1−j,k−j+1})z^{−k−n−1}  (11.97)

= \frac{1}{2} ∑_{k=1}^{l}(− : ψ^{+,k}(z)∂_{z}^{n}ψ^{−,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(z)ψ^{−,k}(z) :). (11.98)

Here (11.96) is given by (11.94) and (5.45), (11.97) is obtained by shifting the indices from j − k − 1 to j in the second part of the right hand side of (11.96), and (11.98) is obtained by taking n-th derivatives of (11.93) with respect to w. It is clear that (11.98) is the same as (11.95). □

For λ = (m_{1}, ⋯, \overline{m}_{l}) ∈ Σ(D) where m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{l} = 1, we let e(λ) be the set of exponents m_{k} (k = 1, …, i) with multiplicity 1 and the exponent 1 with multiplicity l − i; for λ = (m_{1}, ⋯, m_{l}) ∈ Σ(D) where

m_{1} ≥ … m_{i} > m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0 (j < l),

we let e(λ) be the set of exponents m_{k} (k = 1, …, i) of multiplicity 1, exponent 1 of multiplicity j − i; for (m_{1}, ⋯, m_{l}) ⨂ det ∈ Σ(D) where

m_{1} ≥ … m_{i} > m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0 (j < l),

we let e(λ) be the set of exponents m_{k} (k = 1, …, i) of multiplicity 1, exponent 1 of multiplicity 2l − i − j. We will simply write (0, ⋯, 0) and (0, ⋯, 0) ⨂ det as 0 and det respectively.

Theorem 11.2 O(2l) and D̂^{+} form a dual pair on F^{⨂ l}. Moreover we have the following (O(2l), D̂^{+})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(D)} V (O(2l); λ) ⊗ L (D̂^{+}; e(λ), l) .  (11.99)

Proof. By Theorem 6.1 the d_{∞}-module L (d_{∞}; Λ^{𝔡𝔡}(λ), l) regarded as a D̂^{+}-module via the pullback by ϕ̂_{0} remains irreducible. Then this theorem follows from Theorem 11.1 once we determine the corresponding Δ(x) for this D̂^{+}-module. It follows from the definition of Λ^{𝔡𝔡}(λ) by using Proposition 6.6 that for λ = (m_{1}, ⋯, \overline{m}_{l}) ∈ Σ(D) where m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{l} = 1,

Δ(x) = ∑_{k=1}^{i} \frac{cosh(m_{k}x) + (l − i) cosh x + \frac{1}{2}((l − i) − (l − i))}{2 sinh(x/2)} − \frac{l}{2 sinh(x/2)}.

The computations of Δ(x) for the remaining λ ∈ Σ(D) are similar. □

We have an immediate corollary.

Corollary 11.1 The space of invariants of O(2l) in the Fock space F^{⨂ l} is naturally isomorphic to the irreducible module L(d_{∞}; 2l ^{d}Λ̂_{0}) of central charge l with highest weight vector |0⟩, or equivalently to the irreducible module L (D̂^{+}; e(0), l).

Remark 11.1 The Dynkin diagram of d_{∞} admits an automorphism of order 2 denoted by σ. σ induces naturally an automorphism of order 2 of d_{∞}, which is denoted again by σ by abuse of notation. σ acts on the set of highest weights of d_{∞} by mapping λ = ^{d}h_{0} ^{d}Λ̂_{0} + ^{d}h_{1} ^{d}Λ̂_{1} + ∑_{i≥2} ^{d}h_{i} ^{d}Λ̂_{i} to σ(λ) = ^{d}h_{1} ^{d}Λ̂_{0} + dh_{0} ^{d}Λ̂_{1} + ∑_{i≥2} ^{d}h_{i} ^{d}Λ̂_{i}. In this way one can obtain an irreducible module of the semi-product σ ⋉ d_{∞} on L(d_{∞}; λ) ⨁ L(d_{∞}; σ(λ)) if σ(λ) ≠ λ and on L(d_{∞}; λ) if σ(λ) = λ.

It was noted in [W2] that (SO(2l), σ ⋉ d_{∞}) form a dual pair on F^{⨂ l}. In particular the space of invariants of F^{⨂ l} under the action fo SO(2l) is isomorphic to the d_{∞}-module L(d_{∞}; 2l ^{d}Λ̂_{0}) ⨁ L(d_{∞}; 2l ^{d}Λ̂_{1}) or equivalently the D̂^{+}-module L (D̂^{+}; e(0), l) ⨁ L (D̂^{+}; e(det), l).

Now we consider the case \underline{ℤ} = ℤ. It follows from (11.89) that

\begin{matrix}∑_{i,j∈ℤ}(E_{ij} − E_{−j,−i})z^{i}w^{−j} = \\ ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :). \end{matrix}  (11.100)

One can show that E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) span b̃_{∞}. The action of the horizontal subalgebra 𝔰𝔬(2l) can be integrated into an action of Spin(2l) (cf. e.g. [BtD] for more on spin groups) and then extended naturally to P in(2l). P in(2l) and b̃_{∞} form a dual pair on F^{⨂ l} [W2].

We define a map Λ^{𝔡𝔟} from Σ(P in) (see (9.84) for notation) to b_{∞}_{0}^{∗} by sending

λ = (m_{1}, …, \overline{m}_{l}), m_{1} ≥ m_{2} ≥ … ≥ m_{l} ≥ 0

to

Λ^{𝔡𝔟}(λ) = (2l − 2j)^{b}Λ̂_{0} + ∑_{k=1}^{j}^{b}Λ̂_{m_{k}}

if m_{1} ≥ … m_{j} > m_{j+1} = … = m_{l} = 0.

The following theorem was proved in [W2].

Theorem 11.3 We have the following (P in(2l), b̃_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(P in)} V (P in(2l); λ) ⊗ L (b̃_{∞}; Λ^{𝔡𝔟}(λ), l)

where V (P in(2l); λ) is the irreducible P in(2l)-module parametrized by λ ∈ Σ(P in), and L (b̃_{∞}; Λ^{𝔡𝔟}(λ), l) is the irreducible highest weight b̃_{∞}-module of highest weight Λ^{𝔡𝔟}(λ) and central charge l.

We define the action of D̂^{+} on F^{⨂ l} by the composition of the action of b̃_{∞} and the homomorphism ϕ̂_{−1/2} given by (5.45). It follows that the action of P in(2l) commutes with that of D̂^{+}. Introduce the following generating function

W_{1/2}^{n}(z) = ∑_{k∈ℤ} W_{k}^{n,1/2}z^{−k−n}  (11.101)

It follows from Proposition 5.2 that the representation of D̂^{+} on F^{⨂ l} has central charge l and

ϕ̂_{−1/2} (e^{xD} − e^{−xD}) = ϕ_{−1/2} (e^{xD} − e^{−xD}) − l tanh(x/4).  (11.102)

Therefore

ϕ̂_{−1/2}(W_{0}^{n,1/2}) = ϕ_{−1/2}(W_{0}^{n,1/2}) − α_{n}l

for some constant α_{n} determined by (11.102) depending on n only. Similarly as in Lemma 11.1 one can recast the action of D̂^{+} in terms of W_{1/2}^{n}(z) as

\begin{matrix}W_{1/2}^{n}(z) \\ = & \frac{1}{2} ∑_{k=1}^{l} (: ∂_{z}^{n}ψ^{−,k}(z)ψ^{+,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(z)ψ^{−,k}(z) :) − α_{n}lz^{−n}. \end{matrix} (11.103)

Given λ = (m_{1}, …, \overline{m}_{l}) ∈ Σ(P in) where m_{1} ≥ m_{2} ≥ … ≥ m_{l} ≥ 0, we let e(λ) be the set of exponents m_{k} + \frac{1}{2} (k = 1, …, j) of multiplicity 1 and exponent \frac{1}{2} of multiplicity l − j.

