Duality in infinite dimensional Fock representations

Weiqiang Wang

Max-Planck Institut fur Mathematik, 53225 Bonn, Germany ∗ E-mail: wqwang@mpim-bonn.mpg.de

Abstract

We construct and study in detail various dual pairs acting on some Fock spaces between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to a dual pair, present explicit formulas for the common highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.

0 Introduction

As simple and elegant as any fundamental concept should be, the theory of dual pairs of Howe has been very successful in the study of representation theory of reductive groups (cf. [H1, H2, KV] and references therein). The idea roughly goes as follows: assume that there is a maximally commuting pair of Lie groups/algebras acting natually on a vector space which in turn by itself usually is a “minimal” module of some other larger groups/algebras. Then one gets a decomposition of the vector space into a direct sum of isotypic subspaces which are irreducible under the joint action of these two commuting Lie groups/algebras. Often time one can obtain all the unitary highest weight representations of a fixed reductive Lie group by varying dual pair partners and minimal modules accordingly. Among many applications, we mention that the applications to branching laws and to a decomposition of a tensor product of two modules, cf. [H2].

It is not too surprising that a fundamental principle such as dual pairs applies to representation theory of infinite dimensional Lie algebras and superalgebras as well. The purpose of this paper is to present a systematic study of dual pairs between a finite dimensional Lie group and a completed infinite rank affine algebra of Kac-Moody type associated to an infinite affine Cartan matrix acting on some infinite dimensional Fock spaces (namely representations of infinite dimensional Clifford/Heisenberg algebras in plain language). One of our main motivations behind this paper comes from the study of quasifinite highest weight modules of Lie subalgebras of W_{1+∞} in forthcoming papers with Yan [WY] which generalize the earlier works on the study of representation theory of W_{1+∞} algebra [KR1, FKRW, KR2] and on the tensor categories of modules over 𝔤̂𝔩 and W_{1+∞} [W]. A duality result involving a particular dual pair of this sort between a finite dimensional general linear group and an infinite dimensional general linear Lie algebra 𝔤̂𝔩 in a fermionic Fock space was first obtained by I. Frenkel [F1] by other methods than invoking the principle of dual pairs.

∗On leave from Department of Mathematics, Yale University, New Haven, CT 06520, USA

Dual pairs in infinite dimensional Fock spaces are intimately related to the classical dual pairs in finite dimensional cases. However they are not simply direct limits of some classical dual pairs. Rather there is some sort of semi-infinite twist in the Fock space which, in our opinion, makes things more interesting. As a consequence, the infinite dimensional Lie algebras appearing in these dual pairs have non-trivial central extensions and the irreducible representations appearing in the Fock space decomposition are of highest weight. The central charge of these representations corresponds to the rank of the Lie group in the corresponding dual pair. To some extend, the existence of dual pairs in our infinite dimensional setting ensures the stability properties of classical dual pairs.

It turns out that the Fock spaces on which we have constructed various dual pairs are those which A. Feingold and I. Frenkel used to realize the level ±1 representations of classical affine algebras [FF]. A key observation is that the Lie algebras of the Lie groups appearing in our dual pairs are the horizontal subalgebras of these classical affine algebras accordingly, both untwisted and twisted cases included.

There will be various dual pairs, depending on whether a Fock space is built out of fermionic or bosonic fields, whether these fields are indexed by ℤ or \frac{1}{2} + ℤ, whether there is an additional neutral field or not, whether the related affine Kac-Moody algebra is untwisted or twisted. We list below in two tables the dual pairs which we will construct in this paper in infinite dimensional Fock spaces. The first (resp. second) table lists those dual pairs appearing in a Fock space of l pairs of fermionic (resp. bosonic ghost) fields possibly together with an extra fermionic or bosonic field.

Some remarks on notations are in order. We use F^{⨂ ∗} to denote various Fock spaces throughout this paper. The plus or minus sign in the tensor power of a Fock space means the Fock space is fermionic or bosonic. The number l in the tensor power of a Fock space indicates it is a Fock space of l pairs of fields which are fermions or bosonic ghosts depending on the sign. ±\frac{1}{2} in the tensor power

F^{⨂ l}  F^{⨂ l+ \frac{1}{2}}  F^{⨂ l− \frac{1}{2}}
Untwisted \frac{1}{2} + ℤ (O(2l), d_{∞}) (O(2l + 1), d_{∞})
Untwisted ℤ  (P in(2l), b̃_{∞}) (Spin(2l + 1), b̃_{∞})
Twisted \frac{1}{2} + ℤ (Sp(2l), c_{∞})  (Osp(1, 2l), c_{∞})
Twisted  ℤ  (P in(2l), b_{∞}) (Spin(2l + 1), b_{∞})

Table 1: dual pairs in (mostly) fermionic Fock spaces

F^{⨂ −l}  F^{⨂ −l+ \frac{1}{2}}  F^{⨂ −l− \frac{1}{2}}
Untwisted \frac{1}{2} + ℤ (Sp(2l), d_{∞}) (Osp(1, 2l), d_{∞})
Untwisted ℤ  (Sp(2l), b̃_{∞}) (Osp(1, 2l), b̃_{∞})
Twisted \frac{1}{2} + ℤ (O(2l), c_{∞})  (O(2l + 1), c_{∞})
Twisted  ℤ  (Sp(2l), b_{∞}) (Osp(1, 2l), b_{∞})

Table 2: dual pairs in (mostly) bosonic Fock spaces

indicates the appearance of an extra neutral field, fermionic or bosonic depending on the sign again. For instance, F^{⨂ l} denotes the Fock space of l pairs of fermionic fields. F^{⨂ −l+ \frac{1}{2}} denotes the Fock space of l pairs of bosonic ghost fields and a neutral fermionic field.

The entry ℤ or \frac{1}{2} + ℤ in the two tables means the Fourier components are indexed by ℤ or \frac{1}{2} + ℤ. Untwisted (resp. twisted) in the two tables means that the Lie algebra of the finite dimensional Lie group appearing in the corresponding dual pair is the horizontal Lie subalgebra of an untwisted (resp. twisted) classical affine algebra acting on the same Fock space. For instance, the dual pair (O(2l + 1), d_{∞}) in the first table appears in a row starting with “Untwisted” and \frac{1}{2} + ℤ and in a column starting with F^{⨂ l+ \frac{1}{2}} . This tells us that there is an untwisted affine algebra (which is indeed 𝔰̂𝔬(2l + 1) in this case) acting on the Fock space F^{⨂ l+ \frac{1}{2}} of l pairs of fermionic fields and a neutral fermionic field which are indexed by \frac{1}{2} + ℤ. The action of O(2l + 1) in the dual pair is given by integrating the action of the horizontal subalgebra 𝔰𝔬(2l + 1) of 𝔰̂𝔬(2l + 1). b_{∞}, b̃_{∞}, c_{∞} and d_{∞} appearing in the tables are Lie subalgebras of 𝔤̂𝔩 of B, C, D types as defined in Section 1, where 𝔤̂𝔩 is the central extension of the Lie algebra of infinite size matrices (a_{ij})_{i,j∈ℤ} with finitely many non-zero diagonals.

The dual pair between a general linear group GL(l) and 𝔤̂𝔩 is not put in these two tables. Indeed we have a dual pair (GL(l), 𝔤̂𝔩) acting on the Fock spaces F^{⨂ l} and F^{⨂ −l} regardless the other constraints on the tables, such as twisted or untwisted, ℤ or \frac{1}{2} + ℤ. Dual pair (GL(l), 𝔤̂𝔩) acting on the Fock space F^{⨂ −l} was treated earlier by Kac and Radul [KR2], where the decomposition of F^{⨂ −l} into isotypic subspaces was given. However the calculation of the highest weights in isotypic spaces was performed there in some indirect way without appealing to explicit formulas for the highest weight vectors in isotypic spaces.

In each case listed in the two tables, we completely determine the Fock space decomposition into a direct sum of isotypic subspaces with respect to a corresponding dual pair, and give explicit formulas for the highest weight vectors in these isotypic subspaces and calculate their highest weights. The highest weights can be read off from the explicit formulas of these highest weight vectors. We will see that these Fock spaces usually provides natural models on which every finite dimensional irreducible representation of a finite dimensional Lie group in the corresponding dual pair appears and every unitary highest weight module of an infinite dimensional Lie algebra in the corresponding dual pair also appears.

While the theory of classical dual pairs has many far-reaching applications [H1, H2], we confine ourselves to the discussion of one particular application in the paper. We establish a number of reciprocity laws associated to see-saw pairs arising in these Fock spaces. We construct semisimple tensor categories of certain highest weight modules of d_{∞} (resp. b_{∞}, b̃_{∞}, c_{∞}) and establish the equivalence of tensor categories with some suitable tensor categories of representations of finite dimensional Lie groups with various ranks. Similar results on tensor categories were studied earlier by the author in the case of dual pair (GL(l), 𝔤̂𝔩) [W].

One may have various formulations of our duality results just as in the case of classical dual pairs. We do not intend to do so since it is quite clear how to do them by imitating [H2] and doing that will increase the length of the paper considerably.

The paper is organized as follows. In Section 1 we review Lie algebra 𝔤̂𝔩 and its subalgebra b_{∞}, b̃_{∞}, c_{∞}, d_{∞} of B, C, D types. In Section 2 we give a parametrization of finite dimensional irreducible representations of classical Lie groups and Spin(2l), P in(2l), Osp(1, 2l) which appear in our Fock space decompositions. In Section 3 we study the dual pair actions in the fermionic Fock space F^{⨂ l}. In Section 4 we study the dual pair action in the Fock space F^{⨂ l± \frac{1}{2}} . In Section 5 we work out in detail duality in the bosonic Fock space F^{⨂ −l}. In Section 6 we treat dual pairs in the Fock space F^{⨂ −l± \frac{1}{2}} . In Section 7 we outline a number of reciprocity laws and a study of various tensor categories.

Conventions: ℤ is the set of integers; ℤ_{+} is the set of non-negative integers; ℕ is the set of positive integers; \underline{ℤ} is either ℤ or \frac{1}{2} + ℤ. \underline{ℤ}_{+} is \frac{1}{2} + ℤ_{+} if \underline{ℤ} = \frac{1}{2} + ℤ, and ℕ if \underline{ℤ} = ℤ. All classical Lie groups and algebras are over the complex field ℂ unless otherwise specified. An irreducible representation of a finite dimensional Lie group/algebra always means to be finite dimensional. Sometimes we will simply use λ to denote the highest weight representation with highest weight λ when no ambiguity may arise. Acknowledgement. I am grateful to C. H. Yan for her collaboration in [WY] which is one of the main stimuli for this paper. I am also grateful to R. Howe for very helpful correspondences and remarks. I also thank Max-Planck-Institut for finaicial support and excellent working environment.

1 Lie algebras 𝔤̂𝔩, b_{∞}, b̃_{∞}, c_{∞} and d_{∞}

In this section we review and fix notations on Lie algebras 𝔤̂𝔩 and its various Lie subalgebras of B, C, D types, cf. e.g. [K].

1.1 Lie algebra 𝔤̂𝔩

Let us denote by 𝔤𝔩_{f} the Lie algebra of all matrices of infinite size (a_{ij})_{i,j∈ℤ} with finitely many non-zero entries. Denote by 𝔤𝔩 the Lie algebra of all matrices (a_{ij})_{i,j∈ℤ} with only finitely many nonzero diagonals. Obviously 𝔤𝔩_{f} is a Lie subalgebra of 𝔤𝔩. Putting weight E_{ij} = j − i defines a ℤ–principal gradation 𝔤𝔩 = ⨁_{j∈ℤ} 𝔤𝔩_{j}. Denote by 𝔤̂𝔩 = 𝔤𝔩 ⨁ ℂC the central extension given by the following 2–cocycle with values in ℂ [DJKM]:

C(A, B) = Tr ([J, A]B)  (1.1)

where J = ∑_{j≤0} E_{ii}. The ℤ–gradation of Lie algebra 𝔤𝔩 extends to 𝔤̂𝔩 by putting weight C = 0. In particular, we have a triangular decomposition

𝔤̂𝔩 = 𝔤̂𝔩_{+} ⨁ 𝔤̂𝔩_{0} ⨁ 𝔤̂𝔩_{−}

where

𝔤̂𝔩_{±} = ⨁_{j∈ℕ} 𝔤̂𝔩_{±j}, 𝔤̂𝔩_{0} = 𝔤𝔩_{0} ⊕ ℂC.

Denote by E_{ij} the infinite matrix with 1 at (i, j) place and 0 elsewhere. Denote by ϵ_{i} the linear function on 𝔤𝔩_{0}, s.t. ϵ_{i}(E_{jj}) = δ_{ij}(i, j ∈ ℤ). Then the root system of 𝔤𝔩 is Δ = {ϵ_{i} − ϵ_{j}, (i, j ∈ ℤ, i ≠ j)}. The compact anti-involution ω is defined as ω(E_{ij}) = E_{ji}.

Given c ∈ ℂ and Λ ∈ 𝔤𝔩_{0}^{∗}, we let

λ_{i}^{a} = Λ(E_{ii}), i ∈ ℤ,
H_{i}^{a} = E_{ii} − E_{i+1,i+1} + δ_{i,0}C,
h_{i}^{a} = Λ(H_{i}^{a}) = λ_{i} − λ_{i+1} + δ_{i,0}c.

The superscript a here denotes 𝔤̂𝔩 which is of A type.

Denote by L(𝔤̂𝔩; Λ, c) (or simply L(𝔤̂𝔩; Λ) when the central charge is obvious from the text) the highest weight 𝔤̂𝔩–module with highest weight Λ and central charge c. Easy to see that L(𝔤̂𝔩; Λ, c) is quasifinite (namely having finite dimensional graded subspaces according to the principal gradation of 𝔤̂𝔩) if and only if all but finitely many h_{i}, i ∈ ℤ are zero. A quasifinite representation of 𝔤̂𝔩 is unitary if an Hermitian form naturally defined with respect to ω is positive definite.

Define Λ_{j}^{a} ∈ 𝔤𝔩_{0}^{∗}, j ∈ ℤ as follows:

Λ_{j}^{a}(E_{ii}) = \{ \begin{matrix}1, & for 0 < i ≤ j \\ −1, & for j < i ≤ 0 \\ 0, & otherwise. \end{matrix}  (1.2)

Define Λ̂_{0}^{a} ∈ 𝔤̂𝔩_{0}^{∗} by

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

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

Λ̂_{j}^{a} = Λ_{j}^{a} + Λ̂_{0}^{a}, j ∈ ℤ

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

It is not difficult to prove that L(𝔤̂𝔩; Λ, c) is unitary if and only if Λ = Λ̂_{m_{1}}^{a} + … + Λ̂_{m_{k}}^{a}, c = k ∈ ℤ_{+} by using a method due to Garland [G].

