%Paper: hep-th/9411069
%From: J.vandeLeur <J.vandeLeur@math.utwente.nl>
%Date: Thu, 10 Nov 1994 10:05:42 +0100


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\topmatter
\title   The  $W_{1+\infty}(gl_s)$--symmetries of the $S$--component KP
hierarchy
\endtitle
\author
Johan van de Leur
\endauthor
\thanks
The research of Johan van de Leur is
financially supported by the ``Stichting Fundamenteel Onderzoek der
Materie (F.O.M.)''. E-mail: vdleur\@math.utwente.nl
\endthanks
\abstract
Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice
hierarchies
a formula which translates  the action of the
vertex operator on tau--functions to an action of a vertex operator
of pseudo-differential
operators on  wave functions. This  relates  the additional
symmetries of the KP and Toda lattice hierarchy
to the $W_{1+\infty}$--, respectively $W_{1+\infty}\times
W_{1+\infty}$--algebra
symmeties.  In this paper we generalize the
results to the $s$--component KP hierarchy. The vertex operators
generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of
$W_{1+\infty}$. Since the Toda lattice hierarchy is equivalent to the
$2$--component KP hierarchy, the results of this paper uncover in that
particular
case a much richer structure than the one obtained by Adler, Shiota
and van Moerbeke.




\endabstract
\endtopmatter

\vskip 10pt

\subheading{\S 0. Introduction}
\vskip 10pt
\noindent The KP hierarchy is the set of deformation equations
$${\partial L\over\partial t_k}=[(L^n)_+,L],$$
for the first order pseudo-differential operator
$$L\equiv
L(x,t)=\partial+u_1(x,t)\partial^{-1}+u_2(x,t)\partial^{-2}+\cdots,$$
here $\partial={\partial\over\partial x}$ and $t=(t_1,t_2,\ldots)$.
It is well--known that $L$ dresses as $L=P\partial P^{-1}$ with
$$\aligned
P\equiv
P(\tau,x,t)&=1+a_1(x,t)\partial^{-1}+a_2(x,t)\partial^{-2}+\cdots\\
&={\tau(x,t-[\partial^{-1}])\over \tau(x,t)},
\endaligned
$$
where $\tau $ is the famous $\tau$--function, introduced by the Kyoto
group [DJKM1-3] and $[z]=(z,{z^2\over 2},{z^3\over 3},\ldots)$.

The wave or Baker--Akhiezer function
$$\Psi\equiv \Psi(\tau,x,t,z)= W(\tau,x,t,\partial)e^{zx},$$
where
$$W\equiv W(\tau,x,t,z)=P(\tau,x,t)e^{\xi(t)}\qquad
\text{with}\quad\xi(t)=\sum_{k=1}^\infty
t_k\partial^k$$
is an eigenfunction of L, viz.,
$$L\Psi=z\Psi\qquad\text{and}\quad {\partial\Psi\over\partial
t_k}=(L^k)_+\Psi.$$
{}From this point of view, the introduction by Orlov and  Schulman [OS]
of another pseudo-differential operator $M\equiv M(x,t)=WxW^{-1}$ which action
on $\Psi$ amounts to
$$M\Psi={\partial\Psi\over\partial z}$$
is rather natural.

Recently, Adler, Shiota and van Moerbeke [ASV1], [ASV2] proved a
conjecture of Orlov and Schulman, viz.,  that there
exists a relation between $(M^\ell L^{k+\ell})_-$ acting on $\Psi$
and the generators $W_k^{(\ell+1)}\sim -t^{k+\ell}({\partial\over\partial
t})^{\ell}$
of the $W_{1+\infty}$--algebra  acting on the $\tau$--function. More
explicitly, let
$$Y(\tau,y,w)=\sum_{\ell=0}^\infty {(y-w)^\ell\over \ell !}\sum_{k\in\Bbb
Z}M^\ell L^{k+\ell} w^{-k-\ell-1}$$
be the generating series of the $M^\ell L^{k+\ell}$ and let
$$\aligned
W(y,w)&=\sum_{\ell=0}^\infty {(y-w)^\ell\over \ell !}\sum_{k\in\Bbb
Z}W^{(\ell+1)}_k w^{-k-\ell-1}\\
&={1\over y-w}e^{x(y-w)+\sum_{k=1}^\infty t_k(y^k-w^k)}
e^{-\sum_{k=1}^\infty {\partial\over\partial t_k}{y^{-k}-w^{-k}\over
k}}
\endaligned
\tag{0.1}
$$
be the vertex operator of the KP hierarchy, then one has the following
formula [ASV1]:
$$-Y(\tau,y,w)_-\Psi(\tau,x,t,z)=\Psi(\tau,x,t,z)(e^{-\sum_{k=1}^\infty
{\partial\over\partial t_k}{z^{-k}\over k}}-1)\left ( {W(y,w)\tau(x,t)\over
\tau(x,t)}\right ).\tag{0.2}$$
Dickey gave another proof of this formula [D]. The ``geometric
interpretation'' of this Adler--Shiota--van Moerbeke  formula is as follows.
The transformation $e^{\lambda W(y,w)}=1+\lambda W(y,w)$ is a symmetrie
transformation
or a kind of auto-B\"acklund
transformation of the KP hierarchy. If one rewrites (0.2) as
$$
-\lambda Y(\tau,y,w)_-\Psi(\tau,x,t,z)=\Psi(\tau,x,t,z)(e^{-\sum_{k=1}^\infty
{\partial\over\partial t_k}{z^{-k}\over k}}-1)\left ({e^{\lambda
W(y,w)}\tau(x,t)\over
\tau(x,t)}\right ),\tag{0.3}$$
then one easily sees that  (0.2) is in fact a formula that relates this
B\"acklund transformation to the so--called additional
symmetries of the KP hierarchy. To be more precise, let $\sigma$ be
the new solution of the KP hierarchy which one obtains from $\tau$ by
this B\"acklund transformation, i.e., $\sigma =e^{\lambda
W(y,w)}\tau$, then
$$\aligned
\Psi(\sigma,x,t,z)&=\sigma(x,t)^{-1}e^{-\sum_{k=1}^\infty
{\partial\over\partial t_k}{z^{-k}\over
k}}\sigma(x,t)e^{\xi(t)}e^{zx}\\
&={\tau(x,t)\over\sigma(x,t)}e^{-\sum_{k=1}^\infty
{\partial\over\partial t_k}{z^{-k}\over k}}\left ({e^{\lambda
W(y,w)}\tau(x,t)\over
\tau(x,t)}\right )\Psi(\tau,x,t,z)\\
&=\Psi(\tau,x,t,z)+{\tau(x,t)\over\sigma(x,t)}(e^{-\sum_{k=1}^\infty
{\partial\over\partial t_k}{z^{-k}\over k}}-1)\left ({e^{\lambda
W(y,w)}\tau(x,t)\over
\tau(x,t)}\right )\Psi(\tau,x,t,z).
\endaligned$$
Now using (0.3) one obtains
$$\Psi(\sigma,x,t,z)=\left (
1-\lambda{\tau(x,t)\over\sigma(x,t)}Y(\tau,y,w)_-\right
)\Psi(\tau,x,t,z).$$
Hence $Y(\tau,y,w)_-$ produces, as a consequence of formula (0.2), the
B\"acklund transformation of the wave function corresponding to $\tau$.