The proof of the following theorem is obtained in an analogous way as for Theorem 11.2 by using Theorems 6.1, 11.3 and Proposition 6.7.

Theorem 11.4 We have the following (P in(2l), D̂^{+})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(P in)} V (P in(2l); λ) ⊗ L (D̂^{+}; e(λ), l) .

12 FFR’s of QHWM’s over D̂^{+} with c ∈ \frac{1}{2} + ℤ_{+}

Introduce a neutral fermionic field ϕ(z) = ∑_{n∈\underline{ℤ}} ϕ_{n}z^{−n− \frac{1}{2} +ϵ} which satisfies the following commutation relations:

[ϕ_{m}, ϕ_{n}]_{+} = δ_{m,−n}, m, n ∈ \underline{ℤ}.

Denote by F^{⨂ \frac{1}{2}} the Fock space of ϕ(z) generated by a vacuum vector |0⟩, which is annihilated by ϕ_{n}, n ∈ \underline{ℤ}_{+}. Denote by F^{⨂ l+ \frac{1}{2}} the ℤ_{2}-graded tensor product of F^{⨂ \frac{1}{2}} and the Fock space F^{⨂ l} of l pairs of fermionic fields ψ^{±,k}(z)(k = 1, …, l).

Denote by

e^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n−1+2ϵ} = : ψ^{−,p}(z)ϕ(z) :
e_{∗}^{p}(z) ≡ ∑_{n∈ℤ} e_{∗}^{p}(n)z^{−n−1+2ϵ} = : ψ^{+,p}(z)ϕ(z) : (p = 1, …, l). (12.104)

Then the Fourier components of e^{p}(z), e_{∗}^{p}(z), together with generating functions (11.90), (11.91) and (11.92)

e_{∗}^{pq}(n), e^{pq}(n)(p ≠ q), e_{∗∗}^{pq}(n)(p ≠ q), e^{p}(n), e_{∗}^{p}(n) (n ∈ ℤ, p, q = 1, …, l)

generate an affine algebra 𝔰̂𝔬(2l + 1) of level 1 [F1, KP]. e^{pq}(0) (p ≠ q), e_{∗∗}^{pq}(0) (p ≠ q), e_{∗}^{pq}(0), e^{p}(0), e_{∗}^{p}(0) (p, q = 1, …, l) generate the horizontal subalgebra 𝔰𝔬(2l + 1) of 𝔰̂𝔬(2l + 1). In particular, we identify the Borel subalgebra 𝔟(𝔰𝔬(2l + 1)) with the one generated by e_{∗∗}^{pq}(0)(p ≠ q), e_{∗}^{pq}(0)(p ≤ q), e_{∗}^{p}(0), p, q = 1, ⋯, l.

¿From now on we need to consider the two cases \underline{ℤ} = \frac{1}{2} + ℤ and ℤ separately. First consider the case \underline{ℤ} = \frac{1}{2} + ℤ. Introduce a generating function

\begin{matrix}∑_{i,j∈ℤ}(E_{ij} − E_{1−j,1−i})z^{i−1}w^{−j} \\ = ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :)+ : ϕ(z)ϕ(w) : . \end{matrix}

One can show that this defines an action of d_{∞} on F^{⨂ l+ \frac{1}{2}} with central charge l + \frac{1}{2}. The action of the horizontal subalgebra 𝔰𝔬(2l + 1) can be lifted to an action of O(2l + 1). The action of O(2l + 1) commutes with that of d_{∞} generated by E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ l+ \frac{1}{2}} .

Define a map Λ^{𝔟𝔡} from Σ(B) to d_{∞}_{0}^{∗} by sending

λ = (m_{1}, m_{2}, …, m_{l})

to

Λ^{𝔟𝔡} = (2l + 1 − i − j)^{d}Λ̂_{0} + (j − i)^{d}Λ̂_{1} + ∑_{k=1}^{i}^{d}Λ̂_{m_{k}}

and sending

λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det

to

Λ^{𝔟𝔡} = (j − i)^{d}Λ̂_{0} + (2l + 1 − i − j)^{d}Λ̂_{1} + ∑_{k=1}^{i}^{d}Λ̂_{m_{k}}

assuming that

m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0.

The following theorem is quoted from [W2].

Theorem 12.1 We have the following (O(2l + 1), d_{∞})-module decomposition:

F^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(B)} V (O(2l + 1); λ) ⊗ L (d_{∞}; Λ+^{𝔟𝔡}(λ), l + 1/2)

where V (O(2l+1); λ) is the irreducible O(2l+1)-module parametrized by λ ∈ Σ(B) and L (d_{∞}; Λ^{𝔟𝔡}(λ), l + 1/2) is the irreducible highest weight d_{∞}-module with highest weight Λ^{𝔟𝔡}(λ) and central charge l + 1/2.

The action of D̂^{+} on F^{⨂ l+ \frac{1}{2}} is given by the composition of the action of d_{∞} on F^{⨂ l+ \frac{1}{2}} and the homomorphism ϕ̂_{0}. Similarly as in Lemma 11.1 we can show that

\begin{matrix}W^{n}(z) = \\ \frac{1}{2} ∑_{k=1}^{l} (: ∂_{z}^{n}ψ^{−,k}(z)ψ^{+,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(z)ψ^{−,k}(z) :) + \frac{1}{2} : ∂_{z}^{n}ϕ(z)ϕ(z) : . \end{matrix} (12.105)

For λ = (m_{1}, m_{2}, …, m_{l}) ∈ Σ(B) we let e(λ) be the set of exponents m_{k} (k = 1, …, i) of multiplicity 1, the exponent 1 of multiplicity j − i (the exponent 0 has multiplicity l − j + \frac{1}{2}); and for λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det we let e(λ) be the set of exponents m_{k} (k = 1, …, i) of multiplicity 1, the exponent 1 of multiplicity 2l + 1 − i − j (the exponent 0 has multiplicity −l + j − \frac{1}{2}), where

m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0.

The following theorem can be proved in an analogous way as Theorem 11.2 by using Theorems 6.1, 12.1 and Proposition 6.6.

Theorem 12.2 We have the following (O(2l + 1), D̂^{+})-module decomposition:

F^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(B)} V (O(2l + 1); λ) ⊗ L (D̂^{+}; e(λ), l + 1/2) .

The following corollary is immediate.