1.2 Lie algebra d_{∞}

Now consider the vector space ℂ[t, t^{−1}] and take a basis v_{i} = t^{i}, i ∈ ℤ. The Lie algebra 𝔤𝔩 acts on this vector space naturally, namely E_{ij}v_{k} = δ_{jk}v_{i}. We denote by \overline{d}_{∞} the Lie subalgebra of 𝔤𝔩 preserving the following symmetric bilinear form (cf. [K]):

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

Namely we have

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

Denote by d_{∞} = \overline{d}_{∞} ⨁ ℂC the central extension given by the 2-cocycle (1.1) restricted to \overline{d}_{∞}. Then d_{∞} has a natural triangular decomposition induced from 𝔤̂𝔩:

d_{∞} = d_{∞}_{+} ⨁ d_{∞}_{0} ⨁ d_{∞}_{−}

where d_{∞}_{±} = d_{∞} ∩ 𝔤̂𝔩_{±} and d_{∞}_{0} = d_{∞} ∩ 𝔤̂𝔩_{0}.

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

\begin{matrix}Π^{∨} = \{α_{0}^{∨} = E_{00} + E_{−1,−1} − E_{2,2} − 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}

Given Λ ∈ d_{∞}_{0}^{∗}, we let

λ_{i}^{d} = Λ(E_{ii} − E_{1−i,1−i}) (i ∈ ℕ),
H_{i}^{d} = E_{ii} + E_{−i,−i} − E_{i+1,i+1} − E_{−i+1,−i+1} (i ∈ ℕ),
h_{i}^{d} = Λ(H_{i}^{d}) = λ_{i} − λ_{i+1} (i ∈ ℕ),
H_{0}^{d} = E_{0,0} + E_{−1,−1} − E_{2,2} − E_{1,1} + 2C,
c = \frac{1}{2}(h_{0}^{d} + h_{1}^{d}) + ∑_{i≥2} h_{i}^{d}.

The superscript d denotes d_{∞} which is of D type. Then we denote by Λ̂_{i}^{d} the i-th fundamental weight of d_{∞}, namely Λ̂_{i}^{d}(H_{j}^{d}) = δ_{ij}. Denote by L(d_{∞}; Λ, c) (or L(d_{∞}; Λ) when the central charge is obvious) the highest weight d_{∞}–module with highest weight Λ and central charge c. Such a representation is unitary if an Hermitian form naturally defined with respect to the compact anti-involution ω on 𝔤̂𝔩 when restricted to d_{∞} is positive definite. It is not difficult to show that L(d_{∞}, Λ) is unitary if and only if Λ = Λ̂_{m_{1}}^{d} + … + Λ̂_{m_{k}}^{d}.

1.3 Lie algebras b_{∞} and b̃_{∞}

Let us consider the following symmetric bilinear forms B(v_{i}, v_{j}) = (−1)^{i}δ_{i,−j} and B̃(v_{i}, v_{j}) = δ_{i,−j}, i, j ∈ ℤ. Denote by \overline{b}_{∞} (resp. \overline{b^{}}_{˜ ∞}) the Lie subalgebra of 𝔤𝔩 which preserves the bilinear form B (resp. B̃), namely we have

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

Denote by b_{∞} = \overline{b}_{∞} ⨁ ℂC (resp. b̃_{∞} = \overline{b^{}}_{˜ ∞} ⨁ ℂC) the central extension of \overline{b}_{∞} (resp. \overline{b^{}}_{˜ ∞}) given by the 2-cocycle (1.1) restricted to \overline{b}_{∞} (resp. \overline{b^{}}_{˜ ∞}). Clearly these two Lie algebras b_{∞} and b̃_{∞} share the same Cartan subalgebras. Then b_{∞} (resp. b̃_{∞}) inherits from 𝔤̂𝔩 a natural triangular decomposition:

b_{∞} = b_{∞}_{+} ⨁ b_{∞}_{0} ⨁ b_{∞}_{−}, b̃_{∞} = b̃_{∞,+} ⨁ b_{∞}_{0} ⨁ b̃_{∞,−}

where b_{∞,±} = b_{∞} ∩ 𝔤̂𝔩_{±}, b_{∞}_{0} = b_{∞} ∩ 𝔤̂𝔩_{0}, and b̃_{∞,±} = b̃_{∞} ∩ 𝔤̂𝔩_{±}.

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

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

Given Λ ∈ b_{∞}_{0}^{∗}, we let

λ_{i}^{b} = Λ(E_{ii} − E_{−i,−i}),
H_{i}^{b} = E_{ii} + E_{−i−1,−i−1} − E_{i+1,i+1} − E_{−i,−i},
H_{0}^{b} = 2(E_{−1,−1} − E_{1,1}) + 2C,
h_{i}^{b} = Λ(H_{i}^{b}) = λ_{i} − λ_{i+1}, i ∈ ℕ,
c = \frac{1}{2}h_{0}^{b} + ∑_{i≥1} h_{i}^{b}.

The superscript b here denotes b_{∞} and b̃_{∞} which are of B type. Denote by Λ̂_{i}^{b} the i-th fundamental weight of b_{∞} as well as b̃_{∞}, namely Λ̂_{i}^{b}(H_{j}^{b}) = δ_{ij}. Denote by L(b_{∞}; Λ) (resp. L(b̃_{∞}; Λ)) the highest weight module over b_{∞} (resp. b̃_{∞}) with highest weight Λ and central charge c. Such a representation is unitary if an Hermitian form naturally defined with respect to the compact anti-involution ω on 𝔤̂𝔩 when restricted to b_{∞} (resp. b̃_{∞}) is positive definite. It is not difficult to show that L(b_{∞}, Λ) (resp. L(b̃_{∞}; Λ)) is unitary if and only if Λ = Λ̂_{m_{1}}^{b} + … + Λ̂_{m_{k}}^{b}.

1.4 Lie algebra c_{∞}

Let us consider the skew-symmetric bilinear forms C(v_{i}, v_{j}) = (−1)^{i}δ_{i,−j+1}, i, j ∈ ℤ. Denote by \overline{c}_{∞} the Lie subalgebra of 𝔤𝔩 which preserves the bilinear form C, namely we have

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

Denote by c_{∞} = \overline{c}_{∞} ⨁ ℂC the central extension of \overline{c}_{∞} given by the 2-cocycle (1.1) restricted to \overline{c}_{∞}. Then c_{∞} inherits from 𝔤̂𝔩 a natural triangular decomposition:

c_{∞} = c_{∞}_{+} ⨁ c_{∞}_{0} ⨁ c_{∞}_{−}

where c_{∞}_{±} = c_{∞} ∩ 𝔤̂𝔩_{±}, c_{∞}_{0} = c_{∞} ∩ 𝔤̂𝔩_{0}.

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 = 1, 2, ...\}. \end{matrix}

Given Λ ∈ c_{∞}_{0}^{∗}, and c ∈ ℂ, we let

λ_{i} = Λ(E_{ii} − E_{1−i,1−i}),
H_{i}^{c} = E_{ii} + E_{−i,−i} − E_{i+1,i+1} − E_{1−i,1−i},
H_{0}^{c} = E_{0,0} − E_{1,1} + C,
h_{i}^{c} = Λ(H_{i}^{c}) = λ_{i} − λ_{i+1}, i ∈ ℕ,
c = ∑_{i≥0} h_{i}^{c}.

The superscript c here denotes c_{∞} which is of C type. Then we denote by Λ̂_{i}^{c} the i-th fundamental weight of c_{∞}, namely Λ̂_{i}^{c}(H_{j}^{c}) = δ_{ij}. Denote by L(c_{∞}; Λ, c) the highest weight module over c_{∞} with highest weight Λ and central charge c. Such a representation is unitary if an Hermitian form naturally defined with respect to the compact anti-involution ω on 𝔤̂𝔩 when restricted to c_{∞} is positive definite. It is not difficult to show that L(c_{∞}, Λ) is unitary if and only if Λ = Λ̂_{m_{1}}^{c} + … + Λ̂_{m_{k}}^{c}.

2 Parametrization of irreducible representations of classical groups

In this section we give a parametrization of irreducible modules of finite dimensional Lie groups appearing in our Fock space decompositions in later sections. See [BtD] for more detail on Spin(n), P in(n) and other classical groups.

2.1 O(2l)

We define O(2l) = {g ∈ GL(2l);^{t}gJg = J} with J equal to

[ \begin{matrix}0 & I_{l} \\ I_{l} & 0 \end{matrix} ] .

Lie group GL(l) can be identified as a subgroup of O(2l) consisting of matrices of the form

[ \begin{matrix}g & 0 \\ 0 & ^{t}g^{−1} \end{matrix} ]

where g is an l × l non-singular matrix. Here and below ^{t}g denotes the transpose of the matrix g. Lie algebra 𝔰𝔬(2l) of SO(2l) consists of matrices of the form

[ \begin{matrix}α & β \\ γ & −^{t}α \end{matrix} ]  (2.3)

where α, β, γ are l × l matrices and β, γ are skew-symmetric. Lie algebra 𝔤𝔩(l) is identified with the subalgebra of 𝔰𝔬(2l) consisting of matrices of the form (2.3) by putting β and γ to be 0. We take a Borel subalgebra 𝔟(𝔰𝔬(2l)) of 𝔰𝔬(2l) to be the intersection of 𝔰𝔬(2l) with the set of upper triangular matrices of 𝔤𝔩(2l) and a Cartan subalgebra 𝔥(𝔰𝔬(2l)) consisting of the diagonal matrices diag(t_{1}, …, t_{l}, −t_{1}, …, −t_{l}), t_{i} ∈ ℂ. The subalgebra 𝔤𝔩(l) of 𝔰𝔬(2l) share the same Cartan subalgebra with 𝔰𝔬(2l).

An irreducible representation of GL(l) is parametrized by its highest weight with respect to the chosen Cartan subalgebra by

Σ(A) ≡ {(m_{1}, m_{2}, …, m_{l}), m_{1} ≥ m_{2} ≥ … ≥ m_{l}, m_{i} ∈ ℤ}.

An irreducible representation of SO(2l) is parametrized by its highest weight with respect to 𝔥(𝔰𝔬(2l)) in {(m_{1}, m_{2}, …, m_{l}), m_{1} ≥ m_{2} ≥ … ≥ m_{l−1} ≥ |m_{l}|, m_{i} ∈ ℤ}.

O(2l) is a semi-direct product of SO(2l) by ℤ_{2}. Denote by τ the 2l × 2l matrix

[ \begin{matrix}A & B \\ B & A \end{matrix} ]  (2.4)

with A = diag(1, …, 1, 0), B = diag(0, …, 0, 1). Then τ ∈ O(2l)−SO(2l) normalizes the Borel subalgebra 𝔟. If λ is a representation of SO(2l) of highest weight (m_{1}, m_{2}, …, m_{l}), then τ.λ has highest weight (m_{1}, m_{2}, …, −m_{l}). It follows that the induced representation of (m_{1}, m_{2}, …, m_{l}) (m_{l} ≠ 0) 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} > 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} .

2.2 O(2l + 1)

We take O(2l + 1) to be the form

O(2l + 1) = {g ∈ GL(2l + 1);^{t}gJg = J},  (2.5)
SO(2l + 1) = {g ∈ O(2l + 1); det g = 1},  (2.6)

where J is the following (2l + 1) × (2l + 1) matrix

[ \begin{matrix}0 & I_{l} & 0 \\ I_{l} & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

Denote by 𝔤𝔩(2l + 1) the Lie algebra of GL(2l + 1). The Lie algebra 𝔰𝔬(2l + 1) is the Lie subalgebra of 𝔤𝔩(2l + 1) consisting of (2l + 1) × (2l + 1) matrices of the form

[ \begin{matrix}α & β & δ \\ γ & −^{t}α & h \\ −^{t}h & −^{t}δ & 0 \end{matrix} ]  (2.7)

where α, β, γ are l × l matrices and β, γ skew-symmetric. The Borel subalgebra 𝔟(𝔰𝔬(2l + 1)) consists of matrices of the form (2.7) by putting γ and h to be 0 and α to be upper triangular. The Cartan subalgebra 𝔥(𝔰𝔬(2l + 1)) consists of diagonal matrices of the form diag(t_{1}, …, t_{l}; −t_{1} … − t_{l}; 0), t_{i} ∈ ℂ. An irreducible module of SO(2l + 1) is parametrized by its highest weight (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 representation λ of SO(2l + 1) extends to two different representations λ and λ ⨂ det of O(2l + 1), and all irreducible representations of O(2l + 1) is obtained this way. Then we can parametrize irreducible representatiions of O(2l + 1) by (m_{1}, …, m_{l}) and (m_{1}, …, m_{l}) ⨂ det. We denote

Σ(B) = {(m_{1}, …, m_{l}), (m_{1}, …, m_{l}) ⨂ det | m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ} .

2.3 Spin(n) and P in(n)

The Pin group P in(n) is the double covering group of O(n), namely we have

1 ⟶ ℤ_{2} ⟶ P in(n) ⟶ O(n) ⟶ 1.

We then define the Spin group Spin(n) to be the inverse image of SO(n) under the projection from P in(n) to O(n). Then we have the following exact sequence of Lie groups:

1 ⟶ ℤ_{2} ⟶ Spin(n) ⟶ SO(n) ⟶ 1.

Case n = 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 do not factor to SO(2l) is an irreducible representation of 𝔰𝔬(2l) parametrized by its highest weight

λ = \frac{1}{2}1_{l} + (m_{1}, m_{2}, …, m_{l})  (2.8)

or

λ = \frac{1}{2} ¯1_{l} + (m_{1}, m_{2}, …, −m_{l})  (2.9)

where m_{1} ≥ … ≥ m_{l} ≥ 0, m_{i} ∈ ℤ.

The Pin group P in(2l) is not connected. There are two possibilities. First, an irreducible representation of P in(2l) factors to that of O(2l), then we can use the parametrization of irreducible representations of O(2l) to parametrize these representations of P in(2l).