Adler, Shiota and van Moerbeke also treated in [ASV2] the Toda lattice
hierarchy of Ueno and Takasaki [UT] and showed that an analogous
formula also holds. In their treatment they considered two vertex
operators, each depending on a different time flow
$t^{(j)}=(t^{(j)}_1, t^{(j)}_2,\dots)$, $j=1,2$,
of a form similar to that of (0.1). Hence The $W_{1+\infty}$--algebra
of the KP hierarchy is replaced by two copies of this algebra.

Using $2\times 2$--matrix pseudo--differential operators instead of
infinite shift operators, one can show that the Toda lattice hierarchy
is equivalent to the 2--component KP hierarchy as treated by Kac and
the author in [KV]. In that case there are however more vertex
operators than only the ones of the form (0.1), viz., one also has
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
$$W^{(ab)}(y,w)={C^{(ab)}(y,w)\over(y-w)^{\delta_{ab}}}e^{x(y-w)+\sum_{k=1}^\infty (t^{(a)}_ky^k-t^{(b)}_kw^k)}
e^{-\sum_{k=1}^\infty{1\over k} ({\partial\over\partial t^{(a)}_k}y^{-k}-
{\partial\over\partial t^{(b)}_k}w^{-k})},$$
here $C^{(ab)}(y,w)$ are operators that act on the twisted group
algebra of the root lattice of $sl_2$. A natural question now is: Are
there also matrix pseudo--differential operators such that a formula
as (0.2) hold for these $W^{(ab)}(y,w)$?  In this paper we show that
a similar result holds, not only for the 2--component KP hierarchy, but
in general for the $s$--component KP hierarchy. One finds that the
natural generalization of $W_{1+\infty}$ is not $(W_{1+\infty})^s$ but
$W_{1+\infty}(gl_s)$, the central extension of the algebra of
differential operators on $(\Bbb C[t,t^{-1}])^s$. Hence one can
conclude that the results of [ASV2] for the Toda lattice hierarchy are
not complete, but that the structure is richer.

It is a pleasure to thank Gerard Helminck, Peter van den Heuvel
and Takahiro Shiota,  for usefull discussions.


\vskip 10pt
\subheading{\S 1.  $a_{\infty}$ and the KP hierarchy in
the fermionic picture [KV]}

\vskip 10pt
\noindent
Consider the infinite dimensional complex  Lie algebra
$a_{\infty} := \overline{gl_{\infty}}\bigoplus {\Bbb C}c$, where
%
$$\overline{gl_{\infty}} = \{ a = (a_{ij})_{i,j \in {\Bbb Z}+\frac{1}{2}}
|\  a_{ij}=0\  \text{if}\   |i-j|>>0\},$$
%
with Lie bracket defined by
%
$$[a+\alpha c,b+\beta c]=ab-ba+\mu (a,b)c , \tag{1.1}$$
%
for $a,b\in \overline{gl_{\infty}}$ and $\alpha ,\beta\in{\Bbb C}$.
Here $\mu$ is the following 2--cocycle:
%
$$\mu (E_{ij},E_{kl})=\delta_{il}\delta_{jk}(\theta(i)-\theta(j)),
\tag{1.2}$$
%
with $E_{ij}$ the matrix that has a $1$ on the $(i,j)$-th entry and zeros
elsewhere and $\theta :{\Bbb R}\to {\Bbb C}$ the function  defined by
%
$$\theta(i):=\cases 0 & \text{if}\ i>0,\\
                    1 & \text{if}\ i\le 0.\endcases \tag{1.3}$$
%
Let ${\Bbb C}^{\infty} = \bigoplus_{j \in {\Bbb Z}+\frac{1}{2}} {\Bbb
C} v_{j}$ be the infinite dimensional complex vector space with fixed
basis $\{ v_{j}\}_{j \in {\Bbb Z}+\frac{1}{2}}$. The Lie algebra $a_{\infty}$
acts linearly on
${\Bbb C}^{\infty}$ via the usual formula:
%
$$E_{ij} (v_{k}) = \delta_{jk} v_{i}.$$
We introduce,
following [KP2], the corresponding semi-infinite wedge space $F =
\Lambda^{\frac{1}{2}\infty} {\Bbb C}^{\infty}$, this is the vector space
with a basis consisting of all semi-infinite monomials of the form
$v_{i_{1}} \wedge v_{i_{2}} \wedge v_{i_{3}} \ldots$, where $i_{1} >
i_{2} > i_{3} > \ldots$ and $i_{\ell +1} = i_{\ell} -1$ for $\ell >>
0$.
In order to describe representations of the Lie algebra on this space,
 we find it convenient to define  wedging and contracting
operators $\psi^{-}_{j}$ and $\psi^{+}_{j}\ \ (j \in {\Bbb Z} +
\frac{1}{2})$ on $F$ by
%
$$\align
&\psi^{-}_{j} (v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots ) = \cases 0
& \text{if}\ j = -i_{s} \text{ for some}\ s \\
(-1)^{s} v_{i_{1}} \wedge v_{i_{2}} \cdots \wedge v_{i_{s}} \wedge
v_{-j} \wedge v_{i_{s+1}} \wedge \cdots &\text{if}\ i_{s} > -j >
i_{s+1}\endcases \\
&\psi^{+}_{j} (v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots ) = \cases 0
&\text{if}\ j \neq i_{s}\ \text{for all}\ s \\
(-1)^{s+1} v_{i_{1}} \wedge v_{i_{2}} \wedge \cdots \wedge
v_{i_{s-1}} \wedge v_{i_{s+1}} \wedge \cdots &\text{if}\ j = i_{s}.
\endcases
\endalign
$$
%%
These  wedging and contracting operators satisfy the following relations
$(i,j \in {\Bbb Z}+\frac{1}{2}, \lambda ,\mu = +,-)$:
%
$$\psi^{\lambda}_{i} \psi^{\mu}_{j} + \psi^{\mu}_{j}
\psi^{\lambda}_{i} = \delta_{\lambda ,-\mu} \delta_{i,-j}, \tag{1.4}$$
hence they generate a Clifford algebra, which we denote by ${\Cal C}\ell$.