Corollary 12.1 The space of invariants of O(2l + 1) in F^{⨂ l+ \frac{1}{2}} is naturally isomorphic to the irreducible d_{∞}-module L(d_{∞}; (2l + 1) ^{d}Λ̂_{0}) or equivalently to the irreducible D̂^{+}-module L (D̂^{+}; e(0), l + 1/2).

Remark 12.1 (SO(2l + 1), σ ⋉ d_{∞}) form a dual pair on F^{⨂ l+ \frac{1}{2}} . In particular the space of invariants of F^{⨂ l+ \frac{1}{2}} with respect to SO(2l + 1) is isomorphic to the d_{∞}- module L(d_{∞}; (2l + 1) ^{d}Λ̂_{0}) ⨁ L(d_{∞}; (2l + 1) ^{d}Λ̂_{1}) or equivalently the D̂^{+}-module L(D̂^{+}; e(0), l + 1/2) ⨁ L(D̂^{+}; e(det), l + 1/2).

Now we consider the case \underline{ℤ} = ℤ. Introduce the following generating function

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − E_{−j,−i})z^{i}w^{−j} \\ = & ∑_{k=1}^{l} (: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :) + : ϕ(z)ϕ(w) : . \end{matrix}

This defines a representation of b̃_{∞} on F^{⨂ l+ \frac{1}{2}} of central charge l + 1/2.

The Fock space F^{⨂ l+ \frac{1}{2}} splits into a sum of two subspaces F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} , where F_{e}^{⨂ l+ \frac{1}{2}} consists all even vectors while F_{o}^{⨂ l+ \frac{1}{2}} consists all odd vectors according to the ℤ_{2} gradation on the vector superspace F^{⨂ l+ \frac{1}{2}} . The action of 𝔰𝔬(2l + 1) can be lifted to Spin(2l) on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} respectively. Spin(2l) and b_{∞} form a dual pair on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} [W2].

Define a map Λ^{𝔟𝔟} from Σ(P B) to b_{∞}_{0}^{∗} by sending λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}) to

Λ^{𝔟𝔟}(λ) = (2l + 1 − 2j)^{b}Λ̂_{0} + ∑_{k=1}^{j}^{b}Λ̂_{m_{k}}

if m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0.

The following theorem was proved in [W2].

Theorem 12.3 We have the following (Spin(2l + 1), b̃_{∞})-module decomposition:

F_{e}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (b̃_{∞}; Λ^{𝔟𝔟}(λ), l + \frac{1}{2})
F_{o}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (b̃_{∞}; Λ^{𝔟𝔟}(λ), l + \frac{1}{2})

where V (O(2l + 1); λ) is the irreducible Spin(2l + 1)-module parametrized by λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}) and L (b̃_{∞}; Λ^{𝔟𝔟}(λ), l + \frac{1}{2}) is the irreducible highest weight b̃_{∞}-module with central charge l + \frac{1}{2}.

Now the action of D̂^{+} on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} can be obtained by the composition of the action of b̃_{∞} and the homomorphism ϕ̂_{−1/2} given by (5.45). In terms of W_{1/2}^{n}(z) = ∑_{k∈ℤ} W_{k}^{n,1/2}z^{−k−n} we obtain in an anagolous way as in (11.103) that

W_{1/2}^{n}(z) = \frac{1}{2} ∑_{k=1}^{l} (: ∂_{z}^{n}ψ^{−,k}(z)ψ^{+,k}(z) : + : ∂_{z}^{n}ψ^{+,k}(z)ψ^{−,k}(z) :)

+\frac{1}{2} : ∂_{z}^{n}ϕ(z)ϕ(z) : −α_{n}(l + 1/2)z^{−n}.

For λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l}) ∈ Σ(P B) where

m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0,

we let e(λ) be the set of exponents m_{k} + \frac{1}{2} (k = 1, …, j) of multiplicity 1 and the exponent \frac{1}{2} of multiplicity l + 1/2 − j. The following theorem can be proved in an analogous way as Theorem 11.2 by using instead Theorems 6.1, 12.3 and Proposition 6.7.

Theorem 12.4 We have the following (Spin(2l + 1), D̂^{+})-module decomposition:

F_{e}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (D̂^{+}; e(λ), l + \frac{1}{2})
F_{o}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (D̂^{+}; e(λ), l + \frac{1}{2}) .

13 FFR’s of QHWM’s over D̂^{+} with c ∈ −\frac{1}{2}ℕ

13.1 Case c ∈ −ℕ

Let us take a pair of bosonic ghost fields

γ^{±}(z) = ∑_{n∈ \frac{1}{2} +ℤ} γ_{n}^{±}z^{−n− \frac{1}{2}} .

Equivalently we have the following operator product expansions

γ^{+}(z)γ^{−}(w) ∼ \frac{1}{z − w}, γ^{+}(z)γ^{+}(w) ∼ 0, γ^{−}(z)γ^{−}(w) ∼ 0.

We take l pairs of bosonic ghost fields γ^{±,p}(z) (p = 1, …, l) and consider the corresponding Fock space F^{⨂ −l}.

Introduce the following generating functions

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1}w^{−j} = − ∑_{p=1}^{l} : γ^{+,p}(z)γ^{−,p}(w) :  (13.106)
e_{∗∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗∗}^{pq}(n)z^{−n−1} =: γ^{+,p}(z)γ^{+,q}(z) : (p ≠ q)
e^{pq}(z) ≡ ∑_{i,j∈ℤ} e^{pq}(n)z^{−n−1} =: γ^{−,p}(z)γ^{−,q}(z) : (p ≠ q)  (13.107)
e_{∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗}^{pq}(n)z^{−n−1} =: γ^{+,p}(z)γ^{−,q}(z) : (p, q = 1, …, l)

where the normal ordering :: means that the operators annihilating |0⟩ are moved to the right.

It is well known that the operators E_{ij} (i, j ∈ ℤ) form a representation in F^{⨂ −l} of the Lie algebra 𝔤̂𝔩 with central charge −l; the operators

e^{pq}(n), e_{∗}^{pq}(n), e_{∗∗}^{pq}(n) (p, q = 1, …, l, n ∈ ℤ)

form an affine algebra 𝔰̂𝔭(2l) with central charge −1 [FF]. The operators e^{pq}(0), e_{∗}^{pq}(0), e_{∗∗}^{pq}(0) (p, q = 1, ⋯ l) span the horizontal subalgebra 𝔰𝔭(2l) in 𝔰̂𝔭(2l). In particular, operators e_{∗}^{pq} (p, q = 1, ⋯ l) form a Lie subalgebra 𝔤𝔩(l) in the horizontal subalgebra 𝔰𝔭(2l). We identify the Borel subalgebra 𝔟(𝔰𝔭(2l)) with the one generated by e_{∗∗}^{pq}(0), e_{∗}^{pq}(0)(p ≤ q), p, q = 1, ⋯, l. The action of the horizontal subalgebra 𝔰𝔭(2l) can be lifted to an action of Lie group Sp(2l) on F^{⨂ −l}.

It follows from (13.106) that

∑_{i,j∈ℤ}(E_{ij} − E_{1−j,1−i})z^{i−1}w^{−j} = ∑_{p=1}^{l}(: γ^{+,p}(w)γ^{−,p}(z) : − : γ^{+,p}(z)γ^{−,p}(w) :).