Secondly, an irreducible representation of P in(2l) is induced from an irreducible representation of Spin(2l) with highest weight of (2.8) or (2.9). When restricted to Spin(2l), it will decompose into a sum of the two irreducible representations of highest weights (2.8) and (2.9). We will use λ = \frac{1}{2}|1_{l}| + (m_{1}, m_{2}, …, \overline{m}_{l}), m_{l} ≥ 0 to denote this irreducible representation of Spin(2l). Denote by

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

Case n = 2l + 1. An irreducible representation of Spin(2l + 1) which do 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.  (2.10)

Denote by

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

2.4 Osp(1, 2l) and Sp(2l)

Denote by ℂ^{1|2l} the ℤ_{2}-graded vector space with ℂ as the even subspace and ℂ^{2l} as the odd subspace. We denote by GL(1, 2l) the general linear Lie supergroup on the superspace ℂ^{1|2l}, and its Lie superalgebra by 𝔤𝔩(1, 2l). Consider the following supersymmetric bilinear form

[ \begin{matrix}1 & 0 & 0 \\ 0 & 0 & I_{l} \\ 0 & −I_{l} & 0 \end{matrix} ]

which is symmetric on the even subspace ℂ and skew-symmetric on the odd subspace ℂ^{2l}. Define Osp(1, 2l) to be the Lie sub-supergroup of GL(1, 2l) which preserves the above bilinear form. Its Lie superalgebra 𝔬𝔰𝔭(1, 2l) consists of (2l + 1) × (2l + 1) matrices of the following form:

[ \begin{matrix}0 & x & y \\ ^{t}y & a & b \\ ^{t}x & c & −^{t}a \end{matrix} ]  (2.11)

where a, b, c are l × l matrices, b, c are symmetric.

We define 𝔟(𝔬𝔰𝔭(1, 2l)) to be the Borel subalgebra consisting of matrices of the form (2.11) with x and c to be zero and a to be upper triangular. The Cartan subalgebra 𝔥(𝔬𝔰𝔭(1, 2l)) consists of diagonal matrices diag (0; t_{1}, …, t_{l}, −t_{1}, …, −t_{l}).

Lie supergroup Osp(1, 2l) shares complete reducibility property of an ordianry Lie group which makes it distinguished from other Lie supergroups. So we will treat Osp(1, 2l) just as a Lie group. An irreducible representation of Osp(1, 2l) can be parametrized by its highest weight with respect to the Borel and Cartan subalgebra chosen above in

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

The symplectic group Sp(2l) can be taken as the subgroup of Osp(1, 2l) consisting of matrices in Osp(1, 2l) with 1 at the left-upper corner and 0 at all other entries in the first row and column. We also similarly define 𝔟(𝔰𝔭(2l)) and 𝔥(𝔰𝔭(2l)) to be the Borel and Cartan subalgebra of 𝔰𝔭(2l) consisting of those matrices with zeros in the first row and column in 𝔟(𝔬𝔰𝔭(1, 2l)) and 𝔥(𝔬𝔰𝔭(1, 2l)) respectively. An irreducible representation of Sp(2l) can be parametrized by its highest weight with respect to the chosen Borel and Cartan subalgebras by

Σ(C) = {(m_{1}, m_{2}, …, m_{l}), m_{1} ≥ m_{2} ≥ … ≥ m_{l} ≥ 0}.

3 Duality in the fermionic Fock space F^{⨂ l}

3.0 Fock space F^{⨂ l}

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}^{−}]_{+} = ψ_{m}^{+}ψ_{n}^{−} + ψ_{n}^{−}ψ_{m}^{+} = δ_{m+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 generating series

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1+2ϵ}w^{−j} = ∑_{p=1}^{l} : ψ^{+,p}(z)ψ^{−,p}(w) :,  (3.12)
e^{pq}(z) ≡ ∑_{n∈ℤ} e^{pq}(n)z^{−n−1+2ϵ} =: ψ^{−,p}(z)ψ^{−,q}(z) : (p ≠ q),  (3.13)
e_{∗∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{+,q}(z) : (p ≠ q),  (3.14)
e_{∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{−,q}(z) : +δ_{p,q}ϵ  (3.15)

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 that the operators E_{ij} (i, j ∈ ℤ) generate 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 a representation of the affine Kac-Moody algebra ŝo(2l) with central charge 1 [F1, F, KP]. Denote

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

The operators e^{pq}, e_{∗}^{pq}, e_{∗∗}^{pq} (p, q = 1, ⋯, l) form the horizonal subalgebra 𝔰𝔬(2l) in 𝔰̂𝔬(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 generated by e_{∗∗}^{pq} (p ≠ q), e_{∗}^{pq} (p ≤ q), p, q = 1, ⋯, l.

Lemma 3.1 The action of 𝔤𝔩(l) generated by e_{∗}^{pq} (p, q = 1, ⋯, l) and that of 𝔤̂𝔩 generated by E_{ij} (i, j ∈ ℤ) on F^{⨂ l} commute with each other.

Proof. One can prove [e_{∗}^{pq}, E_{ij}] = 0 by direct computation starting from the definition of e_{∗}^{pq} and E_{ij}. We will prove here the theorem in a more conceptual way by invoking the Wick’s theorem. We will do it in the case \underline{ℤ} = \frac{1}{2} + ℤ. The case \underline{ℤ} = ℤ can be treated similarly by some slight modification.

The statement 1) is equivalent to

[∑_{k=1}^{l} : ψ^{+,k}(z)ψ^{−,k}(w) :, ∫ : ψ^{+,p}(u)ψ^{−,q}(u) : du] = 0, p, q = 1, … l. (3.16)

In order to prove (3.16) we calculate some operator product expansions (OPE) as follows. Since

ψ^{+,m}(z)ψ^{−,n}(w) ∼ \frac{δ_{m,n}}{z − w}, ψ^{−,m}(z)ψ^{+,n}(w) ∼ \frac{δ_{m,n}}{z − w}.

we obtain by the Wick theorem

(∑_{k=1}^{l} : ψ^{+,k}(z)ψ^{−,k}(w) :) (: ψ^{+,p}(u)ψ^{−,q}(u) :)

∼ \frac{: ψ^{+,p}(z)ψ^{−,q}(u) :}{w − u} + \frac{: ψ^{−,q}(w)ψ^{+,p}(u) :}{z − u}.

But for local fields a(z) and ψ^{−}(z) with OPE a(z)b(u) ∼ ∑_{j} c_{j}(z)/(z − u)^{j} we have [a(z), ∫ b(u) du] = −c_{1}(z). Hence the left-hand side of (3.16) is equal to : ψ^{+,p}(z)ψ^{−,q}(w) : +ψ^{−,q}(w)ψ^{+,p}(z) := 0. □

In the remaining part of this section, we will divide into three cases: untwisted with \underline{ℤ} = \frac{1}{2} + ℤ, untwisted with \underline{ℤ} = ℤ, and twisted cases.

3.1 Untwisted case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (O(2l), d_{∞})

Let

\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}  (3.17)

We have the following lemma.

Lemma 3.2 The action of the horizontal subalgebra 𝔰𝔬(2l) and that of d_{∞} generated by E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ l} commute with each other.

Proof. A similar argument to the proof of (3.16) shows that

[∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :),

∫ : ψ^{+,p}(u)ψ^{+,q}(u) : du] = 0,

[∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :),

∫ : ψ^{−,p}(u)ψ^{−,q}(u) : du] = 0

where p, q = 1, … l, p ≠ q. The lemma now follows from the formula (3.17). □

Remark 3.1 We may introduce a natural ℤ_{+}/2-gradation on F^{⨂ l} by the eigenvalues of a degree operator d on F^{⨂ l} which satisfies

[d, ψ_{−i}^{±,j}] = iψ_{−i}^{±,j}, d|0⟩ = 0.

Each weight space of F^{⨂ l} is finite dimensional. Clearly an element in the horizontal subalgebra 𝔰𝔬(2l) has weight 0. Thus as a representation of 𝔰𝔬(2l), each weight space of F^{⨂ l} is preserved by the action 𝔰𝔬(2l). So as a representation of 𝔰𝔬(2l), F^{⨂ l} is decomposed into a direct sum of finite dimensional irreducible representations. As a representation of 𝔰𝔬(2l), F^{⨂ l} is isomomorphic to ∧^{∗}(V ⨂ ℂ^{ℕ}), where V ≅ ℂ^{2l} carries a natural symmetric bilinear form and ℂ^{ℕ} is the direct limit of ℂ^{N} as N tends to +∞. The action of the Lie algebra 𝔰𝔬(2l) can be lifted to an action of SO(2l) and naturally extends to O(2l).

Note that τ ∈ O(2l) − SO(2l) as defined in (2.4) commutes with the Fourier components of ψ^{±,k}(z), k = 1, …, l −1, and sends ψ^{+,l}(z) (resp. ψ^{−,l}(z)) to ψ^{−,l}(z) (resp. ψ^{+,l}(z)). It follows that τ commutes with the Fourier components of the generating function ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :) which span d_{∞} on F^{⨂ l} by equation (3.17). Since g and SO(2l) generate O(2l), the following lemma follows from Lemma 3.2.

Lemma 3.3 The action of O(2l) commutes with the action of d_{∞} on F^{⨂ l}.

Now we need to quote a lemma from the classical invariant theory (cf. e.g. [H1]).

Lemma 3.4 Let G be the orthogonal group O(k), and let U and W be G-modules formed by taking direct sums of the natural module for G. Consider the resulting action on the tensor product S(U) ⨂ ∧(V) of the symmetric tensor S(U) and antisymmetric tensor ∧(V). Then the algebra of G-invariants is generated by the invariants of degree 2.

It is easy to check that all the invariants of degree 2 in F^{⨂ l} are precisely vectors obtained by letting elements of d_{∞} acting on the vacuum vector |0⟩. So we have a dual pair (O(2l), d_{∞}) in the sense of Howe acting on F^{⨂ l} (also cf. [KR2] for an appropriate however straightforward adaption of classical dual pairs to infinite dimensional Fock representation cases). It can be argued similarly that (GL(l), 𝔤̂𝔩) also form a dual pair on F^{⨂ l}.

Denote

Ξ_{i}^{+,m} ≡ ψ_{−m+ \frac{1}{2}}^{+,i} ⋯ ψ_{− \frac{3}{2}}^{+,i}ψ_{− \frac{1}{2}}^{+,i},  (3.18)
Ξ_{i}^{−,m} ≡ ψ_{−m+ \frac{1}{2}}^{−,i} ⋯ ψ_{− \frac{3}{2}}^{−,i}ψ_{− \frac{1}{2}}^{−,i},  (3.19)
Ξ_{i}^{det} ≡ ψ_{− \frac{1}{2}}^{+,i+1}ψ_{− \frac{1}{2}}^{−,i+1}ψ_{− \frac{1}{2}}^{+,i+2}ψ_{− \frac{1}{2}}^{−,i+2} … ψ_{− \frac{1}{2}}^{+,l}ψ_{− \frac{1}{2}}^{−,l}.  (3.20)

We take the convention that Ξ_{i}^{±,0} = 1. Define a map Λ+^{𝔞𝔞} : Σ(A) ⟶ 𝔤̂𝔩_{0}^{∗}:

λ = (m_{1}, ⋯, m_{l}) ⟼ Λ+^{𝔞𝔞}(λ)

to be

Λ+^{𝔞𝔞}(λ) = Λ̂_{m_{1}} + ⋯ + Λ̂_{m_{l}}.  (3.21)

We now recall a duality theorem for the dual pair (GL(l), 𝔤̂𝔩) which was earlier proved in [F1, F2] (also see [FKRW]). For the sake of completeness, we sketch a proof here by invoking the dual pair principle in the spirit of this paper. Similar arguments will be needed for the proofs of other duality theorems in this paper.

Theorem 3.1 1) We have the following (GL(l), 𝔤̂𝔩)-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(A)} V (𝔤𝔩(l); λ) ⊗ L (𝔤̂𝔩; Λ+^{𝔞𝔞}(λ), l)

where V (GL(l); λ) is the irreducible GL(l)-module of highest weight λ, and L(𝔤̂𝔩; Λ+^{𝔞𝔞}(λ), l) is the irreducible highest weight 𝔤̂𝔩-module of highest weight Λ+^{𝔞𝔞}(λ) and central charge l.

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(A), assume that

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

Then the highest weight vector corresponding to the weight λ ∈ Σ(A) is

v_{Λ} = Ξ_{1}^{+,m_{1}} … Ξ_{i}^{+,m_{i}}Ξ_{j}^{−,−m_{j}} … Ξ_{l}^{−,−m_{l}}|0⟩.  (3.22)

Proof. First we can easily check that the vector v_{Λ} in (3.22) is indeed a highest weight vector for GL(l) and 𝔤̂𝔩 respectively. Another direct calculation shows that the highest weight of v_{Λ} for GL(l) is (m_{1}, …, m_{l}).

The highest weight of v_{Λ} for 𝔤̂𝔩 can be read off from a table below conveniently. Similar tables will be used throughout this paper so we make some general remarks and conventions here. In the first row k decreases one by one from left to right while in the second row the corresponding weights when acting on E_{k,k} are listed. Weights are the same within each box in the second row. For k greater than the first entry (which is m_{1} in this case) or less than the last entry in the first row (which is m_{l} − 1 in this case), the weight on the corresponding E_{k,k} is zero. We further assume that m_{1} > … > m_{i} > m_{i+1} = … = m_{j} = 0 > m_{j+1} > … > m_{l} for the sake of simplicity of presenting the following table as a general convention used throughout the paper:

\begin{matrix}k & m_{1}, & … & . & m_{i} & … & 0, & …, & m_{j+1} − 1 & . & m_{l−1}, & … & m_{l} − 1 \\ E_{kk} & 1, & … & . & i & … & j − l, & …, & j − l & . & −1, & … & −1 \end{matrix}

The general case m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{j} = 0 > m_{j+1} ≥ … ≥ m_{l} can be treated in a similar way and the highest weight for the highest weight with respect to 𝔤̂𝔩 can be shown easily to be also Λ+^{𝔞𝔞}(λ).

Now the part 1) follows from the general principle of dual pairs since we have found in 2) highest weight vectors of all the irreducible representations of GL(l).

□

Remark 3.2 The irreducible representations L(𝔤̂𝔩; Λ+^{𝔞𝔞}(λ), l) exhaust all irreducible unitary representations of 𝔤̂𝔩 of central charge l as λ ranges over Σ(A).

