\nopagenumbers
\
\vskip .5cm
\centerline{\bf LEVEL ONE REPRESENTATIONS OF THE TWISTED AFFINE ALGEBRAS
$A_n^{(2)}$ AND $D_n^{(2)}$}
\vskip 1cm
\centerline {FONS TEN KROODE\footnote *{Part of this research was done while the author was a visiting professor at the  Department of Mathematics of the  University of Illinois at Urbana Champaign.}}
\vskip .1cm
\centerline {\sl Nieuwe Prinsengracht $77^1$,}
\centerline {\sl 1018 VR Amsterdam, The Netherlands}
\vskip .5cm
\centerline {JOHAN VAN DE LEUR\footnote {**}{E-mail:
vdleur@math.ruu.nl
\hfil \break
The research of J. van de Leur has been made possible by a fellowship
of the Royal Netherlands Academy of Arts and Sciences.}} 
\vskip .1cm
\centerline {\sl Mathematical Institute,}
\centerline {\sl University of Utrecht,}
\centerline {\sl P.O. Box 80.010, 3508~TA, Utrecht, The Netherlands}
\vskip 2cm
\centerline {\bf Abstract}
\vskip .1cm
\midinsert\narrower\narrower\narrower\narrower
\noindent We give an explicit description of the Kac-Peterson-Lepowsky construction of the
level one representations for the twisted affine Lie algebras $A_n^{(2)}$ and 
$D_n^{(2)}$. 
We assign to any conjugacy class of the Weyl group of  $A_n$ and $D_n$ 1) an 
equivalent class of maximal Heisenberg subalgebras of the corresponding twisted
affine Lie algebras, and 2) multicomponent charged and
neutral free fermionic fields. The boson-fermion correspondence for
these fields provides us with fermionic vertex operators, whose `normal
ordered products' give the (twisted) vertex operators of the
Kac-Peterson-Lepowsky construction.
\overfullrule=0pt
\vskip 1cm
\noindent{\bf Keywords}:\quad affine algebra, fermionic fields, Heisenberg
algebra, Kac-Moody algebra, Kac-Peterson-Lepowsky construction,
vertex operator, Virasoro algebra, Weyl group
\vskip .5cm
\overfullrule=5pt
\noindent{\bf AMS classification}:\quad 17B65, 17B67, 22E65
\endinsert













\baselineskip=22pt plus.1pt
\count30=0                      %chapterno = 0
\count31=1                      %sectionno = 1
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count33}

\def\vac{\mid \overline 0 >}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}


\def\so{so_{n}({\bf C})}
\def\sotw{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}
\def\nug{\nu_{\underline g}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_n({\bf C})}
\def\hgl{\underline h^{gl_n}}
\def\hso{\underline h^{so_n}}
\def\gltw{gl_{2n}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde g l_{2n}({\bf C})}
\def\glh{\hat g l_{2n}({\bf C})}
\def\sot{\tilde s o_{2n}({\bf C})}
\def\soh{\hat s o_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}




\beginsection \chapno . {\bf  Introduction and summary}
\par
In this paper we present a collection of explicit vertex operator
representations of the twisted affine Lie algebras $\hat{sl}_n({\bf
C})^{(2)}, n\ge 3$  and $\hat{so}_{2n}({\bf C})^{(2)}, n\ge 4$. Some
isolated cases of our construction were known before [DJKM,\ F80,\ F81,\ FF,\ FLM,\ KKLW,\ KP81,\ JM,\ \dots] but most
of them are new. The paper is a natural sequel to earlier publications
[KrL91a,b,c] in which we consider the untwisted affine algebras
$\hat{sl}_n({\bf C}), \hat{so}_{2n}({\bf C})$ and $\hat{so}_{2n+1}({\bf C})$.


A unified treatment of vertex operator constructions for affine Lie
algebras was introduced by Kac and Peterson [KP85] and Lepowsky and Wilson
[LW84, L] (see also [FLM]). The different vertex operator constructions are
directly related to the different realizations of an affine Lie algebra as
an algebra of loops in a simple, finite dimensional Lie algebra $\underline
g$, twisted by a finite order automorphism of $\underline g$. The origin of
this work therefore lies in the connection between affine Lie algebras and
finite order automorphisms, which was found by Kac [K69] and can be found
in detail in [K, H]. The result we need is as follows:

\proclaim Lemma  \prono .
Let $\underline g$ be a simple, finite dimensional Lie algebra, $\sigma:
{\underline g}\mapsto{\underline g}$ an automorphism of order $N$ and 
${\underline g}=\oplus_{{\overline\jmath}\in{\bf Z}_{N}}{\underline g}_
{\overline\jmath}$ be the associated ${\bf Z}_{N}$-gradation of $\underline
g$. Let $k\in\{1,2,3\}$ be the least integer such that $\sigma^k$ is inner 
and let $\mu_k$ be an automorphism of order $k$ induced by the Dynkin
diagram of $\underline g$. Define $L({\underline
g},\sigma):=\oplus_{j\in{\bf Z}}t^j{\underline g}_{\overline\jmath}$ with
commutation relations
$$[t^lx,t^my]=t^{l+m}[x,y],\qquad\qquad\forall l,m\in{\bf Z},x\in
{\underline g}_{\overline l},y\in{\underline g}_{\overline m}\forno$$
\noindent then there exists an isomorphism $L({\underline g},\sigma)\cong
L({\underline g},\mu_k)=:\tilde{\underline g}^{(k)}$. Denote by $\hat
L({\underline g},\sigma)$ the universal central extension of $L({\underline
g},\sigma)$ by a one dimensional center, then the isomorphism can be
extended to an isomorphism $\hat L({\underline g},\sigma)\cong\hat
L({\underline g},\mu_k)=:\hat{\underline g}^{(k)}$.

The main ingredient for a (level one) vertex operator representation is a
Heisenberg subalgebra (HSA) $\hat{\underline s}$ of ${\underline{\hat
g}}^{(k)}$, i.e. an algebra on a basis $\{p_i,q_i\}_{i\in{\bf N}}$ and the
canonical central element $c$ with commutation relations
$[p_i,q_j]=\delta_{ij}c$. If $V$ is a representation of ${\underline {\hat
g}}^{(k)}$ such that for all $v\in V$ there exists an $N$ such that
$p_{i_1}p_{i_2}\ldots p_{i_l}(v)=0$ whenever $i_1+i_2+\cdots +i_l>N$, it is
completely reducible with respect to the action of $\hat{\underline s}$. The
only irreducible $\hat{\underline s}$ module satisfying this condition is
the ring of polynomials ${\bf C}[x_i]$ in infinitely many variables
$x_1,x_2,\ldots$ with the assignments $p_i\mapsto {\partial\over \partial
x_i},q_i\mapsto x_i,c\mapsto 1$. Therefore, $V$ can be identified with
$V^+\otimes {\bf C}[x_i]$, where the `vacuum space' $V^+$ is defined as the
set of all vectors, which are annihilated by the $p_i$'s. In this way one
obtains for every HSA $\hat{\underline s}$ a realization for a given (level
one) ${\underline {\hat g}}^{(k)}$-module V. Such realizations can look very
different, which is exemplified by the principal [LW78] (see also [KKLW]) and homogeneous
[FK, S] realization of the basic representation of the simplest affine Lie
algebra $\hat{sl}_2({\bf C})^{(1)}$.

Two HSA's $\hat{\underline s}_1$ and $\hat{\underline s}_2$ of 
${\underline{\hat g}}^{(k)}$ are called
equivalent if there exists an isomorphism $\psi:{\underline{\hat
g}}^{(k)}\rightarrow{\underline{\hat g}}^{(k)}$ such that
$\psi(\hat{\underline s}_1)= \hat{\underline s}_2$. Clearly, we only get
different realizations of a given representation if we consider inequivalent
HSA's.

Our main motivation to study the vertex operator representations associated
to inequivalent HSA's lies in the theory of soliton equations. Since the
pioneering work of the Kyoto school [DJKM] in the early eighties it is well
known that there is an intimate connection between these equations and
affine Lie algebras, which can be formulated in many ways. One of these
formulations, the zero curvature formalism, considers the flows on
an orbit of the Kac-Moody group $\hat G^{(k)}$ generated by the element
${\rm exp}\left(\sum_{i\ge 1}t_ip_i\right),t_i\in{\bf C}$. The commutativity
of these flows then gives rise to an infinite family of soliton equations.
For an alternative formulation, the Hirota formalism, we refer to [K, KP86].
It turns out that one finds a different family of soliton equations for each
realization of a given representation. For the principal and homogeneous
realizations of the basic representation of $\hat{sl}_2({\bf C})^{(1)}$
mentioned above one finds the KdV-family [K] and the AKNS-family [BKr],
respectively. This gives a firm representation theoretical basis to the
well known fact that these two integrable systems have the same infinite
dimensional symmetry algebra [N].
For these and other examples we also refer to the paper [KW89].

In the sequel let $\underline g$ be either $sl_n({\bf C})$ or $so_{2n}({\bf
C})$ and denote by $G$ the corresponding Lie groups $SL_n({\bf C})$ and 
$SO_{2n}({\bf C})$. Let $\sigma$ be an arbitrary outer automorphism of
order $N$. (In the case ${\underline g}=so_8({\bf C})$ we will also require
that $\sigma^2$ is inner, to exclude the exceptional case $\hat{so}_8({\bf
C})^{(3)}$.) Lemma 1.1 then provides us with a realization 
$\hat L({\underline g},\sigma)$ of the twisted affine algebra ${\underline
{\hat g}}^{(2)}$ for any such $\sigma$. 

We will now fix a standard outer automorphism of order two, $\nu_{\underline g}$. The
realization $\hat L({\underline g},\nu_{\underline g})$ will serve as our reference
model for ${\underline {\hat g}}^{(2)}$. In the case ${\underline
g}=sl_n({\bf C})$ we take for $\nu_{\underline g}$ the Cartan involution, which is outer
for $n\ge 3$: $\nu_{\underline g}(X)=-X^t\quad \forall X\in sl_n({\bf C})$.
For ${\underline g}=so_{2n}({\bf C})$, we define $\nu_{\underline g}(X)=O_{2n}XO_{2n}^{-1}\quad \forall X\in so_{2n}({\bf
C})$, where $O_{2n}$ is the orthogonal matrix representing the
permutation $\pi\in S_{2n}$ defined by $\pi(i)=i {\ \rm if\ }1\le i\le
2n-2,\pi(2n-1)=2n,\pi(2n)=2n-1$. It is well known [J], that any outer
automorphism $\sigma$ ($\sigma^2$ inner) can be written as 
$$\sigma={\rm Ad}\,g\circ \nu_{\underline g}\qquad\qquad {\rm for\ some\ }g\in G\forno$$



To construct HSA's we choose a Cartan
subalgebra $\underline h$, which is left invariant by $\sigma$. The ${\bf
Z}_N$ gradation of $\underline g$ then induces a similar gradation on
$\underline h$ and we define
$$\hat{\underline s}:=\bigoplus_{j\in{\bf Z},j\ne 0}t^j\underline
h_{\overline\jmath}\oplus{\bf C}c\forno$$

\noindent a subalgebra of $\hat L({\underline g},\sigma)$, which is easily
seen to be a HSA (on behalf of the nondegeneracy of the Killing form on
$\underline h$). In this way we obtain a HSA for every element of the set
$\{({\underline h},\sigma)\mid {\underline h} {\rm\ is\ a\ Cartan\
subalgebra\ of\ }{\underline g}, \sigma: {\underline h}\rightarrow
{\underline h}\}$. Clearly, if one is only interested in inequivalent
HSA's, there is a lot of redundancy in this construction; the pairs 
$({\underline h},\sigma)$ and $({\underline h}^\prime:={\rm
Ad}\,g({\underline h}),\sigma^\prime:={\rm Ad}\,g\circ\sigma\circ{\rm
Ad}\,g^{-1})$ give rise to equivalent HSA's, the first one in 
$\hat L({\underline g},\sigma)$, the second one in 
$\hat L({\underline g},\sigma^\prime)$. The equivalence is provided by the
isomorphism ${\rm Ad}\,g:\hat L({\underline g},\sigma)\rightarrow
\hat L({\underline g},\sigma^\prime)$.

So to describe the inequivalent HSA's obtained in this manner, it is
sufficient to choose an arbitrary Cartan subalgebra $\underline h$ and study
all finite order outer automorphisms $\sigma$ of $\underline g$ ($\sigma^2$
inner), which leave this $\underline h$ invariant. Using this freedom we fix
a Cartan subalgebra which is left invariant by the standard outer
automorphism $\nu_{\underline g}$. Because of (1.2) the remaining freedom in $\sigma$ is
then determined by the normalizer $N({\underline h}):=\{g\in G\mid {\rm
Ad}\,g({\underline h})={\underline h}\}$ of $\underline h$ in $G$. It is
well known that the mapping $g\in N({\underline h})\mapsto{\rm
Ad}\,g_{|{\underline h}}$ is a projection on the Weyl group $W$ of
$\underline g$ with kernel the Cartan subgroup $H={\rm exp}\, ({\underline
h})$ of $G$. One can prove that the equivalence class of the HSA associated
to $\sigma={\rm Ad}\,g\circ\nu_{\underline g}, g\in N({\underline h})$ depends only on
the conjugacy class of the element $w:={\rm Ad}\,g_{|{\underline h}}$ in $W$.

Summarizing, we have a sequence 
$$[w]\rightarrow w={\rm Ad}\,
g_{|{\underline h}}\rightarrow {\rm Ad}\,g;g\in N({\underline
h})\rightarrow \sigma={\rm Ad}\,g\circ\nu_{\underline g}\rightarrow \hat{\underline
s}_{[w]}:=\oplus_{j\in{\bf Z}}t^j{\underline h}_{\overline\jmath}\forno$$

\noindent and
we have argued that the various choices do not affect the equivalence class
of $\hat{\underline s}_{[w]}$. Clearly, $\hat{\underline s}_{[w]}$ is a
subalgebra of $\hat L({\underline g},\sigma)$. For our construction it will
be more convenient however, to have an equivalent HSA in the reference model
$\hat L({\underline g},\nu_{\underline g})$. Since the two realizations of
$\hat{\underline g}^{(2)}$ are isomorphic this HSA can be obtained by
constructing an explicit isomorphism and computing the image of
$\hat{\underline s}_{[w]}$. This is done as follows. We will show that
there exists an orthogonal matrix $S_{[w]}\in G$ and an antisymmetric
matrix $H_{[w]}\in {\underline g}$ satisfying $\nu_{\underline g}(H_{[w]})=H_{[w]}$
such that:
$$\sigma={\rm Ad}\,S_{[w]} \circ e^{2\pi i{\rm ad}\,H_{[w]}}\circ\nu_{\underline g} \circ
{\rm Ad}\,S_{[w]}^{-1}\forno$$

\noindent (cf. [K69, H, K] for a more general statement). The isomorphism between
$\hat L({\underline g},\sigma)$ and  $\hat L({\underline g},\nu_{\underline g})$ then
consists of reading the parameter $t$ in (1.1) as $e^{i\theta/N}$ and
applying the mapping ${\rm exp}(-i\theta{\rm ad}\,H_{[w]}){\rm
Ad}\,S_{[w]}^{-1}$. The image of $\hat{\underline s}_{[w]}$ becomes the
collection of loops:
$$\bigoplus_{j\in{\bf Z},j\ne 0}e^{-i\theta{\rm ad}\,H_{[w]}}e^{i{j\over
N}\theta}{\rm Ad}\,S_{[w]}^{-1}\left({\underline
h}_{\overline\jmath}\right)\forno$$

\noindent By abuse of notation this HSA of $\hat L({\underline g},\nu_{\underline g})$
will also be denoted by $\hat{\underline s}_{[w]}$.

\vskip.75cm
After this general introduction on the importance of HSA's, we will now formulate the main results of this paper for the case $\hat{sl}_n({\bf C})^{(2)}$. The results for $\hat{so}_{2n}({\bf C})^{(2)}$ are similar and can be found in the main text.

It will be convenient to consider $\hat{sl}_n({\bf C})^{(2)}$ as a subalgebra of $\hat{gl}_n({\bf C})^{(2)}$ and construct a representation of this latter algebra. The restriction to  $\hat{sl}_n({\bf C})^{(2)}$ is straightforward and will be left to the reader.  
Let us therefore start to recall some well known facts about $\gl$. The usual basis for $\gl$ is the collection 
$\{E_{ij}\}_{1\le i,j\le n}$ where $E_{ij}$ is the matrix with a $1$ on
the $(i,j)^{th}$ entry and zeros elsewhere.
As usual we take the subspace of diagonal matrices as standard Cartan subalgebra $\underline h^{gl_n}$ of $\gl$. Define  $\epsilon_j:=E_{jj}$, $1\le j\le n$,  then these elements form an orthonormal basis of $\hgl$ with respect to the  
trace form $(A|B):={\rm trace}(AB)$, $A,B\in\gl$. 


The Weyl group $W(\gl)$ consists of all permutations  
of the $\epsilon_i$'s. 
It is easy to show that the conjugacy classes in this group are in one--one correspondence with the partitions of $n$. 
Any partition of the number $n$ in $s$ different parts 
$n_1,\ldots , n_s$ determines a direct sum decomposition 
${\bf C}^{n}\cong {\bf C}^{n_1}\oplus{\bf C}^{n_2}
\oplus\ldots{\bf C}^{n_s}$, 
and an associated block decomposition of a $n\times n$-matrix.
In accordance with this we write:
$$E^{ab}_{jk}:=
  E_{n_1+n_2+\cdots n_{a-1}+j,n_1+n_2+\cdots n_{b-1}+k} \qquad 1\le a,b\le s, 1\le j\le n_a,1\le k\le n_b\forno$$

Now choose a representative 
$w\in W(\gl)$ for each conjugacy class $[w]$ and a matrix $\hat w\in GL_n ({\bf C})$ such that
${\rm Ad}(\hat w)\lower 3pt\hbox{$|{\underline h^{gl_n}}$} = w$.
 The next step is to calculate the matrix $S_{[w]}$ such that  ${\rm Ad}(\hat w)\circ\nu_{\underline g}={\rm Ad} (S_{[w]})\times$ ${\rm exp}(2\pi i {\rm ad}
H_{[w]})\nu_{\underline g}{\rm Ad}( S_{[w]})^{-1}$.
This calculation takes up most of section 2. Although we do need the explicit form of $S_{[w]}$ in section 4, the calculation in itself is not very interesting and highly technical. The reader is advised to skip it on first reading. 

As sugested by (1.6), we introduce
a  new Cartan subalgebra  ${\underline h}_{[w]}^{gl_n}:=S_{[w]}^{-1}{\underline
h^{gl_n}}S_{[w]}$ and new root vectors
$A^{ab}_{jk}=S_{[w]}^{-1}E^{ab}_{jk}S_{[w]}$.
Let $N$ be the order of  ${\rm Ad}(\hat w)\circ \nu_{\underline g}$ and let $\gl=
  \oplus_{\overline\jmath\in{\bf Z}_N}\gl_{\overline \jmath}$
be the ${\bf Z}_N$-gradation associated to $e^{2\pi i\ad H_{[w]}}\circ \nu_{\underline g}$. The set
$\{\left(A^{ab}_{jk}\right)_{\overline l} \mid 1\le a,b\le s,  1\le j\le n_a, 
  1\le k\le n_b, \overline l\in{\bf Z}_N\}$ 
is clearly the most natural spanning set for $\gl$ 
with respect to the automorphism $e^{2\pi i {\rm ad }H_{[w]}}\circ\nu_{\underline g}$. Similarly, the elements
$$\hat A^{ab}_{jk}(l):= 
  e^{-i\theta{\rm ad}H_{[w]}}e^{i{l\over N}\theta}
  \left( A^{ab}_{jk}\right)_{\overline l}-\delta_{l0}
  (H_{[w]}|A^{ab}_{jk})c\forno$$

\noindent together with a central element $c$ span the affine 
algebra $\hat{gl}_{n}({\bf C})^{(2)}$ (see section 3 or [KP85]). 

As usual one introduces the following generating series in a 
formal indeterminate $z$:
$$\hat A^{ab}_{jk}(z):=
  \sum_{l\in{\bf Z}}\hat A^{ab}_{jk}(l)z^{-l}
\forno$$

\noindent Constructing a unitary representation for 
$\hat{gl}_n({\bf C})^{(2)}$ now amounts to finding operators 
on a (pre-)Hilbert space representing the action of these  
$\hat A^{ab}_{jk}(z)$'s.

In order to achieve this, we choose an embedding of 
$\hat{gl}_{n}({\bf C})^{(2)}$ into the infinite rank affine algebra 
$B_\infty$, if $n$ is odd, and into $D_\infty$, if $n$ is even, and make use of the well known spin representation 
of the latter algebras. This representation can be described as follows; 
let $\epsilon=0$ for $B_\infty$ and $1\over 2$  for $D_\infty$ and  let $Cl^\epsilon$ be the Clifford algebra on generators 
$\phi(i), i\in {\epsilon+\bf Z}$, satisfying the anticommutation relations
$$\{\phi(i),\phi(j)\}:=\phi(i)\phi(j)+\phi(j)\phi(i)=\delta_{i,-j}\forno$$

\noindent The spin module $V^\epsilon$ (also called fermionic Fock space) is 
the unique irreducible $Cl^\epsilon$-module generated by a vacuum vector 
$|{\overline 0}>$, which is annihilated by all $\phi(i)$'s with $i< 0$:
$$\phi(i)|{\overline 0}>=0\qquad\forall i< 0\forno$$

\noindent $V^\epsilon$ is thus simply the vector space on a basis 
$\{\phi(i_1)\phi(i_2)\ldots \phi(i_k)|{\overline 0}>
\mid i_1>i_2>\cdots >i_k\ge 0\}$. 
It can be made into a (pre-)Hilbert space by introducing an inner 
product such that:
$$\eqalign{&(|{\overline 0}>,|{\overline 0}>)
  \equiv <{\overline 0}\mid {\overline 0}>=1\cr
  &\phi(i)^\dagger=\phi(-i)\cr}\forno$$

It is well known that the so called normal ordered products 
$:\phi(i)\phi(j):,\ i,j\in \epsilon+{\bf Z}$ define a representation of 
$B_\infty$ or $D_\infty$ on $V^\epsilon$. Moreover, this representation breaks into two 
irreducible components:
$$V^\epsilon=V_{\overline 0}^\epsilon\oplus V_{\overline 1}^\epsilon\forno$$

\noindent where $ V_{\overline 0}^\epsilon$ ($ V_{\overline 1}^\epsilon$) is the 
subspace of $V^\epsilon$ generated by the action of an even (odd) number of 
fermions on the vacuum $|{\overline 0}>$.

The embedding $\hat{gl}_{n}({\bf C})^{(2)}\hookrightarrow B_\infty\hbox{ or } D_\infty$ and 
the spin representation of $B_\infty$ and $D_\infty$ on $V^0$, $V^{1\over 2}$, respectively  now provides 
a fermionic realization of the fields.
It turns out that these fields can --up to a multiple of the identity-- 
be written as the normal ordered product of two fermionic fields. 
This  fermionic realization of $\hat{gl}_{n}({\bf C})^{(2)}$ on $V^\epsilon$ is 
the first main result of this paper;

\vskip.3cm
\noindent{\bf Main result A.}

{\sl \noindent Let $\lambda_{[{n\over 2}]}$ 
be the ${[{n\over 2}]}^{th}$ fundamental weight of $so_{n}({\bf C})$. On $V^\epsilon$ ($\epsilon=0$ if $n$ is odd and $\epsilon={1\over 2}$ if $n$ is even) we
have:} $$\hat
A^{ab}_{jk}(z)+(H_{[w]}-{\textstyle{1\over 2}}\lambda_{[{n\over 2}]}\vert
A^{ab}_{jk})=C_aC_b:\phi_a^{(-)^j}(e^{-2\pi ij/ N}z)\phi_b^{(-)^{k+1}}(e^{-2\pi ik/ N}z):\forno$$
{\sl\noindent The $C_a$'s in this formula are constants,  which
will be defined in section 4.4. The fermionic fields  $\phi_a^\lambda(z),
\break 1\le a\le s,\lambda=\pm$  have the following formal
expansions $$\phi_a^\lambda(z):=\cases{\displaystyle{\sum_{k\in
{1\over 2}+{\bf Z}}\phi_a^\lambda(k)z^{{N\over n_a}k}}&if\ $n_a$ even\cr\cr\displaystyle{ \sum_{k\in{\bf Z}}\phi_a^\lambda(k)z^{{N\over
2n_a}k}}&if\ $n_a$ odd\cr}\forno$$

\noindent where the fermions $\phi_a^\lambda (l)$ can be expressed in 
terms of (linear combinations of) the old fermions $\phi(i)$, 
determined by the partition 
$[n_1n_2\ldots n_s]$ 
(see section 4.4). Moreover, the fermion $\phi_a^\lambda(l)$ is a creation, 
annihilation and a combination of a creation and annihilation operator for
$l>0$, $l<0$, $l=0$, respectively.}
\vskip .2cm
So our construction associates fermionic fields 
$\phi_a^\lambda(z)$ to each cycle in the conjugacy class $[w]$. 
The distinction between even and odd cycles will be 
apparent from the different properties of the associated fermions. 
Define the following two subsets of $\{1,2,\cdots,s\}$:
$CF:=\{a\in \{1,2,\cdots,s\}|\  n_a\  {\rm even}\}$ and $NF:=\{a\in \{1,2,\cdots,s\}|\  n_a\  {\rm odd}\}$. We will show that  $\phi_a^+(l)^\dagger = \phi_a^-(-l)$ for $a\in CF$,
 or equivalently:
$\phi_a^+(z)^\dagger=\phi_a^-(z)$. The $\phi_a^+(l)$'s and $\phi_a^-(l)$'s are
 independent. They satisfy the anticommutation relations:
$$\eqalign{&\{ \phi_a^+ (k),\phi_b^+(l)\} =0= \{ \phi_a^- (k),\phi_b^-(l)\}\cr
  &\{ \phi_a^+ (k),\phi_b^-(l)\} =\delta_{ab}\delta_{k,-l}\cr}
  \qquad\qquad a,b\in CF,\ k,l\in {\textstyle{1\over 2}}+{\bf Z}\forno$$

\noindent In physics the fermions $\phi_a^+ (k)$ and $\phi_a^- (k)$ 
are associated to two fermionic particles of opposite charge. 
Hence, one speaks of charged  free fermions in this case.