The operators E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) span the Lie algebra d_{∞} with central charge −l. The actions of Sp(2l) and of d_{∞} on F^{⨂ −l} commutes with each other and form a dual pair. We now define a map Λ^{𝔠𝔡} from Σ(C) to d_{∞}_{0}^{∗} which maps λ = (m_{1}, …, m_{l}) to

Λ^{𝔠𝔡}(λ) = (−2l − m_{1} − m_{2})^{d}Λ̂_{0} + ∑_{k=1}^{l}(m_{k} − m_{k+1})^{d}Λ̂_{k}

with the convention m_{l+1} = 0 here and below. The following theorem was proved in [W2].

Theorem 13.1 We have the following (Sp(2l), d_{∞})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (d_{∞}; Λ^{𝔠𝔡}(λ), −l)

where V (Sp(2l); λ) is the irreducible highest weight Sp(2l)-module of highest weight λ and L (d_{∞}; Λ^{𝔠𝔡}(λ), −l) the irreducible highest weight d_{∞}-module of highest weight Λ^{𝔠𝔡}(λ) and central charge −l.

The action of D̂^{+} on F^{⨂ −l} is given by the composition of the action of d_{∞} and the homomorphism ϕ̂_{−1/2}. A similar argument as in Lemma 11.1 shows that

W^{n}(z) = \frac{1}{2} ∑_{p=1}^{l} (: ∂_{z}^{n}γ^{+,p}(z)γ^{−,p}(z) : − : γ^{+,p}(z)∂_{z}^{n}γ^{−,p}(z) :) .  (13.108)

Given λ = (m_{1}, …, m_{l}) ∈ Σ(C), we let e(λ) be the set of exponents k with multiplicity m_{k} − m_{k+1} (k = 1, …, l) where m_{l+1} = 0 (the exponent 0 has multiplicity −l − m_{1}).

Theorem 13.2 We have the following (Sp(2l), D̂^{+})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (D̂^{+}; e(λ), −l) .

Proof. By Theorem 6.1 the d_{∞}-module L (d_{∞}; Λ^{𝔠𝔡}(λ), −l) regarded as a D̂^{+}- module via the pullback by ϕ̂_{0} remains irreducible. To prove the thereom it suffices to determine the corresponding Δ(x) for this D̂^{+}-module. It follows from the definition of Λ^{𝔠𝔡}(λ) by using Proposition 6.6 that for λ = (m_{1}, …, m_{l}) ∈ Σ(C)

Δ(x) = \frac{∑_{k=1}^{l}(m_{k} − m_{k+1}) cosh(kx) + \frac{1}{2}((−2l − m_{1} − m_{2}) − (m_{1} − m_{2})) − (−l)}{2 sinh(x/2)}
= \frac{∑_{k=1}^{l}(m_{k} − m_{k+1}) cosh(kx) + (−l − m_{1})}{2 sinh(x/2)} − \frac{(−l)}{2 sinh(x/2)}.
□

The following corollary is immediate.

Corollary 13.1 The space of invariants of Sp(2l) in the Fock space F^{⨂ −l} is naturally isomorphic to the irreducible module L(d_{∞}; −2l^{d}Λ̂_{0}) or equivalently to the irreducible D̂^{+}-module L (D̂^{+}; e(0), −l).

13.2 Case c ∈ −ℕ + 1/2

We denote by F^{⨂ −l+ \frac{1}{2}} the tensor product of the Fock space F^{⨂ −l} of l pairs of bosonic ghost fields and the Fock space F^{⨂ \frac{1}{2}} of a neutral fermionic field ϕ(z). It is known [FF] that the Fourier components of the generating functions e^{pq}(z), e_{∗}^{pq}(z), e_{∗∗}^{pq}(z) in (13.107) and the following generating functions

e^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n−1} = : γ^{−,p}(z)ϕ(z) :
ẽ_{∗}^{p}(z) ≡ ∑_{n∈ℤ} ẽ_{∗}^{p}(n)z^{−n−1} = : γ^{+,p}(z)ϕ(z) : (p = 1, …, l)

span the affine superalgebra 𝔬̂𝔰𝔭(1, 2l) of level −1. The operators e^{pq}(0), e_{∗}^{pq}(0), e_{∗∗}^{pq}(0), ẽ^{p}(0), ẽ_{∗}^{p}(0) (p, q = 1, …, l) generate the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) of the affine superalgebra 𝔬̂𝔰𝔭(1, 2l). We identify the Borel subalgebra 𝔟(𝔬𝔰𝔭(1, 2l)) with the one generated by e_{∗}^{pq}(0)(p ≤ q), e_{∗∗}^{pq}(0), ẽ_{∗}^{p}(0), p, q = 1, ⋯, l. Introduce the generating function

\begin{matrix}∑_{i,j∈ℤ} (E_{ij} − E_{1−j,1−i}) z^{i−1}w^{−j} \\ = ∑_{k=1}^{l} (: γ^{+,k}(z)γ^{−,k}(w) : − : γ^{+,k}(w)γ^{−,k}(z) :) + : ϕ(z)ϕ(w) : . \end{matrix} (13.109)

This defines a representation of d_{∞} of central charge −l + \frac{1}{2} on F^{⨂ −l+ \frac{1}{2}} . It is known [W2] the action of the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) commute with that of Lie algebra d_{∞} on F^{⨂ −l+ \frac{1}{2}} . (𝔬𝔰𝔭(1, 2l), d_{∞}) form a dual pair on F^{⨂ −l+ \frac{1}{2}} .

We define a map Λ^{𝔬𝔰𝔭𝔡} from Σ(osp) to d_{∞}_{0}^{∗} by sending λ = (m_{1}, …, m_{l}) to

Λ^{𝔬𝔰𝔭𝔡}(λ) = (−2l + 1 − m_{1} − m_{2})^{d}Λ̂_{0} + ∑_{k=1}^{l}(m_{k} − m_{k+1})^{d}Λ̂_{k}.

The following duality theorem was quoted from [W2].

Theorem 13.3 We have the following (𝔬𝔰𝔭(1, 2l), d_{∞})-module decomposition

F^{⨂ −l+ \frac{1}{2}} = ⨁_{λ∈Σ(Osp)} V (𝔬𝔰𝔭(1, 2l); λ) ⊗ L (d_{∞}; Λ^{𝔬𝔰𝔭𝔡}(λ), −l + 1/2)

where V (𝔬𝔰𝔭(1, 2l); λ) is the irreducible module of 𝔬𝔰𝔭(1, 2l)(1, 2l) of highest weight λ, and L (d_{∞}; Λ^{𝔬𝔰𝔭𝔡}(λ), −l + 1/2) is the irreducible highest weight d_{∞}-module of highest weight Λ^{𝔬𝔰𝔭𝔡}(λ) and central charge −l + 1/2.