We define a map Λ^{𝔡𝔡} : Σ(D) ⟶ d_{∞}_{0}^{∗} (see Section 2.1 for the definition Σ(D)) by sending λ = (m_{1}, ⋯, \overline{m}_{l}) (m_{l} > 0) to

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

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

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

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

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

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

Theorem 3.2 1) We have the following (O(2l), d_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(D)} I_{λ} ≡ ⨁_{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.

2) With respect to (𝔰𝔬(2l), d_{∞}),

a) the isotypic subspace I_{λ} is decomposed into a sum of two irreducible representations with highest weight vectors

Ξ_{1}^{+,m_{1}} ⋯ Ξ_{l−1}^{+,m_{l−1}}Ξ_{l}^{+,m_{l}}|0⟩  (3.23)

and

Ξ_{1}^{+,m_{1}} ⋯ Ξ_{l−1}^{+,m_{l−1}}Ξ_{l}^{−,m_{l}}|0⟩  (3.24)

in the case λ = (m_{1}, …, \overline{m}_{l}) ∈ Σ(D), m_{l} > 0. The highest weight of (3.23) for 𝔰𝔬(2l) is (m_{1}, …, m_{l−1}, m_{l}) while that of (3.24) for 𝔰𝔬(2l) is (m_{1}, …, m_{l−1}, −m_{l});

b) the isotypic subspace I_{λ} is irreducible with highest weight vector

Ξ_{1}^{+,m_{1}} ⋯ Ξ_{l−1}^{+,m_{j}}|0⟩  (3.25)

in the case λ = (m_{1}, ⋯, m_{j}, 0, …, 0), m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0, j < l;

c) the isotypic subspace I_{λ} is irreducible with highest weight vector

Ξ_{1}^{+,m_{1}} ⋯ Ξ_{j}^{+,m_{j}}Ξ_{j}^{det}|0⟩  (3.26)

in the case λ = (m_{1}, ⋯, m_{j}, 0, …, 0) ⨂ det, m_{1} ≥ … ≥ m_{j} > 0, m_{j+1} = … = m_{l} = 0, j < l.

Proof. We prove first that the vectors given in part 2) above indeed are of highest weight. The vectors in 2a) and 2b) are of highest weight for 𝔤̂𝔩 and so they are also highest weight vectors for the Lie subalgebra d_{∞}. One can also prove by a direct computation that the vector in 2c) is also a highest weight vector for d_{∞} (note that it is not a highest weight vector for 𝔤̂𝔩 so we cannot use the argument for 2a) and 2b)). On the other hand, these vectors are of highest weight for GL(l) ⊂ O(2l) by Theorem 3.1. Therefore to prove they are of highest weight for 𝔰𝔬(2l) it suffices to show that they are also annihilated by

e_{∗∗}^{pq} = ∑_{n∈ \frac{1}{2} +ℤ} : ψ_{−n}^{+,p}ψ_{n}^{+,q} :, p, q = 1, …, l, p ≠ q.

This is so for the vector (3.23) since both ψ_{−n}^{+,p} and ψ_{n}^{+,q} anticommute with any ψ_{i}^{+,k} in (3.23) and thus one can move (up to a sign) the annihilator between ψ_{−n}^{+,p} and ψ_{n}^{+,q} to the right to kill the vacuum vector |0⟩. Noting that (ψ_{−n}^{+,p})^{2} = 0, we also easily check that each term in the sum e_{∗∗}^{pq} = ∑_{n∈ \frac{1}{2} +ℤ} : ψ_{−n}^{+,p}ψ_{n}^{+,q} : (p, q = 1, …, l − 1) annihilates the vector (3.24) and so does e_{∗∗}^{pq}. The case of (3.25) is proved in the same way as in the case of (3.23). The case of (3.26) also follows by noting that (: ψ_{−n}^{+,p}ψ_{n}^{+,q} :)ψ_{− \frac{1}{2}}^{+,i}ψ_{− \frac{1}{2}}^{−,i}|0⟩ = 0 for p ≠ q, p, q, i = 1, …, l.

It is easy to check case by case that the corresponding highest weights of these vectors in 2) above with respect to the Cartan subalgebra 𝔥(𝔰𝔬(2l)) of 𝔰𝔬(2l) are given as in the theorem.

On the other hand, in the case of (3.25), with m_{1} ≥ … m_{i} ≥ m_{i+1} = … = m_{j} = 1 > m_{j+1} = … = m_{l} = 0, we can easily calculate the highest weight when acting on E_{i,i} as in the following table.

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{i}, & …, & 2 & 1 \\ E_{kk} & 1, & … & 2, & … & … & i, & …, & i & j \end{matrix}

Then we easily read off the highest weight with respect to d_{∞} from the above table (see subsection 1.2 for the definition of Λ̂_{i}^{d}):

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

The case of (3.23) can be treated as a special case j = l by using the preceding argument. Therefore we read off the corresponding highest weight for d_{∞} as

Λ^{𝔡𝔡}(λ) = (l − i)Λ̂_{0}^{d} + (l − i)Λ̂_{1}^{d} + ∑_{a=1}^{i} Λ̂_{m_{a}}^{d}.  (3.27)

Case of (3.24) is divided into two subcases: first if m_{1} ≥ … ≥ m_{l−1} ≥ 2 ≥ −m_{l} > 0, we have the following table:

k  m_{1}, … m_{2}, . . . . . . m_{l−1}, …, 1 0, …, 1 − m_{l}
E_{kk} 1, … 2, . . . . . . l − 1, …,  −1, …, −1

Secondly, if m_{1} ≥ … ≥ m_{i} > m_{i+1} = … = m_{l} = 1, we have the following table:

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{i}, & …, & 2 & 1 & 0 \\ E_{kk} & 1, & … & 2, & … & … & i, & …, & i & l − 1 & -1 \end{matrix}

In either subcase, we easily obtain the highest weight with respect to d_{∞} from the above two tables in the uniform formula (3.27).

In the case of (3.26), we have

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{i}, & …, & 2 & 1 & 0 \\ E_{kk} & 1, & … & 2, & … & … & i, & …, & i & l & j − l \end{matrix}

Then we read off the highest weight in this case:

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

Now the decomposition in part 1) follows from the general abstract nonsense of dual pair theory since we have found highest weight vectors of all the irreducible representations of O(2l) in 2). □

Remark 3.3 The irreducible representations L(d_{∞}; Λ+^{𝔡𝔡}(λ), l) exhaust all irreducible unitary representations of d_{∞} of central charge l as λ ranges over Σ(D).

We have the following corollary.

Corollary 3.1 The invariant of O(2l) in the Fock space F^{⨂ l} is the irreducible d_{∞}-module L(d_{∞}; 2lΛ̂_{0}^{d}) of central charge l.

Remark 3.4 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 od notation. σ acts on the set of highest weights of d_{∞} by mapping λ = h_{0}^{d}Λ̂_{0}^{d} + h_{1}^{d}Λ̂_{1}^{d} + ∑_{i≥2} h_{i}^{d}Λ̂_{i}^{d} to σ(λ) = h_{1}^{d}Λ̂_{0}^{d} + h_{0}^{d}Λ̂_{1}^{d} + ∑_{i≥2} h_{i}^{d}Λ̂_{i}^{d}. In this way one can obtains an irreducible module of the semi-product σ ∝ d_{∞} on L(d_{∞}; λ) ⨁ L(d_{∞}; σ(λ)).

Then it follows from Theorem 3.2 that the isotypic subspace of F^{⨂ l} with respect to the joint action (SO(2l), σ ∝ d_{∞}) is irreducible and (SO(2l), σ ∝ d_{∞}) form a dual pair on F^{⨂ l}. In particular the invariant of F^{⨂ l} under the action fo SO(2l) is isomorphic to L(d_{∞}; 2lΛ̂_{0}^{d}) ⨁ L(d_{∞}; 2lΛ̂_{1}^{d}).

3.2 Untwisted case \underline{ℤ} = ℤ: dual pair (P in(2l), b̃_{∞})

It follows from formula (3.12) that

\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) :) . \end{matrix}

Lemma 3.5 The action of the horizontal subalgebra 𝔰𝔬(2l) and that of b̃_{∞} generated by E_{ij} − E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ l} commute with each other.

Proof. We have shown in Lemma 3.1 that E_{ij}, i, j ∈ ℤ commutes with the Lie subalgebra 𝔤𝔩(l) of 𝔰𝔬(2l). So it remains to prove that E_{ij} commutes with e_{∗∗}^{pq}, e^{pq}, p, q = 1, …, l.

[E_{ij}, e_{∗∗}^{pq}] = [∑_{k=1}^{l} : ψ_{−i}^{+,k}ψ_{j}^{−,k} :, ∑_{m∈ℤ} : ψ_{−m}^{+,p}ψ_{m}^{+,q} :]

= [∑_{k=1}^{l} ψ_{−i}^{+,k}ψ_{j}^{−,k}, ∑_{m∈ℤ} ψ_{−m}^{+,p}ψ_{m}^{+,q}]
= [ψ_{−i}^{+,p}ψ_{j}^{−,p}, ψ_{−j}^{+,p}ψ_{j}^{+,q}] + [ψ_{−i}^{+,q}ψ_{j}^{−,q}, ψ_{j}^{+,p}ψ_{−j}^{+,q}]
= ψ_{−i}^{+,p}ψ_{j}^{+,q} − ψ_{−i}^{+,q}ψ_{j}^{+,p}.

It follows immediately that [E_{ij} − E_{−j,−i}, e_{∗∗}^{pq}] = 0. Similarly we can prove that [E_{ij} − E_{−j,−i}, e^{pq}] = 0. □

Remark 3.5 As a representation of 𝔰𝔬(2l), the Fock space F^{⨂ l} is isomorphic to ∧(ℂ^{l}) ∧ (ℂ^{2l} ⨂ ℂ^{ℕ}), where ∧(ℂ^{l}) is the sum of two half-spin representations and 𝔰𝔬(2l) acts on ℂ^{2l} ⨂ ℂ^{ℕ} naturaly by the left action on ℂ^{2l}. The action of 𝔰𝔬(2l) can be lifted to Spin(2l) which extends naturally to P in(2l). It follows that any irreducible representation of Spin(2l) appearing in F^{⨂ l} cannot factor to SO(2l). A similar argument to the classical dual pair case [H2] shows that P in(2l) and b_{∞} form a dual pair on F^{⨂ l}.

Denote

Σ_{i}^{+,m} ≡ ψ_{−m}^{+,i} ⋯ ψ_{−2}^{+,i}ψ_{−1}^{+,i},
Σ_{i}^{−,m} ≡ ψ_{−m}^{−,i} ⋯ ψ_{−1}^{−,i}ψ_{0}^{−,i}, m ≥ 0

with the convention Σ_{i}^{+,0} = 1. We define a map Λ^{𝔡𝔟} from Σ(P in) to b_{∞}_{0}^{∗} by sending

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

to

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

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

Theorem 3.3 1) We have the following (P in(2l), b̃_{∞})-module decomposition:

F^{⨂ l} = ⨁_{λ∈Σ(P in)} I_{λ} ≡ ⨁_{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.

2) With respect to (Spin(2l), b̃_{∞}), the isotypic subspace I_{λ} is decomposed into a sum of two irreducible representations with highest weight vectors

Σ_{1}^{+,m_{1}} … Σ_{l−1}^{+,m_{l−1}}Σ_{l}^{+,m_{l}}|0⟩  (3.28)

and

Σ_{1}^{+,m_{1}} … Σ_{l−1}^{+,m_{l−1}}Σ_{l}^{−,m_{l}}|0⟩  (3.29)

where λ = \frac{1}{2}|1_{l}| + (m_{1}, …, m_{l}) ∈ Σ(P in), m_{1} ≥ … ≥ m_{l−1} ≥ m_{l} ≥ 0. The highest weight of (3.28) for Spin(2l) is \frac{1}{2}1_{l} +(m_{1}, …, m_{l}) while that of (3.29) for Spin(2l) is \frac{1}{2} ¯1_{l} + (m_{1}, …, m_{l−1}, −m_{l}).

Sketch of a proof. Proof is similar to that of Theorem 3.2. Below we caculate the highest weights of the vectors (3.28) and (3.29) for b̃_{∞}. We list the weight on E_{k,k} for the vector (3.28) in the following table:

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{j}, & …, & 1 \\ E_{kk} & 1, & … & 2, & … & … & j, & …, & j \end{matrix}

if m_{1} ≥ ⋯ ≥ m_{j} > m_{j+1} = ⋯ = m_{l} = 0. Then we can easily read off the highest weight of the vector (3.28) for b̃_{∞} as follows:

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

Case of (3.29) is divided into two subcases: first if m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0, j < l, we have the following table:

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{j}, & …, & 1 & 0 \\ E_{kk} & 1, & … & 2, & … & … & j, & …, & j & −1 \end{matrix}

From this, we can read off the highest weight of the vector (3.29) for b̃_{∞} as follows:

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

Secondly, if m_{l} ≥ 1, we have the following table:

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{l−1}, & …, & 1 & 0, & … & −m_{l} \\ E_{kk} & 1, & … & 2, & … & … & l − 1, & …, & l − 1 & −1, & … & −1 \end{matrix}

From this, we can see that the highest weight of the vector (3.29) for b̃_{∞} to be Λ^{𝔡𝔟}(λ) = ∑_{k=1}^{l} Λ̂_{m_{k}}^{b}. □

Remark 3.6 The irreducible representations L(b̃_{∞}; Λ+^{𝔡𝔟}(λ), l) exhaust all irreducible unitary representations of b̃_{∞} of central charge l as λ ranges over Σ(P in).

3.3 Twisted case: dual pairs (Sp(2l), c_{∞}) and (P in(2l), b_{∞})

It is well known [FF] that the Fourier components of the following “twisted” generating functions

ẽ^{pq}(z) ≡ ∑_{n∈ℤ} ẽ^{pq}(n)z^{−n−1+2ϵ} =: ψ^{−,p}(z)ψ^{−,q}(−z) :,
ẽ_{∗∗}^{pq}(z) ≡ ∑_{n∈ℤ} ẽ_{∗∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{+,q}(−z) :,  (3.30)
e_{∗}^{pq}(z) ≡ ∑_{n∈ℤ} e_{∗}^{pq}(n)z^{−n−1+2ϵ} =: ψ^{+,p}(z)ψ^{−,q}(z) : +δ_{p,q}ϵz^{−1}, p, q = 1, …, l

span a representation of the twisted affine algebra 𝔤l^{(2)}(2l) of type A_{2l−1}^{(2)} with central charge 1. Denote

ẽ^{pq} ≡ ẽ^{pq}(0), e_{∗}^{pq} ≡ e_{∗}^{pq}(0), ẽ_{∗∗}^{pq} ≡ ẽ_{∗∗}^{pq}(0), p, q = 1, ⋯, l.  (3.31)

Now we divide into two cases according to \underline{ℤ} = \frac{1}{2} + ℤ or ℤ.