For $a\in NF$
we have
$\phi_a^+ (l)^\dagger=\phi_a^-(-l)=
  (-)^l\phi_a^+(-l)$ or equivalently:
$\phi_a^+(z)^\dagger=\phi_a^-(z)=
  \phi_a^+(e^{2\pi i n_a/N}z)$, so there is only one (neutral) particle in this
case.   The anticommutation relations are
$$\{\phi_a^+(k),\phi_a^+(l)\}=
  (-)^l\delta_{ab}\delta_{k,-l}\qquad\qquad a,b\in NF,\ k,l\in{\bf Z}\forno$$

\noindent These fermions are usually called neutral free fermions.

\vskip.75cm
\noindent For the second main result of this paper, 
a bosonization procedure for the fermionic fields $\phi_a^\lambda(z)
, 1\le a \le s,\ \lambda=\pm$, we need an alternative construction of the HSA $\hat{\underline s}_{[w]}$. Introduce
$$\hat\alpha_a(z)=\cases{:\phi_a^+(z)\phi_a^-(z):&if $a\in CF$\cr
           {1\over 2}:\phi_a^+(z)\phi_a^-(z):
           &if $a\in NF$ \cr}\forno$$

\noindent With the formal expansions (1.15) it is immediate that 
these $\hat\alpha_a(z)$'s have the following expansions:
$$\hat\alpha_a(z)=\cases{\sum_{k\in{\bf Z}} 
  z^{-{N\over n_a}k}\hat\alpha_a( {N\over n_a}k)
  &if $a\in CF$\cr\sum_{k\in {1\over 2}+{\bf Z}}
  z^{-{N\over n_a}k}\hat\alpha_a({N\over n_a}k)&if $a\in NF$ \cr}\forno$$

\noindent From (1.14) we see that  $\hat\alpha_a(z)$ is 
--up to multiples of the identity and multiplicative factors-- 
equal to the field $\hat A^{aa}_{n_an_a}(+,-)(z)$. 
Formulas (1.6) and (1.8) show that the modes $\hat\alpha_a({N\over n_a}k)$ generate $\hat{\underline s}_{[w]}$. They satisfy 
$$[\hat\alpha_a({N\over n_a}k) ,
  \hat\alpha_b({N\over n_b}l)]
  =k\delta_{ab}\delta_{k,-l}\forno$$
 the commutation relations of 
 bosonic oscillators.
 We can now formulate

\vskip.3cm
\noindent{\bf Main result B.}

\noindent{\sl Let $a\in CF$, $b\in NF$, $1\le d,e\le s$ and $\lambda =+(1),-(1)$, then
$$\eqalign{\phi_a^\lambda(z)=&Q_a^\lambda z^{{N\over
n_a}\left(\lambda\hat\alpha_a(0)
           +{1\over 2}\right)}
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           \cr
           \phi_b^\lambda(z)=&Q_b
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           \cr}
           \forno$$
where the operators $Q_d :V^\epsilon\rightarrow V^\epsilon$ are defined by:
$$\eqalign{Q_d\phi_e^\lambda(k)&=-\phi_e^\lambda(k)Q_d\quad\hbox{if } 
           e\ne d\cr
           Q_a\phi_a^\lambda(k)&=\phi_a^\lambda(k+\lambda)Q_a\cr
           Q_b\phi_b^\lambda(k)&=\phi_b^\lambda(k)Q_b\cr
	   Q_d\sigma(k)&=-\sigma(k)Q_d\cr
           }
  \forno$$
and
$$Q_d\vac=
           \cases{\phi_d^+({1\over2})\vac
                  &if $d\in CF$\cr\cr
                  \phi_d^+(0)\vac
                  &if $d\in NF$\cr}
  \forno$$
Moreover,
$$\eqalign{Q_a^\dagger &=Q_a^{-1}\cr
           Q_b^\dagger &=Q_b ={1\over 2}Q_b^{-1}\cr}
  \forno$$
and 
$$[\a dk,Q_e]=\delta_{de}\delta_{k0}Q_e\forno$$
The following anticommutation relations are satisfied:
$$\{Q_d,Q_e\}=0
  \quad\hbox{for } d\ne e\forno$$}




The exponential dependence of the fermionic fields on the oscillators 
is of course quite familiar. 
Important is the appearance of the fermionic operators $Q_a$. 
There is again a big difference between $Q$'s associated to even and odd cycles; formula (1.22) shows that the operators 
$Q_a$ for $a\in CF$ have infinite period. For this reason we 
will call them fermionic translation operators. 
For $a\in NF$ however, (1.24) shows that we have $Q_a^2={1\over 2}$. 
We will call these $Q$'s fermionic chirality operators.


Combining results A and B, we find for each conjugacy class 
$[w]$ a realization of $\hat{gl}_{n}({\bf C})^{(2)}$ on the spin module 
$V^\epsilon$, 
in terms of the oscillators $\hat\alpha_a({N\over n_a}k)$ and the operators $Q_a^\lambda,\ 1\le a\le s,\lambda=\pm 1$.
The action of the  $Q_a$'s generate the vacuum space
$V^{\epsilon,+}$ of $\hat{\underline s}_{[w]}$ in $V^\epsilon$. In section 7 we
will show that $V^{\epsilon,+}$ has the structure of the tensor product of the
group algebra of a lattice and a finite Grassmann algebra.

All these
realizations are isomorphic as $\hat {gl}_{n}({\bf C})^{(2)}$-modules. In fact, by
considering the so called principal realization,  i.e. the one associated to the
partition $[n]$ we will be able to prove the following

\vskip.3cm 
{\noindent\bf Corollary C.}

{\noindent\sl The irreducible $B_\infty$-modules 
$V_{\overline 0}^0$ and $V_{\overline 1}^0$  ($D_\infty$-modules 
$V_{\overline 0}^{1\over 2}$ and $V_{\overline 1}^{1\over 2}$)  remain irreducible when 
restricted to $\hat{gl}_{2m+1}({\bf C})^{(2)}\subset B_\infty$  ($\hat{gl}_{2m}({\bf C})^{(2)}\subset D_\infty$). 
They can be identified with the fundamental 
$\hat{gl}_{2m+1}({\bf C})^{(2)}$-modules  $L(\Lambda_0)$ 
($\hat{gl}_{2m}({\bf C})^{(2)}$-modules $L(\Lambda_0)$ and  $L(\Lambda_1)$, respectively).}

















\baselineskip=22pt plus.1pt

\count30=1                      %chapterno = 1
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{n}({\bf C})}
\def\sotw{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}
\def\nug{\nu_{\underline g}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_n({\bf C})}
\def\hgl{\underline h^{gl_n}}
\def\hso{\underline h^{so_n}}
\def\gltw{gl_{2n}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde g l_{2n}({\bf C})}
\def\glh{\hat g l_{2n}({\bf C})}
\def\sot{\tilde s o_{2n}({\bf C})}
\def\soh{\hat s o_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}



\beginsection \chapno .  The  Lie algebras $\gl$ and $\sotw$\par
\beginsection \secno .  Preliminairies on the algebra $\gl$\par
Let $\gl$ be the Lie algebra of all $n\times n$-matrices with complex
entries. The usual basis for this algebra is the collection 
$\{E_{ij}\}_{1\le i,j\le n}$ where $E_{ij}$ is the matrix with a $1$ on
the $(i,j)^{th}$ entry and zeros elsewhere. The commutation relations
for these basis elements are:
$$[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{kj}\forno$$

The trace form $(A|B)_{gl_n}:=\hbox{trace}(AB)$ defines a nondegenerate invariant
bilinear form on $\gl$. Its restriction to the subspace of diagonal
matrices, the Cartan subalgebra  $\hgl$ of $\gl$, is nondegenerate.
Let $\epsilon_j:=E_{jj}$, $1\le j\le n$, then $\hgl =
\bigoplus_{j=1}^n {\bf C}\epsilon_j$ and
$$(\epsilon_i|\epsilon_j)_{gl_n}=\delta_{ij}\forno$$
From the
commutation relations (2.1.1), one easily deduces the
eigenspace decomposition of $\gl$ with respect to $\hgl$:
$$\gl=\bigoplus_{j=1}^n{\bf C}\epsilon_j \bigoplus\bigoplus_{1\le j\ne k\le n}
  {\bf C}E_{jk}\forno$$
where
$$[H,E_{ij}]=(\epsilon_i -\epsilon_j|H)_{gl_n}E_{ij}\qquad\hbox{for $H\in\hgl$}\forno$$
We call this the {\it root space decomposition of} $\gl$, and define 
the {\it set of roots}:
$$\Delta^{gl_n}:=\{\epsilon_i-\epsilon_j |1\le i\ne j\le n\}\forno$$

In section 3 we want to describe the twisted affine Lie algebra  $\hat{gl}_n({\bf C})^{(2)}$. From early work of Kac [K69] (see also [K] and [H]) one knows that the affine Lie algebra of type $\hat{sl}_n({\bf C})^{(2)}$
can be constructed using an arbitrary outer automorphism of $sl_n({\bf C})$.
Instead of choosing the usual diagram automorphism, i.e.,  an automorphism induced by the involution of the Dynkin diagram of type $A_{n-1}$, we choose the Cartan involution $\Omega$; 
$$\Omega (X)=-X^t\qquad\hbox{for all }X\in\gl\forno$$
where $X^t$ stands for the transpose of the matrix $X$.
This involution induces a ${\bf Z}_2\equiv {\bf Z}/2{\bf Z}$ gradation of 
the Lie algebra $\gl$,viz.
$$\gl =\bigoplus_{\overline k\in {\bf Z}_2} \gl_{\overline k}\forno$$
where the eigenspace $\gl_{\overline k}$ is defined by
$$\gl_{\overline k}=\{y\in \gl |\Omega (y)= (-1)^k y\}\forno$$

The fixed point set $\gl_{\overline 0}$ is isomorphic to $so_n({\bf C})$, the set  of skew symmetric $n \times n$ matrices.
The elements of $\gl_{\overline 1}$ are the symmetric $n\times n$ matrices.

The matrices $E_{ij}$ 
act on the space of $n$-dimensional
column vectors, which we denote by ${\bf C}^{n}$.
As a basis for this vector space ${\bf C}^{n}$ we take the collection
$\{ e_i\}_{1\le i\le n}$ where $e_i$ is the vector with a one on the
$i^{th}$ entry and zeros elsewhere.
In the sequel it will be convenient to have another basis of ${\bf C}^{n}$;  
we define for $\lambda =+$ or $-$  (we sometimes also write $\lambda = +1$
or $-1$ instead of $+$ or $-$, respectively):
$$f_j(\lambda ) :={1\over \sqrt 2}(e_{2j-1}-i\lambda e_{2j})\forno$$
The collection $\{ f_j(+), f_j(-) (, e_n \hbox{ if }$n$ \hbox{ is odd})\}_{1\le j\le
[{n\over 2}]}$
is a new basis of ${\bf C}^{n}$.

Let $\delta_j :=i(E_{2j-1,2j}-E_{2j,2j-1})$, $1\le j \le [{n\over 2}]$, then the subspace 
$\hso=\oplus_{j=1}^{[{n\over 2}]} {\bf C}\delta_j$
is a Cartan subalgebra of $\so$.
We want to describe the eigen space decomposition of $\gl_{\overline 0}=\so$ 
and $\gl_{\overline 1}$
with respect to the adjoint action of
$\hso$. For this purpose it will be convenient to divide the trace 
form by $2$ and define
$(A\vert B)_{so_n}={1\over 2}(A|B)_{gl_n}={1\over 2}\trace (AB)$. The restriction 
of this bilinear form to each $\gl_{\overline k}$ space is nondegenerate;
its restriction to $\hso$ is also nondegenerate:
$$(\delta_i \vert \delta_j)_{so_n}= \delta_{ij} \forno$$
Introduce the following elements of $\gl$ 
($1\le j,k\le [{n\over 2}]$, $\kappa, \lambda ,\mu \in \{ +,-\}$):
$$F_{jk}(\kappa ;\lambda ,\mu ) := 
           f_j(\lambda )f_k(\mu )^t-\kappa f_k(\mu )f_j(\lambda
)^t\forno$$
and if $n$ is odd:
$$\eqalign{F_j(\kappa;\lambda) &:=
           f_j(\lambda)e_n^t-\kappa e_nf_j(\lambda)^t
           \cr
           F(\kappa) &:=e_ne_n^t-\kappa e_ne_n^t\cr}
           \forno$$
where $v^t$ denotes the transpose of a vector $v\in {\bf C}^{n}$.
These elements are of course not linearly independent;
$F(+)=0$ and
$F_{jk}(\kappa ;\lambda ,\mu  )=-\kappa F_{kj}(\kappa;\mu ,\lambda )$,
hence $F_{jj}(+;\lambda ,\lambda )=0$. 
Notice that $\delta_j = F_{jj}(+;+,-)$. 
These elements satisfy the following commutation relations with
$H \in \hso$:
$$\eqalign{[H,F_{jk}(\kappa ;\lambda ,\mu)]&=
   (\lambda\delta_j +\mu\delta_k
           \vert H)_{so_n}F_{jk}(\kappa ;\lambda ,\mu)
           \cr
           [H,F_j(\kappa ;\lambda )]&= (\lambda\delta_j \vert H)_{so_n}
           F_j(\kappa ;\lambda )
           \cr
           [H,F(\kappa)]&=0\cr}
           \forno$$
Then
$$\gl_{\overline 0}=\so =\cases{{\displaystyle
  \bigoplus_{1\le j\le {n\over 2}} {\bf C}\delta_j \bigoplus
  \bigoplus_{\scriptstyle 1\le j<k\le {n\over 2}
             \atop\scriptstyle\lambda ,\mu =+,-}
  {\bf C}F_{jk}(+ ;\lambda ,\mu)}&if $n$ is even\cr\cr
              {\displaystyle\bigoplus_{1\le j \le [{n\over 2}]}{\bf C}\delta_j 
  \bigoplus \bigoplus_{\scriptstyle 1\le j<k\le [{n\over 2}]
             \atop\scriptstyle\lambda ,\mu =+,-}{\bf C}F_{jk}(+ ;\lambda ,\mu)
  \bigoplus \bigoplus_{\scriptstyle 1\le j\le [{n\over 2}]
                       \atop\scriptstyle\lambda =+,-}
   {\bf C}F_j(+;\lambda)}&if $n$ is odd\cr}\forno$$
We call this the {\it root space decomposition of} $\so$ and define  
the {\it set of roots}: 
$$\Delta^{so_n} :=\cases{\{ \pm (\delta_i \pm \delta_j) \vert 1 \le i\not= j \le
{n\over 2}\}& if $n$ is even\cr\cr
                 \{\pm \delta_j, \pm (\delta_i \pm \delta_j) 
\vert 1 \le i\not= j \le [{n\over 2}]\}&if $n$ is odd\cr}
  \forno$$
The eigenspace $\gl_{\overline 1}$ coincides with the sum of irreducible $\so$-modules $L(2\lambda_1)\oplus L(0)\cong S^2({\bf C}^n)$, the collection of symmetric tensors over ${\bf C}^n$ of degree two. Its {\it weight space decomposition} is given by:
$$\gl_{\overline 1}=\cases{{\displaystyle
  \bigoplus_{\scriptstyle 1\le j\le k\le {n\over 2}
             \atop\scriptstyle\lambda ,\mu =+,-}
  {\bf C}F_{jk}(- ;\lambda ,\mu)}&if $n$ is even\cr\cr
              {\displaystyle 
  \bigoplus_{\scriptstyle 1\le j\le k\le [{n\over 2}]
             \atop\scriptstyle\lambda ,\mu =+,-}
  {\bf C}F_{jk}(- ;\lambda ,\mu)
  \bigoplus\bigoplus_{\scriptstyle 1\le j\le [{n\over 2}]
             \atop\scriptstyle\lambda =+,-} 
  {\bf C}F_j(-;\lambda)\bigoplus {\bf C}F(-)}&if $n$ is odd\cr}\forno$$
The corresponding {\it set of weights} is
$$P(2\lambda_1)\cup P(0):=
\cases{\{0, \pm 2\delta_j, \pm (\delta_i \pm \delta_j) \vert 1 \le i\not= j \le
{n\over 2}\}& if $n$ is even\cr\cr
                 \{0, \pm2\delta_j, \pm \delta_j, \pm (\delta_i \pm \delta_j) 
\vert 1 \le i\not= j \le [{n\over 2}]\}&if $n$ is odd\cr}
  \forno$$

\beginsection \secno . Preliminaries on the algebra $\sotw$\par
In this subsection we describe an outer automorphism of $\sotw$,
in order to define in section  3 the affine Lie algebras 
of type $\hat{so}_{2n}({\bf C})^{(2)}$. This automorphism is slightly different from the one chosen in [K69] and [H].

Let $$ O_{2n}=E_{2n-1,2n}+E_{2n,2n-1}+\sum_{j=1}^{2n-2}E_{jj}\forno$$ and consider the (outer) automorphism $\Ad(O_{2n})$. Here $\Ad(Y)$ is defined as usual:
$\Ad(Y)(X):=YXY^{-1}$.
Clearly,  $\Ad(O_{2n})$ induces a ${\bf Z}_2$-gradation 
$\sotw =\sotw_{\overline 0}\bigoplus\sotw_{\overline 1}$. Now, $\sotw_{\overline
0}\cong so_{2n-1}({\bf C})$ and $\sotw_{\overline 1}$ is isomorphic to the irreducible 
$so_{2n-1}$-module $L(\lambda_1)\cong {\bf C}^{2n-1}$. We know from the previous section that the elements $F_{jk}(+;\lambda,\mu)$, for $1\le j,k\le n$ and $\lambda,\mu=\pm$, span $\sotw$.
It will be convenient to define also some other spanning set, which is better adapted to the ${\bf Z}_2$-gradation defined by $\Ad(O_{2n})$. We introduce the following elements ($1\le j,k\le n$, $\lambda ,\mu =\pm$):
$$G_{jk}(\lambda ,\mu)
  :=g_j(\lambda)g_k(\mu)^t-g_k(\mu)g_j(\lambda)^t\forno$$
where 
$$g_j(\lambda):=\cases{{1\over \sqrt 2}(e_{2j-1}-i\lambda e_{2j})&if $1\le j\le n-1$\cr\cr
{1\over \sqrt 2}(e_{2n-1}+\lambda e_{2n})&if $j=n$\cr}\forno$$
Define $\delta_j:=G_{jj}(+,-)$ ($1\le j<n$) then
$$\eqalign{\sotw_{\overline 0}&=so_{2n-1}({\bf C}) =
              \bigoplus_{1\le j \le n-1}{\bf C}\delta_j 
  \bigoplus\bigoplus_ {\scriptstyle 1\le j<k\le n-1
             \atop\scriptstyle\lambda ,\mu =+,-}{\bf C}G_{jk}(\lambda ,\mu)
  \bigoplus \bigoplus_{\scriptstyle 1\le j\le n-1
                       \atop\scriptstyle\lambda =+,-}
   {\bf C}G_{jn}(\lambda,+)\cr
\sotw_{\overline 1}&=\bigoplus_{\scriptstyle 1\le j\le n-1  \atop\scriptstyle\lambda =+,-} 
  {\bf C}G_{jn}(\lambda,-)\bigoplus {\bf C}G_{nn}(+,-)\cr}\forno$$

The corresponding {\it set of roots of $so_{2n-1}({\bf C})$}, respectively {\it set of weights} is
$$\eqalign{\Delta^{so_{2n-1}} &=\{\pm \delta_j, \pm (\delta_i \pm \delta_j) 
\vert 1 \le i\not= j \le n-1\}
 \cr
P(\lambda_1) &=
  \{0, \pm \delta_j 
\vert 1 \le j \le n-1\}\cr}\forno$$
\beginsection\secno .  The Weyl group of $\gl$ and $\sotw$
\par
Let $\underline g =\gl$ or $\sotw$.
To every $\alpha \in \Delta^{\underline g}$ 
we associate a linear transformation 
of $\underline h^{\underline g}$ :
$$r_\alpha (h) = h-(\alpha \vert h)_{\underline g}\,\alpha 
  \qquad h \in \underline h^{\underline g}
  \forno$$
Then $r_\alpha$ is a reflection in the hyperplane
$H_\alpha = \{ x \in \underline h^{\underline g} \vert (\alpha \vert
x)_{\underline g} = 0\}$.
Denote by $W(\underline g)$ the subgroup of $GL(\underline h^{\underline g})$
generated by
all reflections $r_\alpha$, $\alpha \in \Delta^{\underline g}$.
$W(\underline g)$ is called the {\it Weyl group} of $\underline g$.
It plays an important role in the sequel. In this section we want to describe the conjugacy classes in $W(\underline g)$.

First we consider the case $\underline g =\gl$. The elements of $W(\gl)$
act on the orthonormal basis $\{\epsilon_i\}_{1\le i\le n}$ of 
$\hgl$ by permuting the basisvectors. So $W(\gl)$ is isomorphic to
$S_n$, the symmetric group on $n$ letters. It is rather straightforward to
describe the conjugacy classes of $W(\gl)$. Every $w\in W(\gl)$ can be
expressed as a product of $s$ disjoint cycles of length $n_1, n_2,\cdots ,
n_s$,  with $n_1+n_2+\cdots +n_s=n$.  We will write $(k_1 k_2 \cdots k_l)$ for the cycle
$k_1\mapsto k_2\mapsto\cdots\mapsto k_l\mapsto k_1$. 
The action of the corresponding $W(\gl)$-element on $\hgl$ is then given by:
$$w(\epsilon_{k_i} )=
  \cases{\epsilon_{k_{i + 1}} &if $1\le i<l$\cr
         \epsilon_{k_1} &if $i=l$\cr}$$ Since disjoint cycles commute, we can choose any ordering of the integers $n_1, n_2,\cdots ,
n_s$. It will be convenient to choose this ordering as follows: $n_i$ even for $1\le i\le r$, $n_i$  odd for $i>r$, $n_1\ge n_2\cdots\ge n_r$ and $n_{r+1}\ge n_{r+2}\cdots\ge n_s$. The set of integers 
$[n_1 \ n_2 \cdots n_r \ n_{r+1} \ n_{r+2}\cdots n_s]$ is called the {\it cycle type } of $W(\gl)$. Now two elements are conjugate in $W(\gl)$ if  and only if they have the same cycle
type. In this way we obtain a bijection between the partitions of $n$ and
the conjugacy classes of $W(\gl)$. We associate to the cycle type 
$[n_1 \ n_2 \cdots n_r \ n_{r+1} \ n_{r+2}\cdots n_s]$  the following representative of the corresponding conjugacy class of $W(\gl)$:
$$w(\epsilon_{n_1 +\cdots + n_i + p}) =
           \cases{\epsilon_{n_1 +\cdots + n_i + p +1}&if $1\le p<n_{i+1}$
                  \cr
                  \epsilon_{n_1 +\cdots + n_i + 1}&if $p= n_{i+1}$
                  \cr
}\forno$$ 

Now, let $\underline g =\sotw$.
The elements of $W(\sotw)$ act on the orthonormal basis
$\{ \delta_i \}_{1\le i \le n}$ of $\underline h^{so_{2n}}$
by permuting the basis vectors and changing the signs of an arbitrary
even subset of them. So $W(\sotw)$ is isomorphic to the semidirect product of
$({\bf Z}/2{\bf Z})^{n-1}$ and $S_n$. Carter [C] has classified all conjugacy classes of the Weyl
group of an arbitrary simple Lie algebra. In order to describe his result
for $\sotw$, we introduce some terminology and notation.
If we ignore the sign-changes, then each 
element $w \in W(\sotw)$ determines a permutation in $S_n$.
Again we express such a permutation  as a product of $s$ disjoint
cycles. 
The action of a cycle  $(k_1 k_2 \cdots k_l)$ on $\underline h^{so_{2n}}$ is then given by:
$$w(\delta_{k_i} )=
  \cases{\pm \delta_{k_{i + 1}} &if $1\le i<l$\cr
         \pm \delta_1 &if $i=l$\cr}$$
We now adjoin a sign to the cycle $(k_1 k_2 \cdots k_l)$ and call it {\it positive} 
(resp. {\it negative}) if $w^l(\delta_{k_1} )=\delta_{k_1}$
(resp. $w^l(\delta_{k_1} )= -\delta_{k_1}$).
Observe that a positive (resp. negative) cycle changes the sign
of an even (resp. odd) number of $\delta_i$'s.
Hence the number of negative cycles of $w\in W(\sotw)$ is always even.
Let $w$ be the
product of $r$ disjoint positive cycles of length $n_i$ ($1\le i\le r$) and
$s-r$ disjoint negative cycles of length $n_j$ ($r<j\le s$). Since disjoint
cycles commute, we can assume that $n_1\ge n_2\ge\cdots\ge n_r$ and $n_{r+1}
\ge n_{r+2}\ge\cdots\ge n_s$. Hence, to each  $w\in W$ we can associate a
so-called {\it signed cycle type}
$$[n_1 n_2 \cdots n_r \overline n_{r+1} \overline n_{r+2}
  \cdots \overline n_s] \qquad \sum_{i=1}^sn_i=n,\qquad  n_1\ge n_2\ge\cdots
n_r, n_{r+1} \ge n_{r+2}\ge\cdots \ge n_s\forno$$
 