We define the action of D̂^{+} on F^{⨂ −l+ \frac{1}{2}} to be the composition of the action of d_{∞} and the homomorphism ϕ̂_{0} given by (5.45). In a similar way as obtaining (13.108) we have

W^{n}(z) = \frac{1}{2} ∑_{k=1}^{l} (: γ^{+,k}(z)∂_{z}^{n}γ^{−,k}(z) : − : γ^{−,k}(z)∂_{z}^{n}γ^{+,k}(z) :)
+\frac{1}{2} : ϕ(z)∂_{z}^{n}ϕ(z) : .  (13.110)

Then of course the action of the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) commute with that of Lie algebra D̂^{+} on F^{⨂ −l+ \frac{1}{2}} .

Given λ = (m_{1}, …, m_{l}) ∈ Σ(Osp), we let e(λ) be the set of exponents k of multiplicity m_{k} − m_{k+1} (k = 1, …, l) where m_{l+1} = 0 (the exponent 0 has multiplicity −l − m_{1} + \frac{1}{2}). We obtain the following theorem in a similar way as Theorem 13.2 by using Theorems 6.1, 13.3, and Proposition 6.6.

Theorem 13.4 We have the following (𝔬𝔰𝔭(1, 2l), D̂^{+})-module decomposition

F^{⨂ −l+ \frac{1}{2}} = ⨁_{λ∈Σ(Osp)} V (𝔬𝔰𝔭(1, 2l); λ) ⊗ L (D̂^{+}; e(λ), −l + 1/2) .

The following corollary is immediate.

Corollary 13.2 The space of invariants of 𝔬𝔰𝔭(1, 2l) in the Fock space F^{⨂ −l+ \frac{1}{2}} is naturally isomorphic to the irreducible module L(d_{∞}; (−2l + 1)^{d}Λ̂_{0}) or equivalently to the irreducible D̂^{+}-module L (D̂^{+}; 0, −l + 1/2).

14 Vertex algebra associated to D̂^{+}

14.1 Vertex algebra structure on the vacuum module of D̂^{+}

In Example 3.1 we have constructed the vacuum D̂^{+}-modules M_{c} and V_{c} (c ∈ ℂ). We want to show that M_{c} and V_{c} carry a natural structure of a vertex algebra, cf. [B, FLM, DL, K2].

Denote by |0⟩ the highest weight vector of M_{c} and V_{c}. M_{c} (resp. V_{c}) is spanned by the vectors

W_{−k_{j}−n_{j}−1}^{n_{j}} ⋯ W_{−k_{1}−n_{1}−1}^{n_{1}}|0⟩, n_{i} ∈ 2ℤ_{+} + 1, k_{i} ∈ ℤ.  (14.111)

By the definition of M_{c} the vector |0⟩ is annihilated by P, namely W_{k}^{n}|0⟩ = 0 for k + n ≥ 0. Operators from P is often referred to as annihilation operators and the operators W_{k}^{n} from D̂^{+} − P as creation operators. We have known that W_{k}^{1} = −t^{k}(D + (k + 1)/2) (k ∈ ℤ) form the Virasoro algebra with central charge c when acting on M_{c} or V_{c}. Also it follows by a direct computation using (2.16) that

[W_{−1}^{1}, W_{k}^{n}] = −(n + k)W_{k−1}^{n}

or equivalently

[W_{−1}^{1}, W^{n}(z)] = ∂_{z}W^{n}(z).  (14.112)

Denote T = W_{−1}^{1}. Define a linear map Y (·, z) : M_{c} ⟶ (End M_{c})[[z, z^{−1}]] by

Y (W_{−n−1}^{n}|0⟩, z) = W^{n}(z) (n ∈ 2ℤ_{+} + 1)

and in general we associate to a vector a of the form (14.111) the following generating function

Y (a, z) =: ∂_{z}^{(k_{j})}W^{n_{j}}(z) ⋯ ∂_{z}^{(k_{1})}W^{n_{1}}(z) :

where ∂_{z}^{(k)} denotes ∂_{z}^{k}/k! and :: denotes the standard normal ordering from the right to left which moves the annihilators to the right. Clearly Y (a, z)|0⟩ |_{z=0}= a.

Proposition 14.1 (V_{l}, |0⟩, T, Y (·, z)) is a vertex algebra isomorphic to the space of invariants of F^{⨂ l} for \underline{ℤ} = \frac{1}{2} + ℤ with respect to O(2l). The generating fields W^{n}(z) are given by formula (11.94).

By Corollary 11.1, V_{l} and the space of invariants of F^{⨂ l} with respect to O(2l) are naturally isomorphic as D̂^{+}-modules. Since F^{⨂ l} has a natural vertex algebra structure, the vertex algebra structure on V_{l} is ensured by the well-known fact that the space of invariants of an automorphism group of a vertex algebra is a vertex algebra. We will give a direct proof in Lemma 14.2 below that the generating fields W^{n}(z) (n ∈ 2ℤ_{+} + 1) are closed under operator product expansions.

Consider the case of F^{⨂ l} with \underline{ℤ} = \frac{1}{2} + ℤ. Denote

Ψ^{m,n}(z) = ∑_{k=1}^{l} (: ∂^{m}ψ^{+,k}(z)∂^{n}ψ^{−,k}(z) : + : ∂^{m}ψ^{−,k}(z)∂^{n}ψ^{+,k}(z) :) , m, n ∈ ℤ_{+}.

Note the obvious symmetry Ψ^{m,n}(z) = −Ψ^{n,m}(z).

Lemma 14.1 Ψ^{m,n}(z) is a linear combination of ∂^{i}W^{m+n−i}(z), 0 ≤ i ≤ m+n and i ≡ m + n − 1 mod 2.

Proof. We prove by induction on m + n. When m + n = 0 or 1, the statement is obvious.

Assume that for m + n = 2k − 1 (k ∈ ℕ) the statement is true. Then by this induction assumption, ∂Ψ^{2k−1−m,m}(z) (m = 0, …, k − 1) is a linear combination of ∂^{i}W^{2k+2−i}(z) (0 ≤ i ≤ 2k − 1 and i odd). Since

∂Ψ^{2k−1−m,m}(z) = Ψ^{2k−m,m}(z) + Ψ^{2k−1−m,m+1}(z)

it follows by a little algebra that the linear span of Ψ^{2k−m,m}(z), m = 0, …, k − 1 is equal to the linear span of ∂Ψ^{2k−1−m,m}(z), m = 0, …, k − 1. This proves that the statement is true for m + n = 2k.

Therefore it follows that ∂Ψ^{2k−m,m}(z), m = 0, …, k − 1 is a linear combination of ∂^{i}W^{2k+1−i}(z) (0 ≤ i ≤ 2k + 1 and i even). Since

∂Ψ^{2k−m,m}(z) = Ψ^{2k+1−m,m}(z) + Ψ^{2k+1−m,m}(z)

by a little linear algebra it is easy to show that the linear span of Ψ^{2k+1−m,m}(z) (m = 0, …, k) is equal to the linear span of ∂Ψ^{2k−m,m}(z) (m = 0, …, k − 1) and W^{2k+1}(z). This proves that the statement is true for m + n = 2k + 1. □

Lemma 14.2 ∂^{i}W^{j}(z), i ∈ ℤ_{+}, j ∈ 2ℤ_{+} + 1 are closed under the operator product expansions.