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

It is easy to check that ẽ^{pq} = ẽ^{qp} and ẽ_{∗∗}^{pq} = ẽ_{∗∗}^{qp}. The horizontal subalgebra of 𝔤l^{(2)}(2l) spanned by the operators ẽ^{pq}, e_{∗}^{pq}, ẽ_{∗∗}^{pq}, (p, q = 1, ⋯ l) is isomorphic to Lie algebra 𝔰𝔭(2l). In particular, the operators e_{∗}^{pq} (p, q = 1, ⋯ l) form a subalgebra 𝔤𝔩(l) in the horizontal 𝔰𝔭(2l). We identify the Borel subalgebra 𝔟(𝔰𝔭(2l)) with the one generated by e_{∗}^{pq} (p ≤ q), ẽ_{∗∗}^{pq}, 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) :) . \end{matrix}

The following lemma is straightforward.

Lemma 3.6 The action of the horizontal subalgebra 𝔰𝔭(2l) and that of c_{∞} generated by E_{ij} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ l} commute with each other.

Remark 3.7 The action of 𝔰𝔭(2l) can be integrated to an action of Sp(2l) on F^{⨂ l}. Sp(2l) and c_{∞} form a dual pair on F^{⨂ l}.

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_{j}}

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

Theorem 3.4 1) We have the following (Sp(2l), c_{∞})-module decomposition:

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

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

2) With respect to (Sp(2l), c_{∞}), the highest weight vector in the isotypic subspace I_{λ} is:

Ξ_{1}^{+,m_{1}} … Ξ_{l}^{+,m_{l}}|0⟩.  (3.32)

Proof. Proof is similar to that of Theorem 3.2. Below we calculate the highest weight of the vector (3.32) for c_{∞} from the following table:

\begin{matrix}k & m_{1}, & … & m_{2}, & … & … & m_{j}, & …, & 1 \\ E_{kk} & 1, & … & 2, & … & … & j, & …, & j \end{matrix}

From this we easily see that the highest weight of the vector (3.32) for c_{∞} is equal to Λ^{𝔠𝔠}(λ). □

Remark 3.8 The irreducible representations L(c_{∞}; Λ+^{𝔠𝔠}(λ), l) exhaust all irreducible unitary representations of c_{∞} of central charge l as λ ranges over Σ(C).

II. Case \underline{ℤ} = ℤ: dual pair (Pin(2l), b_{∞})

In this case it is easy to check that e^{pq} = −e^{qp} and e_{∗∗}^{pq} = −e_{∗∗}^{qp}. 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 𝔰𝔭(2l). We identify the Borel subalgebra 𝔟(𝔰𝔬(2l)) with the one generated by e_{∗}^{pq} (p ≤ q), e_{∗∗}^{pq}, p, q = 1, ⋯, l. 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}

Lemma 3.7 The action of the horizontal subalgebra 𝔰𝔬(2l) and that of b_{∞} generated by E_{ij} − (−1)^{i+j}E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ l} commute with each other.

Proof is similar to that of Lemma 3.5.

Remark 3.9 The action of 𝔰𝔬(2l) can be integrated to an action of Spin(2l) and extends naturally to P in(2l) on F^{⨂ l}. Furthermore, P in(2) and b_{∞} form a dual pair on F^{⨂ l}. We can also obtain a duality theorem for the dual pair (P in(2l), b_{∞}). Indeed the theorem can be stated in the same way as Theorem 3.3 by replacing b̃_{∞} in Theorem 3.3 by b_{∞}. We omit the detail here. The dual pair (P in(2l), b_{∞}) is related to the dual pair (P in(2l), b̃_{∞}) by choosing a different symmetric bilinear form in defining the action on F^{⨂ l} of an infinite dimensional Lie subalgebra of 𝔤̂𝔩 of B type (cf. Section 1.3).

4 Duality in the Fock space F^{⨂ l±\frac{1}{2}}

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 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.  (4.33)

Then the Fourier components of e^{p}(z), e_{∗}^{p}(z), together with generating fuctions (3.13), (3.14) and (3.13)

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 [F, KP]. Put

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

Then e^{pq}, e_{∗}^{pq}(p ≠ q), e_{∗∗}^{pq}(p ≠ q), e^{p}, e_{∗}^{p} (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}(p ≠ q), e_{∗}^{pq}(p ≤ q), e_{∗}^{p}, p, q = 1, ⋯, l.

4.1 Untwisted case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (O(2l + 1), d_{∞})

Lemma 4.1 1) Putting

\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}

defines an action of d_{∞} on F^{⨂ l+ \frac{1}{2}} with central charge l + \frac{1}{2}.

2) The action of the horizontal subalgebra 𝔰𝔬(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}} commute with each other.

Proof. Part 1) can be proved by a direct calculation.

In the same way as we proved Lemma 3.2 we see that

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

commute with the action of d_{∞}. So it remains to check that e^{p}, e_{∗}^{p}(p, q = 1, …, l) also commute with the action of d_{∞}. This is equivalent to showing that

[∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :)+ : ϕ(z)ϕ(w) :,

∫ : ψ^{+,p}(u)ϕ(u) : du] = 0,  (4.34)

[∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :)+ : ϕ(z)ϕ(w) :,

∫ : ψ^{−,p}(u)ϕ(u) : du] = 0  (4.35)

where p = 1, … l.

We will only prove (4.34) since the proof of (4.35) goes parallely.

In order to prove (4.34) we calculate some operator product expansions (OPE). Since

ψ^{+,m}(z)ψ^{−,n}(w) ∼ \frac{δ_{m,n}}{z − w}, ϕ(z)ϕ(w) ∼ \frac{δ_{m,n}}{z − w}.

we have by using the Wick theorem,

( ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w)− : ψ^{+,k}(w)ψ^{−,k}(z) :) : + : ϕ(z)ϕ(w) : ) ·

(: ψ^{+,p}(u)ϕ(u) :)

∼ \frac{: ψ^{+,p}(z)ϕ(u) :}{w − u} − \frac{: ψ^{+,p}(w)ϕ(u) :}{z − u} − \frac{: ϕ(z)ψ^{+,p}(u) :}{w − u} + \frac{: ϕ(w)ψ^{+,p}(u) :}{z − u}.

But for local fields a(z) and b(z) with OPE a(z)b(u) ∼ ∑_{j} c_{j}(z)/(z − u)^{j} we have [a(z), ∫ b(u) du] = −c_{1}(z). Hence the left-hand side of (4.34) is equal to

: ψ^{+,p}(z)ϕ(w) : − : ψ^{+,p}(w)ϕ(z) : − : ϕ(z)ψ^{+,p}(w) : + : ϕ(w)ψ^{+,p}(z) :

which is equal to 0. □

Remark 4.1 The Fock space F^{⨂ l+ \frac{1}{2}} is isomorphic to ∧^{∗} (ℂ^{2l+1} ⨂ ℂ^{ℕ}) as a representation of 𝔰𝔬(2l + 1). We can lift the action of 𝔰𝔬(2l + 1) to SO(2l + 1) and extends to O(2l + 1) naturally. A particular element g ∈ O(2l + 1) − SO(2l + 1) is diag(1, …, 1, −1). It commutes with the Fourier components of ψ^{±,k}(z) (k = 1, …, l), and sends ϕ(z) to −ϕ(z).

It follows that g also commutes with the Fourier components of ∑_{k=1}^{l}(: ψ^{+,k}(z)ψ^{−,k}(w) : − : ψ^{+,k}(w)ψ^{−,k}(z) :)+ : ϕ(z)ϕ(w) : which span d_{∞} on F^{⨂ l+ \frac{1}{2}} . Since g ans SO(2l + 1) generate O(2l + 1), the following lemma follows from Lemma 4.1.

Lemma 4.2 The action of O(2l + 1) commutes with the action of d_{∞} on F^{⨂ l+ \frac{1}{2}} .

Remark 4.2 We can easily check that indeed all the invariants of degree 2 in F^{⨂ l+ \frac{1}{2}} are exactly the vectors obtained by letting elements of d_{∞} acting on the vacuum vector |0⟩ of F^{⨂ l+ \frac{1}{2}} . So by results on classical invariant theory [H1] we have a dual pair (O(2l + 1), d_{∞}) acting on F^{⨂ l+ \frac{1}{2}} .

We define a map Λ^{𝔟𝔡} from Σ(B) (see Section 2 for the definition of Σ(B)) to d_{∞}_{0}^{∗} by sending

λ = (m_{1}, m_{2}, …, m_{l})

to

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

and sending

λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det

to

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

assuming that

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

Theorem 4.1 1) We have the following (O(2l + 1), d_{∞})-module decomposition:

F^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(B)} I_{λ} ≡ ⨁_{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.

2) With respect to (so(2l +1), d_{∞}), the isotypic subspace I_{λ} is decomposed into a sum of two irreducible representations with highest weights

Ξ_{1}^{+,m_{1}} … Ξ_{l}^{+,m_{l}}|0⟩ for λ = (m_{1}, m_{2}, …, m_{l}),  (4.36)

and

Ξ_{1}^{+,m_{1}} … Ξ_{j}^{+,m_{j}}Ξ_{j}^{det}ϕ_{− \frac{1}{2}} |0⟩ for λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det.  (4.37)

Sketch of a proof. Proof is again similar to that of Theorem 3.2. We indicate below only the points which are different from that of Theorem 3.2. The determination of the highest weight of the vector (4.36) for d_{∞} is the same as before. E_{i,i} − E_{1−i,1−i} (i ∈ ℕ) acts on the vector by the operator ϕ_{−i+ \frac{1}{2}} ϕ_{i− \frac{1}{2}} . Then it is easy to see that (E_{i,i} − E_{1−i,1−i})ϕ_{− \frac{1}{2}} |0⟩ = δ_{i,1}ϕ_{− \frac{1}{2}} |0⟩. From this and the computation for the case of (4.36), we obtain the highest weight of the vector (4.37). □

We immediately have the following corollary

Corollary 4.1 The invariant of O(2l + 1) in F^{⨂ l+ \frac{1}{2}} is isomorphic to the irreducible d_{∞}-module L(d_{∞}; (2l + 1)Λ̂_{0}^{d}) of central charge l + \frac{1}{2}.

Remark 4.3 The irreducible representations L(d_{∞}; Λ+^{𝔟𝔡}(λ), l + 1/2) exhaust all irreducible unitary representations of d_{∞} of central charge l + 1/2 as λ ranges over Σ(B).

A similar remark to Remark 3.4 also holds. Namely, (SO(2l + 1), σ ∝ d_{∞}) form a dual pair on F^{⨂ l}. In particular the invariant of F^{⨂ l+ \frac{1}{2}} under the action of SO(2l + 1) is isomorphic to L(d_{∞}; (2l + 1)Λ̂_{0}^{d}) ⨁ L(d_{∞}; (2l + 1)Λ̂_{1}^{d}).

4.2 Untwisted case \underline{ℤ} = ℤ: dual pair (Spin(2l + 1), b̃_{∞})

We first have the following lemma.

Lemma 4.3 Putting

\begin{matrix}∑_{i,j∈ℤ}(E_{i,j} − E_{−j,−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}

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

Proof is straightforward.

Lemma 4.4 The action of the horizontal subalgebra 𝔰𝔬(2l + 1) and that of b̃_{∞} generated by E_{ij} − E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ l+ \frac{1}{2}} commute with each other.

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)Λ̂_{0}^{b} + ∑_{k=1}^{j} Λ̂_{m_{k}}^{b}

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

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}} . Each subspace is clearly invariant under the action of 𝔰𝔬(2l + 1).

Remark 4.4 The action of 𝔰𝔬(2l + 1) can be integrated to Spin(2l + 1) on F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} respectively. One has indeed dual pair (Spin(2l + 1), b̃_{∞}) acting on each F_{e}^{⨂ l+ \frac{1}{2}} and F_{o}^{⨂ l+ \frac{1}{2}} .

Theorem 4.2 1) 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}) , (4.38)
F_{o}^{⨂ l+ \frac{1}{2}} = ⨁_{λ∈Σ(P B)} V (Spin(2l + 1); λ) ⊗ L (b̃_{∞}; Λ^{𝔟𝔟}(λ), l + \frac{1}{2}) (4.39)

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}.

2) With respect to (Spin(2l + 1), b̃_{∞}), the highest weight vectors corresponding to the weight λ ∈ Σ(P B) in (4.38) and (4.39) are respectively

Σ_{1}^{+,m_{1}} … Σ_{l}^{+,m_{l}}|0⟩,  (4.40)
Σ_{1}^{+,m_{1}} … Σ_{l}^{+,m_{l}}ϕ_{0}|0⟩.  (4.41)

Remark 4.5 The irreducible representations L(b̃_{∞}; Λ+^{𝔟𝔟}(λ), l + 1/2) exhaust all irreducible unitary representations of b̃_{∞} of central charge l + 1/2 as λ ranges over Σ(P B).

4.3 Twisted cases: (Osp(1, 2l), c_{∞}) and (Spin(2l + 1), b_{∞})

I. Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (Osp(1, 2l), c_{∞})

Introduce 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 generating functions ẽ^{pq}(z), ẽ_{∗∗}^{pq}(z) and e_{∗}^{pq}(z) defined in (3.30) together with the following generating functions

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

span a representation of the affine algebra 𝔤𝔩^{(2)}(1, 2l) of type A_{0,2l−1}^{(2)} with central charge 1. Denote

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

Easy to check that the horizonal subalgebra in 𝔤𝔩^{(2)}(1, 2l) spanned by the operators ẽ^{p}, ẽ_{∗}^{p}, ẽ^{pq}, e_{∗}^{pq}, ẽ_{∗∗}^{pq}, (p, q = 1, ⋯, l) is isomorphic to Lie superalgebra 𝔬𝔰𝔭(1, 2l). In particular, the operators e_{∗}^{pq} (p, q = 1, ⋯, l) form a subalgebra 𝔤𝔩(l) in the horizontal 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.

Lemma 4.5 Putting

\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}

defines a representation of c_{∞} on F^{⨂ l− \frac{1}{2}} of central charge l − \frac{1}{2}.

Proof is straightforward.