Introduce the following elements of $F$ $(m \in {\Bbb Z})$:
%
$$|m\rangle = v_{m-\frac{1}{2} } \wedge v_{m-\frac{3}{2} } \wedge
v_{m-\frac{5}{2} } \wedge \cdots .$$
It is clear that $F$ is an irreducible ${\Cal C}\ell$-module such that
%
$\psi^{\pm}_{j} |0\rangle = 0 \ \text{for}\ j > 0 . $
%
Define  a representation $\hat r$ of $a_{\infty}$ on $F$ by
%
$$ \hat r(E_{ij})=:\psi^-_{-i}\psi^+_j:,\quad \hat r(c)=I,\tag{1.5}
$$
%
where $:\ :$ stands for the {\it normal ordered product} defined in
the usual way $(\lambda ,\mu = +$ or $-$):
%
$$:\psi^{\lambda (i)}_{k} \psi^{\mu (j)}_{\ell}: = \cases \psi^{
\lambda (i)}_{k}
\psi^{\mu (j)}_{\ell}\ &\text{if}\ \ell \ge k \\
-\psi^{\mu (j)}_{\ell} \psi^{\lambda (i)}_{k} &\text{if}\ \ell <
k.\endcases \tag{1.6}$$

Define the {\it charge decomposition}
%
$$F = \bigoplus_{m \in {\Bbb Z}} F^{(m)} \tag{1.7}$$
by letting
%

$$\text{charge}(|0\rangle ) = 0\ \text{and charge} (\psi^{\pm}_{j}) =
\pm 1. \tag{1.8}$$
%
It  is easy to see that each $F^{(m)}$ is an  irreducible $a_{\infty}$--highest
weight module with
highest weight vector $|m\rangle$.

We are now able to define the {\it KP hierarchy in the fermionic
picture}, it is the equation
$$\sum_{k \in {\Bbb Z}+\frac{1}{2}} \psi^{+}_{k} \tau \otimes \psi^{-}_{-k}
\tau = 0, \tag{1.9}$$
for
$\tau\in F^{(0)}$. One can prove (see e. g. [KP2] or [KR]) that
this equation characterizes  the group orbit of the vacuum vector
$|0\rangle$  for the  the group $GL_{\infty}$.
Since the group does not play an important role in this paper,
we will not introduce it here.
\vskip 10pt

\subheading{\S 2. $W_{1+\infty}(gl_s)$ as subalgebra of $a_{\infty}$}

\vskip 10pt
\noindent Let $e_i$, $1\le i\le s$ be a basis of ${\Bbb C}^s$.
By identifying $({\Bbb C}[t,t^{-1}])^s$ with ${\Bbb C}^{\infty}$,
we can embed   $W_{1+\infty}(gl_s)$ into  $a_{\infty}$. This,
however, can be done in many different ways, the simplest
one is the following.
We put ($1\le a \le s,\ j\in {\Bbb Z} + \frac{1}{2})$:
%
$$\align
v^{(a)}_{j}&=v_{sj+\frac{1}{2}(s-2a+1)} =t^{-j-{1\over 2}}e_a,\\
\psi^{\pm (a)}_{j }&=\psi^{\pm }_{sj\pm\frac{1}{2}(s-2a+1)}.
\tag{2.1}
\endalign$$
%
Notice that with this relabeling we have:
%
$\psi^{\pm (a)}_{k}|0\rangle = 0\ \text{for}\ k > 0.$
%
We introduce the  generating series of the fermions, the so--called
fermionic fields $(z \in {\Bbb C}^{\times})$:
%
$$\psi^{\pm (a)}(z) \overset{\text{def}}\to{=} \sum_{k \in {\Bbb
Z}+\frac{1}{2}} \psi^{\pm
(a)}_{k} z^{-k-\frac{1}{2}}.\tag{2.2}$$
%
We also rewrite the $E_{jk}$'s:
$$E^{(ab)}_{j,k}=E_{sj+\frac{1}{2}(s-2a+1),sk+\frac{1}{2}(s-2b+1)},
\tag{2.3}$$
%
then  $\hat r(E^{(ab)}_{jk})=:\psi^{-(a)}_{-j}\psi^{+(b)}_k:$.