\noindent  (A bar means that the cycle is negative.)  Conversely, we define an element $w\in W(\sotw)$ associated to the 
signed cycle type (2.3.3) by:
$$w(\delta_{n_1 +\cdots + n_i + p}) =
           \cases{\delta_{n_1 +\cdots + n_i + p +1}&if $1\le p<n_{i+1}$
                  \cr
                  \delta_{n_1 +\cdots + n_i + 1}&if $p= n_{i+1} 
                  \hbox{and } 1\le i <r$
                  \cr
                  -\delta_{n_1+\cdots +n_i+1}
                  &if $p=n_{i+1} \hbox{and } r\le i<s$
                  \cr}\forno$$ 
Conjugate elements in $W(\sotw)$ have the same signed cycle type, but
the converse is  not quite true. Carter proved the following 
proposition [C]:
\proclaim Proposition  \prono.
          There is a one-one correspondence between the
          conjugacy classes in $W(\sotw)$ and the signed cycle types,
          except when all cycles are positive and even. 
          In this case there are two distinct conjugacy classes
          which have the same signed cycle type.\par
It is not hard to find representatives for the two distinct conjugacy
classes in the case of signed cycle type 
$[n_1 n_2\cdots n_s]$ with all $n_i$ even; the element $w$
described by (2.3.4) is a representative of one of the two conjugacy
classes, while the element $w'$ defined by
$$w'(\delta_i)=
  \cases{w(\delta_i)&if $i\not= n_1-1\hbox{, } n_1$
         \cr
         -\delta_{n_1}&if $i=n_1-1$
         \cr
         -\delta_1&if $i=n_1$
         \cr}\forno$$
is a representative of the other class.
\beginsection \secno. Lifting $w\in W(\underline g)$ 

Let $w$ be an arbitrary element of $W(\underline g)$. Such an element
can be lifted to the group $Aut (\underline g)$. In the forthcoming sections
we will construct for any representative $w$ of the form (2.3.2) and (2.3.4)
 an orthogonal $\hat w\in G$ ($ G=GL_n({\bf C})$
or $SO_{2n}({\bf C})$), not unique, such that
$$\Ad(\hat w)(h)=w(h)\qquad\hbox{for }h\in\underline h^{\underline
g}\forno$$
So  $\Ad (\hat w)$  lifts the action of $w$ on $\underline h$ to the
full algebra $\underline g$. $\Ad (\hat w)$ and --by abuse of language also $\hat w$-- will be called lifts of $w$.
Because $\hat w$ is an orthogonal matrix, it can be written as:
$$\hat w=e^{2\pi i X_{[w]}}\qquad\hbox{for some }X_{[w]}\in\underline g\forno$$
\noindent It is not hard to find such a matrix $X_{[w]}$. There exists an orthonormal basis $\{s_i\}$ such
that
$$\eqalign{\hat w(s_{2k-1})&=\phantom{-}\cos(-\theta_k) s_{2k-1}+\sin(-\theta_k)
s_{2k}\cr \hat w(s_{2k})&=-\sin(-\theta_k) s_{2k-1}+\cos(-\theta_k) s_{2k}\cr
}\forno$$
showing that, with respect to this basis, $\hat w$ has the form
${\rm exp}\{2\pi i{\rm ad}H_{[w]}\}$, where
$H_{[w]}=\sum_{k=1}^n\theta_k\delta_k$. In other words: if
$S_{[w]}$ is the orthogonal matrix with columns $s_i$, we have
$$\hat w = S_{[w]}e^{2 \pi i H_{[w]}}S_{[w]}^{-1}=e^{2 \pi i S_{[w]}H_{[w]}S_{[w]}^{-1}}\forno$$



This $X_{[w]}$ or rather $H_{[w]}$ is not unique; our choice of $H_{[w]}$ will be related to the choice of an embedding of the twisted affine Lie algebra $\hat{\underline g}^{(2)}$ in an infinite matrix algebra (see section 4).

Now let $w$ be the representative of the conjugacy class of $W(\underline g)$ (see (2.3.2) and (2.3.4)).
We shall first consider the case of $\gl$. So let $[n_1 n_2 \cdots n_r  n_{r+1} 
 n_{r+2}\cdots  n_s ]$ be the corresponding 
 cycle type.
 A quick glance at (2.3.2) shows that the matrix $\hat w$ should have the
following block diagonal form:
$$\pmatrix{\hat w_1\cr&\hat w_2\cr&&\ddots\cr&&&\hat w_r\cr&&&&\hat w_{r+1}\cr&&&&&\ddots\cr&&&&&&\hat w_s\cr}$$
\noindent where $\hat w_i\in O_{n_i}({\bf C})$. Related to this block form is the following partition of
 $n\times n$ matrices:
$$\bordermatrix{&{\scriptstyle n_1\atop\scriptstyle\longleftrightarrow}
                &{\scriptstyle n_2\atop\scriptstyle\longleftrightarrow}
                &&{\scriptstyle n_r\atop\scriptstyle\longleftrightarrow}
                &{\scriptstyle n_{r+1}\atop\scriptstyle\longleftrightarrow}
                &&{\scriptstyle n_s\atop\scriptstyle\longleftrightarrow}
                \cr
                {\scriptstyle n_1\updownarrow }&B_{11}&B_{12}&\ldots&B_{1r}
                &B_{1,r+1}&\ldots&B_{1,s}
                \cr
                {\scriptstyle n_2\updownarrow }&B_{21}&B_{22}&\ldots&B_{2r}
                &B_{2,r+1}&\ldots&B_{2,s}
                \cr
                &\vdots&\vdots&\ddots&\vdots&\vdots&&\vdots
                \cr
                {\scriptstyle n_r\updownarrow }&B_{r1}&B_{r2}&\ldots&B_{rr}
                &B_{r,r+1}&\ldots&B_{r,s}
                \cr
                {\scriptstyle n_{r+1}\updownarrow }&B_{r+1,1}&B_{r+1,2}&\ldots
                &B_{r+1,r}&B_{r+1,r+1}&\ldots&B_{r+1,s}
                \cr
                &\vdots&\vdots&&\vdots&\vdots&\ddots&\vdots
                \cr
                {\scriptstyle n_s\updownarrow }&B_{s,1}&B_{s,2}&\ldots
                &B_{s,r}&B_{s,r+1}&\ldots&B_{s,s}
                \cr}\forno$$ \vskip.1cm
\noindent where $B_{ij}$ is a block of size $n_i\times n_j$.  
With this blockform in mind we rephrase the elements $e_j$ and define 
$$e_p^i:=e_{n_1+\cdots +n_{i-1}+p}\quad\qquad\hbox{for }1\le p\le n_i\forno$$
We also define  ($1\le i,j \le s$):
$$\eqalign{E^{ij}_{pq}&:= e_p^ie_q^{jt}\cr
\epsilon^i_p  &:=
 \epsilon_{n_1+\cdots +n_{i-1}+p}\cr}
\forno$$

The block structure for $\underline g=\sotw$ is similar to the one described above. To be more precise, let $[n_1 n_2 \cdots n_r \overline n_{r+1} \overline n_{r+2}
  \cdots \overline n_s]$ be the signed cycle type, the associated partition of $2n\times 2n$ skew symmetric matrices is also given by (2.4.5), but now $B_{ij}$ is a block of size $2n_i\times 2n_j$ and $B_{ij}^t=-B_{ji}$. The elements $e_p^i$ and $E_{pq}^{ij}$ are defined as in (2.4.6-7) but now with all $n_i$'s replaced by $2n_i$. We also rephrase the elements $f_i(\lambda)$:
$$ f_p^i(\lambda ):=f_{n_1+\cdots +n_{i-1}+p}(\lambda )\qquad \hbox{for }1\le p\le n_i\forno$$
and define ($1\le i,j \le s$):
            $$\eqalign{F_{pq}^{ij}(+;\lambda, \mu)&:=
            f_p^i(\lambda )f_q^j(\mu )^t-
                             f_q^j(\mu )f_p^i(\lambda )^t\cr
\delta^i_p &:= \delta_{n_1+\cdots +n_{i-1}+p}\cr}\forno$$






\baselineskip=22pt plus.1pt


\count30=2                      %chapterno = 1
\count31=4                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{n}({\bf C})}
\def\sotw{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_{n}({\bf C})}
\def\hgl{\underline h^{gl_n}}
\def\hso{\underline h^{so_n}}
\def\gltw{gl_{2n}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde g l_{2n}({\bf C})}
\def\glh{\hat g l_{2n}({\bf C})}
\def\sot{\tilde s o_{2n}({\bf C})}
\def\soh{\hat s o_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\ug{\underline g}
\def\omegag{\omega_{\ug}}



\beginsection \secno .  The example of one cycle for $\gl$\par
In this section we will describe the case that $w\in W(\gl)$ corresponds to
one cycle. We assume that $w$ is the element of the Weyl
group defined by (2.3.2), hence
$$w(\epsilon_i) =
           \cases{\epsilon_{i +1}&if $1\le i<n$
                  \cr\cr
                  \epsilon_1&if $i=n$
                  \cr}\forno$$ 
We choose the following lift $\hat w\in SO_n({\bf C})\subset GL_n({\bf C})$ of $w$:
$$\hat w:= (-1)^{n+1} E_{1,n}+
                          \sum_{j=1}^{n-1} E_{j+1,j}\forno$$   
This matrix is complex diagonalizable; it has eigenvalues $\omega^{2(k-\epsilon(n))}$
for $1\le k\le n$, where $\omega :=e^{2\pi i\over 2n}$ and
$$\epsilon(n):=\cases{{1\over 2}&if $n$ is even\cr\cr
               0&if $n$ is odd\cr}\forno$$
Each eigenvalue has multiplicity one; its corresponding eigenvectors are
$$x_{2(k-\epsilon(n))}:={1\over\sqrt n}\sum_{j=1}^n \omega^{-2j(k-\epsilon(n))} e_j\forno$$
However, we are not interested in the complex diagonal form of this matrix, 
but rather in its real simple form. It is well known how to obtain this form;
 define for $1\le k\le [{n\over 2}]$:
 $$\eqalign{v_{2k-1}:&={1\over\sqrt 2}(x_{n+2k-1}+x_{n-2k+1})
  \cr  v_{2k}:&={-i\over\sqrt 2}(x_{n+2k-1}-x_{n-2k+1})\cr}\forno$$
In the case of odd $n$ these definitions are supplemented by:
$$v_{n}:=x_{2n}\forno$$
Then
$$\eqalign{\hat w(v_{2k-1})&=\cos (\textstyle{{(n-2k+1)\pi\over n}})v_{2k-1}+
                             \sin (\textstyle{{(n-2k+1)\pi\over n}})v_{2k}\cr
           \hat w(v_{2k})&=-\sin (\textstyle{{(n-2k+1)\pi\over n}})v_{2k-1}+
                             \cos (\textstyle{{(n-2k+1)\pi\over n}})v_{2k}\cr}\forno$$ 
So relative to the basis $\{ v_k\}_{1\le k\le n}$
$\hat w$ has the form
$$\exp\{ 2\pi i(\sum_{k=1}^{[{n\over 2}]} 
  {n-2k+1\over 2n}\delta_k)\}\forno$$
To be more precise, we have:
\proclaim Lemma  \prono.
Let $S$ be the following orthogonal matrix in $Gl_{n}({\bf C})$:
$$S:={\sqrt 2\over\sqrt n}\sum_{j=1}^n\Bigl[\sum_{k=1}^{[{n\over 2}]}
  \Bigl\{ \cos (\textstyle{{j(n-2k+1)\over n}\pi}) 
                                   E_{j,2k-1}
                                  +\sin (\textstyle{{j(n-2k+1)\over
n}\pi})E_{j,2k}
                           \Bigr\} +{1-2\epsilon(n)\over \sqrt 2}E_{jn}\Bigr] 
  \forno$$
then
$\hat w = e^{2\pi i X}$,
with
$X = S H S^{-1}$
and
$$ H=\sum_{k=1}^{[{n\over 2}]} {n-2k+1\over 2n}\delta_k\forno$$
\par
 

The mapping
$\Ad(S^{-1}):\gl\to\gl$ is an isomorphism.  It maps
the standard Cartan subalgebra $\underline h^{gl_n}$ to another
Cartan subalgebra which we denote by $\underline h^{gl_n}_{[n]}$. Since $S$ is orthogonal, $\Ad(S^{-1})$ commutes with the Cartan involution $\Omega$ on $\gl$. (This can also be formulated by saying that $\Ad(S^{-1})$ leaves the subspaces $\gl_{\overline k}$, $\overline k\in {\bf Z}_2$ invariant.) With these remarks and the fact that $\Omega$ leaves $\underline h^{gl_n}$ invariant, we have the following diagram:
\def\normalbaselines{\baselineskip20pt
       \lineskip3pt \lineskiplimit3pt }
  \def\mapright#1{\smash{
      \mathop{\longrightarrow}\limits^{#1}}}
  \def\rmapdown#1{\Big\downarrow
      \rlap{$\vcenter{\hbox{$\textstyle#1$}}$}}
  \def\lmapdown#1{\Big\downarrow
      \llap{$\vcenter{\hbox{$\textstyle#1\quad$}}$}}
 $$\matrix{\underline h^{gl_n}&\mapright{e^{2\pi i{\rm ad} X}\circ\Omega}&\underline h^{gl_n}\cr
          \lmapdown{{\rm Ad }(S^{-1})}&&\rmapdown{{\rm Ad} (S^{-1})}\cr
           \underline h^{gl_n}_{[n]}&\mapright{e^{2\pi i{\rm ad} H}\circ\Omega}
            &\underline h^{gl_n}_{[n]}\cr}\forno$$
\vskip .2cm 
We now want to determine the root vectors with respect to this new
Cartan subalgebra. Of course they are 
$$A_{jk}:=S^{-1}E_{jk}S\forno$$
Let $\hat e_j:=S^{-1}e_j$, then:
$$A_{jk}=\hat e_j\hat e_k^t\forno$$
\proclaim Lemma  \prono . Let $\omega =e^{2\pi i\over 2n}$, then
$$\hat e_j={1\over\sqrt n}\Bigl\{\sum _{q=1}^{[{n\over 2}]}\sum_{\mu=\pm}  \omega^{\mu j(n-2q+1)}f_q(\mu)
  +(1-2\epsilon(n))e_n\Bigr\}\forno$$
and
$$\eqalign{A_{jk}={1\over 2n}&\sum_{\kappa=\pm}\Bigl\{\sum_{q,r=1}^{[{n\over 2}]}
\sum_{\lambda ,\mu =\pm }
\omega^{\lambda j(n-2q+1) +\mu k(n-2r+1) }F_{qr}(\kappa ;\lambda , \mu)\cr
&+(1-2\epsilon(n))\Bigl(\sum_{q=1}^{[{n\over 2}]}\sum_{\lambda =\pm}
\omega^{\lambda j(n-2q+1)}F_q(\kappa ;\lambda)
-\sum_{r=1}^{[{n\over 2}]}\sum_{\mu =\pm}
\kappa\omega^{\mu k(n-2r+1)}F_r(\kappa ;\mu)+F(\kappa)\Bigr)\Bigr\}\cr}
\forno$$
\par
\proof {$$\eqalign{S^{-1}e_j&=
          {\sqrt2\over\sqrt n}
          \sum_{r=1}^n\Bigl\{\sum_{q=1}^{[{n\over 2}]}
          \Bigl(\cos (\textstyle{{r(n-2q+1)\over n}\pi})E_{2q-1,r}+          
          \sin (\textstyle{{r(n-2q+1)\over n}\pi})E_{2q,r}\Bigr)
          +{1-2\epsilon(n)\over \sqrt2}E_{nr}\Bigr\}e_j\cr
                                      &=
          {\sqrt 2 \over\sqrt{n}}\Bigl\{\sum_{q=1}^{[{n\over 2}]}\Bigl( 
          \cos (\textstyle{{j(n-2q+1)\over n}\pi}){f_q(+)+f_q(-)\over\sqrt 2}+
          \sin (\textstyle{{j(n-2q+1)\over n}\pi})i{f_q(+)-f_q(-)\over\sqrt 2}\Bigr)
          +{1-2\epsilon(n)\over\sqrt 2}e_n\Bigr\}\cr
                                      &=
         {1\over\sqrt n}\Bigl\{\sum_{q=1}^{[{n\over 2}]}\Bigl( 
         \omega^{j(n-2q+1)}f_q(+)+\omega^{j(n-2q+1)}f_q(-)\Bigr)
         +(1-2\epsilon(n))e_n\Bigr\}\cr}$$
The formula for $A_{jk}$ follows immediately from this result.}
\beginsection\secno .  The general case for $\gl$
\par
In this section we generalize the results of the previous section
to the case of an arbitrary cycle type. From now on we assume 
that the cycle type is
$[n_1 n_2 \cdots n_r  n_{r+1} n_{r+2}
  \cdots  n_s] $ with $n_j$ even if $j\le r$ and odd if $j>r$.
We  start by giving some definitions.
Denote by $w_a$ the following element of the Weyl group:
$$w_a(\epsilon_j^b )=(1-\delta_{ab})\epsilon_j^b +
           \delta_{ab}(1-\delta_{j,n_a})\epsilon_{j+1}^a+
           \delta_{ab}\delta_{j,n_a}\epsilon_1^a\forno$$
Define:
$$\omega_a:=e^{2\pi i\over 2n_a}\forno$$
$$\hat w_a:= (-1)^{n+1} E_{1,n}^{aa}+
                          \sum_{j=1}^{n_a-1} E_{j+1,j}^{aa}\forno$$ 
$$\eqalign{S_a:=&{\sqrt 2\over\sqrt {n_a}}\sum_{j=1}^{n_a}\Bigl[\sum_{k=1}^{[{n_a\over 2}]}
  \Bigl\{ \cos ({\textstyle{{j(n_a-2k+1)\over n_a}\pi}}) 
                                   E_{j,2k-1}^{aa}
                                  +\sin ({\textstyle{{j(n_a-2k+1)\over
n_a}\pi}})E_{j,2k}^{aa}
                           \Bigr\} +{1-2\epsilon(n_a)\over \sqrt 2}E_{jn_a}^{aa}\Bigr]\cr
T_a:=&{\sqrt 2\over\sqrt {n_a}}\sum_{j=1}^{n_a}\Bigl[{1-2\epsilon(n_a)\over \sqrt 2}E_{j1}^{aa}+\sum_{k=1}^{[{n_a\over 2}]}
  \Bigl\{ \cos ({\textstyle{{j(n_a-2k+1)\over n_a}\pi}}) 
                                   E_{j,2k}^{aa}
                                  +\sin ({\textstyle{{j(n_a-2k+1)\over
n_a}\pi}})E_{j,2k+1}^{aa}
                           \Bigr\}\Bigr]\cr} 
  \forno$$
 where $$\epsilon(m):=\cases{{1\over2}&if $m$ is even\cr
               0&if $m$ is odd\cr}\forno$$
$$ H_a=\sum_{k=1}^{[{n_a\over 2}]} {n_a-2k+1\over 2n_a}\delta_{[{n_1+\cdots+n_{a-1}+1\over 2}]+k}\forno$$

\proclaim Lemma  \prono.
Let
$[w]=[n_1 n_2 \cdots n_r  n_{r+1}  n_{r+2}
  \cdots  n_s] $
be the signed cycle type of $w$, with $n_a$ even if $a\le r$ and odd if $a>r$. Then the element
$$w:=\prod_{a=1}^s w_a\forno$$
is a representative of the conjugacy class of the Weyl group that
corresponds to this  cycle type. 
The element
$$\hat w:=\sum_{a=1}^s \hat w_a\forno$$
is a lift of $w$.
Define the orthogonal matrix
$$S_{[w]}:=\sum_{a=1}^r S_a +\sum_{a=1}^{[{s-r\over 2}]}( S_{r+2a-1}+T_{r+2a})+(1-2\epsilon(n))S_s\forno$$
then $\hat w=e^{2\pi iX_{[w]}}$ with $X_{[w]}=S_{[w]}H_{[w]}S_{[w]}^{-1}\in\gl$ 
and
$$H_{[w]}:=\sum_{a=1}^s H_a\forno$$
Define
$$A_{jk}^{ab}:=S_{[w]}^{-1}E_{jk}^{ab}S_{[w]}\forno$$
then
$$A_{jk}^{ab}=\hat e_j^a \hat e_k^{bt}\forno$$
where  
$$\hat e_j^a={1\over\sqrt {n_a}}\Bigl\{\sum _{q=1}^{[{n\over 2}]}\sum_{\mu=\pm}  \alpha_j^{aq} (\mu)f_q(\mu)
  +(1-2\epsilon(n))\delta_{as}e_{n}\Bigr\}\forno$$
and
$$\eqalign{A_{jk}^{ab}={1\over \sqrt{4n_an_b}}&\sum_{\kappa=\pm}\Bigl\{\sum_{q,t=1}^{[{n\over 2}]}
\sum_{\lambda ,\mu =\pm }\alpha_j^{aq}(\lambda)\alpha_k^{bt}(\mu)
F_{qt}(\kappa ;\lambda , \mu)\cr
&+(1-2\epsilon(n))\Bigl(\sum_{q=1}^{[{n\over 2}]}\sum_{\lambda =\pm}
\alpha_j^{aq}(\lambda)\delta_{bs}F_q(\kappa ;\lambda)
-\sum_{t=1}^{[{n\over 2}]}\sum_{\mu =\pm}
\kappa\delta_{as}\alpha_k^{bt}(\mu)
F_t(\kappa ;\mu)+\delta_{as}\delta_{bs}F(\kappa)\Bigr)\Bigr\}\cr}
\forno$$
with
$$\alpha_j^{aq}(\mu):=\cases{\omega_a^{\mu j(n_a-2t+1)}& if $q=[{n_1+\cdots +n_{a-1}+1\over 2}]+t$, $1\le t\le [{n_a\over 2}]$\cr\cr
{1\over\sqrt 2}&if $q=[{n_1+\cdots +n_a+1\over 2}]$ and $0<a-r$ odd\cr\cr
{i\mu\over\sqrt 2}&if $q=[{n_1+\cdots +n_{a-1}+1\over 2}]$ and $0<a-r$ even\cr\cr
0&in all other cases\cr}\forno$$
\par
\beginsection\secno .  The example of one positive cycle for $\sotw$\par
In this section we will describe the case that $w\in W(\sotw)$ corresponds to
one positive cycle. We assume that $w$ is the element of the Weyl
group defined by (2.3.4), hence
$$w(\delta_i) =
           \cases{\delta_{i +1}&if $1\le i<n$
                  \cr
                  \delta_1&if $i=n$
                  \cr}\forno$$ 
Let $\delta=0$ or $1\over 2$. We choose the lifts $\hat w,\ \hat w^\delta\in \SO$ of $w$ (see also [KrL91b,c]):
$$\eqalign{\hat w:=&  E_{1,2n-1}+E_{2,2n}+
                          \sum_{j=1}^{2n-2} E_{j+2,j}\cr
     \hat w^\delta:=&(4\delta-1)\cos ({\textstyle{n+1\over n}}\pi )(E_{1,2n-1}+ E_{2,2n} +
                   \sum_{k=1}^{n-1}(E_{2k+2,2k} +E_{2k+1,2k-1}))+\cr
                  &\qquad(4\delta-1)\sin ({\textstyle{n+1\over n}}\pi )(E_{2,2n-1}-E_{1,2n} +
                   \sum_{k=1}^{n-1}(E_{2k+2,2k-1}-E_{2k+1,2k}))\cr}
\forno$$   
Notice that 
$$\hat w O_{2n}=E_{1,2n}+E_{2,2n-1}+ \sum_{j=1}^{2n-2} E_{j+2,j}\forno$$  
For reasons that will be clear later on, we shall describe both the
automorphisms $\Ad (\hat w^\delta)$ and $\Ad(\hat wO_{2n})
$. Again, we transform these matrices to their real simple form.  Notice that  $\Ad(\hat wO_{2n})|_{\underline h^{so_{2n}}}$ has become a negative cycle. Similar calculations as in the
example of one cyle for $\gl$ lead to the following result (see also [KrL91b]):
\proclaim Lemma  \prono.
Let $S_{ w}$, $T_{ w}$ be the orthogonal matrices
$$\eqalign{S_w:=&{1\over\sqrt n}\sum_{j,k=1}^n \cos ({\textstyle {2jk\over n}}\pi ) 
                                   (E_{2j-1,2k-1}+E_{2j,2k})
                                  +\sin ({\textstyle {2jk\over n}}\pi) 
                                   (E_{2j,2k-1}-E_{2j-1,2k})\cr
T_w:=&{1\over\sqrt n}\Bigl\{\sum_{j=1}^{n}\sum_{k=1}^{n-1} \cos ({\textstyle {j(n-k)\over n}}\pi ) 
                   ( E_{2j-1,2k-1}+(-1)^{n-k}E_{2j,2k-1})
                   +\sin ({\textstyle{j(n-k)\over n}}\pi)(E_{2j-1,2k}+(-1)^{n-k}E_{2j,2k})\cr
                   &\qquad +\sum_{j=1}^{n }{\textstyle {1+(-1)^j\over 2} }E_{2j-1,2n-1}+{\textstyle {1-(-1)^j\over 2}}E_{2j-1,2n}+{\textstyle {1+(-1)^{n+j}\over 2}}E_{2j,2n-1}+{\textstyle {1-(-1)^{n+j}\over 2}}E_{2j,2n}\Bigr\}\cr}\forno$$
Then $S_w^{-1}\hat w^\delta S_w= e^{2\pi i  H^\delta}$, with
$$ H^\delta=\sum_{k=1}^n {(1-\delta)2n-k+1\over 2n}\delta_k\forno$$
And $T_w^{-1}\hat wO_{2n}T_w=e^{2\pi i H}O_{2n}$, with 
$$ H=\sum_{k=1}^n {n-k\over 2n}\delta_k\forno$$
Moreover, $ H O_{2n}=O_{2n} H$.\par