Lemma 4.6 The action of the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) and that of c_{∞} generated by E_{ij} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ l− \frac{1}{2}} commute with each other.

Remark 4.6 The action of the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) can be integrated to Osp(1, 2l). Osp(1, 2l) and c_{∞} form a dual pair on F^{⨂ l− \frac{1}{2}} .

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

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

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

Theorem 4.3 1) We have the following (Osp(1, 2l), c_{∞})-module decomposition:

F^{⨂ l− \frac{1}{2}} = ⨁_{λ∈Σ(Osp)} I_{λ} ≡ ⨁_{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 Osp(1, 2l)-module 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.

2) With respect to (Osp(1, 2l), c_{∞}), the isotypic subspace I_{λ} is irreducible with highest weight vector

Ξ_{1}^{+,m_{1}} … Ξ_{l}^{+,m_{l}}|0⟩.

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

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

[φ_{m}, φ_{n}]_{+} = (−1)^{m}δ_{m,−n}, m, n ∈ ℤ.  (4.42)

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

: φ(z)φ(−z) :, : ψ^{−,p}(z)φ(−z) :, : ψ^{+,p}(z)φ(z) :

together with generating functions ẽ^{pq}(z), ẽ_{∗∗}^{pq}(z), e_{∗}^{pq}(z) defined in (3.30) form 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).

One can prove the following lemmas by a direct computation.

Lemma 4.7 Putting

\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}

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

Lemma 4.8 The action of the horizontal subalgebra 𝔰𝔬(2l +1) commutes with that of b_{∞} generated by E_{i,j} − (−1)^{i+j}E_{−j,−i}), i, j ∈ ℤ on F^{⨂ l+ \frac{1}{2}} commute with each other.

We can also show that the action of 𝔰𝔬(2l + 1) can be integrated to Spin(2l + 1). Spin(2l+1) and b_{∞} form a dual pair on F^{⨂ l+ \frac{1}{2}} . We then obtain a duality theorem for the dual pair (Spin(2l + 1), b_{∞}). Indeed it can be stated literally as Theorem 4.2 by simply replacing b̃_{∞} there by our b_{∞} so we will not repeat here. The dual pair (Spin(2l + 1), b_{∞}) is related to the dual pair (Spin(2l + 1), b̃_{∞}) by choosing a different symmetric bilinear form in defining the action on F^{⨂ l+ \frac{1}{2}} of an infinite dimensional Lie subalgebra of 𝔤̂𝔩 of B type (cf. Section 1.3).

5 Duality in a bosonic Fock space F^{⨂ −l}

5.0 Fock space F^{⨂ −l}

Let us take a pair of bosonic ghost fields

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

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

Introduce the following generating functions

E(z, w) ≡ ∑_{i,j∈ℤ} E_{ij}z^{i−1+2ϵ}w^{−j} = − ∑_{p=1}^{l} : γ^{+,p}(z)γ^{−,p}(w) :,  (5.43)
e_{∗∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗∗}^{pq}(n)z^{i−1+2ϵ} =: γ^{+,p}(z)γ^{+,q}(z) : (p ≠ q),
e^{pq}(z) ≡ ∑_{i,j∈ℤ} e^{pq}(n)z^{i−1+2ϵ} =: γ^{−,p}(z)γ^{−,q}(z) : (p ≠ q),  (5.44)
e_{∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗}^{pq}(n)z^{i−1+2ϵ} =: 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]. Denote

e^{pq} ≡ e^{pq}(0), e_{∗}^{pq} ≡ e_{∗}^{pq}(0), e_{∗∗}^{pq} ≡ e_{∗∗}^{pq}(0).

Then the operators e^{pq}, e_{∗}^{pq}, e_{∗∗}^{pq} (p, q = 1, ⋯ l) form the horizontal subalgebra 𝔰𝔭(2l) in 𝔰̂𝔭(2l). In particular, operators e_{∗}^{pq} (p, q = 1, ⋯ l) form a subalgebra 𝔤𝔩(l) in the horizontal subalgebra 𝔰𝔭(2l). We identify the Borel subalgebra 𝔟(𝔰𝔭(2l)) with the one generated by e_{∗∗}^{pq}, e_{∗}^{pq}(p ≤ q), p, q = 1, ⋯, l.

Lemma 5.1 The action of 𝔤𝔩(l) generated by e_{∗}^{pq} (p, q = 1, ⋯, l) and that of 𝔤̂𝔩 generated by E_{ij} (i, j ∈ ℤ) on F^{⨂ −l} commute with each other.

In the remaining part of this section we will divide into three cases.

5.1 Untwisted case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (Sp(2l), d_{∞})

First we have the following lemma.

Lemma 5.2 The action of the horizontal subalgebra 𝔰𝔭(2l) and that of d_{∞} generated by E_{ij} − E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ −l} commute with each other.

Proof is similar to the fermionic case. We omit it here.

A similar remark to Remark 3.1 holds in our bosonic case. So as a representation of 𝔰𝔭(2l), F^{⨂ −l} is decomposed into a direct sum of finite dimensional irreducible representations. Furthermore as a representation of 𝔰𝔭(2l), F^{⨂ −l} is isomomorphic to the polynomial algebra P(ℂ^{2l} ⨂ ℂ^{ℕ}), where 𝔰𝔭(2l) acts on ℂ^{2l} ⨂ ℂ^{ℕ} naturally on the left of ℂ^{2l}. The action of the Lie algebra 𝔰𝔭(2l) (resp. 𝔤𝔩(l)) can be integrated to an action of Sp(2l) (resp. GL(l)). The action of Sp(2l) commutes with the action of d_{∞} on F^{⨂ −l}. Indeed they form a dual pair. Similarly GL(l) and 𝔤̂𝔩 form a dual pair on F^{⨂ −l}.

Denote by Γ_{i}^{+,m} the m-th power of the determinant of the following i×i matrix:

[ \begin{matrix}γ_{−1/2}^{+,1} & γ_{−3/2}^{+,1} & ⋯ & γ_{−i+1/2}^{+,1} \\ γ_{−1/2}^{+,2} & γ_{−3/2}^{+,2} & ⋯ & γ_{−i+1/2}^{+,2} \\ ⋯ & ⋯ & ... & ⋯ \\ γ_{−1/2}^{+,i} & ⋯ & ⋯ & γ_{−i+1/2}^{+,i} \end{matrix} ] .  (5.45)

We also denote by Γ_{l+1−i}^{−,m} the m-th power of the determinant of the following i × i matrix:

[ \begin{matrix}γ_{−1/2}^{−,l} & γ_{−3/2}^{−,l} & ⋯ & γ_{−i+1/2}^{−,l} \\ γ_{−1/2}^{−,l−1} & γ_{−3/2}^{−,l−1} & ⋯ & γ_{−i+1/2}^{−,l−1} \\ ⋯ & ⋯ & ... & ⋯ \\ γ_{−1/2}^{−,l−i+1} & ⋯ & ⋯ & γ_{−i+1/2}^{−,l−i+1} \end{matrix} ] .  (5.46)

We define a map Λ−^{𝔞𝔞} : Σ(A) ⟶ 𝔤̂𝔩_{0}^{∗} by sending λ = (m_{1}, ⋯, m_{l}) to

Λ−^{𝔞𝔞}(λ) = −m_{i+1}Λ̂_{i−l}^{a} + ∑_{k=1+i−l}^{−1}(m_{l+k} − m_{l+k+1})Λ̂_{k}^{a}

+ (m_{l} − m_{1} − l)Λ̂_{0}^{a} + ∑_{k=1}^{i−1}(m_{k} − m_{k+1})Λ̂_{k}^{a} + m_{i}Λ̂_{i}^{a}.  (5.47)

Theorem 5.1 1) We have the following (GL(l), 𝔤̂𝔩)-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(A)} I_{λ} ≡ ⨁_{λ∈Σ(A)} V (GL(l); λ) ⊗ L (𝔤̂𝔩; Λ−^{𝔞𝔞}(λ), −l)  (5.48)

where V (GL(l); λ) is the irreducible GL(l)-module of highest weight λ and L (𝔤̂𝔩; Λ−^{𝔞𝔞}(λ), −l) is the irreducible highest weight module of 𝔤̂𝔩 with highest weight Λ−^{𝔞𝔞}(λ) and central charge −l.

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(A), fix i, j such that m_{1} ≥ ⋯ ≥ m_{i} ≥ m_{i+1} = ⋯ = m_{j} = 0 > m_{j+1} ≥ ⋯ ≥ m_{l}. Then the corresponding highest weight vector in I_{λ} with respect to (GL(l), 𝔤̂𝔩) is

Γ_{1}^{+,m_{1}−m_{2}} … Γ_{i}^{+,m_{i}}Γ_{j+1}^{−,−m_{j+1}}Γ_{j+2}^{−,m_{j+1}−m_{j+2}} … Γ_{l}^{−,m_{l−1}−m_{l}}|0⟩.  (5.49)

Proof. We first prove that the vector (5.49) is a highest weight vector for 𝔤𝔩(l). Indeed We see that the action of e_{∗}^{pq}(p < q, p, q = 1, …, l) on the vector(5.49) has the effect by replacing a row in the matrices (5.45) and (5.46) by another row up in the same matrix, so the determinant of the new matrix is zero.

On the other hand, the action of E_{i,j}(i ≤ 0 < j, i, j ∈ ℤ) on the vector (5.49) has the effect by replacing a column in the matrices (5.45) and (5.46) by another column with same superscripts but some positive subscripts. Then we see that the new entries in the replaced column will commute with all other operators in the expression of (5.49), so we can move it over to the right to kill the vacuum vector |0⟩. The action of E_{i,j}(0 < i < j, or i < j < 0, i, j ∈ ℤ) on the vector(5.49) has the effect by replacing a column in matrices (5.45) and (5.46) by another column to the left in the same matrix, so the determinant of the new matrix is zero again. Thus the vector (5.49) is also a highest weight vector for 𝔤̂𝔩.

One moment’s thought shows that the weight of the vector Γ_{i}^{+,k}|0⟩ with respect to 𝔥(𝔤𝔩(l)) is (k, …, k, 0, …, 0), where the number of k (resp. 0) is i (resp. l−i). It is also easy to see that the weight of the vector Γ_{i}^{−,k}|0⟩ with respect to 𝔥(𝔰𝔭(2l)) is (0, …, 0, −k, …, −k), where the number of −k is i. Then we easily get the highest weight of the vector (5.49) is (m_{1}, m_{2}, …, m_{l}).

Easy to see E_{a,a}.Γ_{b}^{−,m} = 0, for a > 0; E_{a,a}.Γ_{b}^{+,m} = mΓ_{b}^{+,m} if 0 ≤ a ≤ b, = 0 if a > b. Also we have E_{a,a}.Γ_{b}^{+,m} = 0, for a ≤ 0; E_{a,a}.Γ_{b}^{−,m} = −mΓ_{b}^{−,m} if b−l ≤ a ≤ 0, = 0 if −a > b. From these data, we can calculate the following table for the vector (5.49):

\begin{matrix}k & i & i − 1 & … & 1 & 0 & −1 & … & j − l + 1 \\ E_{kk} & m_{i} & m_{i−1} & … & m_{1} & m_{l} & m_{l−1} & … & m_{j+1} \end{matrix}

Now we can easily see that the highest weight of the vector (5.49) is indeed given by Λ−^{𝔞𝔞}(λ).

Since λ ranges over all Σ(A), the decomposition in 1) follows by general nonsense of dual pair theory. □

Remark 5.1 The first part of the theorem was proved in [KR2]. The most difficult part of their proof is to determine the highest weight Λ̂^{𝔞𝔞}(λ) for the vector (5.49) is rather indirect and based on a series of combinatorial lemmas. As we see from our proof, this follows from the explicit formula of the highest weight vector (5.49) fairly easily.

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

with the convention here and below m_{l+1} = 0.

Theorem 5.2 1) We have the following (Sp(2l), d_{∞})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(C)} I_{λ} ≡ ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (d_{∞}; Λ^{𝔠𝔡}(λ), −l)  (5.50)

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

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(C), with respect to (Sp(2l), d_{∞}) the isotypic subspace I_{λ} is irreducible with highest weight vector

Γ_{1}^{+,m_{1}−m_{2}}Γ_{2}^{+,m_{2}−m_{3}} … Γ_{l}^{+,m_{l}−m_{l+1}}|0⟩.  (5.51)

Proof. Since the vector (5.51) is a highest weight vector for 𝔤̂𝔩, it is so for the Lie subalgebra d_{∞} of 𝔤̂𝔩. On the other hand, e_{∗∗}^{pq}(p, q = 1, …, l) annihilates the vector (5.51) since it commutes with all Γ_{i}^{+,k} (i = 1, …, l) and it annihilates |0⟩. Thus the vector (5.51) is also a highest weight vector for Sp(2l).

The highest weight of the vector (5.51) for Sp(2l) is already calculated as in Theorem 5.1 since 𝔰𝔭(2l) and 𝔤𝔩(l) share the same Cartan subalgebra. The highest weight of the vector (5.51) with respect to d_{∞} can be seen from the following table:

\begin{matrix}k & i & i − 1 & … & 1 \\ E_{kk} & m_{i} & m_{i−1} & … & m_{1} \end{matrix}

Then we can read off the highest weight of the vector (5.51) to be Λ^{𝔠𝔡}(λ).

The decomposition in 1) now follows from general nonsense of dual pair theory since all irreducible representations of Sp(l) already appear in the decomposition.

□

We immediately have

Corollary 5.1 The invariant of Sp(2l) in the Fock space F^{⨂ −l} is the irreducible module L(d_{∞}; −2lΛ̂_{0}^{a}) of central charge −l.

5.2 Untwisted case \underline{ℤ} = ℤ: dual pair (Sp(2l), b̃_{∞})

First note that the operators E_{ij} − E_{−j,−i} (i, j ∈ ℤ) acting on F^{⨂ −l} as defined by the formula (5.43) span the Lie algebra b̃_{∞} with central charge −l.

Lemma 5.3 The action of the horizontal subalgebra 𝔰𝔭(2l) and that of b̃_{∞} generated by E_{ij} − E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ −l} commute with each other. The action of 𝔰𝔭(2l) can be integrated to Sp(2l). Sp(2l) and c_{∞} form a dual pair on F^{⨂ −l}.