We can associate to $({\Bbb C}[t,t^{-1}])^s$  the Lie algebra
of differential operators on this space,
it has as basis the operators:
%
$$-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ij},
\quad\text{for}\ k\in {\Bbb Z},\ \ell\in{\Bbb Z}_+,\ 1\le i,j\le s.$$
%
We will denote this Lie algebra by $D(gl_s)$. We can embed this algebra via
(2.1)
into $\overline{gl_{\infty}}$
and also into $a_{\infty}$, one finds
%
$$-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ij}
\mapsto \sum_{m\in{\Bbb
Z}}-m(m-1)\cdots(m-\ell+1)E^{(ij)}_{-m-k-\frac{1}{2},-m-\frac{1}{2}}.
\tag{2.4}
$$
%
It is straightforward, but rather tedious, to calculate the corresponding
2--cocycle, the result is as follows (see also [KP1], [Ra] and [KRa]).
Let $f(t), g(t)\in {\Bbb C}[t,t^{-1}]$ and $a,b\in gl_s$ then
%
$$\mu(f(t)({\partial\over\partial t})^{\ell}a,g(t)({\partial\over\partial t})^m
b)
={{\ell !m!}\over(\ell+m+1)!}\text{Res}_{t=0}dt\
f^{(m+1)}(t)g^{(\ell)}(t)\text{trace}(ab).
$$
Hence in this way we get a central extension $W_{1+\infty}(gl_s)=D(gl_s)\oplus
{\Bbb C}c$ of $D(gl_s)$
with Lie bracket
%
$$\align
&[f(t)({\partial\over\partial t})^{\ell}a+\alpha c,g(t)({\partial\over\partial
t})^m b+\beta c]=\\
&f(t)({\partial\over\partial t})^{\ell}g(t)({\partial\over\partial
t})^m
 ab-
g(t)({\partial\over\partial t})^m f(t)({\partial\over\partial t})^{\ell}ba
+\mu(f(t)({\partial\over\partial t})^{\ell}a,g(t)({\partial\over\partial t})^m
b)c.
\tag{2.5}\endalign
$$
%
Since we have the representation $\hat r$ of $a_{\infty}$, we find that
%
$$\hat r (-t^{k+\ell}({\partial\over\partial t})^{\ell}e_{ab})=
\sum_{m\in{\Bbb Z}}m(m-1)\cdots (m-\ell+1):\psi^{+(a)}_{-m-\frac{1}{2}}
\psi^{-(b)}_{m+k+\frac{1}{2}}:.
$$
%
In terms of the fermionic fields (2.2), we find
%
$$\sum_{k\in{\Bbb Z}}\hat r (-t^{k+\ell}({\partial\over\partial
t})^{\ell}e_{ab})z^{-k-\ell-1}=
:{{\partial^{\ell}\psi^{+(a)}(z)}\over \partial z^{\ell}}
\psi^{-(b)}(z):. \tag{2.6}
$$
%
Now define
%
$$\aligned
W^{(ab,\ell+1)}_k:&=\hat r (-t^{k+\ell}({\partial\over\partial
t})^{\ell}e_{ab}),\\
W^{(ab,\ell+1)}(z)&=\sum_{k\in{\Bbb Z}}W^{(ab,\ell+1)}_k z^{-k-\ell-1},
\endaligned
\tag{2.7}$$
then
$$\aligned W^{(ab)}(y,z):&=:\psi^{+(a)}(y)
\psi^{-(b)}(z):\\ &=\sum_{\ell =0}^\infty {(y-z)^\ell \over \ell !}
W^{(ab,\ell+1)}(z)\\
&= \sum_{\ell =0}^\infty {(y-z)^\ell \over \ell !}\sum_{k\in\Bbb Z}
W^{(ab,\ell+1)}_kz^{-k-\ell-1}.\endaligned
\tag{2.8}$$

\subheading{\S 3. The $s$--component boson fermion correspondence}

\vskip 10pt
\noindent
Using a bosonization one can rewrite (1.9) as a system
of partial
differential equations and express the basis elements of
$W_{1+\infty}(gl_s)$ in terms of vertex operators.
 We begin by introduce bosonic fields $(1 \leq i \leq s)$:
%
$$\alpha^{(i)}(z) \equiv \sum_{k \in {\Bbb Z}} \alpha^{(i)}_{k} z^{-k-1}
\overset{def}\to{=} :\psi^{-(i)}(z) \psi^{+(i)}(z):, \tag{3.1}$$
Since $\alpha^{(i)}(z)=W^{(ii,1)}(z)$, one easily checks  that the operators
$\alpha^{(i)}_{k}$ satisfy
the canonical commutation relation of the associative
oscillator algebra,  which we
denote by ${\frak a}$:
%
$$[\alpha^{(i)}_{k},\alpha^{(j)}_{\ell}] =
k\delta_{ij}\delta_{k,-\ell},\tag{3.2}$$
and one has
%
$$\alpha^{(i)}_{k}|m\rangle = 0 \ \text{for}\ k > 0.\tag{3.3}$$


It is easy to see that restricted to $\hat{gl}_{s}$, which is the subalgebra
generated
by the elements $W^{(ij,1)}_k$,
$F^{(0)}$ is its basic highest weight representation (see [K, Chapter
12]).

We will now
describe the $s$-component boson-fermion
correspondence (see [KV]).    Let $L$ be a lattice with a basis
$\delta_{1},\ldots
,\delta_{s}$ over ${\Bbb Z}$ and the symmetric bilinear form
$(\delta_{i}|\delta_{j}) = \delta_{ij}$, where $\delta_{ij}$ is the
Kronecker symbol.  Let
%
$$\varepsilon_{ij} = \cases -1 &\text{if $i > j$} \\
1 &\text{if $i \leq j$.} \endcases \tag{3.4}$$
%
Define a bimultiplicative function $\varepsilon :\ L \times L @>>> \{
\pm 1 \}$ by letting
%
$$\varepsilon (\delta_{i}, \delta_{j}) = \varepsilon_{ij}.
\tag{3.4}$$
Let $\delta = \delta_{1} + \ldots + \delta_{s},\  Q= \{ \gamma \in
L|\ (\delta | \gamma ) = 0\}$, $\Delta = \{ \alpha_{ij} :=
\delta_{i}-\delta_{j}| i,j = 1,\ldots ,s,\ i \neq j \}$.  Of course
$Q$ is the root lattice of $sl_{s}({\Bbb C})$, the set $\Delta$
being the root system.

Consider the vector space ${\Bbb C}[L]$ with basis $e^{\gamma}$,\
$\gamma \in L$, and the following twisted group algebra product:
%
$$e^{\alpha}e^{\beta} = \varepsilon (\alpha ,\beta)e^{\alpha +
\beta}. \tag{3.6}$$ Let ${\Bbb C}[t]$ be the space of polynomials in
indeterminates $t = \{ t^{(i)}_{k}\},\ k = 1,2,\ldots ,\ i =
1,2,\ldots ,s$ and denote by $B = {\Bbb C}[t] \otimes_{\Bbb C} {\Bbb C}[L]$ be
the tensor
product of these algebras.  Then the $s$-component boson-fermion
correspondence is the vector space isomorphism
%
$$\sigma :F @>\sim >> B, \tag{3.7}$$
given by $\sigma (|0\rangle )=1$ and
$$\sigma\psi^{\pm (a)}(z) \sigma^{-1}=e^{\pm \delta_a}z^{\pm \delta_a}
\exp(\pm\sum_{k=1}^{\infty}t_k^{(a)}z^k)\exp(\mp\sum_{k=1}^{\infty}
{\partial\over \partial t_k^{(a)}}{z^{-k}\over k}),\tag {3.8}$$
%
where
$$\delta_a (p(t) \otimes e^{\gamma}) =
(\delta_{a}|\gamma ) p(t) \otimes e^{\gamma}.\tag{3.9}$$
%
The transported charge then is as follows:
%
$$
\text{charge}(p(t)\otimes e^{\gamma}) = (\delta |\gamma).
$$
We denote the transported charge decomposition by
%
$B = \bigoplus_{m \in {\Bbb Z}} B^{(m)}$, then
the transported action of the operators $\alpha^{(i)}_{m}$ is given by
%
$$\cases
\sigma \alpha^{(j)}_{-m}\sigma^{-1}(p(t) \otimes e^{\gamma}) =
mt^{(j)}_{m}p(t)\otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{m} \sigma^{-1}(p(t) \otimes e^{\gamma}) =
 \frac{\partial
p(t)}{\partial t_{m}} \otimes e^{\gamma},\ \text{if}\ m > 0, &\  \\
\sigma \alpha^{(j)}_{0} \sigma^{-1} (p(t) \otimes e^{\gamma}) =
(\delta_{j}|\gamma ) p(t) \otimes e^{\gamma}
. & \
\endcases \tag{3.10}
$$
%