Since the matrices $S_w$ and $T_w$ are orthogonal,  the mappings
$\Ad(S_w^{-1})$ and  $\Ad (T_w^{-1})$ are isomorphisms of $\sotw$. 
Define
$$\eqalign{\hat D_{jk}(\lambda ,\mu )&:=S_w^{-1}F_{jk}(+;\lambda ,\mu )S_w\cr
\tilde D_{jk}(\lambda, \mu)&:=T_w^{-1}F_{jk}(+;\lambda ,\mu )T_w\cr}\forno$$
and let $\hat f_j(\lambda):=S_w^{-1}f_j(\lambda)$ and
$\tilde f_j(\lambda):=T_w^{-1}f_j(\lambda)$. Then:
$$\eqalign{\hat D_{jk}(\lambda,\mu)=& \hat f_j(\lambda)\hat f_k(\mu)^t-\hat f_k(\mu)\hat f_j(\lambda)^t\cr
\tilde D_{jk}(\lambda ,\mu )=&\tilde f_j(\lambda )\tilde f_k(\mu )^t-
                        \tilde f_k(\mu )\tilde f_j(\lambda )^t\cr}\forno$$
\proclaim Lemma  \prono.
Let $\omega =e^{2\pi i\over 2n}$  then
$$\eqalign{\hat f_j(\lambda )=&{1\over\sqrt n}\sum _{p=1}^n
                     \omega^{-2\lambda jp}f_p(\lambda )\cr
           \tilde f_j(\lambda)=&{1\over \sqrt {2n}}\sum_{\mu=\pm}\Bigl\{ {\textstyle {1-i\lambda (-1)^{{1-\mu\over 2}n}\over\sqrt 2}}\omega^{{1-\mu\over 2}jn}g_n(\mu)  +\sum_{k=1}^{n-1}{\textstyle {1-i\lambda (-1)^{n-k}\over\sqrt 2}}\omega^{\mu j(n-k)}g_k(\mu)\Bigr\}\cr}\forno$$
and
$$\eqalign{\hat D_{jk}(\lambda ,\mu )&={1\over n}\sum_{p,q=1}^n
  \omega^{-2\lambda jp -2\mu kq }F_{pq}(+;\lambda , \mu )\cr
  \tilde D_{jk}(\lambda,\mu)&={1\over 2n} \Bigl\{\sum_{\alpha ,\beta =\pm}\Bigl(
\sum_{p,q=1}^{n-1}{\textstyle {1-i\lambda (-1)^{n-p}\over\sqrt 2} }     {\textstyle {1-i\mu(-1)^{n-q}\over\sqrt 2} } \omega^{\alpha j(n-p)+\beta k(n-q)}G_{pq}(\alpha ,\beta )\cr
&\qquad\qquad + \sum_{p=1}^{n-1}{\textstyle {1-i\lambda (-1)^{n-p}\over\sqrt 2} }{\textstyle {1-i\mu(-1)^{{1-\beta\over 2}n}\over\sqrt 2} } \omega^{\alpha j(n-p)+{1-\beta \over 2}kn}G_{pn}(\alpha,\beta)\cr
&\qquad\qquad +\sum_{q=1}^{n-1}{\textstyle {1-i\lambda (-1)^{{1-\alpha\over 2}n}\over\sqrt 2} }{\textstyle {1-i\mu(-1)^{n-q}\over\sqrt 2} } \omega^{{1-\alpha\over 2} jn+\beta k(n-q)}G_{nq}(\alpha ,\beta )\Bigr)\cr
&\qquad\qquad +{\textstyle {1-i\lambda(-1)^{n}\over\sqrt 2} }{\textstyle {1-i\mu\over\sqrt 2} }\omega^{jn}G_{nn}(-,+)+{\textstyle {1-i\lambda\over\sqrt 2} }{\textstyle {1-i\mu(-1)^{n}\over\sqrt 2} }\omega^{kn} G_{nn}(+,-) \Bigr\}\cr }\forno$$
\par
\proof {A straightforward calculation.}
If $n$ is even, we have seen in section 2.3 that there is  another conjugacy
class associated to the cycle type $[n]$. A representative  of this class is the element $w'$ (see (2.3.5)).
Lifts of this element can be easily obtained from the lift 
$\hat w^\delta$, respectively $\hat w$ of $w$, viz. 
$$\eqalign{\hat w^\delta\!\! \hbox{ $'$} &=O_{2n}\hat w^\delta O_{2n}^{-1}\cr
           \hat w' &=O_{2n}\hat w O_{2n}^{-1}\cr}\forno$$
So $\hat w^\delta\!\! \hbox{ $'$} =O_{2n}S_we^{2\pi iH^\delta}S_w^{-1}O_{2n}^{-1}$ and   $\hat w'O_{2n} =O_{2n}T_we^{2\pi iH}O_{2n}T_w^{-1}O_{2n}^{-1}$ Now
$$O_{2n}^{-1}f_j(\lambda )=\cases{f_j(\lambda )& if $j\not =n$\cr
                       -i\lambda f_n(-\lambda )& if $j=n$\cr}\forno$$
hence
$$\eqalign{\hat f_j(\lambda)':=&S_w^{-1}O_{2n}^{-1}f_j(\lambda )
  =\cases {{1\over\sqrt n}\sum_{p=1}^n \omega^{-2pj\lambda} f_p(\lambda )&
           if $j\not =n$\cr\cr
           -i\lambda {1\over\sqrt n}\sum_{p=1}^n f_p(-\lambda )&
           if $j=n$\cr}\cr
\tilde f_j(\lambda)':=&T_w^{-1}O_{2n}^{-1}f_j(\lambda)\cr
=&\cases{{1\over \sqrt {2n}}\sum_{\mu=\pm}\Bigl\{ {\textstyle {1-i\lambda (-1)^{{1-\mu\over 2}n}\over\sqrt 2}}\omega^{{1-\mu\over 2}jn}g_n(\mu)  +\sum_{k=1}^{n-1}{\textstyle {1-i\lambda (-1)^{n-k}\over\sqrt 2}}\omega^{\mu j(n-k)}g_k(\mu)\Bigr\}&if $j\ne n$\cr\cr
-i\lambda{1\over \sqrt {2n}}\sum_{\mu=\pm}\Bigl\{ {\textstyle {1+i\lambda (-1)^{{1-\mu\over 2}n}\over\sqrt 2}}(-1)^{{1-\mu\over 2}}g_n(\mu) +\sum_{k=1}^{n-1}{\textstyle {1+i\lambda (-1)^{n-k}\over\sqrt 2}}(-1)^{\mu (n-k)}g_k(\mu)\Bigr\}&if $j=n$\cr}\cr}\forno$$
With the definitions 
$$\eqalign{\hat D_{jk}(\lambda ,\mu )':&=S_w^{-1}O_{2n}^{-1}G_{jk}(+;\lambda ,\mu )O_{2n}S_w\cr
\tilde D_{jk}(\lambda ,\mu )':&=T_w^{-1}O_{2n}^{-1}G_{jk}(+;\lambda ,\mu )O_{2n}T_w\cr}\forno$$
we find
$$\eqalign{\hat D_{jk}(\lambda, \mu )'
       =&(1-\delta_{jn})(1-\delta_{kn})\hat D_{jk}(\lambda ,\mu )
         -i\lambda (1-\delta_{kn})\delta_{jn}\hat D_{nk}(-\lambda ,\mu )\cr
        &\qquad -i\mu\delta_{kn} (1-\delta_{jn})\hat D_{jn}(\lambda ,-\mu )
         -\lambda\mu \delta_{jn}\delta_{kn}\hat D_{nn}(-\lambda ,-\mu )\cr}\forno$$
and  a similar expression for $\tilde D(\lambda ,\mu)'$,  with the hat  replaced by  the tilde.
\beginsection\secno .  The general case for $\sotw$
\par
In this section we generalize the results of the previous section to the case of an arbitrary signed cycle type. From now on we assume 
that the signed cycle type is
$[n_1 n_2 \cdots n_r \overline n_{r+1} \overline n_{r+2}
  \cdots \overline n_s] $.
 Notice that if $r=s$ we only have positive cycles, and we can generalize the results of the previous section. Now asume that $r\ne s$. Let $ w$ be a representative of the signed cycle type (2.3.3). We want to describe $\hat w O_{2n}$ where $\hat w$ is a lift of $w$. Restricted to the Cartan subalgebra $\underline h^{so_{2n}}$ $\hat w O_{2n}$ consists of $r+1$ positive cycles and $s-r-1$ negative cycles. So we are again in a similar situation as in the previous section. Remember that  $\Ad(\hat wO_{2n})|_{\underline h^{so_{2n}}}$ was a negative cycle.

We  start by giving some definitions.
Denote by $w_a$, $w'_s$ and $w_{ab}$ the following elements of the Weyl group:
$$\eqalign{w_a(\delta_j^b )=&(1-\delta_{ab})\delta_j^b +
           \delta_{ab}(1-\delta_{j,n_a})\delta_{j+1}^a+
           \delta_{ab}\delta_{j,n_a}\delta_1^a\cr
           w_{ab}(\delta_j^c )=&(1-\delta_{ac})(1-\delta_{bc})\delta_j^c+
           \delta_{ac}(1-\delta_{j,n_a})\delta_{j+1}^a-
           \delta_{ac}\delta_{j,n_a}\delta_1^a\cr
           &\qquad+\delta_{bc}(1-\delta_{j,n_b})\delta_{j+1}^b-
           \delta_{bc}\delta_{j,n_b}\delta_1^b\cr
           w'_s(\delta_j^a)=&(1-\delta_{sa})\delta_j^a +
           \delta_{sa}(1-\delta_{j,n_s-1})(1-\delta_{j,n_s})\delta_{j+1}^s-
          \delta_{sa}\delta_{j,n_s-1}\delta_{n_s}^s-
          \delta_{sa}\delta_{j,n_s}\delta_1^s\cr}\forno$$
Define
$$\eqalign{\hat w_a^\delta:=&(4\delta-1)\cos ({\textstyle{n_a+1\over n_a}}\pi )
                      (E_{1,2n_a-1}^{aa}+ E_{2,2n_a}^{aa} +
                      \sum_{k=1}^{n_a-1}
                      (E_{2k+2,2k}^{aa} +E_{2k+1,2k-1}^{aa}))+\cr
                     &\qquad(4\delta-1)\sin ({\textstyle{n_a+1\over n_a}}\pi )
                      (E_{2,2n_a-1}^{aa}-E_{1,2n_a}^{aa} +
                      \sum_{k=1}^{n_a-1}
                      (E_{2k+2,2k-1}^{aa}-E_{2k+1,2k}^{aa}))\cr
 \hat w_a:=&E_{1,2n_a}^{aa}+E_{2,2n_a-1}^{aa}+
                          \sum_{j=1}^{2n_a-2} E_{j+2,j}^{aa}\cr}\forno$$
$$\eqalign{\hat w_{ab}:=&\sum_{p=a,b} (E_{1,2n_p}^{pp}+E_{2,2n_p-1}^{pp}+
                             \sum_{j=1}^{2n_p-2} E_{j+2,j}^{pp})\cr
}
\forno$$   
$$S_a:={1\over\sqrt n_a}\sum_{j,k=1}^{n_a} \cos ({\textstyle{2jk\over {n_a}}}\pi ) 
                                   (E_{2j-1,2k-1}^{aa}+E_{2j,2k}^{aa})
                                  +\sin ({\textstyle{2jk\over {n_a}}}\pi) 
                                   (E_{2j,2k-1}^{aa}-E_{2j-1,2k}^{aa})\forno$$
$$\eqalign{T_a:=&{1\over\sqrt {n_a}}\Bigl\{\sum_{j=1}^{n_a}\sum_{k=1}^{n_a-1} \cos ({\textstyle {j(n_a-k)\over n_a}}\pi )
( E_{2j-1,2k-1}^{aa}+(-1)^{n_a-k}E_{2j,2k-1}^{aa})\cr
&\qquad\qquad
                   +\sin ({\textstyle{j(n_a-k)\over n_a}}\pi)(E_{2j-1,2k}^{aa}+(-1)^{n_a-k}E_{2j,2k}^{aa})\cr
                   &\qquad +\sum_{j=1}^{n_a }{\textstyle {1+(-1)^j\over 2} }E_{2j-1,2n_a-1}^{aa}+{\textstyle {1-(-1)^j\over 2}}E_{2j-1,2n_a}^{aa}+{\textstyle {1+(-1)^{n_a+j}\over 2}}E_{2j,2n_a-1}^{aa}+{\textstyle {1-(-1)^{n_a+j}\over 2}}E_{2j,2n_a}^{aa}\Bigr\}\cr}\forno$$
$$\eqalign{S_{ab}:=&{1\over\sqrt {n_a}}\sum_{i=1}^{n_a}\Biggl\{{1\over\sqrt 2}
              \Bigl((-1)^iE_{2i-1,1}^{aa} +(-1)^{n_a+i}E_{2i,1}^{aa} +
              E_{2i-1,2n_b-1}^{ab} +E_{2i,2n_b-1}^{ab}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=2}^{n_a}\Bigl(\cos ({\textstyle{i(n_a-j+1)\over {n_a}}}\pi )
              (E_{2i-1,2j-1}^{aa} +(-1)^{n_a-j+1}E_{2i,2j-1}^{aa})+\cr
             &\qquad\qquad\qquad\sin ({\textstyle{i(n_a-j+1)\over {n_a}}}\pi )
              (E_{2i-1,2j}^{aa} +(-1)^{n_a-j+1}E_{2i,2j}^{aa})
             \Bigr)\Biggr\}+\cr
             &{1\over\sqrt {n_b}}\sum_{i=1}^{n_b}\Biggl\{{1\over\sqrt 2}
              \Bigl((-1)^iE_{2i-1,2}^{ba} +(-1)^{n_b+i}E_{2i,2}^{ba} +
              E_{2i-1,2n_b}^{bb} +E_{2i,2n_b}^{bb}\Bigr)+\cr
             &\qquad\qquad
              \sum_{j=1}^{n_b-1}\Bigl(\cos ({\textstyle{i(n_b-j)\over {n_b}}}\pi )
              (E_{2i-1,2j-1}^{bb} +(-1)^{n_b-j}E_{2i,2j-1}^{bb})+\cr
             &\qquad\qquad\qquad\sin ({\textstyle{i(n_b-j)\over {n_b}}}\pi )
              (E_{2i-1,2j}^{bb} +(-1)^{n_b-j}E_{2i,2j}^{bb})
             \Bigr)\Biggr\}\cr}
  \qquad(a\ne b)\forno$$
$$\eqalign{T_{ab}:=&{1\over\sqrt {n_a}}\Bigl\{\sum_{j=1}^{n_a}\sum_{k=1}^{n_a-1} \cos ({\textstyle {j(n_a-k)\over n_a}}\pi )
( E_{2j-1,2k-1}^{aa}+(-1)^{n_a-k}E_{2j,2k-1}^{aa})\cr
&\qquad\qquad
                   +\sin ({\textstyle{j(n_a-k)\over n_a}}\pi)(E_{2j-1,2k}^{aa}+(-1)^{n_a-k}E_{2j,2k}^{aa})\cr
                   &\qquad +\sum_{j=1}^{n_a }{\textstyle {1+(-1)^j\over 2} }E_{2j-1,2n_b-1}^{ab}+{\textstyle {1-(-1)^j\over 2}}E_{2j-1,2n_b}^{ab}+{\textstyle {1+(-1)^{n_a+j}\over 2}}E_{2j,2n_b-1}^{ab}+{\textstyle {1-(-1)^{n_a+j}\over 2}}E_{2j,2n_b}^{ab}\Bigr\}\cr
&+{1\over\sqrt{ n_b}}\sum_{j=1}^{n_b }\Bigl\{
           \cos ({\textstyle {2j\over n_b}}\pi ) 
                                   (E_{2j-1,2n_a-1}^{ba}+E_{2j,2n_a}^{ba})
                                  +\sin ({\textstyle {2j\over n_b}}\pi) 
                                   (E_{2j,2n_a-1}^{ba}-E_{2j-1,2n_a}^{ba})\cr
          & \qquad +\sum_{k=2}^{n_b}\cos ({\textstyle {2jk\over n_b}}\pi ) 
                                   (E_{2j-1,2k-3}^{bb}+E_{2j,2k-2}^{bb})
                                  +\sin ({\textstyle {2jk\over n_b}}\pi)                                   (E_{2j,2k-3}^{bb}-E_{2j-1,2k-2}^{bb})\Bigr\}\cr}\forno$$
$$\eqalign{ H_a^\delta:=&\sum_{k=1}^{n_a} {(1-\delta)2n_a-2k+1\over 2n_a}\delta_k^a\cr
 H_a:=&\sum_{k=1}^{n_a} {n_a-k\over 2n_a}\delta_k^a\cr}
\forno$$
$$\eqalign{ H_{ab}:=&\sum_{k=1}^{n_a}
       {n_a-k+1\over 2n_a}\delta_k^a+
     \sum_{k=1}^{n_b}
       {n_b-k\over 2n_b}\delta_k^b\cr
 H_{ab}^\delta:=&\sum_{k=1}^{n_a-1} {n_a-k\over 2n_a}\delta_k^a
            +{(1-\delta)2n_b-1\over 2n_b}\delta_{n_a}^a+
            \sum_{k=2}^{n_b} {(1-\delta)2n_b-2k+1\over 2n_b}\delta_{k-1}^b\cr}\qquad (a\ne b)     \forno$$
\proclaim Lemma  \prono.
Let
$[w]=[n_1 n_2 \cdots n_r \overline n_{r+1} \overline n_{r+2}
  \cdots \overline n_s] $
be a signed cycle type of $n$, then the element
$$w:=\prod_{a=1}^r w_a \prod_{a=1}^{s-r\over 2} w_{r+2a-1,r+2a}\forno$$
is a representative of the conjugacy class of the Weyl group that
corresponds to this signed cycle type. If $r=s$ and all $n_i$ even, 
then there are two conjugacy classes for $[w]$.
In that case $w$ is a representative of one of these classes, and
$$w':=w'_s\prod_{a=1}^{s -1} w_a \forno   $$
is a representative of the other conjugacy class.
The elements
$$\hat w^\delta O_{2n}:=\cases{\displaystyle \sum_{a=1}^{s-1} \hat w_a^\delta  +\hat w_s&if $r= s$ \cr
\displaystyle \sum_{a=1}^r \hat w_a^\delta  +\sum_{a=1}^{s-r-2\over 2}
\hat w_{r+2a-1,r+2a}+\hat w_{s-1}\hat w_{s}^\delta &if $r\ne s$ \cr}\forno$$
$$\hat w^\delta\!\! \hbox{ $'$}O_{2n}:= O_{2n}\Bigl\{\sum_{a=1}^{s-1} \hat w_a^\delta  +\hat w_s\Bigr\}O_{2n}^{-1}\forno$$
are lifts of $w\Ad (O_{2n})|_{\underline h^{so_{2n}}}$, $w'\Ad (O_{2n})|_{\underline h^{so_{2n}}}$, respectively.
Define an orthogonal matrix
$$S_{[w]}:=\cases{\displaystyle\sum_{a=1}^{s-1} S_a +T_s&if $r=s$\cr
  \displaystyle\sum_{a=1}^r S_a +\sum_{a=1}^{s-r-2\over 2} S_{r+2a-1,r+2a}+T_{s-1,s}&if $r\ne s$\cr}\forno$$
then 
$S_{[w]}^{-1}\hat w^\delta  O_{2n}S_{[w]}
=e^{2\pi iH_{[w]}^\delta }O_{2n}$
and $S_{[w]}^{-1}O_{2n}^{-1}(\hat w^\delta  O_{2n})O_{2n}S_{[w]}
=e^{2\pi iH_{[w]}^\delta }O_{2n}$
where
$$H_{[w]}^\delta :=\cases{\displaystyle\sum_{a=1}^{r-1}\hat H_a^\delta  +\hat H_s&if $r=s$\cr
\displaystyle\sum_{a=1}^r\hat H_a^\delta  +\sum_{a=1}^{s-r-2\over 2} \hat H_{r+2a-1,r+2a}+\hat H_{s-1,s}^\delta &if $r\ne s$\cr}\forno$$
and $H_{[w]}^\delta  O_{2n}=O_{2n}H_{[w]}^\delta $. Let
$$\omega_a=e^{2\pi i\over 2n_a}\forno$$
and define
$$D_{jk}^{ab}(\lambda ,\mu ):=S_{[w]}^{-1}F_{jk}^{ab}(+;\lambda ,\mu )S_{[w]}\forno$$
$$D_{jk}^{ab}(\lambda ,\mu )':=
                       S_{[w]}^{-1}O_{2n}^{-1}F_{jk}^{ab}(+;\lambda ,\mu )O_{2n}S_{[w]}\forno$$
then
$$D_{jk}^{ab}(\lambda ,\mu )=
\hat f_j^a(\lambda)\hat f_k^b (\mu)^t-\hat f_k^b(\mu)\hat f_j^a(\lambda)^t
\forno$$
with
$$\hat f_j^a(\lambda)={1\over\sqrt {2n_a}}                 \sum_{\mu = +,-} \sum_{k=1}^{n}    
                 \alpha_{j}^{ak}(\lambda ,\mu )g_k(\mu )\forno$$
and
$$D_{jk}^{ab}(\lambda ,\mu )={1\over\sqrt{4n_an_b}}\sum_{\xi,\eta=\pm}\sum_{q,t=1}^n \alpha_j^{aq}(\lambda, \xi)\alpha_k^{bt}(\mu,\eta)G_{qt}(\xi, \eta)\forno$$
where
for $1\le a\le r$:
$$\alpha_{j}^{ak}(\lambda ,\mu)=
         \cases{\sqrt 2\omega_a^{-2\lambda jq}&if $1\le q=k-n_1-\cdots -n_{a-1}\le n_a$, $\lambda=\mu$ and ($a\ne r$ or $a=r\ne s$)\cr
\omega_s^{\mu j(n_s-q)}{\textstyle {1-i\lambda (-1)^{n_s-q}\over\sqrt 2}}&if $1\le q=k-n_1-\cdots -n_{s-1}<n_s$ and $a=r=s$\cr
 \omega_s^{{1-\mu\over 2}jn_s}{\textstyle {1-i\lambda (-1)^{{1-\mu\over 2}n_s}\over\sqrt 2}}&if $a=r=s$ and $k=n$\cr
 0&in all other cases\cr}\forno$$
and for $1\le a\le {s-r-2\over 2}$:
$$\eqalign{\alpha_{j}^{r+2a-1,k}(\lambda ,\mu)=& 
  \cases{(-1)^j{1-i\lambda (-1)^{n_{r+2a-1}}\over 2}&if $k=n_1+ \cdots +n_{r+2a-2}+1$\cr
         \omega_{r+2a-1}^{\mu j(n_{r+2a-1}-q+1)}
         {1-i\lambda (-1)^{n_{r+2a-1}-q+1}\over\sqrt 2}
         &if $1<q=k- n_1-\cdots -n_{r+2a-2} \le n_{r+2a-1}$\cr
         1-i\lambda\over 2&if $k=n_1+\cdots +n_{r+2a}$\cr
         0&in all other cases\cr}\cr
  \alpha_{j}^{r+2a,k}(\lambda ,\mu)=& 
  \cases{(-1)^j\mu {i+\lambda (-1)^{n_{r+2a}}\over 2}&if $k=n_1+\cdots +n_{r+2a-2}+1$\cr
         \omega_{r+2a}^{\mu j(n_{r+2a}-q)}
         {1-i\lambda (-1)^{n_{r+2a}-q}\over\sqrt 2}
         &if $1\le q= k-n_1-\cdots -n_{r+2a-1}< n_{r+2a}$\cr
         \mu {i+\lambda\over 2}&if $k=n_1+\cdots +n_{r+2a}$\cr
         0&in all other cases\cr}\cr}\forno$$
and for $s\ne r$:
$$\eqalign{\alpha_{j}^{s-1,k}(\lambda ,\mu):=& 
  \cases{\omega_{s-1}^{{1-\mu\over 2}jn_{s-1}}{1-i\lambda (-1)^{{1-\mu\over 2}n_{s-1}}\over \sqrt 2}&if $k=n$\cr
         \omega_{s-1}^{\mu j(n_{s-1}-q)}{1-i\lambda (-1)^{n_{s-1}-q}\over\sqrt 2}
         &if  $1\le q=k-n_1-\cdots -n_{s-2}\le n_{s-1}-1$\cr
         0&in all other cases\cr}\cr
  \alpha_{j}^{sk}(\lambda , \mu):=& 
  \cases{\sqrt 2\omega_s^{-2\lambda j(q+1)}&if  $0\le q=k-n_1-\cdots -n_{s-1}<n_s$ and $\lambda=\mu$\cr
         0&in all other cases\cr}\cr
}\forno$$
Moreover
$$D_{jk}^{ab}(\lambda ,\mu )'=
\cases{D_{jk}^{ab}(\lambda ,\mu )&if $(a,j)$, $(b,k)\not = (s,n_s)$\cr
       -i\lambda D_{n_s,k}^{sb}(-\lambda ,\mu )&if
       $(a,j)=(s,n_s)$ and $(b,k)\not =(s,n_s)$\cr
       -i\mu D_{j,n_1}^{a1}(\lambda ,-\mu )&if 
       $(a,j)\not =(s,n_s)$ and $(b,k)=(s,n_s)$\cr
       -\lambda \mu D_{jk}^{ab}(-\lambda ,-\mu )& if
       $(a,j)=(b,k)=(s,n_s)$\cr}\forno$$
\par
\proof {A  generalization of the results of the previous section.}

 




\baselineskip=22pt plus.1pt



\count30=2                      %chapterno = 2
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{n}({\bf C})}
\def\sotw{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}
\def\nug{\nu_{\underline g}}
\def\ug{\underline g}
\def\ugt{\ug^{(2)}}
\def\tugt{\tilde{\ug}^{(2)}}
\def\hugt{\hat{\ug}^{(2)}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_n({\bf C})}
\def\hgl{\underline h^{gl_n}}
\def\hso{\underline h^{so_n}}
\def\gltw{gl_{2n}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde {gl}_{n}({\bf C})}
\def\glh{\hat {gl}_{n}({\bf C})}
\def\sot{\tilde s o_{2n}({\bf C})}
\def\soh{\hat s o_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}




\beginsection \chapno .  The affine algebra $\hat{\underline g}^{(2)}$\par
\beginsection \secno .  Introduction
\par
In this section we will briefly review the realizations of the affine
algebras $\hat{gl}_n({\bf C})^{(2)}$ and $\hat{so}_{2n}({\bf C})^{(2)}$ as algebras of loops twisted by outer  automorphisms $\tw$ (see [K]).
We will also introduce so-called vertex operators as
generating series in a formal indeterminate $z$, whose coefficients describe
the action of the affine algebra in an integrable highest weight
representation. In section 6 we will see that these operators are
precisely the vertex operators one encounters in physics.