Let us denote by Π_{i}^{+,m} the m-th power of the determinant of the following i × i matrix:

[ \begin{matrix}γ_{−1}^{+,1} & γ_{−2}^{+,1} & ⋯ & γ_{−i}^{+,1} \\ γ_{−1}^{+,2} & γ_{−2}^{+,2} & ⋯ & γ_{−i}^{+,2} \\ ⋯ & ⋯ & ... & ⋯ \\ γ_{−1}^{+,i} & ⋯ & ⋯ & γ_{−i}^{+,i} \end{matrix} ] .  (5.52)

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

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

where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0. Then we have the following duality theorem whose proof we omit.

Theorem 5.3 1) We have the following (Sp(2l), b̃_{∞})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(C)} I_{λ} ≡ ⨁_{λ∈Σ(C)} V (Sp(2l); λ) ⊗ L (b̃_{∞}; Λ^{𝔠𝔟}(λ))

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

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(C), m_{1} ≥ ⋯ ≥ m_{l} ≥ 0, the highest weight vector in the isotypic subspace I_{λ} is

Γ_{1}^{+,m_{1}−m_{2}} … Γ_{l−1}^{+,m_{l−1}−m_{l}}Γ_{l}^{+,m_{l}}|0⟩.

5.3 Twisted case: dual pairs (O(2l), c_{∞}) and (Sp(2l), b_{∞})

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+2ϵ} =: γ^{+,p}(z)γ^{+,q}(−z) : (p ≠ q),
e^{pq}(z) ≡ ∑_{i,j∈ℤ} e^{pq}(n)z^{−n−1+2ϵ} =: γ^{−,p}(z)γ^{−,q}(−z) : (p ≠ q),  (5.53)
e_{∗}^{pq}(z) ≡ ∑_{i,j∈ℤ} e_{∗}^{pq}(n)z^{−n−1+2ϵ} =: γ^{+,p}(z)γ^{−,q}(z) : +ϵδ_{p,q}, p, q = 1, …, l

span an affine algebra 𝔤𝔩^{(2)}(2l) of type A_{2l−1}^{(2)} of central charge −1.

I. Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pairs (O(2l), c_{∞})

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). Put

∑_{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)) .

Proof of the following lemma is straightforward.

Lemma 5.4 The action of the horizontal subalgebra 𝔰𝔬(2l) and that of c_{∞} generated by the operators E_{ij} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ −l} commute with each other.

The action of 𝔰𝔬(2l) can be integrated to an action of the Lie group SO(2l) and extends to O(2l) naturally. Indeed O(2l) and c_{∞} form a dual pair on F^{⨂ −l}.

Define Γ_{j}^{det} to be the determinant of the following (2l − j) × (2l − j) matrix M:

[ \begin{matrix}γ_{−1/2}^{+,1} & γ_{−1/2}^{+,2} & ⋯ & γ_{−1/2}^{+,l} & γ_{−1/2}^{−,j+1} & ⋯ & γ_{−1/2}^{−,l} \\ γ_{−3/2}^{+,1} & γ_{−3/2}^{+,2} & ⋯ & ⋯ & −γ_{−3/2}^{−,j+1} & ⋯ & −γ_{−3/2}^{−,l} \\ ⋯ & ⋯ & ... & ⋯ & ⋯ & ⋯ & ⋯ \\ γ_{j−2l+1/2}^{+,1} & γ_{j−2l+1/2}^{+,2} & ⋯ & γ_{j−2l+1/2}^{+,l} & (−1)^{j+1}γ_{j−2l+1/2}^{−,j+1} & ⋯ & (−1)^{j+1}γ_{j−2l+1/2}^{−,l} \end{matrix} ] .

Note that in the last l − j columns there is a sign (−1)^{i+1} in the i-th row.

Denote by Γ̃_{l}^{+,m} the m-th power of the deteminant of the matrix obtained from the matrix (5.45) for i = l by replacing the last row with the vector

(γ_{−1/2}^{−,i}, −γ_{−3/2}^{−,i}, …, (−1)^{i+1}γ_{−i+1/2}^{−,i}) .

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

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

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

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

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

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

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

Theorem 5.4 1) We have the following (O(2l), c_{∞})-module decomposition:

F^{⨂ −l} = ⨁_{λ∈Σ(D)} I_{λ} ≡ ⨁_{λ∈Σ(D)} V (O(2l); λ) ⊗ L (c_{∞}; Λ^{𝔡𝔠}(λ), −l)  (5.54)

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.

2) With respect to (𝔰𝔬(2l), c_{∞}),

a) the isotypic subspace I_{λ} is decomposed into a sum of two irreducible representations with highest weight vectors

Γ_{1}^{+,m_{1}−m_{2}} ⋯ Γ_{l−1}^{+,m_{l−1}−m_{l}}Γ_{l}^{+,m_{l}}|0⟩  (5.55)

and

Γ_{1}^{+,m_{1}−m_{2}} ⋯ Γ_{l−1}^{+,m_{l−1}−m_{l}} Γ̃_{l}^{+,m_{l}}|0⟩  (5.56)

in the case λ = (m_{1}, …, \overline{m}_{l}) ∈ Σ(D), m_{l} > 0. The highest weight of (5.55) for 𝔰𝔬(2l) is (m_{1}, …, m_{l−1}, m_{l}) while that of (5.56) for 𝔰𝔬(2l) is (m_{1}, …, m_{l−1}, −m_{l});

b) the isotypic subspace I_{λ} is irreducible with highest weight vector

Γ_{1}^{+,m_{1}−m_{2}} ⋯ Γ_{l−1}^{+,m_{l−1}−m_{l}}|0⟩  (5.57)

in the case λ = (m_{1}, ⋯, m_{l−1}, 0) ∈ Σ(D) with m_{l} = 0;

c) the isotypic subspace I_{λ} is irreducible with highest weight vector

Γ_{1}^{+,m_{1}−m_{2}} ⋯ Γ_{j}^{+,m_{j−1}−m_{j}}Γ_{j}^{+,m_{j}−1}Γ_{j}^{det}|0⟩  (5.58)

in the case λ = (m_{1}, ⋯, m_{j}, 0, …, 0) ⨂ det ∈ Σ(D), m_{1} ≥ … ≥ m_{j} > 0, m_{j+1} = … = m_{l} = 0, 0 ≤ j < l.

Proof. One can prove part 2a) and 2b) in a similar way as in the proofs of Theorems 5.1 and 5.2. In the case of 2c), it suffices to check that Γ_{0}^{det}|0⟩ is a highest weight vector for 𝔰𝔬(2l) since the highest weight vectors in Bosonic Fock space form a semi-group. Indeed an action of the element ∑_{m∈ \frac{1}{2} +ℤ} : γ_{−m}^{+,p}γ_{m}^{−,q} (p < q) (resp. ∑_{m∈ \frac{1}{2} +ℤ} : γ_{−m}^{+,p}γ_{m}^{+,q}) in the Borel subalgebra of 𝔰𝔬(2l) on Γ_{0}^{det}|0⟩ has the effect of replacing the q-th (resp. (l + q)-th) column of the matrix M by its p-th column. So the determinant of the matrix thus obtained is zero.

Consider the element

E_{i,j} − (−1)^{i+j}E_{1−j,1−i} ∑_{p=1}^{l} : γ_{−i+1/2}^{+,p}γ_{j−1/2}^{−,p} : −(−1)^{i+j} : γ_{j−1/2}^{+,p}γ_{−i+1/2}^{−,p} :

in the Borel of c_{∞}, where i ≤ 0 < j, i, j ∈ ℤ. By applying : γ_{−i+1/2}^{+,p}γ_{j−1/2}^{−,p} : to Γ_{0}^{det}|0⟩, we obtain up to a sign the determinant of the (2l − j − 2) × (2l − j − 2) submatrix of M by deleting the (−i + 1)-th and j-th rows and p-th and (l + p)- th column. The sign can be determined to be −(−1)^{i}(−1)^{j+p+(−i+1)+(l+p−1)} = (−1)^{j+l+1}. By applying : γ_{j−1/2}^{+,p}γ_{−i+1/2}^{−,p} : to Γ_{0}^{det}|0⟩, we obtain up to a sign the determinant of the same submatrix of M. The sign can be determined to be −(−1)^{j−1}(−1)^{(−i+1)+p+(j−1)+(l+p−1)} = (−1)^{i+l+1}. Thus E_{i,j} −(−1)^{i+j}E_{1−j,1−i} acting on Γ_{0}^{det}|0⟩ is zero due to cancellations.

On the other hand, E_{i,j} − (−1)^{i+j}E_{1−j,1−i} (0 < i < j) acts on Γ_{0}^{det}|0⟩ has the effect by replacing the j-th row by i-th row in M, so the determinant of the matrix thus obtained is obvious zero. The case for i < j < 0 is similar.

One can determine the highest weights for the vectors in 2a) and 2b) with respect to 𝔰𝔬(2l) and c_{∞} in a similar way as in the proofs of Theorems 5.1 and 5.2. A simple calculation shows that the highest weights of vectors in 2) with respect to 𝔰𝔬(2l) are indeed given by λ ∈ Σ(D). The highest weight of Γ_{j}^{det}|0⟩ with respect to 𝔰𝔬(2l) is (1, …, 1, 0, …, 0) where the number of 1 is equal to j. The highest weight of the vector (5.55) and (5.57) with respect to c_{∞} can be read off from the following table which is equal to Λ^{𝔡𝔠}(λ):

\begin{matrix}k & i & i − 1 & … & 1 & 0 & −1 & … & j − l + 1 \\ E_{kk} & m_{i} & m_{i−1} & … & m_{1} & m_{l} & m_{l−1} & … & m_{j+1} \end{matrix}

The vector (5.56) is obtained from (5.55) by an action of the element τ of O(2l) as defined in 2.4. So it has the same highest weight with respect to c_{∞} since O(2l) commutes with c_{∞}. The highest weight of Γ_{j}^{det}|0⟩ can be read off to be (−l − 1)Λ̂_{0}^{c} + Λ̂_{2l−j} from the following table:

\begin{matrix}k & < 2l − j & 2l − j, & …, & 1 \\ E_{kk} − E_{1−k,1−k} & 0 & 1, & …, & 1 \end{matrix}

The highest weights with respect to c_{∞} form a semigroup. From this we can see that the highest weight of (5.58) is indeed given as in the theorem. □

II. Case \underline{ℤ} = ℤ: dual pair (Sp(2l), b_{∞})

In this case the horizontal subalgebra of 𝔤𝔩^{(2)}(2l) is isomophic to 𝔰𝔭(2l). The Borel subalgebra of 𝔰𝔭(2l) is taken to be the one spanned by e_{∗}^{pq}(p ≤ q), e_{∗∗}^{pq}, p, q = 1, …, l. Put

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

Lemma 5.5 The action of the horizontal subalgebra 𝔰𝔭(2l) and that of b_{∞} generated by E_{ij} − (−1)^{i+j}E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ −l} commute with each other.

Remark 5.2 The action of 𝔰𝔭(2l) can be integrated Sp(2l). Sp(2l) and b̃_{∞} form a dual pair on F^{⨂ −l}. We can also obtain a duality theorem for the dual pair (Sp(2l), b_{∞}). Indeed the theorem can be stated in the same way as Theorem 5.3 by replacing b̃_{∞} in Theorem 5.3 by b̃_{∞}. We omit the detail here. The dual pair (Sp(2l), b_{∞}) is related to the dual pair (Sp(2l), b̃_{∞}) by choosing a different symmetric bilinear form in defining the action on F^{⨂ −l} of an infinite dimensional Lie subalgebra of 𝔤̂𝔩 of B type (cf. Section 1.3).

6 Duality in the Fock space F^{⨂ −l±\frac{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. It is known [FF] that the Fourier components of the generating functions e^{pq}(z), e_{∗}^{pq}(z), e_{∗∗}^{pq}(z) in (5.44) 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. Put

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

6.1 Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pair (Osp(1, 2l), d_{∞})

Then the operators e^{pq}, e_{∗}^{pq}, e_{∗∗}^{pq}, ẽ^{p}, ẽ_{∗}^{p} (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}(p ≤ q), e_{∗∗}^{pq}, ẽ_{∗}^{p}, p, q = 1, ⋯, l.

The following lemma can be proved similarly as in the fermionic Fock space case, cf. Lemma 4.1.

Lemma 6.1 Putting

\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}  (6.59)

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

The following lemma can also be proved similarly as in the fermionic Fock space case.

Lemma 6.2 The action of the Lie supergroup Osp(1, 2l) by integrating the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) commute with that of Lie algebra d_{∞} generated by E_{i,j} − E_{1−j,1−i}(i, j ∈ ℤ) on F^{⨂ −l+ \frac{1}{2}} .

Osp(1, 2l) and 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})Λ̂_{0}^{d} + ∑_{k=1}^{l}(m_{k} − m_{k+1})Λ̂_{k}^{d}.

We obtain the following duality theorem.

Theorem 6.1 1) 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; λ) is the irreducible module of Osp(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.

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(Osp), the isotypic subspace I_{λ} is irreducible with highest weight vector

Γ_{1}^{+,m_{1}−m_{2}} … Γ_{l}^{+,m_{l}−m_{l+1}}|0⟩.

6.2 Case \underline{ℤ} = ℤ: dual pair (Osp(1, 2l), b̃_{∞})

Let

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

Lemma 6.3 The action of the Lie supergroup Osp(1, 2l) by integrating the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) commute with that of Lie algebra b̃_{∞} generated by E_{i,j} − E_{−j,−i}(i, j ∈ ℤ) on F^{⨂ −l+ \frac{1}{2}} .

Osp(1, 2l) and b̃_{∞} form a dual pair on F^{⨂ −l+ \frac{1}{2}} . We define a map Λ^{𝔬𝔰𝔭𝔟} from Σ(Osp) to b_{∞}_{0}^{∗} by sending λ = (m_{1}, …, m_{l}) to

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

Theorem 6.2 1) We have the following (Osp(1, 2l), b_{∞})-module decomposition:

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

where V (𝔒𝔰𝔭(1, 2l); λ) is the irreducible module of 𝔒𝔰𝔭(1, 2l) of highest weight λ, and L (b̃_{∞}; Λ^{𝔬𝔰𝔭𝔟}(λ), −l + 1/2) is the irreducible highest weight b̃_{∞}-module of highest weight Λ^{𝔬𝔰𝔭𝔟}(λ) and central charge −l + 1/2.

2) Given λ = (m_{1}, …, m_{l}) ∈ Σ(Osp), with respect to (Osp(1, 2l), b_{∞}) the isotypic subspace I_{λ} is irreducible with highest weight vector

Π_{1}^{+,m_{1}−m_{2}} … Π_{l−1}^{+,m_{l−1}−m_{l}}Π_{l}^{+,m_{l}}|0⟩.