If one substitutes (3.8) into (2.8), one obtains  the following
vertex operator expression for the generating
series of the fields $W^{(ab,\ell+1)}(z)$:
$$\aligned W^{(ab)}(y,z)
&={1\over (y-z)^{\delta_{ab}}}
(X^{(ab)}(y,z)-\delta_{ab}),\quad\text{where}\\
X^{(ab)}(y,z):&=\\
\epsilon(\delta_a,\delta_b)&e^{\delta_a-\delta_b}y^{\delta_a}z^{-\delta_b}
\exp(\sum_{k=1}^{\infty}(t_k^{(a)}y^k-t_k^{(b)}z^k))
\exp(-\sum_{k=1}^{\infty}(
{\partial\over \partial t_k^{(a)}}{y^{-k}\over k}-
{\partial\over \partial t_k^{(b)}}{z^{-k}\over k})
)\endaligned \tag {3.11}$$

The fields for which $\ell=0$, give the
vertex operator realization of the homogeneous
realization of $\hat{gl}_s$, which was first found by
Frenkel and Kac [FK] and independently by Segal [Se](see also [TV] for more
details).

Using the isomorphism $\sigma$ we can reformulate the KP hierarchy
(1.9) in the bosonic picture.
We start by observing that (1.9) can be rewritten as follows:
%
$$\text{Res}_{z=0}\ dz ( \sum^{s}_{j=1} \psi^{+(j)}(z)\tau
\otimes \psi^{-(j)}(z)\tau ) = 0,\ \tau \in F^{(0)}.
\tag{3.12}$$
%
Notice that for $\tau \in F^{(0)},\ \sigma (\tau) = \sum_{\gamma \in Q}
\tau_{\gamma}(t)e^{\gamma}$.
 Here and further  we write $\tau_{\gamma}(t)e^{\gamma}$ for
$\tau_{\gamma} (t)\otimes
e^{\gamma}$.  Using  (3.8), equation (3.12) turns under $\sigma
\otimes \sigma :\ F \otimes F \overset\,\,\sim\to\longrightarrow
{\Bbb C}[t,t^{\prime }]
\otimes ({\Bbb C}[L^{}] \otimes {\Bbb C}[L^{ \prime}])$ into the
following set of equations;
 for all $\alpha ,\beta \in L$ such that $(\alpha
|\delta ) = -(\beta |\delta ) = 1$ we have:
%
$$\aligned
&\text{Res}_{z=0} ( dz
 \sum^{s}_{j=1} \varepsilon (\delta_{j}, \alpha-\beta)
z^{(\delta_{j}|\alpha - \beta - 2\delta_{j})}  \\
 &\times \exp
(\sum^{\infty}_{k=1} (t^{(j)}_{k} - t^{(j)^{
\prime}}_{k})z^{k})
\exp (-\sum^{\infty}_{k=1} (\frac{\partial}{\partial t^{(j)}_{k}}
 - \frac {\partial}{\partial t^{(j)^{
\prime}}_{k}})\frac{z^{-k}}{k}) \\
& \tau_{\alpha -
\delta_{j}}(t)(e^{\alpha})\tau_{\beta + \delta_{j}}(t^{
\prime})(e^{\beta})^{\prime})
= 0  . \endaligned \tag{3.13}$$
%
Notice that if $s=2$, the set of equations (3.13) are equivalent
(for more general $\tau$) to the Toda lattice hierarchy of Ueno and Takasaki
[UT].
For this reason,
{\it we assume from now on that $\tau=\sum \tau_\alpha e^\alpha$ is any
solution
of} (3.13). Hence the results of this paper also hold for an
(alternative definition) of the Toda lattice hierarchy.

In order to define the equations (3.13) in terms of formal
pseudo--differential operators it will be convenient to replace
$t_k^{(j)}$ by $t_k^{(j)}+\delta_{k,1}x$ and to introduce the notations
$$\aligned &\xi^{(j)}(t,z)=\sum_{i=1}^{\infty}t^{(j)}_iz^i,\quad
 \xi^{(j)}(x,t,z)=zx+\xi^{(j)}(t,z)\\ &\text{ and}\qquad \eta^{(j)}(t,z)=
\sum_{i=1}^{\infty}{\partial\over\partial t^{(j)}_i}{z^{-i}\over
i.}\endaligned$$
%
Next we replace $\alpha$ resp. $\beta$ by $\alpha+\delta_i$ and
$\beta-\delta_k$
then for all $\alpha,\beta\in Q$ and $1\le i,k\le s$ (3.13) turns into
$$\aligned
&\text{Res}_{z=0}  dz
 (\sum^{s}_{j=1} \varepsilon (\delta_{j}, \alpha+\delta_i-\beta+\delta_k)
z^{(\delta_{j}|\alpha - \beta +\delta_i+\delta_k-
2\delta_{j})}e^{\xi^{(j)}(x,t,z)
-\xi^{(j)}(x',t',z)}\\
&\times e^{-\eta^{(j)}(t,z)+\eta^{(j)}(t',z)}\tau_{\alpha +\delta_i-
\delta_{j}}(x,t)
(e^{\alpha+\delta_j})\tau_{\beta + \delta_{j}-\delta_k}
(x',t^\prime )(e^{\beta-\delta_k})^\prime)
= 0  . \endaligned \tag{3.14}$$