\beginsection \secno .  Twisted realizations of $\glt^{(2)}$ and $\sot^{(2)}$
\par
Let $\underline g=\gl$ or $\sotw$ and let  $\nug $ be the outer automorphism defined by
$$\nug :=\cases{\Omega& if $\ug=\gl$\cr\cr
\Ad (O_{2n})&if $\ug=\sotw$\cr}\forno$$
Using this automorphism, one
defines the twisted loopalgebra $\tugt$ of $\ug$ as
the collection of (polynomial) loops in $\ug$ twisted by $\nug$:
$$\eqalign{
\tugt\equiv L(\ug,\nug )&:=
\{f:[0,2\pi]\to \ug | f(\theta+2\pi)=\nug(f(\theta))\hbox{ and }f\hbox{ polynomial}\}\cr
&\cong \bigoplus_{k\in {\bf Z}} e^{ik\theta \over2}\ug_{\overline k}\cr}
\forno$$
Here $\overline k\in {\bf Z}_2$; the parameter $\theta$ is a local coordinate
on the circle. The restriction to polynomial mappings means that we only consider elements of the form 
$$\tilde x(\theta)=\sum_{k=M}^N e^{ik\theta\over 2}x_k,\qquad x_k\in \ug_{\overline k}$$
It is  expressed by the direct sum sign in (3.2.2).
The algebra $\tugt$ has a well known central extension $\hugt=\tugt\bigoplus {\bf C}c$ defined by the commutation relations:
$$[\tilde x+\alpha c,\tilde y+\beta c]:=[\tilde x,\tilde y]
  +\nu(\tilde x,\tilde y)c\forno$$
where the two cocycle $\nu :\tugt\times\tugt\rightarrow{\bf C}$ is
given by:
$$\nu (\tilde x,\tilde y):={1\over 2\pi i}\int^{2\pi}_0
  d\theta\left({d\tilde x(\theta )\over d\theta}\vert
  \tilde y(\theta )\right)_{\ug}\forno$$
The twisted loop algebras $\tugt$ and $\hugt$ can also be constructed by means of the (outer) automorphism 
$$\tw :=\cases{\exp(2\pi i\ad H_{[w]})\Omega& if $\ug=\gl$\cr\cr
\exp(2\pi i\ad H_{[w]}^\delta)\Ad (O_{2n})&if $\ug=\sotw$\cr}\forno$$
Let $N$ be the order of $\tw$ and 
$$\ug=\bigoplus_{\overline k\in {\bf Z}_N}\ug_{\overline k}\forno$$
the ${\bf Z}_N\equiv {\bf Z}/N{\bf Z}$-gradation of $\ug$ induced by $\tw$. Define $\lambda:=e^{ik\theta\over N}$, then 
$$L(\ug,\tw ):= 
\bigoplus_{k\in {\bf Z}}\lambda^k\ug_{\overline k}\forno$$
with  commutation relations
$$[\lambda^kx,\lambda^jy]:=\lambda^{k+j}[x,y]\qquad\hbox{for all } 
  x\in\ug_{\overline k},y\in\ug_{\overline j}\forno$$ The corresponding central extension is induced by the same cocycle $\nu$ of (3.2.4); $\hat L(\ug,\tw ):=L(\ug,\tw )\bigoplus {\bf C}c$
with commutation relations
$$[\lambda^ky+\alpha c,\lambda^lz+\beta c]
=\lambda^{k+l}[y_
{\overline k},z_{\overline l}]+{k\over N}\delta_{k+j,0}(y\vert
z)_{\underline g}c\qquad\hbox{for all } 
  x\in\ug_{\overline k},y\in\ug_{\overline j}\forno$$

It is well known [K, H] that $\tugt$ and
$L(\ug,\tw )$  are isomorphic; an isomorphism is given by:
$$\Phi_{[w]}:\lambda^ky\mapsto e^{-i\theta \ad\Hw}e^{i{k\over N}\theta}y
  \qquad \hbox{for all } y\in\ug_{\overline k}\forno$$
This automorphism suggests to introduce the following spanning set of
$\tugt$ (for an analog in the untwisted case see [KP85]):
$$y(k):=e^{-i\theta \ad\Hw} e^{i{k\over N}\theta}y_{\overline
k}\qquad y\in \ug, k\in{\bf Z}\forno$$
where $y_{\overline k}$ denotes the component of $y$, which is
homogeneous of degree $\overline k$ in the ${\bf Z}_N$ gradation of $\ug$ 
induced by $\tw$ (see (3.2.5-6)). The isomorphism
$\Phi_{[w]}:L(\ug,\tw ) \rightarrow \tugt$ can be
extended to an isomorpism $\hat\Phi_{[w]}$ between  
$\hat L(\ug,\tw )$ and $\hugt$:
$$\hat\Phi_{[w]}(\lambda^ky_{\overline k}+\alpha c):=
  \hat y(k)+\alpha c\forno$$
where the elements $\hat y(k)\in\hugt$ are defined by
$$\hat y(k):=y(k)-\delta_{k,0}(H_{[w]}\vert y)_{\underline g}c\forno$$

\beginsection \secno.  Vertex operators
\par
Recall from section 2.6 and 2.8 the vectors $A^{ab}_{jk}$ and $D_{jk}^{ab}(\lambda,\mu)$. It is
clear that the affine algebra $\glh^{(2)}$ ($\soh^{(2)}$) is
spanned by the loops
$$\hat A^{ab}_{jk}(l) \quad (\hat D_{jk}^{ab}(\lambda,\mu)(l))\quad \hbox{for}\quad 1\le j\le n_a,\ 1\le k\le n_b,\ 1\le a,b\le s\hbox{ and }l\in{\bf Z}\forno$$
In the sequel we will be interested in representations
$\rho :\hat {\underline g}^{(2)}\rightarrow {\rm End}\,V$ of the affine algebra on a vector space 
$V$.
In the construction of representations one uses the following formal generating
series:
$$\eqalign{\hat A^{ab}_{jk}(z):&=\sum_{l\in {\bf
Z}}z^{-l}\rho\left(\hat A^{ab}_{jk}(l)\right)\cr
\hat D^{ab}_{jk}(\lambda,\mu)(z):&=\sum_{l\in {\bf
Z}}z^{-l}\rho\left(\hat D^{ab}_{jk}(\lambda,\mu)(l)\right)\cr}
\forno$$
The parameter $z$ in these series is just a formal parameter. If one can
somehow find explicit expressions for the operators
$\hat A^{ab}_{jk}(z)$ and $\hat D^{ab}_{jk}(\lambda,\mu)(z)$, one can --in principle-- 
extract the coefficients of $z^{-l}$ and find the action of
the affine algebra on the representation space $V$.
In fact we will only be interested in integrable highest weight representations
of the affine algebra, in which the central element c is represented by the
identity operator. So from now on we will take $\rho (c)=I$.

Below we give the explicit form of the vertex operators
in terms of the elements $e^{im\theta\over 2 }F_{jk}((-)^m;\lambda ,\mu )$, $e^{im\theta\over 2 }F_{j}((-)^m;\lambda)$, $e^{im\theta\over 2 }F((-)^m)$ and
$e^{i(m+ \delta_{jn}{1-\lambda\over 4}+\delta_{kn}{1-\mu\over 4})\theta }G_{jk}(\lambda ,\mu )$.
$$\eqalign{\hat A_{jk}^{ab}(z)&=\sum_{l\in{\bf Z}}z^{-l}\rho
  \left(e^{i{l\over N}\theta }e^{-i\theta\ad\,\Hw}
  \left(A^{ab}_{jk}\right)_{\overline l}\right)
  -\left(\Hw\vert A^{ab}_{jk}\right)_{gl_n}I\cr
\hat D_{jk}^{ab}(\lambda,\mu)(z)&=\sum_{l\in{\bf Z}}z^{-l}\rho
  \left(e^{i{l\over N}\theta }e^{-i\theta\ad\,\Hw^\delta}
  \left(D^{ab}_{jk}(\lambda,\mu)\right)_{\overline l}\right)
  -\left(\Hw^\delta\vert D^{ab}_{jk}(\lambda,\mu)\right)_{so_{2n}}I\cr}
  \forno$$
Define
$$\eqalign{h_q&:=(\Hw|\delta_q)_{so_n}\cr
h_q^\delta&:=(\Hw^\delta |\delta_q)_{so_{2n}}\cr}\forno$$
then the first terms in the right--hand--sides of   (3.3.3) can be worked out as:
$$\eqalign{2\sqrt{n_an_b}  \sum_{l\in{\bf Z}}z^{-l}\rho&\left(e^{i{l\over N}
  \theta}e^{-i\theta\ad\,\Hw}\left(A^{ab}_{jk}
  \right)_{\overline l}\right)
  \cr=\sum_{m\in{\bf Z}}
\Bigl\{&\sum_{q,t=1}^{[{n\over 2}]}
\sum_{\lambda ,\mu =\pm }z^{-N({m\over 2} +\lambda h_q+\mu h_t)}\alpha_j^{aq}(\lambda)\alpha_k^{bt}(\mu)
\rho\left(e^{im\theta\over 2}F_{qt}((-)^m ;\lambda , \mu)\right)\cr
&+(1-2\epsilon(n))\Bigl(\sum_{q=1}^{[{n\over 2}]}\sum_{\lambda =\pm}z^{-N({m\over 2} +\lambda h_q)}
\alpha_j^{aq}(\lambda)\delta_{bs}\rho\left(e^{im\theta\over 2}F_{q}((-)^m;\lambda )\right)\cr
&\phantom{=(1-2\epsilon(n)) (}-\sum_{t=1}^{[{n\over 2}]}\sum_{\mu =\pm}z^{-N({m\over 2}+\mu h_t)}
(-)^m\delta_{as}\alpha_k^{bt}(\mu)\rho\left(e^{im\theta\over 2}F_{t}((-)^m ; \mu)\right)\cr
&\phantom{=(1-2\epsilon(n)) (}+z^{-N{m\over 2}}\delta_{as}\delta_{bs}\rho\left(e^{im\theta\over 2}F((-)^m)\right)\Bigr)\Bigr\}\cr}
\forno$$
and
$$\eqalign{2\sqrt{n_an_b}&\sum_{l\in{\bf Z}}z^{-l}\rho\left(e^{i{l\over N}
  \theta}e^{-i\theta\ad\,\Hw^\delta}\left(D^{ab}_{jk}(\lambda,\mu)
  \right)_{\overline l}\right)
  \cr=\sum_{m\in{\bf Z}}&\sum_{\xi,\eta=\pm}\sum_{q,t=1}^n
z^{-N(m+\xi h_q^\delta+\eta h_t^\delta +\delta_{qn}{1-\xi\over 4}+\delta_{tn}{1-\eta\over 4})}\alpha_j^{aq}(\lambda ,\xi)\alpha_k^{bt}(\mu,\eta)\rho\left(e^{i(m+\delta_{qn}{1-\xi\over 4}+\delta_{tn}{1-\eta\over 4})\theta}G_{qt}(\xi,\eta)\right)\cr}\forno$$
Using the lemmas 2.6.1 and 2.8.1, it is straightforward, but tedious, to calculate the central term; we give the result in
the following lemma:
\proclaim Lemma  \prono.
$$\eqalign{(\Hw\vert A_{jk}^{ab})_{gl_n}&={1\over \sqrt {n_an_b}}\sum_{q=1}^{[{n\over2}]}\left(\alpha_j^{aq}(+)\alpha_k^{bq}(-)-\alpha_j^{aq}(-)\alpha_k^{bq}(+)\right)h_q\cr
&=\cases{{i(-)^{j-k+1}\over 2n_a}{1\over\sin({(j-k)\pi\over n_a})}&if $a=b$ and $j\ne k$\cr\cr
0&in all other cases\cr}\cr}\forno$$
$$\eqalign{(&\Hw^\delta\vert D_{jk}^{ab}(\lambda ,\mu ))_{so_{2n}}={1\over 2\sqrt {n_an_b}}\sum_{q=1}^{n-1}\left(\alpha_j^{aq}(\lambda,+)\alpha_k^{bq}(\mu,-)-\alpha_j^{aq}(\lambda,-)\alpha_k^{bq}(\mu,+)\right)h_q^\delta\cr
&=\cases{\lambda\delta_{\lambda ,-\mu}({1\over 2} -\delta)
         &if ($1\le a=b\le r-\delta_{rs}$  or $a=s\ne r$) and $j= k$
         \cr & \cr\lambda\delta_{\lambda ,-\mu}
         {1\over {n_a}} 
         {\omega_a^{2\lambda (k-j)}\over {1-\omega_a^{2\lambda (k-j)}}}
         &if ($1\le a=b\le r-\delta_{rs}$ or $a=s\ne r$ and $j\ne k$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over 8\sqrt {n_an_b}}
         \left({\lambda\mu\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)
         &if $r-\delta_{rs}<a=b\le s-1+\delta_{rs}$ and $j\ne k$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over {8\sqrt {n_an_b}}}(-1)
         &if $r+1<a+1=b=r+2c\le s-2$
         \cr & \cr
         (-1)^{j-k}i{(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})
         \over {8\sqrt {n_an_b}}}(1)
         &if $r+1<b+1=a=r+2c\le s-2$
         \cr &\cr
         0&in all other cases
         \cr}}\forno$$




\baselineskip=22pt plus.1pt


\count30=3                      %chapterno = 3
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{n}({\bf C})}
\def\sotw{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\B{{\cal B}}
\def\D{{\cal D}}
\def\X{{\cal X}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}
\def\gl{gl_{n}({\bf C})}
\def\hgl{\underline h^{gl_n}}
\def\hso{\underline h^{so_n}}
\def\gltw{gl_{2n}({\bf C})}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\glt{\tilde{gl}_{n}({\bf C})^{(2)}}
\def\glh{\hat{gl}_{n}({\bf C})^{(2)}}
\def\sot{\tilde{so}_{2n}({\bf C})^{(2)}}
\def\soh{\hat{so}_{2n}({\bf C})^{(2)}}
\def\vac{\mid \overline 0 >}
\def\ug{\underline g}
\def\nug{\nu_{\ug}}
\def\ugt{\tilde{\ug}^{(2)}}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\Bi{B_{\infty}}


\beginsection \chapno .  Fermionic fields\par
\beginsection \secno .  Introduction\par
In this section we will briefly review the Lie algebras $\glib$, $\oib$ ($\epsilon =0,{1\over 2}$)
and their central extensions $\Ai$, $\Bi$ ($\epsilon =0$) or $\Di$ ($\epsilon={1\over 2}$), respectively.
We will also discuss the embeddings $\glt  \hookrightarrow\overline{o_{\epsilon (n)}(\infty)}$ ($\tilde{so}_{2n}({\bf C})^{(2)}\hookrightarrow \overline {o_0(\infty)}$) and
$\glh \hookrightarrow \Bi$ or $\Di$ ($\hat{so}_{2n}({\bf C})^{(2)}\hookrightarrow \Bi$), which, together with the spin
representation of $\Bi$ and $\Di$, will enable us to find expressions for the vertex 
operators 
$\hat A_{jk}^{ab}(z)$ ($\hat D_{jk}^{ab}(\lambda,\mu)(z)$)
in terms of normal ordered products of fermionic fields. Some parts 
of section 4.2 and section 4.3 
have been taken from [K] and [Y].




\beginsection \secno .  Lie algebras of infinite matrices\par
Let $\glib$ be the Lie algebra of complex infinite matrices such that all
non zero entries  are within a finite distance from the main diagonal,
i.e.,
$$\glib := \{ (g_{ij})_{i,j \in {\bf Z}} \mid g_{ij}=0 
  \hbox{ if } \mid i-j\mid >>0\}\forno$$
The elements $\E_{ij}$ (the matrix with the $(i,j)^{th}$ entry $1$ 
and $0$ elsewhere), for $i,j\in {\bf Z}$, form a basis of a subalgebra
$\gli \subset \glib$. The Lie algebra $\glib$ has a universal central extension
$\Ai :=\glib \bigoplus {\bf C}c_{\Ai}$ with the Lie bracket defined by
$$[a + \alpha c_{\Ai},b+\beta c_{\Ai}]=ab-ba+\mu (a,b)c_{\Ai}\forno$$
for $a,b\in \glib$ and $\alpha ,\beta \in {\bf C}$; here $\mu$ is the
following 2-cocycle:
$$\mu (\E_{ij},\E_{kl}) =\delta_{il} \delta_{jk}
      <\theta ,\E_{ii}- \E_{jj}>\forno$$
where the linear mapping $\theta :\bigoplus_{i\in {\bf Z}} {\bf C}\E_{ii}
\to {\bf C}$ is defined by
$$<\theta ,\E_{ii}>:=\cases{0&if $i>0$\cr &\cr 1&if $i\le 0$\cr}\forno$$
Let $\epsilon :=0$ or $1\over 2$, the Lie algebra $\oib$  is defined as a subalgebra of $\glib$:
$$\oib :=\{(g_{ij})\in \glib \mid g_{ij}=-g_{-j+2\epsilon ,-i+2\epsilon}\}\forno$$
We denote by $\oi =\oib \cap \gli$; it is generated by the elements
$\F_{ij}:= \E_{i+\epsilon,-j+\epsilon} -\E_{j+\epsilon,-i+\epsilon}$, for $i,j\in \epsilon+{\bf Z}$. Notice
that $\F_{ij} =-\F_{j,i}$.

The 2-cocycle $\mu$ on $\glib$
induces a 2-cocycle on $\oib$, and hence we can define a central 
extension $\Bi:= \overline{o_0(\infty)} \bigoplus {\bf C}c_0$  and $\Di :=\overline{o_{1\over 2}(\infty)} \bigoplus {\bf C}c_{1\over 2}$ of $\oib$, with
Lie bracket
$$[a+\alpha c_{\epsilon},b+\beta c_{\epsilon}]=ab-ba +{1\over 2}\mu (a,b)c_{\epsilon}\forno$$
for $a,b\in \oib$ and $\alpha ,\beta \in {\bf C}$.
 
We also introduce the vector space ${\bf C}^{\infty}$ of infinite column 
vectors $(w_i)_{i\in {\bf Z}}$ such that all but a finite number of   
$w_i$'s are zero. As a basis for this vector space we take the collection 
$\{ v_i\} _{i\in {\bf Z}}$ where $v_i$ is the vector with a one on the 
$i^{th}$ entry and zeros elsewhere. The Lie algebra $\glib$ clearly acts on the 
vector space ${\bf C}^{\infty}$ by:
$$\E_{ij} v_k =\delta_{jk} v_i\forno$$


Recall from section 3.2 the twisted loop algebra 
$\tilde {\ug}^{(2)} \cong \bigoplus_{k\in {\bf Z}} e^{ik\theta\over 2}\ug_{\overline k}$. In the case of $\ug=\so$, the `natural'
representation space for the twisted loop algebra $\sot$ is of course the twisted loop  space 
$$L({\bf  C}^{2n}, O_{2n}) := \{ x:[0,2\pi ]\to {\bf C}^{2n}\mid
  x(\theta +2\pi)= O_{2n}x(\theta)\quad\hbox{and }x\hbox{ polynomial}\}\forno$$
The action of $\sot$ on $L({\bf  C}^{2n},O_{2n})$ is  given by
$$e^{ik\theta\over 2} A (e^{il\theta\over 2}x) := e^{i(k+l)\theta\over 2}Ax
\qquad\hbox{for } k,l\in {\bf Z}\forno$$

\noindent The vector space  ${\bf C}^{\infty}$ is
isomorphic to $L({\bf  C}^{2n},O_{2n})$; for $\delta=0,{1\over 2}$ we will use the following isomorphisms ($k\in {\bf Z}$, $\lambda=\pm$):
$$\eqalign{e^{-ik\theta}g_j(\lambda)&\mapsto
v_{2n(k+\delta)-\lambda(n-j)}\qquad\qquad 1\le j<n,\ \delta =0,{1\over 2}\cr
e^{-i(k+{1-\lambda\over 4})\theta}g_n(\lambda)&\mapsto 
v_{2n(k+\delta)+{1-\lambda\over 2}n}\qquad\qquad\ \ \delta=0,{1\over 2}\cr}
\forno$$

If  $\ug= gl_{n}({\bf C})$ there is a small obstruction to a similar construction, because $\Omega$ cannot be written as an adjoint transformation. In order to solve this problem, we embed $\gl$ in $gl_{2n}({\bf C})$ in the following way:
$$\phi (E_{ij}):=E_{ij}-E_{n+j,n+i}\forno$$
Notice that $\phi(\Omega(E_{ij}))=\Ad (P_{2n})(\phi(E_{ij}))$, where
$$P_{2n}=\sum_{k=1}^n E_{k,n+k}+E_{n+k,k}\forno$$
i.e., $\Omega$ corresponds to $\Ad (P_{2n})$ on $\phi(\gl)$. We now proceed as in the case of $\sotw$ and introduce the twisted loop space $L({\bf C}^{2n}, P_{2n})$ (see (4.2.8)). $\glt$ has a natural action on this space given by
$$(e^{ik\theta\over 2} A )\cdot (e^{il\theta\over 2}x) := e^{i(k+l)\theta\over 2}\phi (A) x
\qquad\hbox{for } k,l\in {\bf Z}\forno$$
Again, $L({\bf C}^{2n}, P_{2n})\cong {\bf C}^\infty$; we choose the following isomorphism
   $$\eqalign{{\textstyle{(\lambda )^k\over \sqrt 2}}e^{-{ik\theta\over 2}}\{(e_{2j-1}-i\lambda e_{2j})+(-1)^k(e_{n+2j-1}-i\lambda e_{n+2j})\}&\mapsto v_{nk-{\lambda\over 2}(n-2j+1)+{\epsilon (n)}}\qquad 1\le j\le [{n\over 2}]\cr 
  c(k)e^{-{ik\theta\over 2}}\{e_{n}+(-1)^k e_{2n}\}&\mapsto v_{nk}\qquad\qquad\hbox{only if }n \hbox{ is odd}\cr}\forno$$
where 
$$c(k):=\cases{1& if $k$ is even\cr\cr
              i&if $k$ is odd\cr}\forno$$