6.3 Twisted: dual pairs (O(2l + 1), c_{∞}) and (Osp(1, 2l), b_{∞})

I. Case \underline{ℤ} = \frac{1}{2} + ℤ: dual pairs (O(2l + 1), c_{∞})

Recall that a bosonic field χ(z) = ∑_{n∈ \frac{1}{2} +ℤ} χ_{n}z^{−n− \frac{1}{2}} 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 ẽ^{pq}(z), e_{∗}^{pq}(z), ẽ_{∗∗}^{pq}(z) as in (5.53) and the following generating functions

ζ(z) ≡ ∑_{n∈ℤ} ζ(n)z^{−n−1} = : χ(z)χ(−z) :,
ẽ^{p}(z) ≡ ∑_{n∈ℤ} e^{p}(n)z^{−n−1} = : γ^{−,p}(z)χ(−z) :,
ẽ_{∗}^{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. Its horizontal subalgebra is isomorphic to 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}

Lemma 6.4 The action of the horizontal subalgebra 𝔰𝔬(2l + 1) and that of c_{∞} generated by E_{ij} − (−1)^{i+j}E_{1−j,1−i} (i, j ∈ ℤ) on F^{⨂ −l− \frac{1}{2}} commute with each other.

The action of 𝔰𝔬(2l + 1) can be lifted to O(2l + 1). O(2l + 1) and c_{∞} form a dual pair on F^{⨂ −l− \frac{1}{2}} . Define Γ̃_{j}^{det} to be the determinant of the following (2l − j + 1) × (2l − j + 1) matrix M̃:

[ \begin{matrix}γ_{−1/2}^{+,1} & ⋯ & γ_{−1/2}^{+,l} & γ_{−1/2}^{−,j+1} & ⋯ & γ_{−1/2}^{−,l} & χ_{−1/2} \\ γ_{−3/2}^{+,1} & ⋯ & ⋯ & −γ_{−3/2}^{−,j+1} & ⋯ & −γ_{−3/2}^{−,l} & −χ_{−3/2} \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ γ_{j−2l−1/2}^{+,1} & ⋯ & γ_{j−2l−1/2}^{+,l} & (−1)^{j}γ_{j−2l−1/2}^{−,j+1} & ⋯ & (−1)^{j}γ_{j−2l−1/2}^{−,l} & (−1)^{j}χ_{−2l+j−1/2} \end{matrix} ] .

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)Λ̂_{0}^{c} + ∑_{k=1}^{j}(m_{k} − m_{k+1})Λ̂_{k}^{c}

and sending

λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det

to

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

where m_{1} ≥ … ≥ m_{j} > m_{j+1} = … = m_{l} = 0. Proof of the following theorem is similar to that of Theorem 5.4.

Theorem 6.3 1) We have the following (O(2l + 1), c_{∞})-module decomposition:

F^{⨂ −l− \frac{1}{2}} = ⨁_{λ∈Σ(B)} I_{λ} ≡ ⨁_{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.

2) With respect to 𝔰𝔬(2l + 1), the isotypic subspace I_{λ} is decomposed into a sum of two irreducible representations with highest weight vectors

Γ_{1}^{+,m_{1}−m_{2}} … Γ_{j−1}^{+,m_{j−1}−m_{j}}Γ_{j}^{+,m_{j}}|0⟩ for λ = (m_{1}, m_{2}, …, m_{l})  (6.60)

and

Γ_{1}^{+,m_{1}−m_{2}} … Γ_{j−1}^{+,m_{j−1}−m_{j}}Γ_{j}^{+,m_{j}−1}Γ̃_{j}^{det}|0⟩ for λ = (m_{1}, m_{2}, …, m_{l}) ⨂ det

(6.61)

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

II. Case \underline{ℤ} = ℤ: dual pair (Osp(1, 2l), b_{∞})

We recall a fermionic field φ(z) introduced earlier satisfying the commutation relation (4.42). Denote by F^{⨂ −l+ \frac{1}{2}} the tensor product of the Fock space of l pairs of bosonic ghost fields and the Fock space of a fermionic field φ(z). It is known that the Fourier components of the generating functions ẽ^{pq}(z), e_{∗}^{pq}(z), ẽ_{∗∗}^{pq}(z) as in (5.53) and the following generating functions

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

span an affine superalgebra 𝔤𝔩^{(2)}(1, 2l) of type A^{(2)}(0, 2l − 1). Its horizontal subalgebra is isomomorphic to 𝔬𝔰𝔭(1, 2l). Put

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

Lemma 6.5 The action of the horizontal subalgebra 𝔬𝔰𝔭(1, 2l) and that of b_{∞} generated by E_{ij} − (−1)^{i+j}E_{−j,−i} (i, j ∈ ℤ) on F^{⨂ −l+ \frac{1}{2}} commute with each other.

Remark 6.1 The action of 𝔬𝔰𝔭(1, 2l) can be lifted to Osp(1, 2l). Osp(1, 2l) and b_{∞} form a dual pair on F^{⨂ −l+ \frac{1}{2}} . We can thus obtain a duality theorem for the dual pair (Osp(1, 2l), b_{∞}). Indeed the theorem can be stated in the same way as Theorem 6.2 by replacing b̃_{∞} in Theorem 6.2 by b_{∞}. We omit the detail here. The dual pair (Osp(1, 2l), b_{∞}) is related to the dual pair (Osp(1, 2l), b̃_{∞}) by choosing a different symmetric bilinear form in defining the action on F^{⨂ −l+ \frac{1}{2}} of an infinite dimensional Lie subalgebra of 𝔤̂𝔩 of B type.

7 Reciprocity laws and tensor categories of modules over b_{∞}, b̃_{∞}, c_{∞} and d_{∞}

In this section we outline an application of the theory of dual pairs we have developed in this paper, namely we establish certain reciprocity laws as a formal concequence of the so-called “see-saw” pairs (cf. [H2] and references therein). There will be a number of reciprocity laws associated to different (series of) dual pairs given in the two tables in the introduction. We will take a step further to establish an equivalence of two tensor categories as in [W]. One tensor category consists of modules over finite dimensional Lie groups of a fixed type with all ranks.

The tensor product in this category is defined in terms of an induction functor. The other consists of certain modules over an infinite dimensional Lie algebra with the usual tensor product. This formulation of equivalence of tensor categories is particularly natural in our infinite dimensional setting.

7.1 Reciprocity laws associated to see-saw pairs

We formulate in this subsection a theory of reciprocity laws associated to see-saw pairs in a general term following the treatment of [H2].

Given two dual pairs (G, 𝔤′) and (K, 𝔨′) acting on a same vector space V. For our purpose we assume that G and K are compact (or complex) Lie groups and the irreducible representations of G and K are all finite dimensional. Assume that these two dual pairs satisfy the following relations: K ⊂ G and 𝔤′ ⊂ 𝔨′. Such a pair of dual pairs is called a see-saw pair.

A representation ρ of G can be decomposed into a direct sum of irreducible Hmodules σ with multiplicities m_{ρσ}. Denote ρ |_{K}= ∑_{σ} m_{ρσ}σ. On the other hand a representation of 𝔨′ decomposes into a sum of irreducible 𝔤′-modules: σ′ |_{𝔤′} ≅ mσ′ρ′^{′}ρ′.

Now we decompose V as a K × 𝔤′-module. Let σ be an irreducible module of K and ρ′ an irreducible module of 𝔤′. We compute the σ ⨂ ρ′-isotypic component V^{σ ⨂ ρ′} in two different ways and then compare the results. We have

V^{σ ⨂ ρ′} = (V^{σ})^{ρ′} ≅ (σ ⨂ σ′)^{ρ′} ≅ σ ⨂(σ′^{ρ′}) ≅ Hom_{𝔤′}(ρ′, σ′) ⨂(σ ⨂ ρ′)

and

V^{σ ⨂ ρ′} = (V^{ρ′})^{σ} ≅ (ρ ⨂ ρ′)^{σ} ≅ ρ^{σ} ⨂ ρ′ ≅ Hom_{K}(σ, ρ) ⨂(σ ⨂ ρ′).

It follows that

Hom_{K}(σ, ρ) ≅ Hom_{K×𝔤′}(σ ⨂ ρ′, V) ≅ Hom_{𝔤′}(ρ′, σ′).

In particular we have

mσ′ρ′^{′}ρ′ = m_{ρσ}.  (7.62)

7.2 Reciprocity laws and equivalence of tensor categories

We use the dual pair (O(2l), d_{∞}) acting on F^{⨂ l} to demonstrate how the general reciprocity law applies to our situation. To emphasize the rank l, we will write Σ(D_{l}) for Σ(D). Then by Theorem 3.2 we have the following decompositions:

F^{⨂ m} = ⨁_{μ∈Σ(D)} V (O(2m); μ) ⊗ L (d_{∞}; Λ^{𝔡𝔡}(μ), m)  (7.63)
F^{⨂ n} = ⨁_{ν∈Σ(D)} V (O(2n); ν) ⊗ L (d_{∞}; Λ^{𝔡𝔡}(ν), n)  (7.64)
F^{⨂(m+n)} = ⨁_{λ∈Σ(D)} V (O(2m + 2n); λ) ⊗ L (d_{∞}; Λ^{𝔡𝔡}(λ), m + n) . (7.65)

By the obvious isomophism F^{⨂(m+n)} ≅ F^{⨂ m} ⨂ F^{⨂ n}, we see that (O(2m) × O(2n), d_{∞} × d_{∞}) also form a dual pair on F^{⨂(m+n)}. The two dual pairs

\{ \begin{matrix}(O(2m + 2n), & d_{∞}|_{c=m+n}) \\ ↑ & ↓ \\ (O(2m) × O(2n), & d_{∞}|_{c=m} ⨁ d_{∞}|_{c=n}) \end{matrix}

form a see-saw pair, where inclusions of Lie groups/algebras are shown by the arrows and the second inclusion is given by the diagonal imbedding. So as a consequence of (7.62), we have

m (Λ^{𝔡𝔡}(μ) ⊗ Λ^{𝔡𝔡}(ν), Λ^{𝔡𝔡}(λ)) = m(λ |_{O(m)×O(n)}, μ ⊗ ν).  (7.66)

Here we take the convention to use a highest weight to denote the corresponding irreducible highest weight representation. It should be clear by the notations of highest weight which Lie group or algebra acts on.

We call the d_{∞}-modules appearing in the Fock space decomposition (7.64) nprimitive. Denote by ^{𝔡}O the category of all d_{∞}_{0}-diagonalizable, d_{∞}_{+}-locally finite d_{∞}-modules, with n-primitive d_{∞}-modules for every n ∈ ℤ_{+} as all irreducible objects and such that any module in ^{𝔡}O has a Jordan-Holder composition series in terms of n-primitive d_{∞}-modules (n ∈ ℤ_{+}). When n = 0 we have the trivial representation of d_{∞} only.

Denote by ^{𝔡}O_{f}^{n} the category of all representations of O(2n) which decomposes into a sum of finite dimensional irreducibles. Denote by ^{𝔡}O_{f} a direct sum of the categories ^{𝔡}O_{f}^{n} for all n ≥ 0, namely a category whose objects consist of a direct sum of representations of O(2n) in ^{𝔡}O_{f}^{n}, n ≥ 0. We introduce the following tensor product ⨀ on the category ^{𝔡}O_{f}: given a module U ∈ ^{𝔡}O_{f}^{m} and a module V ∈ ^{𝔡}O_{f}^{n}, let U ⨂ V be the outer tensor product of U and V which becomes a O(2m)×O(2n)- module. We define

U ⨀ V = (ind_{O(2m)×O(2n)}^{O(m+n)}U ⨂ V )^{l.f.}

where ind_{H}^{G}W denotes the induced G-module from a module W of H ⊂ G consisting of all continuous functions f : G ⟶ W satisfying f(gh) = h^{−1}f(g), h ∈ H, g ∈ G, and X^{l.f.} means the subset of locally finite vectors of X which are by definition the vectors lying in some finite dimensional O(2m + 2n)-submodule of X. One can prove the following theorem by use of Frobenius reciprocity and the reciprocity law (7.66), cf. Theorem 3.3 of [W].

Theorem 7.1 (^{𝔡}O_{f}, ⨀) and (^{𝔡}O, ⨂) are semisimple abelian tensor categories. Furthermore we have an equivalence of tensor categories between (^{𝔡}O_{f}, ⨀) and (^{𝔡}O, ⨂) by sending V (O(2n), ν) to L_{N} (d_{∞}, Λ^{𝔡𝔡}(ν), n).

Let us consider a second example. Following Theorem 4.1 we have

F^{⨂(n+1/2)} = ⨁_{λ∈Σ(B_{n})} V (O(2n + 1); λ) ⊗ L (d_{∞}; Λ+^{𝔟𝔡}(λ), n + 1/2) .  (7.67)

Hence we have a see-saw pair

\{ \begin{matrix}(O(2m + 2n + 1), & d_{∞}|_{c=m+n+1/2}) \\ ↑ & ↓ \\ (O(2m) × O(2n + 1), & d_{∞}|_{c=m} ⨁ d_{∞}|_{c=n+1/2}) \end{matrix}

acting on F^{⨂(m+n+1/2)}.

We call the d_{∞}-modules appearing in the Fock space decomposition (7.67) (n+ \frac{1}{2})-primitive. Denote by ^{𝔟𝔡}O the category of all d_{∞}_{0}-diagonalizable, d_{∞}_{+}-locally finite d_{∞}-modules, with n-primitive and (n + \frac{1}{2})-primitive d_{∞}-modules for every n ∈ ℤ_{+} as all irreducible objects and such that any module in ^{𝔟𝔡}O has a Jordan-Holder composition series in terms of n-primitive and (n+ \frac{1}{2})-primitive d_{∞}-modules (n ∈ ℤ_{+}). Similarly we define a category ^{𝔟𝔡}O_{f}^{k} consisting of all representations of O(k) which decomposes into a sum of finite dimensional irreducibles. Define category ^{𝔟𝔡}O_{f} to be the direct sum of ^{𝔟𝔡}O_{f}^{k} for all k ≥ 0. Again we can define a tensor product on the category ^{𝔟𝔡}O_{f} by the induction functor and then taking the local finite part. Then again we have an equivalence of tensor categories between 𝔟𝔡O and ^{𝔟𝔡}O_{f}.

One can obtain many other reciprocity laws associated to other dual pairs similarly.

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