\vskip 10pt
\subheading{\S 4. The algebra of formal pseudo-differential operators and the
$s$-component KP hierarchy as a dynamical system }

\vskip 10pt
\noindent We proceed now to rewrite the formulation (3.14) of the
$s$-component KP hierarchy in terms of formal pseudo-differential
operators, generalizing the results of [DJKM1-3]. For more details
see [KV].
For each $\alpha \in \ \text{supp}\ \tau :=\{\alpha\in Q| \tau=\sum_{\alpha\in
Q}\tau_{\alpha}e^{\alpha}, \tau_{\alpha}\ne 0\}$ we define the (matrix
valued) functions
$$\Psi^\pm(\alpha,z)\equiv \Psi^{\pm} (\tau_\alpha ,x,t,z)
= (\Psi^{\pm}_{ij}(\tau_\alpha ,x,t,z))^{s}_{i,j=1}
\tag{4.1}$$
as follows:
%
$$\aligned
\Psi^{\pm}_{ij}(\tau_\alpha ,x,t,z) \overset{\text{def}}\to{=}&
\varepsilon (\delta_{j} , \alpha + \delta_{i})
 z^{(\delta_{j}|\pm \alpha + \delta_{i}-\delta_{j})}e^{\pm \xi^{(j)}(x,t,z)}
e^{\mp \eta^{(j)}(t,z)} \tau_{\alpha  \pm
(\delta_{i}-\delta_{j})}  (x,t)/\tau_{\alpha}(x,t)\\
=&\varepsilon (\delta_{j} , \alpha + \delta_{i})
 z^{(\delta_{j}|\pm \alpha + \delta_{i}-\delta_{j})}
\tau_{\alpha  \pm
(\delta_{i}-\delta_{j})}  (x,t^{(k)}-\delta_{jk}[z^{-1}])/\tau_{\alpha}(x,t)
e^{\pm \xi^{(j)}(x,t,z)},
\endaligned\tag{4.2}
$$
where $[w]=(w,{w^2\over 2},{w^3\over 3},\ldots)$
It is easy to see that equation (3.14) is equivalent to the
following bilinear identity:
%
$$Res_{z=0}\Psi^{+}(\tau_\alpha ,x,t,z)\ ^{t}\Psi^{-}(\tau_\beta
,x^{\prime},t',z)dz = 0\
\text{for all}\ \alpha ,\beta \in Q. \tag{4.3}$$
%
Define $s \times s$ matrices $S^{\pm (m)} (\tau_\alpha ,x,t)$ by the
following generating series (cf. (4.2)):
%
$$\aligned
\sum^{\infty}_{m=0}
S^{\pm (m)}_{ij} (\tau_\alpha ,x,t)(\pm z)^{-m}
=& \varepsilon_{ji}z^{\delta_{ij}-1} e^{\mp\eta(t,z)}
\tau_{\alpha \pm
(\delta_i-\delta_j)} (x,t))/\tau_{\alpha} (x,t)\\
=&\varepsilon_{ji}z^{\delta_{ij}-1}
\tau_{\alpha \pm
(\delta_i-\delta_j)} (x,t^{(k)}-\delta_{jk}[z^{-1}]))/\tau_{\alpha}
(x,t).
\endaligned
\tag{4.4}
$$
We see from (4.2) that $\Psi^{\pm}(\tau_\alpha ,x,t,z)$ can be written in the
following form:
%
$$\Psi^{\pm}(\tau_\alpha ,x,t,z) = (\sum^{\infty}_{m=0}
S^{\pm (m)}(\tau_\alpha ,x,t)R^{\pm}(\alpha ,\pm z)(\pm
z)^{-m})e^{\pm \xi (x,t,z)}, \tag{4.5}$$
where
$$R^{\pm}(\alpha ,z) = \sum^{s}_{i=1}
\varepsilon (\delta_{i}, \alpha ) e_{ii} (\pm z)^{\pm
(\delta_{i}|\alpha )}. \tag{4.6}$$
as before  $e_{ij}$ stands for the $s \times s$ matrix whose
$(i,j)$ entry is $1$ and all other entries are zero. Let
$$\partial = \frac{\partial}{\partial x},$$ we can now rewrite
$\Psi^{\pm}(\tau_\alpha ,x,t,z)$ in
terms of formal pseudo-differential operators,
%
define
$$\aligned &e^{\pm \xi(t,\pm\partial)}=\sum_{j=1}^s e^{\pm
\xi^{(j)}(t,\pm\partial)}
e_{jj}\\
&P^{\pm}(\alpha ) \equiv P^{\pm} (\tau_\alpha ,x,t,\partial ) =
I_{s} + \sum^{\infty}_{m=1} S^{\pm (m)} (\tau_\alpha ,x,t)\partial^{-m},\\
&R^{\pm}(\alpha ) \equiv R^{\pm}(\alpha ,\partial)\ \text{and}\
W^{\pm}(\alpha)\equiv
W^{\pm}(\tau_\alpha,x,t,z)=P^{\pm}(\alpha)R^{\pm}(\alpha)e^{\pm
\xi(t,\pm\partial)}
\endaligned \tag{4.7}$$
then:
$$\Psi^{\pm}(\tau_\alpha ,x,t,z) =W^{\pm}(\alpha)e^{\pm zx}= P^{\pm } (\alpha )
R^{\pm}(\alpha)e^{\pm\xi(t,\pm\partial)}e^{\pm zx}. \tag{4.8}$$

As usual one denotes the differential part of $P(x,t,\partial )$ by
$P_{+}(x,t,\partial ) = \sum_{j\ge 0} P_{j}(x,t) \partial^{j},$
and writes $P_{-} = P-P_{+}$. The  linear  anti-involution   $*$ is defined by
the
following formula:
$$(\sum_{j} P_{j}\partial^{j})^{*} = \sum_{j} (-\partial )^{j}
\circ ^{t}\! P_{j}. \tag{4.9}$$
Here and further $^{t}P$ stands for the transpose of the matrix $P$.
Then one has the following fundamental lemma:

\proclaim{Lemma 4.1} Let $P,Q$ be two formal pseudo--differential
operators, then $$(PQ^{*})_-=\sum_{i<0}R_i\partial^i$$
%
if and only if
$$\text{Res}_{z=0} dz (P(x,\partial) e^{z x})  \ ^{t}
(Q(x^{\prime},\partial^{\prime}) e^{-z x^{\prime}})  = \sum_{i<0}
R_i(x){(x-x')^{-i-1}\over (-i-1)!}.
$$
\endproclaim
\demo {Proof}
Let $y=x-x'$, $P(x,z)=\sum P_i(x)z^i$ and $Q(x,z)=\sum Q_i(x)(-z)^i$,
then
%
$$\aligned &\text{Res}_{z=0} dz(P(x,\partial) e^{z x})  \ ^{t}
(Q(x^{\prime},\partial^{\prime}) e^{-z x^{\prime}})
 =\\
&\text{Res}_{z=0} dz P(x,z)\sum_{k\ge 0}{(-1)^k\over k!}
{\partial^{k\ t}Q(x,-z)\over \partial x^k}y^ke^{zy}=\\
&\text{Res}_{z=0} dz \sum_{k,\ell\ge 0,i,j}{(-1)^k\over k!}
P_i(x){\partial^{k\ t}Q_j(x)\over \partial x^k}{y^{k+\ell}\over\ell !}
z^{i+j+\ell}=\\
&\sum_{k\ge 0,i+j\le -1}{(-1)^k\over k!(-i-j-1)!}
P_i(x){\partial^{k\ t}Q_j(x)\over \partial x^k}y^{k-i-j-1}=\\
&\sum_{k\ge 0,i+j\le -1}{i+j\choose k}
P_i(x){\partial^{k\ t}Q_j(x)\over \partial x^k}{y^{k-i-j-1}\over
(k-i-j-1)!}.
\endaligned\tag{4.10}
$$
Next we calculate
$$\aligned
(P(x,\partial)
Q^{*}(x,\partial))_-&=(\sum_{i,j}P_i(x)\partial^{i+j\ t}Q_j(x))_-\\
&=(\sum_{k\ge 0,i,j}{i+j\choose k}P_i(x){\partial^{k\ t}Q_j(x)
\over\partial x^k}\partial^{i+j-k})_-\\
&=(\sum_{k\ge 0,i+j-k<0}{i+j\choose k}P_i(x){\partial^{k\ t}Q_j(x)
\over\partial x^k}\partial^{i+j-k})_-\\
&=\sum_{k\ge 0,i+j\le -1}{i+j\choose k}
P_i(x){\partial^{k\ t}Q_j(x)\over \partial x^k}\partial^{i+j-k},
\endaligned \tag{4.11}$$
here we have used the fact that ${i+j\choose k}=0$ if $i+j>0$ and $i+j-k<0$.
Now comparing (4.10) and (4.11) gives the desired result.
$\qquad\square$