Proof. For the simplicity of notations, we first assume l = 1 and write ψ^{±}(z) for ψ^{±,1}(z). By Wick’s theorem, we have

W^{m}(z)W^{n}(z) ∼ \frac{1}{4} (: ∂_{z}^{m}ψ^{−}(z)ψ^{+}(z) : + : ∂_{z}^{m}ψ^{+}(z)ψ^{−}(z) :) ·

(: ∂_{w}^{n}ψ^{−}(w)ψ^{+}(w) : + : ∂_{w}^{n}ψ^{+}(w)ψ^{−}(w) :)

∼ \frac{1}{4} (∂_{z}^{m}((z − w)^{−1})) (: ψ^{+}(z)∂_{w}^{n}ψ^{−}(w) : + : ψ^{−}(z)∂_{w}^{n}ψ^{+}(w) :)

+\frac{1}{4} (∂_{w}^{n}((z − w)^{−1})) (: ∂_{z}^{m}ψ^{−}(z)ψ^{+}(w) : + : ∂_{z}^{m}ψ^{+}(z)ψ^{−}(w) :)

+\frac{1}{2}∂_{z}^{m}((z − w)^{−1})∂_{w}^{n}((z − w)^{−1})

∼ \frac{(−1)^{m}m!}{4(z − w)^{m+1}} (: ψ^{+}(z)∂_{w}^{n}ψ^{−}(w) : + : ψ^{−}(z)∂_{w}^{n}ψ^{+}(w) :)

+\frac{n!}{4(z − w)^{n+1}} (: ∂_{z}^{m}ψ^{−}(z)ψ^{+}(w) : + : ∂_{z}^{m}ψ^{+}(z)ψ^{−}(w) :)

+\frac{(−1)^{m}(m + n)!}{2(z − w)^{m+n+2}}.

By taking the Taylor expansions of ψ^{±}(z) and ∂_{z}^{m}ψ^{±}(z) at z = w, we can easily see that all the fields appearing on the right hand side of the above OPE are linear combinations of Ψ^{m,n}(z). Thus our lemma follows from Lemma 14.1. For the general case F^{⨂ l}, it is clear that the only modification in the final OPE formula above is that the central term should be mutiplied by l. □

The following corollary holds in the cases c = l (l ∈ ℕ) by the computation of OPE in Lemma 14.2 since the maximal order of poles at z = w appearing there is m + n + 2. Since the central term depends on the central charge c linearly, the corollary remains true for an arbitrary central charge.

Corollary 14.1 For arbitrary central charge c we have as a formal power series

(z − w)^{m+n+2} [W^{m}(z), W^{n}(w)] = 0.

Now the following theorem follows by a general argument in the theory of vertex algebras, cf. [L], Proposition 3.1 in [FKRW], or Theorem 4.5 in [K2], since all the requirements there are satisfied (also cf. [G]).

Theorem 14.1 (M_{c}, |0⟩, T, Y (·, z)) and (V_{c}, |0⟩, T, Y (·, z)) are vertex algebras.

The following proposition follows from Corollaries 12.1, 13.1, 13.2 and Theorem 14.1.

Proposition 14.2 1) (V_{l+1/2}, |0⟩, ω, Y (·, z)) is a vertex algebra isomorphic to the space of invariants of F^{⨂ l+ \frac{1}{2}} for \underline{ℤ} = \frac{1}{2} + ℤ with respect to O(2l + 1). The generating fields W^{n}(z) are given by formula (12.105).

2) (V_{−l}, |0⟩, ω, Y (·, z)) is a vertex algebra isomorphic to the space of invariants of F^{⨂ −l} with respect to Sp(2l). The generating fields W^{n}(z) are given by formula (13.108).

3) (V_{−l+1/2}, |0⟩, ω, Y (·, z)) is a vertex algebra isomorphic to the space of invariants of F^{⨂ −l+ \frac{1}{2}} with respect to 𝔬𝔰𝔭(1, 2l). The generating fields W^{n}(z) are given by formula (13.110).

Corollary 14.2 The irreducible representations of D̂^{+} appearing in the decompositions of Fock spaces F^{⨂ l}, F^{⨂ l+ \frac{1}{2}} , F^{⨂ −l} and F^{⨂ −l+ \frac{1}{2}} are representations of the vertex algebra associated to D̂^{+} with central charges l, l + 1/2, −l and −l + 1/2 respectively.

Remark 14.1 It is interesting to compare some theorems in [DLM] by Dong-Li-Mason with our Theorems 11.2, 12.2, 13.2 and 13.4 in the framework of vertex algebras. Their results have the virtue of being general while the statements of our theorems are more powerful and precise thanks to being concrete.

14.2 Vertex algebra V_{c} for c ∈ \frac{1}{2}ℤ and W algebras

A particular important class of vertex algebras is the so-called W-algebras. One can associate a W-algebra W𝔤 to an arbitrary complex simple Lie algebra 𝔤 (see [BS, FeF] and references therein). In general such a W-algebra can be defined in terms of intersections of screening operators, cf. e.g. [FeF]. Denote by W(𝔤̂/𝔤, k) the W-algebra arising as the space of invariants of 𝔤 in the vacuum module of the affine algebra 𝔤̂ of level k, where 𝔤 is the horizontal subalgebra of 𝔤̂. In the case when 𝔤 is simply-laced of rank l, W𝔤 with central charge l can be shown to be isomorphic to W(𝔤̂/𝔤, 1) cf. e.g. [BS, F2, BBSS, FKRW].

It is known [F1, KP] that the basic representation of 𝔰̂𝔬(2l) is isomorphic to the even part of the vector superspace F^{⨂ l}. ¿From Remark 11.1 we see that the space of invariants of F^{⨂ l} with respect to SO(2l) is isomorphic to the D̂^{+}- module L (D̂^{+}; 0, l) ⨁ L (D̂^{+}; e(det), l) (see Section 11). The highest weight vector

of L (D̂^{+}; e(det), l) is given by ∏_{k=1}^{l}(ψ_{−1/2}^{+,k}ψ_{−1/2}^{−,k})|0⟩ [W2] which lies in the even part

of F^{⨂ l}. Thus W(D_{l}^{(1)}/D_{l}, 1) is isomorphic to the space of invariants of F^{⨂ l} with respect to SO(2l). In this way we have reached the following conclusion.

Theorem 14.2 The W-algebra WD_{l} with central charge l is isomorphic to a sum of the vertex algebra V_{l} and the irreducible module L (D̂^{+}; e(det), l) of V_{l}.

As in [FKRW], Theorem 14.2 implies the following corollary.

Corollary 14.3 All positive primitive modules over D̂^{+} with a positive integral central charge l are irreducible modules over the vertex algebra V_{l}.

Remark 14.2 It is an important question to determine whether these positive primitive modules are all irreducible modules over the vertex algebra V_{l}.