Because the twisted loop algebra $\tilde{\underline g}^{(2)}$  now acts on $ {\bf C}^\infty$, its elements can be seen as endomorphisms of ${\bf C}^\infty$. In this way we obtain  embeddings $\iota:\tilde {gl}_n({\bf C})^{(2)}\to \glib$ and $\iota_\delta :\tilde{so}_{2n}({\bf C})^{(2)}\to \glib$. Even more is true: the image of $\iota$ lies in $\overline{o_{\epsilon (n)}(\infty)}$ and the image of $\iota_\delta$ lies in $\overline{o_0(\infty)}$.
\proclaim Lemma \prono .
{\bf (a)} The mapping $\iota:\tilde \gl^{(2)}\to\overline{o_{\epsilon (n)}(\infty)}$ defined by
$$\eqalign{\iota :e^{i{l\over 2}\theta}F_{jk}((-)^l;\lambda,\mu)&
\mapsto\sum_{m\in {\bf Z}}\lambda^{m-l}(-\mu)^m\F_{n(m-l)-{\lambda\over 2}(n-2j+1),-nm-{\mu \over 2}(n-2k+1)}\cr\cr
\iota :e^{i{l\over 2}\theta}F_j((-)^l;\lambda)&
\mapsto\sum_{m\in {\bf Z}}\lambda^{m-l}c(m)\F_{n(m-l)-{\lambda\over 2}(n-2j+1),-nm}\quad\qquad(\hbox{only if }n\hbox{ is odd})\cr\cr
\iota :e^{i{l\over 2}\theta}F((-)^l)&
\mapsto\sum_{m\in {\bf Z}}c(m-l)^{-1}c(m)\F_{n(m-l),-nm}\ \quad\qquad\qquad(\hbox{only if }n\hbox{ is odd})\cr}\forno$$
is a Lie algebra homomorphism, mapping $\tilde\gl^{(2)}$ injectively into
$\overline{o_{\epsilon (n)}(\infty)}$.
The 2-cocycle on $\overline{o_{\epsilon (n)}(\infty)}$ induces the following 2-cocycle on $\tilde\gl^{(2)}$
($k,l\in{\bf Z}$ $A\in\gl_{\overline k}$, $B\in\gl_{\overline l}$):
$$\eqalign{
  {1 \over 2}\mu\left(\iota (e^{ik\theta\over 2}A),\iota (e^{il\theta\over 2}B)\right)
&=\nu\left(e^{ik\theta\over 2}A,e^{il\theta\over 2}B\right)+{\delta_{k,-l}\over 2}(\lambda_{[{n\over 2}]}|[A,B])_{gl_n}\cr
  &= \delta _{k,-l}\{{k\over 2}(A\mid B)_{gl_n}+{1\over 2}(\lambda_{[{n\over 2}]}|[A,B])_{gl_n}\}
  \cr}\forno$$
where $\lambda_{[{n\over 2}]}={1\over 2}\sum_{j=1}^{[{n\over 2}]} \delta_j$, the $[{n\over 2}]^{th}$ fundamental weight of $so_n({\bf C})$.
The homomorphism $\iota$ can be extended to a homomorphism $\hat\iota: 
\hat\gl^{(2)}\hookrightarrow \Bi$  ($\Di$) for $n$ odd (even) by defining $\hat {\iota}(c)=c_{\epsilon (n)}$ and
$$\hat\iota(e^{ik\theta\over 2}A) =\iota(e^{ik\theta\over 2}A)+{\delta_{k,0}\over 2}(\lambda_{[{n\over 2}]} |A)_{gl_n}c_{\epsilon (n)}
\forno$$
{\bf (b)} The mapping $\iota_\delta:\tilde so_{2n}({\bf C})^{(2)}\to\overline{o_0(\infty)}$ defined by
$$\iota_\delta :e^{i(l+\delta_{jn}{1-\lambda\over 4}+\delta_{kn}{1-\mu\over 4})\theta}G_{jk}(\lambda,\mu)
\mapsto\sum_{m\in {\bf Z}}\F_{2n(m-l+\delta)-\lambda(n-j)-\delta_{jn}{1-\lambda\over 2}n,-2n(m+\delta)-\mu (n-k)-\delta_{kn}{1-\mu\over 2}n}\forno$$
is a Lie algebra homomorphism, mapping $\tilde{so}_{2n}({\bf C})^{(2)}$ injectively into
$\overline{o_0(\infty)}$.
The 2-cocycle on $\overline{o_0(\infty)}$ induces the following 2-cocycle on $\tilde{so}_{2n}({\bf C})^{(2)}$
($k,l\in{\bf Z}$ $A\in\sotw_{\overline k}$, $B\in\sotw_{\overline l}$):
$$\eqalign{
  {1 \over 2}\mu\left(\iota_\delta (e^{ik\theta\over 2}A),\iota_\delta (e^{il\theta\over 2}B)\right)
&=\nu\left(e^{ik\theta\over 2}A,e^{il\theta\over 2}B\right)+\delta_{k,-l}(1-2\delta)(\lambda_{n-1}|[A,B])_{so_{2n}}\cr
  &= \delta _{k,-l}\{{k\over 2}(A\mid B)_{so_{2n}}+(1-2\delta)(\lambda_{n-1}|[A,B])_{so_{2n}}\}
  \cr}\forno$$
where $\lambda_{n-1}={1\over 2}\sum_{j=1}^{n-1} \delta_j$, the $(n-1)^{th}$ fundamental weight of $so_{2n-1}({\bf C})$.
The homomorphism $\iota_\delta$ can be extended to a homomorphism $\hat\iota_\delta: 
\hat\sotw^{(2)}\hookrightarrow \Bi$   by defining $\hat {\iota}_\delta(c)=c_0$ and
$$\hat\iota_\delta(e^{ik\theta\over 2}A) =\iota_\delta(e^{ik\theta\over 2}A)+\delta_{k,0}(1-2\delta)(\lambda_{n-1} |A)_{so_{2n}}c_{0}
\forno$$ 
\par
\proof{A straightforward calculation.} 

For future use, we calculate the value of $(\lambda_{[{n\over 2}]}|A_{jk}^{ab})_{gl_n}$ and $(\lambda_{n-1}|D_{jk}^{ab}(\lambda,\mu))_{so_{2n}}$:
\proclaim Lemma  \prono .
$$(\lambda_{[{n\over 2}]}|A_{jk}^{ab})_{gl_n}=\cases{
{(1-(-)^{j-k})i\over 2n_a}{1\over\sin({(j-k)\pi\over n_a})}&if $1\le a=b\le r$ and $j\ne k$\cr\cr
{i\over 2n_a}{(-1)^{j-k+1}+\cos({(j-k)\pi\over n_a})\over \sin({(j-k)\pi\over n_a})}&if $r<a=b\le s$ and $j\ne k$\cr\cr
-i\over 2\sqrt{n_an_b}&if $1\le a-r=b-r-1<s$\cr\cr
i\over 2\sqrt{n_an_b}&if $1\le b-r=a-r-1<s$\cr\cr
0& in all other cases\cr}\forno$$
\overfullrule=0pt
$$\eqalign{(&\lambda_{n-1}\vert D_{jk}^{ab}(\lambda ,\mu ))_{so_{2n}}=\cr &\
  \cases{{\lambda \over 2}\delta_{\lambda ,-\mu}         &if ($1\le a=b\le r-\delta_{rs}$ or $a=s\ne r$) and $j=k$\cr & \cr        ( (-1)^{j-k+\delta_{\lambda \mu}n_a}-1)i{(1-i\lambda) (1-i\mu )
         \over 8\sqrt {n_an_b}}
         \left({\lambda\mu\sin ({j-k \over {n_a}}\pi )
         \over {1+\lambda\mu\cos ({j-k\over {n_a}}\pi )}}\right)
         &if $r-\delta_{rs}<a=b\le s-1+\delta_{rs}$ and $j\ne k$
         \cr & \cr
         i{(-1)^{j-k}(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})+(1-i\lambda )(1-i\mu )
         \over {8\sqrt {n_an_b}}}(-1)
         &if $r+1<a+1=b=r+2c\le s-2$
         \cr & \cr
         i{(-1)^{j-k}(1-i\lambda (-1)^{n_a})(1-i\mu (-1)^{n_b})+(1-i\lambda )(1-i\mu )
         \over {8\sqrt {n_an_b}}}(1)
         &if $r+1<b+1=a=r+2c\le s-2$
         \cr &\cr
         0&in all other cases
         \cr}\cr}\forno$$

\overfullrule=5pt

\beginsection \secno .  The spin representation\par
In this subsection we consider highest weight representations for
$\oi$, $\Bi$ and $\Di$. For this purpose we introduce the Clifford algebra 
$Cl^{\epsilon}$ as the associative algebra on generators $\phi (j)$ ($j\in \epsilon+{\bf Z}$) 
with defining relations
$$\{ \phi (i), \phi (j)\} := \phi (i)\phi (j) +\phi (j) \phi (i) 
                           =\delta_{i,-j}\forno$$
Since the $\phi (j)$'s anti-commute, we call them {\it fermions.}
  
We define the {\it spin module} $V^{\epsilon}$ over $Cl^{\epsilon}$ as the irreducible 
module with highest weight vector $\mid \overline 0 >$ satisfying
$$\phi (j)\mid \overline 0 >=0\qquad\hbox{for all }j< 0\forno$$
The vector $\mid \overline 0 >$ is called the {\it vacuum vector}.
The elements
$$\phi (j_p) \phi (j_{p-1})\cdots \phi (j_2) \phi (j_1)\mid \overline 0 >
\qquad \hbox{with}\qquad j_p>j_{p-1}> \cdots>j_2>j_1\ge 0\forno$$
form a basis of $V^{\epsilon}$. 

We use these fermions to define a representation $\tau^{\epsilon}$ of $\oi$ on $V^{\epsilon}$:
$$\eqalign{\tau^{\epsilon} (\F_{ij})&={\textstyle {1\over 2}}
  (\phi (i)\phi (j)-\phi (j)\phi (i))
  \cr
  &=\phi (i)\phi(j)-{\textstyle {1\over 2}}\delta_{i,-j}I\cr}\forno$$
When restricted to $\oi$, the spin module breaks into a direct sum of 
two irreducible modules. To describe this decomposition, we define a
${\bf Z}_2$-gradation on $V^{\epsilon}$ by introducing a {\it chirality operator} $\chi$
satisfying:
$$\chi \vac =\vac\forno$$
$$\{\chi ,\phi (j)\}=0\qquad\hbox{for all  }j\in \epsilon+{\bf Z}\forno$$
Then,
$$V^{\epsilon}=\bigoplus_{\alpha \in {\bf Z}_2} V_{\alpha}^{\epsilon}\forno$$
where 
$$V_{\alpha}^{\epsilon}=\{ v\in V^{\epsilon}\mid \chi v=(-1)^{\alpha} v\} \forno$$
Each module is an irreducible highest weight module, with highest
weight vectors $\mid \overline 0 >$ and $\mid \overline 1> :=\break
\phi (\epsilon)\mid \overline 0 >$ for $V_{\overline 0}^{\epsilon}$, $V_{\overline 1}^{\epsilon}$,
respectively, in the sense that
$$\tau^{\epsilon}(\F_{i,-j})\mid \alpha >=0 \qquad \hbox{for}\qquad i<j,
  \ \alpha\in{\bf Z}_2\forno$$
$$\tau^{0}(\F_{i,-i})|\alpha > =\cases{-{1\over 2}|\alpha>&if $i>0$\cr &\cr
0&if $i=0$\cr &\cr
{1\over 2}|\alpha >&if $i<0$\cr}\forno$$
$$\tau^{{1\over 2}} (\F_{i,-i}) \mid \overline 0 >=\cases{-{1\over 2}\mid \overline 0 >&
  if $i>0$\cr  &\cr {1\over 2}\mid \overline 0>& if $i< 0$\cr}\qquad
  \tau^{{1\over 2}} (\F_{i,-i}) \mid \overline 1 >=\cases{-{1\over 2}\mid \overline 1 >&
  if $i>{1\over 2}$ or $i=-{1\over 2}$\cr &\cr  {1\over 2}\mid \overline 1 >& if $i< -{1\over 2}$ or $i={1\over 2}$\cr}\forno$$
We denote these representations by $\tau_{\alpha}^{\epsilon}$.

If we try to extend the representation $\tau_{\alpha}^{\epsilon}$ to $\oib$, 
we notice that it is not well defined, e.g. ,
$$\tau^{\epsilon} (\sum_{i>0} \F_{i,-i})\mid \overline 0 >=
 (\sum_{i>0} -{1\over 2})\mid \overline 0 >$$ 
We encounter what physicists call an anomaly.
This problem  can be solved as follows. Define
$$\eqalign{\pi_{\alpha}^{\epsilon}(\F_{ij})&:= \tau_{\alpha}^{\epsilon} (F_{ij})
  \phantom{+ {1\over 2}I}\,\qquad
  \hbox{for } i\not= -j\hbox{ or }i=j=0\cr
  \pi_{\alpha}^{\epsilon} (\F_{i,-i})&:=\tau_{\alpha}^{\epsilon}(\F_{i,-i})+{\textstyle {1\over 2}}I
  \qquad
  \hbox{for }i>0\cr 
  \pi_{\alpha}^{\epsilon} (\F_{i,-i})&:=\tau_{\alpha}^{\epsilon}(\F_{i,-i})-{\textstyle {1\over 2}}I
  \qquad
  \hbox{for }i< 0\cr}\forno$$
Extending $\pi_{\alpha}^{\epsilon}$ by linearity, we get a projective representation
of $\oib$ on $V_{\alpha}^{\epsilon}$. By defining $\pi_{\alpha}^{\epsilon}(c_{\epsilon}):=I$,
we obtain a level $1$  representation of the Lie algebras $\Bi$, $\Di$
on $V_{\alpha}^{\epsilon}$ for $\epsilon=0$, $1\over 2$, respectively.
To be more precise $V^0_\alpha \cong L(\Lambda_0)$ as $\Bi$-module and $V^{1\over 2}_{\overline 0}\cong L(\Lambda_0)$, $V^{1\over 2}_{\overline 1} \cong L(\Lambda_1)$ as $\Di$-modules.


In physics the procedure above, i.e., removing the anomaly, is usually formulated in terms of
a normal ordering prescription on the fermions; one writes
$$\pi^\epsilon (\F_{ij})= :\phi (i)\phi (j):\forno$$
where the {\it normal ordered product}  $:\phi (k)\phi (l):$ is defined by:
$$:\phi (k)\phi (l):\ \buildrel {\rm def} \over =\cases{\phi (k) \phi (l) &if $l< 0$\cr &\cr
0&if $k=l=0$\cr &\cr
                         -\phi (l) \phi (k)&in all other cases\cr}\forno$$


\beginsection \secno .  Vertex operators as normal ordered products of fermionic fields\par
\noindent Here we return to the expressions (3.3.3) for the vertex operators
$\hat A_{jk}^{ab}(z)$. For the representation $\rho$ occurring in this formula we take $\rho:=\pi^{\epsilon (n)}\circ \hat {\iota}$.
So we first embed the algebra $\hat\gl^{(2)}$ in $\Bi$ ($\Di$) for $n$ odd (even) by the homomorphism 
$\hat {\iota}$  and then we represent it on the spin module $V^{\epsilon (n)}$ by the representation $\pi^{\epsilon (n)}$.


Recall from (3.3.5) the field $\hat A_{jk}^{ab}(z)$:
$$\eqalign{2\sqrt{n_an_b}\Bigl\{\hat A_{jk}^{ab}(z)&+\left(\Hw\vert A^{ab}_{jk}\right)_{gl_n}\Bigr\}=\cr
\sum_{m\in{\bf Z}}
\Bigl\{&\sum_{q,t=1}^{[{n\over 2}]}
\sum_{\lambda ,\mu =\pm }z^{-N({m\over 2} +\lambda H_q+\mu H_t)}\alpha_j^{aq}(\lambda)\alpha_k^{bt}(\mu)
\rho\left(e^{im\theta\over 2}F_{qt}((-)^m ;\lambda , \mu)\right)\cr
&+(1-2\epsilon(n))\Bigl(\sum_{q=1}^{[{n\over 2}]}\sum_{\lambda =\pm}z^{-N({m\over 2} +\lambda H_q)}
\alpha_j^{aq}(\lambda)\delta_{bs}\rho\left(e^{im\theta\over 2}F_{q}((-)^m;\lambda )\right)\cr
&\phantom{=(1-2\epsilon(n)) (}-\sum_{t=1}^{[{n\over 2}]}\sum_{\mu =\pm}z^{-N({m\over 2}+\mu H_t)}
(-)^m\delta_{as}\alpha_k^{bt}(\mu)\rho\left(e^{im\theta\over 2}F_{t}((-)^m ; \mu)\right)\cr
&\phantom{=(1-2\epsilon(n)) (}+z^{-N{m\over 2}}\delta_{as}\delta_{bs}\rho\left(e^{im\theta\over 2}F((-)^m)\right)\Bigr)\Bigr\}\cr}\forno$$

With the embedding $\hat \iota$ of $\hat\gl^{(2)}$ in $B_\infty$ (for $n$ odd) or $\Di$  (for $n$ even) (see (4.2.12,14)) and the spin representation $\pi^\epsilon :\F_{ij}\mapsto :\phi(i)\phi(j):$, we can rewrite this  formula as:
$$\eqalign{&2\sqrt{n_an_b}\Bigl\{\hat A_{jk}^{ab}(z)+(H_{[w]}-{\textstyle {1\over 2}}\lambda_{[{n\over 2}]}\vert
A_{jk}^{ab})_{gl_n}\Bigr\}=\cr
&:\Bigl(\sum_{l\in{\bf Z}}\Bigl\{\sum_{q=1}^{[{n\over 2}]}\sum_{\lambda=\pm}
  z^{-N({l\over 2}+\lambda  H_q)}\lambda^l\alpha_j^{aq}(\lambda)\phi(-nl-{\textstyle {\lambda\over 2}}(n-2q+1))+(1-2\epsilon(n))z^{-N{l\over 2}}\delta_{as}c(l)^{-1}\phi(-nl)\Bigr\}\Bigr)\times\cr
&\Bigl(\sum_{m\in{\bf Z}}\Bigl\{\sum_{t=1}^{[{n\over 2}]}\sum_{\mu=\pm}
  z^{-N({m\over 2}+\mu H_t)}(-\mu)^m\alpha_k^{bt}(\mu)\phi(-nm-{\textstyle {\mu\over 2}}(n-2t+1))+(1-2\epsilon(n))z^{-N{m\over 2}}\delta_{bs}c(m)\phi(-nm)\Bigr\}\Bigr):\cr}\forno$$
These expressions motivate the introduction of the  auxiliary fields $\phi_a^\lambda(z)$ by:
$$\phi_a^\lambda(z):=\gamma_a\sum_{l\in{\bf Z}}\Bigl\{\sum_{q=1}^{[{n\over 2}]}\sum_{\nu=\pm}z^{-N({l\over 2}+\nu H_q)}(\lambda\nu)^l\alpha_0^{aq}(\nu)\phi(-nl-{\textstyle {\nu\over 2}}(n-2q+1))+(1-2\epsilon(n))z^{-N{l\over 2}}\delta_{as}c(l)^{-\lambda}\phi(-nl)\Bigr\}\forno$$
where
$$\gamma_a:=\cases{1\over\sqrt 2&if $1\le a\le r$\cr\cr
1&if  $r<a\le s$\cr}\forno$$
(Notice that we have extended the definition (2.6.15) of $\alpha_j^{aq}(\nu)$ to $j=0$ in the obvious way.) The next step is to analyze  the modes of these fields $\phi_a^\lambda(z)$. Using (2.6.15), it is staightforward to derive the following result:
\proclaim Lemma  \prono .
Let
$$N_a:=[{n_1+n_2+\cdots+n_a+1\over 2}]\forno$$
Then
$$\phi_a^\lambda (z)=\cases{\displaystyle{\sum_{k\in {1\over 2}+{\bf Z}} \phi_a^\lambda (k)z^{{N\over n_a}k}}&if $1\le a\le r$\cr\cr
\displaystyle{\sum_{k\in{\bf Z}}\phi_a^\lambda (k)z^{{N\over 2n_a}k}}&if $r<a\le s$\cr}\forno$$
where the fermions $\phi_a^\lambda (k)$ are defined as follows:\hfil\break
\noindent for $1\le a\le r$ ($m\in{\bf Z}$, $1\le q\le {n_a\over 2}$):
$$\phi_a^\lambda ({mn_a+2q-1\over 2}):={\lambda^{m+1}\over\sqrt 2}\phi((m+{\textstyle{1\over 2}})n+N_{a-1}+q-{\textstyle{1\over 2}})+{(-\lambda)^m\over\sqrt 2}\phi((m+{\textstyle{1\over 2}})n-N_a+q-{\textstyle{1\over 2}})\forno$$
for $r<a\le s$ ($m\in {\bf Z}$, $1\le q\le [{n_a\over 2}]$ $\nu=\pm$):
$$\eqalign{\phi_a^\lambda (mn_a&+\nu(2q-1)):=(\nu\lambda)^{m+1}\phi((m+\nu{\textstyle{1\over 2}})n+\nu (N_{a-1}
+q-{\textstyle{1\over 2}}))\cr\cr
\phi_a^\lambda (mn_a&):=\cases{\displaystyle{{\lambda^m\over\sqrt 2}\phi((m-{\textstyle{1\over 2}})n+N_a-{\textstyle{1\over 2}})+{(-\lambda)^m\over\sqrt2}\phi((m+{\textstyle{1\over 2}})n-N_a+{\textstyle{1\over 2}})}&if $a-r$ odd and $a\ne s$\cr\cr
c(m)^{-\lambda}\phi(mn)&if $a-r$ odd and $a=s$\cr\cr
\displaystyle{{i\lambda^m\over\sqrt 2}\phi((m-{\textstyle{1\over 2}})n+N_{a-1}-{\textstyle{1\over 2}})-{i(-\lambda)^m\over\sqrt2}\phi((m+{\textstyle{1\over 2}})n-N_{a-1}+{\textstyle{1\over 2}})}&if $a-r$  even\cr}\cr}\forno$$

We can now formulate the following theorem, which provides us with a fermionic realization of $\hat\gl^{(2)}$ on the spin module $V^{\epsilon (n)}$.
\proclaim Theorem  \prono .
Let $\omega=e^{2\pi i\over N}$ and  let  $\phi_a^\lambda(z)$ be defined by (4.4.6-8). Then:
$$\hat A_{jk}^{ab}(z)+(H_{[w]}-{\textstyle{1\over 2}}\lambda_{[{n\over 2}]}\vert
A_{jk}^{ab})_{gl_n}=C_aC_b
:\phi_a^{(-)^j}(\omega^{-j}z)\phi_b^{(-)^{k+1}}(\omega^{-k}z):
\forno$$
where
$$C_a:=\cases{1\over\sqrt{ n_a}&if $1\le a\le r$\cr\cr
              1\over\sqrt{ 2n_a}&if $r<a\le s$\cr}\forno$$

Similar calculations for $\underline g=\sotw$ lead to the following fermionic realization of $\hat{so}_{2n}({\bf C})^{(2)}$ on the spin module $V^0$.
\proclaim  Theorem  \prono .
Let $\omega:=e^{2\pi i/N}$, with $N$ the order of $\tw$,
$$M_a:=n_1+n_2+\cdots+n_a\forno$$
and
$$\eqalign{\nu_(p):&={1-i(-)^p\over \sqrt{2}}\cr
\sigma_a^\delta(p):&={\nu_(p)\over \nu(2\delta n_a)}=\cases{i(-)^{2\delta n_a}&if\ $p+2\delta n_a\in 2{\bf Z}+1$\cr &\cr 1&if\ $p+2\delta n_a\in 2{\bf Z}$\cr}\cr}\forno$$
Define
$$\phi_a^\lambda (z):=\cases{\displaystyle{\sum_{k\in {1\over 2}+{\bf Z}} \phi_a^\lambda (k)z^{{N\over n_a}k}}&if $1\le a\le r-\delta_{rs}$ or $a=s\ne r$\cr\cr
\displaystyle{\sum_{k\in{\bf Z}}\phi_a^\lambda (k)z^{{N\over 2n_a}k}}&if $r-\delta_{rs}<a\le s-1+\delta_{rs}$\cr}\forno$$
where the fermions $\phi_a^\lambda (k)$ are defined as follows:\hfil\break
\noindent for  $1\le a\le r-\delta_{rs}$ or ($a=s$ if  $r\ne s$) $m\in{\bf Z}$, $1\le q\le n_a$:
$$\phi_a^\lambda (mn_a+\lambda(q-{\textstyle{1\over 2}}):=\phi(2n(m+\lambda(1-\delta)) +\lambda(M_{a-1}+q-n-\delta_{as}))\forno$$
for $1\le a\le {s-r\over 2}$, $m\in \delta+{\bf Z}$:
$$\eqalign{&\phi_{r+2a-1}^\lambda(2n_{r+2a-1}m+p):=\cr&\lambda^{p+2\delta n_{r+2a-1}}\sigma_{r+2a-1}^\delta(p)
\cases{\phi(2nm+n-M_{r+2a-1}+p-1+\delta_{r+2a,s})&if\ $1\le p
<n_{r+2a-1}$\cr\cr
\phi(2nm-n +M_{r+2a-1}+p+1-\delta_{r+2a,s})
&if\
$-n_{r+2a-1}< p <0$\cr\cr
{1\over \sqrt{2}}\Bigl\{\phi(2nm+n+M_{r+2a-2}+1)+\cr\qquad\qquad\phi(2nm+n-M_{r+2a-2}-1)\Bigr\}&if\ $p=n_{r+2a-1}$ and $r+2a\ne s$\cr\cr
{1\over \sqrt{2}}\Bigl\{\phi(2nm-n+M_{r+2a})+\cr\qquad\qquad\phi(2nm+n-M_{r+2a})\Bigr\}
&if\ $p=0$ and $r+2a\ne s$\cr\cr
\phi(2nm+n)&if $p=n_{s-1}$ and $r+2a=s$\cr\cr
\phi(2nm)&$p=0$ and $r+2a=s$}
\cr}\forno$$
for $1-\delta_{rs}\le a\le {s-r\over 2}+\delta_{rs}-1$, $m\in \delta+{\bf Z}$:
$$\eqalign{&\phi_{r+2a}^\lambda(2n_{r+2a}m+p):=\cr&\lambda^{p+2\delta n_{r+2a}}\sigma_{r+2a}^\delta(p)
\cases{
\phi(2nm+n-M_{r+2a}+p)&if\ $1-\delta_{a0}\le p
<n_{r+2a}$\cr\cr
\phi(2nm-n+M_{r+2a}+p)&if\
$-n_{r+2a}< p <0$\cr\cr
{i\over \sqrt{2}}
\Bigl\{\phi(2nm+n+M_{r+2a-2}+1)\cr\qquad\qquad-\phi(2nm+n-M_{r+2a-2}-1)\Bigr\}&if\ $p=n_{r+2a}$ and $a\ne 0$\cr\cr
{i\over \sqrt{2}}\Bigl\{\phi(2nm-n+M_{r+2a})\cr\qquad\qquad-\phi(2nm+n-M_{r+2a})\Bigr\}&if\ $p=0$ and $a\ne 0$\cr\cr
\phi(2nm+n)&if $p=n_r$ and $a=0$\cr}\cr}\forno$$
 Then:
$$\eqalign{\hat D^{ab}_{jk}(\lambda,\mu)(z)+(H_{[w]}^\delta-(1-2\delta)\lambda_{n-1}\vert D^{ab}_{jk}(\lambda,\mu))&=C^a_j(\lambda)C^b_k(\mu):\phi_a^\lambda(\omega^{-j}z)\phi_b^\mu(\omega^{-k}z):\cr
}\forno$$
where
$$C^a_j(\lambda):=\cases{{1\over\sqrt{n_a}}\omega^{-{\lambda Nj\over 2n_a}}&if\ $1\le a\le r-\delta_{rs}$ or $a=s\ne r$\cr &\cr (-)^{2\delta j}{\nu(2\delta n_a)^\lambda\over\sqrt{2n_a}}&if\ $r-\delta_{rs}<a\le s-1+\delta_{rs}$\cr}\forno$$


Let us now study the new fermions $\phi_a^\lambda(k)$ in somewhat
more detail. For notational purposes, it will be convenient to introduce the following subsets of $\{1,2,\cdots,s\}$:
$$\eqalign{CF&:=\cases{\{1,2,\cdots,r\}&if $\underline g=\gl$\cr\cr
\{1,2,\cdots,r,s\}&if $\underline g=\sotw$ and $r\ne s$\cr\cr
\{1,2,\cdots,r-1\}\qquad\quad\ &if $\underline g=\sotw$ and $r= s$\cr}\cr\cr
NF&:=\cases{\{r+1,r+2,\cdots,s\}&if $\underline g=\gl$\cr\cr
\{r+1,r+2,\cdots,s-1\}&if $\underline g=\sotw$ and $r\ne s$\cr\cr
\{s\}&if $\underline g=\sotw$ and $r= s$\cr}\cr}\forno$$

First we study the $\phi_a^\lambda(k)$ for $a\in CF$; it is clear from the definitions (4.4.7) and (4.4.14) that $\phi_a^+(k)$ and $\phi_a^-(k)$ are independent. The anticommutation relations are also easily read off from (4.4.7) and (4.4.14). One has:
$$\{\phi_a^\lambda(k),\phi_b^\mu(l)\}=\delta_{ab}\delta_{\lambda,-\mu}\delta_{k,-l}\qquad 1\le a,b\le r, k,l\in{\textstyle {1\over 2}}+{\bf Z}, \lambda,\mu=\pm\forno$$

\noindent This type of fermions is usually called {\it charged free} fermions in physics. The charge conjugation operation will be discussed in the next section.