Remark 14.3 Theorem 14.2 provides a new way to compute the q-character of WD_{l} with central charge l:

ch_{q}WD_{l} = ch_{q}L (D̂^{+}; 0, l) + q^{l}ch_{q}L (D̂^{+}; e(det), l)
= ch_{q}L (d_{∞}, 2l^{d}Λ̂_{0}) + q^{l}ch_{q}L (d_{∞}, 2l^{d}Λ̂_{1}) .  (14.113)

Here q^{l} accounts for the weight l of the highest weight vector ∏_{k=1}^{l}(ψ_{−1/2}^{+,k}ψ_{−1/2}^{−,k})|0⟩ of L (D̂^{+}; e(det), l) in F^{⨂ l}. The q-character formulas of d_{∞}-modules L (d_{∞}, 2l^{d}Λ̂_{0})

and L (d_{∞}, 2l^{d}Λ̂_{1}) can be read from the Appendix. A straightforward computation by using (14.113) yields

ch_{q}WD_{l} = ∏_{1≤i<j≤l}(1 − q^{j−i})(1 − q^{i+j−2})/φ(q)^{l}

= \frac{∏_{i=1}^{l}(∏_{n=1}^{e_{i}}(1 − q^{n}))}{φ(q)^{l}}

where φ(q) = ∏_{i≥1}(1 − q^{i}) and e_{i} = 2i − 1 (i = 1, ⋯, l − 1), e_{l} = l − 1 are the exponents of the simple Lie algebra 𝔰𝔬(2l). The same formula was earlier deduced in [K0] and [BS]. The character formula implies that WD_{l} with central charge l is freely generated by fields W^{i}(z)(i = 1, 3, …, 2l − 3) of conformal weights i + 1 and the field ∏_{i=1}^{l} : ψ^{+,k}(z)ψ^{−,k}(z) : of conformal weight l corresponding to the vector ∏_{k=1}^{l}(ψ_{−1/2}^{+,k}ψ_{−1/2}^{−,k})|0⟩.

Remark 14.4 Combining Corollary 11.1 with Theorem 14.2, we obtain another dual pair (SO(2l), WD_{l}) on the Fock space F^{⨂ l}. When restricting to the even part of F^{⨂ l} which is isomorphic to the basic representation of 𝔰̂𝔬(2l), we recover a duality theorem of I. Frenkel [F2]. Since the language of vertex algebras was not available at the time of [F2], WD_{l} was replaced by the Lie algebra of Fourier components of fields in WD_{l}.

The even part of the fermionic Fock space F^{⨂ l+ \frac{1}{2}} for \underline{ℤ} = \frac{1}{2} + ℤ is isomorphic to the basic representation of 𝔰̂𝔬(2l + 1) [F1, KP]. According to Corollary 12.1, the space of invariants of F^{⨂ l+ \frac{1}{2}} with respect to SO(2l + 1) is isomorphic to the D̂^{+}-module L(D̂^{+}; 0, l + 1/2) ⨁ L(D̂^{+}; e(det), l + 1/2). The highest weight vector of L(D̂^{+}; e(det), l + 1/2) in F^{⨂ l+ \frac{1}{2}} is ∏_{k=1}^{l}(ψ_{−1/2}^{+,k}ψ_{−1/2}^{−,k})ϕ_{−1/2}|0⟩, cf. [W2]. Observe that it is an odd vector in F^{⨂ l+ \frac{1}{2}} . Thus W(B_{l}^{(1)}/B_{l}, 1) is isomorphic to L(D̂^{+}; 0, l+ 1/2). Combining with Proposition 14.2 we have proved the following theorem.

Theorem 14.3 The vertex algebra V_{l+1/2} associated to D̂^{+} with c = l + 1/2 is isomorphic to the W-algebra W(B_{l}^{(1)}/B_{l}, 1).

Corollary 14.4 All positive primitive modules over D̂^{+} with a central charge l + 1/2 ∈ 1/2 + ℤ_{+} are irreducible modules over the vertex algebra V_{l+1/2}.

Remark 14.5 It remains to determine whether these positive primitive modules are all irreducible modules over the vertex algebra V_{l+1/2}.

It follows that the character formula of the W-algebra W(B_{l}^{(1)}/B_{l}, 1) is the same as the q-character formula of L (d_{∞}; (2l + 1)^{d}Λ̂_{0}) (cf. Appendix):

\frac{∏_{i=1}^{l}(∏_{n=1}^{e_{i}}(1 − q^{n}))}{φ(q)^{l}} · \frac{1}{2} ( ∏_{n≥1}(1 + q^{n+l−1/2}) + ∏_{n≥1}(1 − q^{n+l−1/2})) .

where e_{i} = 2i − 1 (i = 1, ⋯, l) are the exponents of the Lie algebra B_{l}. A different method was used in [BS] to obtain the same formula. In particular we have seen that W(B_{l}^{(1)}/B_{l}, 1) lies inside a vertex superalgebra, denoted by WB(0, l), of central charge l−1/2 which is isomorphic to the space of invariants of F^{⨂ l+ \frac{1}{2}} with respect to SO(2l + 1). The q-character of the latter can be calculated by using the q-character formulas of d_{∞}-modules given in the Appendix as

ch_{q}L (d_{∞}; (2l + 1)^{d}Λ̂_{0}) + q^{l+1/2}ch_{q}L (d_{∞}; (2l + 1)^{d}Λ̂_{1})
= \frac{∏_{i=1}^{l}(∏_{n=1}^{e_{i}}(1 − q^{n}))}{φ(q)^{l}} ∏_{n≥1}(1 + q^{n+l−1/2}).  (14.114)

Here q^{l+1/2} accounts for the grading of the highest weight vector of the d_{∞}-module L (d_{∞}; (2l + 1)^{d}Λ̂_{1}) in F^{⨂ l+ \frac{1}{2}} .

Formula (14.114) indicates that vertex superalgebra WB(0, l) is freely generated by the bosonic fields W^{i}(z)(i = 1, 3, …, 2l − 1) of conformal weights i + 1 and the fermionic field ∏_{i=1}^{l} : ψ^{+,k}(z)ψ^{−,k}(z)φ(z) : of conformal weight l + 1/2 corresponding to the vector ∏_{k=1}^{l}(ψ_{−1/2}^{+,k}ψ_{−1/2}^{−,k})φ_{−1/2})|0⟩. The phenomenon that W(B_{l}^{(1)}/B_{l}, 1) lies in such a vertex superalgebra was conjectured in [BS]. Here we find the explicit model for WB(0, l) with central charge l − 1/2. WB(0, l) were also obtained by the quantized Drinfeld-Sokolov reduction in [Ito]. We summarize these in the following theorem which resembles remarkably Theorem 14.2.

Theorem 14.4 The vertex superalgebra WB(0, l) is the sum of the vertex algebra V_{l+1/2} and the irreducible module L (D̂^{+}; e(det), l + 1/2) of V_{l+1/2}.

The following theorem easily follows from Proposition 14.2 and similar argument which leads to Theorems 14.3.

Theorem 14.5 The vertex algebra V_{−l} associated to D̂^{+} with central charge −l is isomorphic to the W-algebra W(C_{l}^{(1)}/C_{l}, −1). The vertex algebra V_{−l+1/2} associated to D̂^{+} with central charge −l + 1/2 is isomorphic to W(B^{(1)}(0, l)/B(0, l), −1).