For $a\in NF$ the situation is quite different. It is immediate from (4.4.8) and (4.4.15-16) that
$$\phi_a^{-\lambda}(k)=(-)^k\phi_a^\lambda(k)\forno$$
hence there is in fact only one {\it neutral free} fermion. The relevant anticommutation relations are:
$$\{\phi_a^+(k),\phi_b^+(l)\}=(-)^k\delta_{ab}\delta_{k,-l}\qquad r< a,b\le s, k,l\in{\bf Z}\forno$$


\noindent All other anticommutators, in particular $\{\phi_a^\lambda(k),\phi_b^\mu(l)\}$ for $a\in CF,\ b\in NF$  are zero.

Recall that, in terms of the fermions $\phi(k)$, $k\in \epsilon+{\bf Z}$ the module $V^\epsilon$ is characterized as the unique irreducible $Cl^\epsilon$-module generated by $\vac$ satisfying $\phi(k)\vac =0$ for $k<0$. In terms of the new fermions $\phi_a^\lambda (k)$ this condition becomes:
$$\eqalign{\phi_a^\lambda(k)&\vac=0\qquad\hbox{for }k<0,\ 1\le a\le s\cr
\{\phi_{r+2a-1}^+(0)-i\phi_{r+2a}^+ (0)\}&\vac=0\qquad\hbox{for }1\le a\le [{s-r\over 2}]\hbox{ and } \underline g=\gl\cr
\{\phi_{r+2a-1}^+(0)+(4\delta-1)i\phi_{r+2a}^+ (0)\}&\vac=0\qquad\hbox{for }1\le a\le {s-r-2\over2}\hbox{ and }\underline g=so_{2n}({\bf C})
\cr}\forno$$

\beginsection \secno .  Hermitian structure and normal ordering
\par
It is well known that the spin module $V^\epsilon$ can be equipped
with an inner product, giving it the structure of a pre-Hilbert space;

\proclaim Theorem  \prono.
There exists a unique positive definite Hermitian form $(\ ,\
):V^\epsilon\times V^\epsilon\rightarrow{\bf C}$ such that:
$$\eqalign{&a)\quad\phi(j)^\dagger=\phi(-j)\cr&b)\quad(\vac,\vac)=1\cr}
\forno$$
\par
Instead of $(\ ,\ )$ we will also use Dirac's `bra-ket' notation;
$$(v,w)\equiv<v\mid w>\forno$$
We will leave out the symbols $\pi$ and $\pi\circ\hat {\iota}$ when
a representation is understood.

With the inner product one formulates the following lemma:
\proclaim Lemma  \prono.
$$\eqalign{
  &a)\quad \phi_a^+(j)^\dagger=\phi_a^-(-j)\cr
  &b)\quad \F_{ij}^\dagger=\F_{-j,-i}\cr
  &c)\quad\left(e^{il\theta\over 2}F_{jk}((-)^l;\lambda ,\mu)\right)^\dagger
 =e^{-{il\theta\over 2}}F_{kj}((-)^l;-\mu ,-\lambda)=-(-)^le^{-{il\theta\over 2}}F_{jk}((-)^l;-\lambda ,-\mu)\cr
&d)\quad \left(e^{il\theta\over 2}F_{j}((-)^l;\lambda)\right)^\dagger=
-(-)^le^{-{il\theta\over 2}}F_{j}((-)^l;-\lambda)\cr
   &e)\quad\left(e^{il\theta\over 2}F((-)^l)\right)^\dagger=
e^{-{il\theta\over 2}}F((-)^l)\cr
&f)\quad \left(e^{i(l+\delta_{jn}{1-\lambda\over 2}+\delta_{kn}{1-\mu\over 2})\theta}G_{jk}(\lambda,\mu)\right)^\dagger=e^{-i(l+\delta_{jn}{1-\lambda\over 2}+\delta_{kn}{1-\mu\over 2})\theta}G_{kj}((2\delta_{kn}-1)\mu,(2\delta_{jn}-1)\lambda)\cr}\forno$$
\par
\proof{A straightforward calculation using (4.4.7-8), (4.4.14-16), (4.3.13) and lemma 4.2.1.}



Notice that relation (4.5.3a) and the definitions (4.4.6) and (4.4.13) imply
$$\phi_a^+(z)^\dagger=\phi_a^-(z)\forno$$


We use the Hermitian structure to define normal ordering for arbitrary
operators on the spin module $V^\epsilon$. For this we note that the
definition (4.3.14) can be rewritten as:
$$:\phi(i)\phi(j): = \phi(i)\phi(j)-<\overline 0\mid \phi(i)\phi(j)\vac
  \forno$$
The notation
$<\overline 0\mid A\vac :=(\vac ,A\vac)$ in this formula is called
the vacuum expectation value of the operator $A$.

Next we extend the normal
ordering prescription to arbitrary operators on the spin module;
\proclaim Definition  \prono.
Let $A$ and $B$ be operators on the spin module $V$, then
the normal ordered product of these operators is defined by:
$$:AB:{\buildrel {\rm def}\over =}AB-<\overline 0\mid AB\vac\forno$$
\par
Of course this definition is different from the usual one, because we
subtract only a $c$-number from the ordinary product 
and no operator structures.
Our motivation for this definition is that it produces the right ordering
prescription for a pair of bosonic oscillators as we will show later on.







\baselineskip=22pt plus.1pt



 

\count30=4                      %chapterno = 4
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}

\def\ug{\underline g}
\def\sotw{so_{2n}({\bf C})}
\def\so{so_{2n}({\bf C})}
\def\SO{SO_{n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}
\def\dim{{\rm dim}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat s_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{n}({\bf C})}
\def\oi{o(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o(\infty )}}
\def\Ai{A_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{n}({\bf C})}
\def\glh{\hat{gl}_{n}({\bf C})}
\def\sot{\tilde{so}_{n}({\bf C})}
\def\soh{\hat{so}_{n}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}_{[w]}}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal F}}
\def\D{{\cal D}}
\def\:{{\scriptstyle{\circ\atop \circ}}}


\beginsection \chapno .  Oscillator and Virasoro algebras
\par
\beginsection \secno .  Introduction
\par
In   this section  we will introduce bosonic and fermionic oscillator algebras. We will use these algebras to construct representations of the
Virasoro algebra. In section 5.4 we will consider the so-called twisted Sugawara construction of the Virasoro algebra.      
These three different constructions of the Virasoro algebra are in fact closely related. We will discuss this relation in section 5.5.

\beginsection \secno . Oscillator algebras
\par 
It is well known how to obtain oscillator algebras
in terms of charged or neutral free fermions  (see e.g.
[KR], [JM] or [GO]). One simply has to take 
the normal ordered product of the fields $\phi^+_a(z)$ 
with their Hermitian conjugate fields $\phi_a^-(z)$.
Using (4.4.6) and (4.4.13) we define
$$\eqalign{\hat\alpha_a(z)=\sum_{k\in{\bf Z}} z^{-{N\over n_a}k}
           \hat\alpha_a( {N\over n_a}k)
           {\buildrel \rm def \over =}
           :\phi_a^+(z)\phi_a^-(z):&\qquad\hbox{if }a\in CF\cr
           \hat\alpha_a(z)=\sum_{k\in {1\over 2}+{\bf Z}}
           z^{-{N\over n_a}k}\hat\alpha_a({N\over n_a}k) 
           {\buildrel \rm def \over =}
           {1\over 2}:\phi_a^+(z)\phi_a^-(z):
           &\qquad\hbox{if }a\in NF\cr}\forno$$
We rewrite these formulas in terms of the modes
$$\hat \alpha_a({N\over n_a}k) 
  =\cases{\displaystyle \sum_{j\in {1\over 2}+{\bf Z} }
           :\phi_a^+(j)\phi_a^-(-j-k):&if $a\in CF$\cr
           \displaystyle {1\over 2}\sum_{j\in {\bf Z}}
           :\phi_a^+(j)\phi_a^-(-j-2k):&if $a\in NF$\cr}\forno$$
 

Using (5.2.1) and the anticommutation relations (4.4.20-22), 
one easily checks the following commutation relations:
$$[\hat\alpha_a( {N\over n_a}k) ,
  \hat\alpha_b( {N\over n_b}l) ]
  =k\delta_{ab}\delta_{k,-l}\forno$$
It is obvious that the set of all such oscillators $\hat\alpha({N\over n_a}k)$ ($1\le a\le s$) together with the central element $c_\epsilon$ form a subalgebra of $B_\infty$ or $D_\infty$; we denote this oscillator (sub)algebra by ${\cal O}_{[w]}$.
Of course we want to know the pull-back of these elements in $\hat{\underline g}^{(2)}$.
 Using (4.4.9) and (4.4.17), we obtain:
$$\hat\alpha_a(z)=\cases{n_a\hat A_{n_an_a}^{aa}(z)&if $\ug=\gl ;\ 1\le a\le s$\cr\cr
n_a\hat D_{n_an_a}^{aa}(+,-)(z)&if $\ug=\sotw ;\ a\in CF$\cr\cr
-n_a\hat D_{n_an_a}^{aa}(+,-)(z)&if $\ug=\sotw ;\ a\in NF$\cr}
  \forno$$
Hence the oscillator algebra ${\cal O}_{[w]}$ can be seen as a subalgebra of $\hat{\underline g}^{(2)}$. If $a\in CF$, there is a   zero-mode  $\hat\alpha_a(0)$; as we will see later on 
this element  is some kind of charge counting operator. If one removes all zero-modes, one obtains the Heisenberg subalgebra $\hat{\underline s}_{[w]}$, introduced in section 1. 

Finally, we give the commutation relations
of the oscillators $\a ak$ with the fermionic fields.

\proclaim Lemma  \prono. 
$$[\a ak ,\phi_b^\lambda(z)]
  =\lambda\delta_{ab}z^{{N\over n_a}k}\phi_b^\lambda(z)\forno$$
\par
\proof{An easy calculation (see also [KrL91c]).}


\beginsection \secno . Bosonic and fermionic oscillator representations of the Virasoro algebra
\par
Recall that the Virasoro algebra is the universal one
dimensional central extension of the conformal algebra. It has a basis
consisting of elements $d_k, k\in{\bf Z}$ and a central element $c_{vir}$,
which satisfy the commutation relations
$$[d_k,d_l]=(k-l)d_{k+l}+{{ 1\over
  12}}\delta_{k+l,0}\left(k^3-k\right)c_{vir}\forno$$
The following proposition was proved in [KrL91b].
\proclaim Proposition  \prono .
The operators  $$\eqalign{B_k:=&\sum_{a\in CF}{1\over 2n_a}
\sum_{j\in {\bf Z}}:\hat\alpha_a(-{N\over n_a}j)\a a{(j+n_ak)}:+{1\over 24}\delta_{k0} {n_a^2-1\over n_a}I\cr
&+\sum_{a\in NF}{1\over 2n_a}
\sum_{j\in {1\over 2}+{\bf Z}}:\hat\alpha_a(-{N\over n_a}j)\a a{(j+n_ak)}:+\delta_{k0}{1\over 16n_a}I+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\cr}\forno$$
and
$$\eqalign{F_k:=&\sum_{a\in CF}\sum_{j\in {1\over 2}+{\bf Z}}
  {1\over n_a}\left(j+{n_ak\over 2}\right)
  :\phi_a^+(j)\phi_a^-(-j-n_ak):+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\cr
  &+{1\over 2}\sum_{a\in NF}\sum_{j\in {\bf Z}}{j\over 2n_a}
  :\phi_a^+(j)\phi_a^-(-j-2n_ak):+\delta_{k0}{1\over 16n_a}I+{1\over 24}\delta_{k0}{n_a^2-1\over n_a}I\cr}       \forno$$
define  a $c_{vir}=n$ bosonic, respectively fermionic oscillator representation of the Virasoro algebra on $V^\epsilon$.
Set $m_a:=n_a$ if $a\in CF$ and $m_a:=2n_a$ if $a\in NF$, then
$$[F_k,\phi_a^\lambda(l)]
  ={1\over m_a}\left(l-{m_ak\over 2}\right)\phi_a^\lambda(l-m_ak)\forno$$
Moreover
$$\eqalign{[B_k,\a al]&=-{l\over n_a}\a a{(l+n_ak)}\cr
[F_k,\hat\alpha_a({N\over n_a}l)]
  &=-{l\over n_a}\hat\alpha_a({N\over
  n_a}(l+n_ak))\cr}\forno$$

\beginsection \secno .  The twisted Sugawara construction of the Virasoro algebra\par
In this section we will present a generalization of the Sugawara construction.
This construction uses a twisted affine Lie algebra and yields a unitary representation of the Virasoro algebra.

Let $\underline g$ be a simple or abelian Lie algebra with non-degenerate symmetric invariant bilinear form $(\cdot|\cdot )$. Consider an involution $\sigma$ on $\underline g$. Then $\underline g =\underline g_0 \oplus \underline g_1$, where
$$\underline g_\alpha =\{x\in\underline g |\sigma (x)=(-)^\alpha x\}\forno$$
Let $d_\alpha=\dim \,\underline g_a$ and  choose a basis $\{u_i^\alpha |i=1,\cdots, d_\alpha\}$ in $\underline g_\alpha$. Let $\{u_\alpha^i |i=1,\cdots, d_\alpha\}$ be the dual basis, i.e.,
$$(u_i^\alpha |u_\beta^j)=\delta_{\alpha \beta}\delta_{ij}\forno$$
Now consider the corresponding twisted affine Lie algebra $\hat{L}(\underline g,\sigma)$ (see also section 3.2). We then have:
\proclaim Proposition \prono .
Let $\ug$ be simple or abelian, $\sigma$ an involution on $\ug$ and  $g$  the dual Coxeter number of $\underline g$. Consider a representation $V$ of $\hat{L}(\underline g ,\sigma)$, such that
$$\eqalign{a)&\quad c=kI \hbox{ with }k\ne -g,\cr
           b)&\quad \hbox{for every }v\in V\ (e^{il\theta}x) v=0\hbox{ for all }x\in\underline g\hbox{ and }l>>0.\cr}$$
Then the operators
$$S_n:={1\over 2(k+g)}\sum_{\alpha=0}^1\sum_{l=1}^{d_\alpha}\sum_{j\in{\bf Z}}
\:(e^{-i(j+{\alpha\over 2})\theta}u_l^\alpha )(e^{i(j+n+{\alpha\over 2})\theta}u^l_\alpha )\:+\delta_{n0}{kd_1\over 16(k+g)}I\forno$$
where the normal ordering $\:\qquad\:$ is defined by:
$$\:(e^{il\theta}x)(e^{im\theta}y)\:=\cases{(e^{il\theta}x)(e^{im\theta}y)&if $m\ge l$\cr\cr
(e^{im\theta}y)(e^{il\theta}x)&if $m<l$\cr}\forno$$
provide a $c_{vir}={k \dim \underline g\over k+g}$ representation of the Virasoro algebra in $V$, satisfying
$$[S_n,e^{il\theta}x]=-le^{i(l+n)\theta}x\forno$$
Moreover, if $V$ is a unitary representation for $\hat{\underline g}^{(2)}$, then we also obtain a unitary representation of the Virasoro algebra.

\noindent The proof of this proposition is completely similar to the proof in the non-twisted case (see e.g. [KR]). These operators were defined  in a slightly different form and in a more general setting, viz. $\sigma$ an arbitrary finite order automorphism, by Kac and Peterson in [KP84].

We will now translate these operators to our situation, viz $\sigma =\nu_{\underline g}$ and $V=V^\epsilon$.


\proclaim Proposition  \prono .
Let $\{u_i^0|1\le i\le {n^2-n\over 2}\}$ ($\{u_i^0|1\le i\le 2n^2-3n+1\}$)
be a basis of $\gl_{\overline 0}$ (resp. $so_{2n}({\bf C})_{\overline 0}$), $\{u_i^1|1\le i\le {n^2+n-2\over 2}\}$ (,$\{u_i^1|1\le i\le 2n-1\}$) be a basis of $sl_n({\bf C})\cap \gl_{\overline 1}$ (resp. $so_{2n}({\bf C})_{\overline 1}$) and let  $\{u_0^i|1\le i\le {n^2-n\over 2}\}$, ($\{u_0^i|1\le i\le 2n^2-3n+1\}$,) $\{u_1^i|1\le i\le {n^2+n-2\over 2}\}$, ( $\{u_1^i|1\le i\le 2n-1\}$)  be the corresponding dual basis with respect to $(\ \mid \ )_{gl_n}$ (resp. $(\ |\ )_{so_{2n}}$). Define 
$u_{n^2+n\over 2}^1=u_1^{n^2+n\over 2}=\sqrt{{n+1\over n}}I_n$ for $\underline g=\gl$.
let $d_0={n^2-n\over 2}$ ($d_0=2n^2-3n+1$) and $d_1={n^2+n\over 2}$ ($d_1=2n-1$) for $\underline g=\gl$  (resp. $\so$), define the operators $ L_k:V^{\epsilon(n)}\rightarrow V^{\epsilon(n)}$  (resp. $L_k:V^0\to V^0$) by:
$$\eqalign{
&L_k:=
\Bigl\{{\textstyle {1\over 2n+2}}\sum_{\alpha=0}^1\sum_{l=1}^{d_\alpha}\sum_{j\in{\bf Z}}
  :\rho(e^{-i(j+{\alpha\over 2})\theta}u_l^\alpha)\rho(
  e^{i(j+k+{\alpha\over 2})\theta}{u^l_\alpha}):\Bigr\}-\rho(e^{ik\theta}\Hw )
+\delta_{k0}\Bigl\{{\textstyle{n\over 16}}+{\textstyle{1\over 2}}\mid \Hw \mid_{gl_n}^2\Bigr\}I\cr\cr
&L_k:=\cr
&\Bigl\{{\textstyle{ 1\over 4n-2}}\sum_{\alpha=0}^1\sum_{l=1}^{d_\alpha}\sum_{j\in{\bf Z}}
  :\rho(e^{-i(j+{\alpha\over 2})\theta}u_l^\alpha)\rho(
  e^{i(j+k+{\alpha\over 2})\theta}{u^l_\alpha}):\Bigr\}-\rho(e^{ik\theta}\Hw^\delta )
+\delta_{k0}\Bigl\{{\textstyle{1+(2-4\delta)(n-1)\over 16}}+{\textstyle{1\over 2}}\mid \Hw^\delta \mid_{so_{2n}}^2\Bigr\}I\cr}\forno$$
where $\rho=\pi^{\epsilon(n)}\circ \hat\iota$ (resp. $\rho=\pi^0\circ \hat\iota_\delta$). Then the assignment $d_k\mapsto L_k$ is a representation of the
Virasoro algebra with $c_{vir}\mapsto nI$. Moreover, we have:
$$[L_k,\hat x(l)]=-{l\over N}\hat x(l+Nk)\forno$$
In particular:
$$[L_k,\hat \alpha_a\left({N\over n_a}l\right)]=-{l\over n_a}\hat \alpha_a\left({N\over n_a}(l+n_ak)\right)\forno$$

\proof{Notice that
$$\sum_{\alpha=0}^1\sum_{l=1}^{d_\alpha}  \:e^{-i(j+{\alpha\over 2})\theta}u_l^\alpha
  e^{i(j+k+{\alpha\over 2})\theta}{u^l_\alpha}\:=\sum_{\alpha=0}^1\sum_{l=1}^{d_\alpha}\Bigl\{  :e^{-i(j+{\alpha\over 2})\theta}u_l^\alpha
  e^{i(j+k+{\alpha\over 2})\theta}{u^l_\alpha}:+\delta_{j0}\delta_{k0}\delta_{\alpha 0}<\overline 0 |u_l^0 u_0^l |\overline 0 >\Bigr \}\forno$$
One easily verifies that
$$\sum_{l=1}^{d_0}<\overline 0 |u_l^0 u_0^l |\overline 0 >=\cases{(n^2-n)/16&for $\ug=\gl$\cr\cr
                      (1-2\delta)(2n^2-3n+1)/4&for $\ug=\so$}\forno$$
The proposition now follows from proposition 5.4.1 (using the decomposition $\gl=sl_n({\bf C})\oplus {\bf C}I_n$)}

\beginsection \secno . Relation between $B_k, F_k$ and $L_k$
\par
\proclaim Theorem \prono .
$$B_k=F_k=L_k\forno$$

\proof{To prove the first relation, we consider the difference $\Delta_k:=F_k-B_k$. Using (5.3.5), we find $[\Delta_k,\hat \alpha({N\over n_a}l)]=0$ and hence $[\Delta_k,B_l]=0$. Consequently, the $\Delta_k$'s span a $c_{vir}=0$ representation of the Virasoro algebra (i.e. a conformal algebra):
$$\eqalign{[\Delta_k,\Delta_l]&=[\Delta_k,F_l]\cr
&=[F_k,F_l]-[B_k,F_l]\cr&=[F_k,F_l]-[B_k,F_l-B_l]-[B_k,B_l]\cr&=(k-l)\Delta_{k+l}\cr}\forno$$

\noindent where we have used the fact that the central charges of the bosonic and fermionic constructions of the Virasoro algebra coincide. It is well known (see [KW88], [GO]) that the only unitary representation of the Virasoro algebra with this propery
 is the trivial one, meaning that $\Delta_k\equiv 0$.

Next, we set $\delta_k:=L_k-B_k$, and, following the same proof as for $\Delta_k$, we conclude thet $\delta_k\equiv 0$.}  

\baselineskip=22pt plus.1pt

\count30=5                      %chapterno = 5

\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}



\def\so{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat {\underline s}_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{2n}({\bf C})}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Ai{A_{\infty}}
\def\Bi{B_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{n}({\bf C})}
\def\glh{\hat{gl}_{n}({\bf C})}
\def\sot{\tilde{so}_{2n}({\bf C})}
\def\soh{\hat{so}_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}_{[w]}} 
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal L}}
\def\D{{\cal D}}

\beginsection \chapno .  Vertex operators
\par
\beginsection \secno .  Introduction
\par
In this section we will show that the fermionic fields $\phi_a^+(z)$
and $\phi_a^-(z)$ can be expressed in terms of so called 
fermionic translation operators $Q_a:V^\epsilon_\alpha\to V^\epsilon_{\alpha +\overline 1}$ for 
$a\in CF$, 
fermionic chirality operators $Q_a:V^\epsilon_\alpha\to V^\epsilon_{\alpha+\overline 1}$
for $a \in NF$
and the bosonic oscillators $\a ak$. This leads to bosonic expressions for the
vertex operators $\hat A_{jk}^{ab}(z)$, $\hat D_{jk}^{ab}(\lambda,\mu)(z)$.
\beginsection \secno . 
Expressions for $\phi_a^+(z)$ and $\phi_a^-(z)$ \par 
Using the commutation relations of the fermions (fermionic fields) with the oscillator algebra and the Virasoro algebra, which can be found in the previous section, one can prove the following theorem.
\proclaim Theorem  \prono .
Let $a\in CF$, $b\in NF$ and $1\le d,e\le s$, then
$$\eqalign{\phi_a^\lambda(z)=&Q_a^\lambda z^{{N\over
n_a}\left(\lambda\hat\alpha_a(0)
           +{1\over 2}\right)}
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_a}k}\a ak\right)
           \cr
           \phi_b^\lambda(z)=&Q_b
           {\rm exp}\left(-\lambda\sum_{k<0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           {\rm exp}\,\left(-\lambda\sum_{k>0}{1\over k}
           z^{-{N\over n_b}k}\a bk\right)
           \cr}
           \forno$$
where the operators $Q_d :V^\epsilon\rightarrow V^\epsilon$ are defined by:
$$\eqalign{Q_d\phi_e^{\lambda}(k)&=-\phi_e^\lambda(k)Q_d\quad\hbox{if } 
           e\ne d\cr
           Q_a\phi_a^\lambda(k)&=\phi_a^\lambda(k+\lambda)Q_a\cr						    Q_b\phi_b^\lambda(k)&=\phi_b^\lambda(k)Q_b\cr
 										           }
  \forno$$
and
$$Q_d\vac=
           \cases{\phi_d^+({1\over2})\vac
                  &if $d\in CF$\cr\cr
                  \phi_d^+(0)\vac
                  &if $d\in NF$\cr}
  \forno$$
Moreover,
$$\eqalign{Q_a^\dagger &=Q_a^{-1}\cr
           Q_b^\dagger &=Q_b ={1\over 2}Q_b^{-1}\cr}
  \forno$$
and 
$$[\a dk,Q_e]=\delta_{de}\delta_{k0}Q_e\forno$$
The following anticommutation relations are satisfied:
$$\{Q_d,Q_e\}=0
  \quad\hbox{for } d\ne e\forno$$
\par

\proof{The proof of this theorem can be found in [KrL91a-b].} 
        


\beginsection \secno .  Vertex operators for $\glh^{(2)}$ and $\soh^{(2)}$

\noindent Consider the normal ordered operator products $:\phi_a^\lambda(u)
\phi_b^\mu(v):$ in two formal variables $u$
and $v$. With the normal ordering definition 4.5.5 these products can be
rewritten as

$$:\phi_a^\lambda(u)\phi_b^\mu(v):=\phi_a^\lambda(u)\phi_b^\mu(v)
  -<\overline 0\mid\phi_a^\lambda (u)\phi_b^\mu (v)\vac
\forno$$

\noindent The contractions $<\overline 0\mid\phi_a^\lambda (u)\phi_b^\mu
(v)\vac$ are formal
series in $u$ and $v$, which can be calculated using (4.4.23). The ordinary products
$\phi_a^\lambda(u)\phi_b^\mu(v)$ can be
rewritten by means of theorem 6.2.1.

\proclaim Proposition  \prono .
Define
$$\eqalign{E_a^{(\pm)}(z)&:={\rm exp}\,\left(-\sum_{\pm k>0}{1\over k}
  z^{-{N\over n_a}k}\a ak\right)\cr\cr
F_{a,\lambda}(z)&:=\cases{Q_a^\lambda 
  z^{{N\over n_a}\left(\lambda\hat\alpha_a(0)+{1\over 2}\right)}&if $a\in CF$\cr\cr Q_a&if $a\in NF$\cr}\cr\cr
C_{ab}(z)&:=\cases{1-\delta_{ab}z^{{N\over n_a}}&if $a\in CF$\cr\cr
{1-\delta_{ab}z^{N\over 2n_a}\over 1+\delta_{ab}z^
{N\over 2n_a}}&if $a\in NF$\cr}\cr
 }\forno$$
Then
$$\phi_a^\lambda(u)\phi_b^\mu(v)=C_{ab}(v/u)^{\lambda\mu}F_{a,\lambda}(u)
F_{b,\mu}(v)E_a^{(-)}(u)^\lambda E_b^{(-)}(v)^\mu E_a^{(+)}(u)^\lambda
E_b^{(+)}(v)^\mu
  \forno$$

\proof{An easy formal calculation shows that
$E_a^{(+)}(u)E_b^{(-)}(v)=C_{ab}(v/u)E_b^{(-)}(v)E_a^{(+)}(u)$. 
The proposition follows immediately from this result and theorem 6.2.1.}

In order to find a bosonic expression for the vertex operators  $\hat
A_{jk}^{ab}(z)$ and $\hat
D_{jk}^{ab}(\lambda,\mu)(z)$, we would like to replace the formal variables $u$
and $v$ in (6.3.1) by $\omega^{-j}z$ and $\omega^{-k}z$, respectively. This has
to be done with some care  (see [KrL91c] for details). The result is:
\proclaim Theorem \prono .
$$\eqalign{\hat
A_{jk}^{ab}&(z)=
C_{ab}(\omega^{j-k})^{(-1)^{j+k+1}}C_aC_b\times\cr
 &\quad F_{a,(-)^j}
(\omega^{-j}z) F_{b,(-)^{k+1}}(\omega^{-k}z)E_a^{(-)}(\omega^{-j}z)^{(-1)^j}
E_b^{(-)}(\omega^{-k}z) ^{(-1)^{k+1}} E_a^{(+)} (\omega^{-j}z)^{(-1)^j}
E_b^{(+)}(\omega^{-k}z)^{(-1)^{k+1}}\cr
\hat D_{jk}^{ab}&(\lambda,\mu)(z)=\cr
&C_{ab}(\omega^{j-k})^{\lambda\mu}C_j^a(\lambda)C_k^b (\mu)F_{a,\lambda}
(\omega^{-j}z) F_{b,\mu}(\omega^{-k}z)E_a^{(-)}(\omega^{-j}z)^\lambda
E_b^{(-)}(\omega^{-k}z) ^\mu E_a^{(+)} (\omega^{-j}z)^\lambda
E_b^{(+)}(\omega^{-k}z)^\mu\cr}\forno$$ 

\noindent In this formula, $C_{ab}(z)$ is of course to be interpreted as a
genuine meromorphic function in $z$. 

We obtain the following important consequence of these results.
\proclaim Theorem  \prono .
{\bf (a).\ }The spin module $V_\alpha^{\epsilon(n)}$ is an irreducible $\hat gl_n({\bf C})^{(2)}$-module, with highest
weight $\Lambda_0$ if $\alpha=\overline 0$ or ($\alpha=\overline 1$ and $n$ is odd) and  $\Lambda_1$ if $\alpha=\overline 1$ and $n$ is even.\hfill\break 
{\bf (b).\ }The spin module $V_\alpha^0$ is an irreducible $\hat {so}_{2n}({\bf C})^{(2)}$-module, with highest
weights $\Lambda_0$ if $\delta={1\over 2}$ and $\Lambda_{n-1}$ if $\delta=0$.
\par
\proof{We only have to show that $V_\alpha^\epsilon$ remains irreducible for one
realization; we choose 
$[w]= [n]$, $[{n-1}\ 1]$, $[n]$ for $\hat{\underline{g}}^{(2)}=\hat gl_n({\bf C})^{(2)}$ and $n$ odd, $\hat{\underline g}^{(2)}=\hat gl_n({\bf C})^{(2)}$ and $n$ even, $\hat{\underline g}^{(2)}=\hat {so}_{2n}({\bf C})^{(2)}$, respectively.
 The action of $B_\infty$ and $D_\infty$ 
on the spin module $V^\epsilon $ is generated by the homogeneous components of the normal ordered product $:\phi_a^+(u) \phi_b^+(v):$ where $a=b=1$, $1\le a,b\le 2$, $a=b=1$, respectively. It is clear from formula (6.3.3) of proposition 6.3.1, that this product can be  expressed in the oscillator algebra ${\cal O}_{[w]}$ if $a=b=1$, and in the oscillator algebra algebra  ${\cal O}_{[w]} $ and  the operator $Q_1Q_2=-Q_2Q_1$ if $1\le a,b\le 2$.  
Now using (6.2.2) and (6.2.3), we calculate: 
$$\eqalign{Q_1Q_2&\phi_1^+(k_1)\phi_1^+(k_2)\cdots\phi_1^+(k_p)
  \phi_2^+(l_1)\phi_2^+(l_2)\cdots\phi_2^+(l_q)\vac  
  \cr &
  =(-1)^{p+q}\phi_1^+(k_1)\phi_1^+(k_2)\cdots\phi_1^+(k_p)
  \phi_2^+(l_1)\phi_2^+(l_2)\cdots\phi_2^+(l_q)Q_1Q_2\vac  
  \cr &
  =(-1)^{p+q+1}\phi_1^+(k_1)\phi_1^+(k_2)\cdots\phi_1^+(k_p)
  \phi_2^+(l_1)\phi_2^+(l_2)\cdots\phi_2^+(l_q)\phi_2^+(0)\phi_1^+(0)\vac  
  \cr &
  =(-1)^{p+q}{i\over 2}\phi_1^+(k_1)\phi_1^+(k_2)\cdots\phi_1^+(k_p)
  \phi_2^+(l_1)\phi_2^+(l_2)\cdots\phi_2^+(l_q)\vac  
  \cr}$$
So $Q_1Q_2={i\over 2}I$ on $V_{\overline 0}^0$ and $=-{i\over 2}I$ on
$V_{\overline 1}^0$, i.e., $Q_1Q_2={i\over 2}\chi$. 

Thus for all three cases  we obtain that the spin module $V_\alpha^\epsilon$ 
remains irreducible when restricted to  ${\cal O}_{[w]} $.
Since  ${\cal O}_{[w]} \subset \hat{\underline g}^{(2)}$, $V_\alpha^\epsilon$ is also an irreducible $\hat{\underline g}^{(2)}$-module.

The calculation of the corresponding highest weight is straightforward; it follows from (4.2.18) and (4.2.22).}


\baselineskip=22pt plus.1pt

\count30=6                      %chapterno = 6
\count31=0                      %sectionno = 0
\count32=0                      %formulano = 0
\count33=0                      %proclaimno= 0

\def\chapno{\global\count31=0 
            \global\advance\count30 by1
            \number\count30}

\def\secno{\global\advance\count31 by1 
           \global\count32=0
           \global\count33=0 
           \number\count30 .\number\count31}

\def\forno{\eqno(\global\advance\count32 by1
            \number\count30 .\number\count31 .\number\count32)}                                                             

\def\prono{\global\advance\count33 by1
            \number\count30 .\number\count31 .\number\count33}


\long\def\proof#1{\noindent{\bf{Proof.}}\quad#1\qedd}
\def\qedd{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}\vskip .5cm}
\def\qed{\hfill\hbox{\vrule\vbox{\hrule\kern2truept\hbox{\kern2truept
\vbox to2pt{\hsize=2pt}\kern2truept}\kern2truept\hrule}\vrule}}



\def\so{so_{2n}({\bf C})}
\def\SO{SO_{2n}({\bf C})}

\def\ad{{\rm ad}}
\def\Ad{{\rm Ad}}
\def\trace{{\rm trace}}

\def\E{{\cal E}}
\def\F{{\cal F}}
\def\tn{\hat \tau_{\underline n}}
\def\tw{\hat \tau_{[w]}}
\def\Hn{H_{\underline n}}
\def\Hw{H_{[w]}}
\def\Sn{S_{\underline n}}
\def\Sw{S_{[w]}}
\def\Xw{X_{[w]}}
\def\sw{\hat {\underline s}_{[w]}}
\def\hw{{\underline h}_{[w]}}

\def\gl{gl_{n}({\bf C})}
\def\oi{o_\epsilon(\infty )}
\def\gli{gl(\infty )}
\def\glib{\overline {gl(\infty )}}
\def\oib{\overline {o_\epsilon(\infty )}}
\def\Bi{B_{\infty}}
\def\Di{D_{\infty}}
\def\glt{\tilde{gl}_{n}({\bf C})}
\def\glh{\hat{gl}_{n}({\bf C})}
\def\sot{\tilde{so}_{2n}({\bf C})}
\def\soh{\hat{so}_{2n}({\bf C})}
\def\vac{\mid \overline 0 >}
\def\sh{\hat {\underline s}}
\def\shw{\hat {\underline s}{[w]}}
\def\a#1#2{\hat\alpha_{#1}({N\over n_{#1}}#2)}
\def\L{{\cal L}}
\def\D{{\cal D}}
\def\Cxrs{{\bf C}[x_k^{(a)};\ 1\le a\le s,\ k\in {\bf N} \hbox{ if }
          a\in CF,\ k\in {\bf N}-{1\over 2} \hbox{ if } a\in NF]}
\def\Cx{{\bf C}[x]}
\def\CL{{\bf C}(L^*)}
\def\Ct{{\bf C}_\epsilon [\theta  ]}
\def\ht{{\rm ht}}

\beginsection \chapno .  Bosonic realization of the spin module,
and $q$-dimension formulas
\par

\beginsection \secno .  Introduction
\par
In this section we realize the spin module $V^\epsilon$ as the tensor
product of a polynomial algebra in infinitely many variables,
a finite dimensional Grassmann algebra and the group algebra of a lattice.
In section  7.3 we will give the $q$-dimension formulas of the 
spin module
that correspond to different realizations.


\beginsection \secno .
  The spin module  as Fock space
\par
Recall from section  5.2 that the elements
$\hat\alpha_a({N\over n_a}k)$ with $k\in {\bf Z}$, $k\ne 0$
if $a\in CF$
and $k\in {1\over 2}+{\bf Z}$ if $a\in NF$ together with $c$  form 
a basis of a Heisenberg subalgebra, which we will denote by  $\sw$. They satisfy the commutation relations
$$[\hat\alpha_a({N\over n_a}k),\hat\alpha_b({N\over n_b}l)]=k\delta_{ab}\delta_{k,-l}c$$
This Heisenberg subalgebra has a canonical representation $\sigma$ in the space of
polynomials in infinitely many variables $\Cx:=\Cxrs$. It is given by:
$$\rho \left(\hat\alpha_a({N\over n_a}k)\right)=
  \cases{{\partial\over\partial x_k^{(a)}}&if $k>0$\cr
         -kx_k^{(a)}&if $k<0$\cr}
  \forno$$

If we consider $V^\epsilon$ as an $\sw$--module, this module contains a collection of 
vectors which are annihilated by all $\hat\alpha_a({N\over n_a}k)$
($k>0$).
This {\it vacuum space} $V^{\epsilon,+}$ for $\sw$ is the linear span of the vectors
$$Q_1^{k_1}Q_2^{k_2}\cdots Q_s^{k_s}\vac
  \forno$$
where $k_a\in {\bf Z}$ if $a\in CF$ and $k_a=0$ or $1$  if $a\in NF$. Unfortunately, these vectors are not linearly independent: we have  (see
(4.4.23) and (6.2.3)):
$$\eqalign{\{Q_{r+2a-1}-iQ_{r+2a}\}\vac&=0\hbox{ for }1\le a\le [{s-r\over 2}]\hbox{ and }\underline g =\gl\cr\{Q_{r+2a-1}+(4\delta-1)iQ_{r+2a}\}\vac&=0\hbox{ for }1\le a\le {s-r-2\over 2}\hbox{ and }\underline g =\so\cr}
\forno$$
Motivated by these relations we introduce operators $q_a$ $a\in {\bf Z}$, $\epsilon\le a\le [{|NF|\over 2}]$ --where $|NF|$ is the order of $NF$-- as follows:
$$q_a:=\cases{{1\over\sqrt 2}\{Q_{r+2a-1}+iQ_{r+2a}\}&if $1\le a\le [{s-r\over 2}]$ and $\underline g =\gl$\cr\cr
{1\over\sqrt 2}\{Q_{r+2a-1}+(1-4\delta)iQ_{r+2a}\}&for $1\le a\le {s-r-2\over 2}$and $\underline g =\so$\cr\cr
Q_s&if $a=0$ and  ($\underline g =\gl$, $n$ odd) or ($\underline g =\so$, $r=s$)\cr\cr
Q_{s-1}&if $a=0$, $\underline g =\so$ and $r\ne s$
\cr}\forno$$
Similarly, we set $q_{-a}:=q_a^\dagger$. The following anticommutation relations are immediate from the definition above:
$$\{q_a,q_b\}=\delta_{a,-b}\forno$$
 We are now able to give a basis for the vacuum space $V^{\epsilon,+}$:
\proclaim Theorem \prono .
The vectors
$$v_{k_1,\cdots,k_{|CF|},l_1,\cdots,l_p}:=
Q_{j_1}^{k_1}\cdots Q_{j_{|CF|}}^{k_{|CF|}}q_{l_1}\cdots q_{l_p}
 \vac \forno$$
with $ 1\le j_1<j_2<\cdots <j_{|CF|}$, $j_p\in CF$ and $k_p\in {\bf Z}$, $\epsilon\le l_1<l_2<\cdots\le [{|NF|\over 2}]$
 form a basis of the vacuum space $V^{\epsilon,+}$.

\noindent For a proof of this theorem see e.g. [KrL91c].\qed
 
Now consider the tensor product ${\bf C}[x]\otimes V^{\epsilon,+}$. It can be
identified in a natural way with a subspace of the spin module $V^\epsilon$. 
From theorem , it is clear that the components $\phi_a^+(k)$,
$\phi_a^-(k)$, of the fields $\phi_a^+(z)$, $\phi_a^-(z)$  leave this
subspace invariant. Hence ${\bf C}[x]\otimes V^{\epsilon,+}$ must be isomorphic to $V^\epsilon$.



Next we will show that the vacuum space $V^{\epsilon,+}$ is isomorphic to the tensor
product of a group algebra of a lattice
and a Grassmann algebra. 
Consider the lattice $L$ in the vector space $\underline h$ 
generated by the $\hat\alpha_i:=\hat\alpha_i(0)$, i.e.,
$$L=\sum_{i\in CF} {\bf Z}\hat\alpha_i\forno$$
The bilinear form $(\ |\ )_{\underline g}$ induces an integral positive definite  bilinear form on $L$:
$$(\hat\alpha_i|\hat\alpha_j)_{\underline g}
  =n_i\delta_{ij}\forno$$
Let $L^*$ be the dual lattice of $L$; if we identify 
$\underline h^*$ with $\underline h$, we obtain as basis for $L^*$
the elements $\alpha_i:=\hat\alpha_i^*={1\over n_i}\hat\alpha_i$.
So
$$L^*=\sum_{i\in CF} {\bf Z}\alpha_i\forno$$
and
$$(\alpha_i|\alpha_j)_{\underline g}={\delta_{ij}\over n_i}\forno$$
showing that $L^*$ is again positive definite, but in general no longer integral.
To the lattice $L^*$ we associate its group algebra $\CL$,
i.e., a ${\bf C}$--algebra 
with basis $e^\gamma$ for $\gamma\in L^*$ and multiplication 
$e^\beta e^\gamma =e^{\beta +\gamma}$. We also define some `height functions' ($i\in CF$):
$$\ht_i(k_1\alpha_{j_1} +k_2\alpha_{j_2} +\cdots +k_{|CF|}\alpha_{j_{|CF|}} ):=
  \sum_{j_p<i} k_p\forno$$
and 
$$\ht (k_1\alpha_{j_1} +k_2\alpha_{j_2} +\cdots +k_{|CF|}\alpha_{j_{|CF|}} ):=
  k_1+k_2+\cdots +k_{|CF|}\forno$$   

Let $\Ct={\bf C}[\theta_i  ;i\in {\bf Z},\ \epsilon\le i\le [{|NF|\over 2}]]$ be the finite Grassmann
algebra, i.e., the $\theta_i$ satisfy the
following anticommutation relations
$$\theta_i\theta_j +\theta_j\theta_i=0\forno$$
One can also introduce the derivations ${\partial\over\partial\theta_i}$on this algebra;
$${\partial\over\partial\theta_i}(\theta_j)=\delta_{ij}-
  \theta_j{\partial\over\partial\theta_i}\forno$$
and hence
$${\partial\over\partial\theta_i}{\partial\over\partial\theta_j}
  +{\partial\over\partial\theta_j}{\partial\over\partial\theta_i}=
0\forno$$






We are now ready to define the isomorphism $\rho$ between the
spin module $V^\epsilon$ and the {\it Fock space} $F^\epsilon$:
$$\rho :V^\epsilon\to F^\epsilon :=\Cx\otimes\CL\otimes\Ct\forno$$
It is given by:
$$\eqalign{\rho :\hat\alpha_{i_1}(-{N\over n_{i_1}}m_1)&\cdots
          \hat\alpha_{i_q}(-{N\over n_{i_q}}m_q)
         v_{k_1,\cdots,k_{|CF|},l_1,\cdots,l_p}
  \mapsto\cr & \Bigl({1\over \sqrt 2}\Bigr)^{\delta_{l_1,0}} m_1\cdots m_q x_{m_1}^{\left( i_1\right)}\cdots
          x_{m_q}^{\left( i_q\right)}
          \otimes
          e^{k_1\alpha_{j_1} +\cdots +k_{|CF|}\alpha_{j_{|CF|}}}
          \otimes
          \theta_{l_1}\cdots \theta_{l_p}\cr}
  \forno$$
and in particular
$$\rho :\vac\mapsto 1\otimes 1\otimes 1\forno$$
This is an isomorphism between vector spaces. Let $p\in\Cx$,
$\gamma\in L^*$ and $q\in\Ct$, then we have the following transported
action
$$\eqalign{\rho Q_i \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht_i(\gamma)}p\otimes e^{\gamma +\alpha_i}\otimes q
           \cr
           \rho Q_i^{-1} \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht_i(\gamma)}p\otimes e^{\gamma -\alpha_i}\otimes q
           \cr
           \rho q_0\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes {1\over \sqrt 2}(\theta_{0}+{\partial\over\partial\theta_0})(q)\cr
           \rho q_j\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-)^{\ht(\gamma)}p\otimes e^\gamma \otimes \theta_jq
           \cr
           \rho q_{-j}\rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (-1)^{\ht(\gamma)}p\otimes e^\gamma \otimes 
           {\partial q\over\partial\theta_j}
           \cr
           \rho \hat\alpha_k(-{N\over n_k}l) \rho^{-1}
           (p\otimes e^\gamma\otimes q)=&
           l x_l^{(k)}p\otimes e^\gamma \otimes q
           \cr
           \rho \hat\alpha_k({N\over n_k}l) \rho^{-1}
           (p\otimes e^\gamma\otimes q)=&
           {\partial p\over \partial x_l^{(k)}}\otimes e^\gamma \otimes q
           \cr
           \rho \hat\alpha_i(0) \rho^{-1}(p\otimes e^\gamma\otimes q)=&
           (\hat\alpha_i|\gamma )p\otimes e^\gamma \otimes q
           \cr
                     \cr}
  \forno$$
here $i\in CF$, $1\le j\le [{|NF|\over 2} ]$, $1\le k\le s$ and $l>0$.


\beginsection \secno .  q-dimension formulas \par
Recall the operator $ L_0=F_0=B_0$ (see theorem 5.5.1). Using (5.3.2-3), (6.2.5) and (7.2.6), one finds:
$$\eqalign{&L_0v_{k_1,\cdots k_{|CF|},l_1\cdots l_{p}}=\cr&
  \left\{\sum_{a\in CF} \Bigl( {n_a^2-1\over 24n_a}\Bigr)+\sum_{a\in NF} \Bigl({1\over 16n_a} +{n_a^2-1\over 24n_a}\Bigr)+{1\over 2}\left(
  {k_1^2\over n_{j_1}}+\cdots {k_{|CF|}^2\over n_{j_{|CF|}}}\right) \right\}v_{k_1,\cdots k_{|CF|},l_1\cdots l_{p}}\cr}
  \forno$$
Combining this with (5.3.5), we find the following
form of the `q-dimension' formula for $V^\epsilon$:

$${\rm trace}\lower 8pt\hbox{$V^\epsilon$}q^{L_0}=
   2^{[{|NF|+1\over 2} ]}q^{n\over 24}
 \prod_{i\in CF} {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i\in NF}  {\eta (q^{1\over n_i})\over  \eta (q^{1\over 2n_i})}\forno$$
where
$$\eta (q)=q^{1\over 24}\prod_{j\in {\bf N}}(1-q^j)\forno$$
is the well--known {\it Dedekind $\eta$--function} and
$$\theta (q)=\sum_{k\in {\bf Z}} q^{k^2\over 2}\forno$$
 is the classical  {\it $\theta$--function}.
Notice that $n$ is the value of the central charge $c_{vir}$. 

Using (7.3.1) and the fact that the chirality operator $\chi$ commutes with
all oscillators, one easily recovers the
$q$-dimension formulas of $V^\epsilon_\alpha$.
\proclaim Proposition  \prono .
If $|NF|=0$:
$${\rm trace}\lower 8pt\hbox{$V^\epsilon_\alpha$}q^{L_0}=
q^{n\over 24} {{\displaystyle \sum_{k_1+\cdots +k_s\in \alpha+2{\bf Z}}q^{{1\over 2}\left(
  {k_1^2\over n_1}+\cdots {k_s^2\over n_s}\right)}} \over {\displaystyle \prod_{i=1}^s  \eta (q^{1\over n_i})}}\forno$$
And 
$${\rm trace}\lower 8pt\hbox{$V^\epsilon_\alpha$}q^{L_0}=  2^{[{|NF|-1 \over 2}]}
q^{n\over 24} 
  \prod_{i\in CF} {\theta (q^{1\over n_i})\over  \eta (q^{1\over n_i})}
  \prod_{i\in NF} 
  {\eta (q^{1\over n_i})\over \eta (q^{1\over 2n_i})}\forno$$
in all other cases.


\vskip 1cm
\noindent{\bf Acknowledgements}: Johan van de Leur would like to thank Prof. Victor Kac for valuable conversations (in  1986-1988), which were crucial for the start of this work.

\hfill\break
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\bye









