Remark 14.6 Theorems 14.2 and 14.4 tell us what is the minimal set of generating fields for V_{c} with c ∈ \frac{1}{2}ℕ (or rather for the vertex (super)algebra which are the ℤ_{2} extension of V_{c}). The question remains open for V_{c} for a general negative halfingeral central charge. A similar question was addressed in the case of W_{1+∞} in [W] (also see [EFH]). The negative central charge case turned out to be more subtle and difficult.

15 Appendix

The following proposition gives the q-character formula (compatible with the ℤgradation of d_{∞} induced from that of 𝔤̂𝔩) of a highest weight representation of d_{∞} with highest weight Λ = ^{d}Λ̂_{n_{1}} +^{d}Λ̂_{n_{2}} + … +^{d}Λ̂_{n_{k}} +^{d}h_{0}^{d}Λ̂_{0}, where n_{1} ≥ … n_{k} ≥ 1, dh_{0} ∈ ℤ_{+}. Denote ^{d}h_{1} to be the number of n_{i}’s which are equal to 1. Then the central charge c = \frac{1}{2}(^{d}h_{0} −^{d}h_{1}) + k. We retain the notation in Section 1.

Proposition 15.1 The q-character formula of L(d_{∞}; Λ) corresponding to the principal gradation of d_{∞} is

ch_{q}L(d_{∞}; Λ) = \frac{∏_{1≤i<j≤k}(1 − q^{n_{i}−n_{j}+j−i})}{∏_{1≤i≤k} φ_{n_{i}+k−i}(q)} · \frac{φ(q^{2})\overline{2c} ∏_{j>0} φ_{2c−2j}(q)}{φ(q)^{[c]−\overline{2c}}} ·

∏_{i=0}^{n_{1}−1} \frac{φ_{2c+i+n_{1}}(q)}{φ_{2c+n_{1}+i−^{d}λ_{i+1}}(q)} · ∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c+j+i−^{d}λ_{i+1}−^{d}λ_{j+1}}}{1 − q^{2c+j+i}} ·

·\frac{1}{2} (∏_{j∈ℕ} (1 + q^{−(Λ+ρ,ϵ_{j})}) + ∏_{j∈ℕ} (1 − q^{−(Λ+ρ,ϵ_{j})})) . (15.115)

Proof. Note that the q-characters of d_{∞}-modules are specialized characters of type (2, 1, 1, …), in contrast to the q-characters of modules of b_{∞}, c_{∞} which are of type (1, 1, …). So the way of computing such a q-character of a d_{∞}-module will be different. We will proceed as follows. By the Weyl-Kac character formula

e(−Λ)chL(Λ) ≡ \frac{N}{D} = \frac{∑_{w∈W} ϵ(w)e(w(Λ + ρ) − (Λ + ρ))}{∏_{α∈Δ_{+}}(1 − e(−α))}.

Here N (resp. D) denotes the numerator (resp. denominator). Given s⃗ = (2, 1, 1, …), define a homomorphism Fs⃗ : ℂ[[e(−α_{i})]] ⟶ ℂ[[q]] by Fs⃗(e(−α_{i}) = q^{s_{i}}. Put t⃗ = (t_{0}, t_{1}, t_{2}, …) where t_{i} = ⟨Λ + ρ, α_{i}^{∨}⟩. Then

Fs⃗(N) = ∑_{w∈W} ϵ(w)q^{⟨Λ+ρ−w(Λ+ρ),_{Λ_{0}^{∨}}+ρ^{∨}⟩}
= q^{⟨Λ+ρ,_{Λ_{0}^{∨}}+ρ^{∨}⟩} ∑_{w∈W} ϵ(w)q^{⟨Λ+ρ,−w(_{Λ_{0}^{∨}}+ρ^{∨})⟩}
= q^{⟨Λ+ρ,_{Λ_{0}^{∨}}+ρ^{∨}⟩}Ft⃗ ( ∑_{w∈W} ϵ(w)e(w(Λ̂_{0} + ρ)))
= q^{⟨Λ+ρ,_{Λ_{0}^{∨}}+ρ^{∨}⟩}Ft⃗ (chL(Λ̂_{0}) · ∑ ϵ(w)e(w(ρ) − ρ) · e(ρ))
= Ft⃗(e(−ρ − Λ̂_{0}))Ft⃗ (chL(Λ̂_{0})) Ft⃗ ( ∏_{α∈Δ_{+}}(1 − e(−α)) Ft⃗(e(ρ))

= Ft⃗ (e(−Λ̂_{0})chL(Λ̂_{0})) · Ft⃗ ( ∏_{α∈Δ_{+}}(1 − e(−α))) .  (15.116)

By Theorem 12.3 (putting l = 0 there), the even part of F^{⨂ \frac{1}{2}} is isomorphic to the irreducible d_{∞}-module with highest weight Λ̂_{0} and central charge \frac{1}{2}, so

Fs⃗(e(−Λ̂_{0})chL(Λ̂_{0})) = \frac{1}{2} (∏_{j∈ℕ}(1 + q^{j− \frac{1}{2}} ) + ∏_{j∈ℕ}(1 − q^{j− \frac{1}{2}} )) , (15.117)

e(−Λ̂_{0})chL(Λ̂_{0}) = \frac{1}{2} (∏_{j∈ℕ}(1 + e(ϵ_{j})) + ∏_{j∈ℕ}(1 − e(ϵ_{j}))) . (15.118)

By combining equations (15.116) and (15.118) we obtain

ch_{q}L(d_{∞}; Λ) = ∏_{α∈Δ_{+}} \frac{1 − q^{(Λ+ρ,α)}}{1 − q^{(_{Λ_{0}}+ρ,α)}} ·

·\frac{1}{2} (∏_{j∈ℕ} (1 + q^{−(Λ+ρ,ϵ_{j})}) + ∏_{j∈ℕ} (1 − q^{−(Λ+ρ,ϵ_{j})}))

= ∏_{1≤i<j} \frac{1 − q^{^{d}λ_{i}−^{d}λ_{j}+j−i}}{1 − q^{j−i}} · ∏_{0≤i<j} \frac{1 − q^{2c−^{d}λ_{i+1}−^{d}λ_{j+1}+j+i}}{1 − q^{j+i+1}} ·

·\frac{1}{2} (∏_{j∈ℕ} (1 + q^{−(Λ+ρ,ϵ_{j})}) + ∏_{j∈ℕ} (1 − q^{−(Λ+ρ,ϵ_{j})})) (15.119)

A little manipulation of algebra shows that the first product on the right hand side of (15.119) is the same as the first term in (15.115). We rewrite the second term in (15.119) into the product of three terms as follows:

∏_{0≤i<j} \frac{1 − q^{2c+j+i}}{1 − q^{j+i+1}} · ∏_{0≤i<n_{1}<j} \frac{1 − q^{2c−^{d}λ_{i+1}−^{d}λ_{j+1}+j+i}}{1 − q^{2c+j+i}} ·

· ∏_{0≤i<j≤n_{1}} \frac{1 − q^{2c−^{d}λ_{i+1}−^{d}λ_{j+1}+j+i}}{1 − q^{2c+j+i}} .

A little further manipulation of algebra will show the first, second and third terms in the above formula are equal to the second, third and fourth terms of (15.115).

□

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