\enddemo

Using this Lemma one deduces the following
%
$$(W^+(\tau_\alpha,x,t,\partial)W^-(\tau_\beta,x,t',\partial))_-=0.\tag{4.12}$$
By putting $t=t'$, one proves in a similar way as in [KV] that
given $\beta \in \text{supp}\ \tau$, all the
pseudo-differential operators $P^{\pm} (\alpha )$, $\alpha \in \text{supp}\
\tau$, are
completely determined by $P^+(\beta )$ from the following equations
$$\align
&R^{-}(\alpha )^{-1} = R^{+}(\alpha
)^{*} ,
\tag{4.13}\\
&P^{-}(\alpha  ) = (P^{+}(\alpha  )^{*})^{-1} ,
\tag{4.14}\\
&(P^{+}(\alpha)R^{+}(\alpha - \beta)P^{+}(\beta)^{-1})_{-} = 0\
\text{for all}\ \alpha ,\beta \in \text{supp}\ \tau . \tag{4.15}
\endalign$$
This and the above lemma can be used to prove the following
proposition which will be crucial later on.
Adler, Shiota and van Moerbeke stated this proposition in
the 1--component case
[ASV2].
\proclaim{Proposition 4.2} Let $\Psi^\pm(\alpha,x,t,z)$ satisfy the
bilinear
identity (4.3) for $\beta=\alpha$ and let $Q(x,t,\partial)$ be an
arbitrary
pseudo--differential operator. Then $Q$ is a differential operator
if and only if
$$\text{Res}_{z=0}dz
Q(x,t,\partial)\Psi^+(\tau_\alpha,x,t,z)\Psi^-(\tau_\alpha,x',t',z)=0\tag{4.16}
$$
\endproclaim
\demo{Proof}
Suppose that $Q$ is a differential operator, then since by lemma 4.1
\break
 $W^+(\tau_\alpha,x,t,\partial)W^-(\tau_\alpha,x,t',\partial)$
is a differential operator
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
$$(Q(x,t,\partial)W^+(\tau_\alpha,x,t,\partial)W^-(\tau_\alpha,x,t',\partial))_-=0.\tag{4.17}$$
Conversely, suppose (4.16) holds, then again by Lemma , (4.17) holds. Now put
$t=t'$, and use (4.13-14), then one deduces that $Q(x,t,\partial)_-=0$.$
\qquad\square$
\enddemo
%
In [KV] Victor Kac and the author also showed
 the following
%
\proclaim{Proposition 4.3}  Consider $\Psi^{+}(\tau_\alpha ,x,t,z)$ and
 $\Psi^{-}(\tau_\alpha ,x,t,z),$ $\alpha \in Q$, of the
form (4.8), then the
bilinear identity (4.3) for all $\alpha ,\beta \in \ \text{supp}\
\tau$ is equivalent to the Sato equation:
$$
\frac{\partial P^+(\alpha)}{\partial t^{(j)}_{k}} = -(P^+(\alpha)e_{jj}
\partial^{k}  P^+(\alpha)^{-1})_{-}  P^+(\alpha), \tag{4.18}$$
for each $\alpha \in\text{supp}\ \tau$ and the matching conditions (4.13-15)
for
all $\alpha ,\beta \in \ \text{supp} \ \tau$.
\endproclaim
\noindent As a consequence of (4.18) one obtains for each
$W^+(\alpha)$:
$$\aligned \frac{\partial W^+(\alpha) }{\partial t^{(j)}_{k}} &=
(P^+(\alpha)e_{jj}
\partial^{k}  P^+(\alpha)^{-1})_{+}W^+(\alpha)
\\
&=(W^+(\alpha)e_{jj}\partial^kW^+(\alpha)^{-1})_+W^+(\alpha)\endaligned$$
  Fix $\tau$ and $\alpha ,\beta\in Q$, introduce the following formal
pseudo-differential
operators $L(\alpha)$, $ \Gamma$, $M(\alpha)$, $N(\alpha,\beta)$,
$\Delta^{(ij)}$, $ C^{(ij)}(\alpha )$  and
differential operators  $B^{(ij)}_{m}(\alpha)$:
$$\aligned
L(\alpha )
  & = W^+(\alpha)\partial W^+(\alpha)^{-1}=
P^{+}(\alpha)  \partial  P^{+}(\alpha)^{-1}, \\
\Gamma&=
x+\sum_{a=0}^s \sum_{k=0}^\infty kt^{(a)}_k\partial^{k-1}e_{aa}\\
M(\alpha)
&=W^+(\alpha)xW^+(\alpha)^{-1}=
P^{+}(\alpha)\Gamma
   P^{+}(\alpha)^{-1},\\
N(\alpha,\beta)&=W^+(\alpha)W^+(\beta)^{-1}
=P^+(\alpha)R^+(\alpha-\beta)P^+(\beta)^{-1}\\
\Delta^{(ij)}&=e^{\xi(t,\partial)}e_{ij}e^{-\xi(t,\partial)}\\
C^{(ij)}(\alpha ) &=
 W^+(\alpha)e_{ij} W^+(\alpha)^{-1}=
P^{+}(\alpha)\Delta^{(ij)}P^{+}(\alpha)^{-1}, \\
B^{(ij)}_{m}(\alpha) &=
( W^+(\alpha)e_{ij}\partial^mW^+(\alpha)^{-1})_+=
(P^{+}(\alpha)e_{ij} \partial^{m}
P^{+}(\alpha)^{-1})_{+}.
\endaligned \tag{4.20}
$$
%
Here we write $x$ for $xI_s$. Denote by
$C^{(i)}(\alpha)=C^{(ii)}(\alpha)$
and $B^{(i)}_m(\alpha)=B^{(ii)}_m(\alpha)$. Then
%
%
$$\aligned
&[L(\alpha),M(\alpha)]=I_s,\
\sum^{s}_{i=1} C^{(i)}(\alpha) = I_{s}, \
C^{(ij)}(\alpha)L(\alpha) = L(\alpha)C^{(ij)}(\alpha),\\
&C^{(ij)}(\alpha)C^{(k\ell)}(\alpha) = \delta_{jk}
C^{(i\ell)}(\alpha),\
L(\alpha)N(\alpha,\beta)=N(\alpha,\beta)L(\beta),\\
&M(\alpha)N(\alpha,\beta)=N(\alpha,\beta)M(\beta),\
C^{(ij)}(\alpha)N(\alpha,\beta)=N(\alpha,\beta)C^{(ij)}(\beta),\\
&N(\alpha,\beta)N(\beta,\gamma)=N(\alpha,\gamma).
\endaligned
 \tag{4.21}$$
\proclaim{Remark 4.4}(i) It is our purpose to describe the general
operators
$$Y^{(ab,\ell+1)}_k(\alpha,\beta)=W^+(\alpha)x^\ell\partial^{k+\ell}e_{ab}
W^+(\beta)^{-1}.$$
One can express them in the operators defined in (4.20), viz.,
$$Y^{(ab,\ell+1)}_k(\alpha,\beta)=M(\alpha)^\ell L(\alpha)^{k+\ell}
C^{(ab)}(\alpha)N(\alpha,\beta).$$
(ii) Notice that (4.14) is equivalent to $N(\alpha,\beta)_-=0$,
so from now on we will assume that $N(\alpha,\beta)$ is a differential
operator.
\endproclaim
\proclaim{Proposition 4.5} If for every $\alpha, \beta \in Q$
the
formal pseudo-differential operators $L(\alpha)$,
 $M(\alpha)$, $
C^{(ij)}(\alpha)$  and the differential operators $N(\alpha,\beta)$ satisfy
conditions (4.21) and if the
equations
$$
\cases
L(\alpha)P^+(\alpha)R^+(\alpha) = P^+(\alpha)R^+(\alpha)\partial \quad(\text{or
equivalently}\ L(\alpha)P^+(\alpha)=P^+(\alpha)\partial)&\ \\
M(\alpha)P^+(\alpha)R^+(\alpha)=P^+(\alpha)R^+(\alpha)\Gamma
&\ \\
N(\alpha,\beta)P^+(\beta)R^+(\beta)=P^+(\alpha)R^+(\alpha)&\ \\
C^{(ij)}(\alpha)P^+(\alpha)R^+(\alpha) = P^+(\alpha)R^+(\alpha)\Delta^{(ij)} &\
\\
\displaystyle{\frac{\partial P^+(\alpha)}{\partial t^{(j)}_{k}} =
-(C^{(j)}(\alpha)L(\alpha)^{k})_{-} P^+(\alpha)}&\
\endcases
\tag{4.22}$$
 have a solution $P^+(\alpha)$ of the form (4.7), then the
differential operators $B^{(j)}_{k}(\alpha)$
satisfies the following
conditions:
%
$$\cases \displaystyle{\frac{\partial L(\alpha)}{\partial t^{(j)}_{k}} =
[B^{(j)}_{k}(\alpha),L(\alpha)],}  & \   \\
\displaystyle{\frac{\partial M(\alpha)}{\partial t^{(j)}_{k}} =
[B^{(j)}_{k}(\alpha),M(\alpha)],}  & \   \\
\displaystyle{\frac{\partial N(\alpha,\beta)}{\partial t^{(j)}_{k}} =
B^{(j)}_{k}(\alpha)N(\alpha,\beta)-N(\alpha,\beta)B^{(j)}_{k}(\beta),}&\
\\
\displaystyle{\frac{\partial C^{(i\ell)}(\alpha)}{\partial t^{(j)}_{k}} =
[B^{(j)}_{k}(\alpha),C^{(i\ell)}(\alpha)],} & \
\  \endcases. \tag{4.23}$$
\endproclaim
\noindent Now (4.23) implies that
$$\frac{\partial Y^{(ab,\ell+1)}_m(\alpha, \beta)}{\partial t^{(j)}_{k}} =
B^{(j)}_{k}(\alpha)Y^{(ab,\ell+1)}_m(\alpha, \beta)
-Y^{(ab,\ell+1)}_m(\alpha, \beta)B^{(j)}_{k}(\beta)$$
Notice that the conditions (4.22) are equivalent to one of the following
conditions:
$$
\cases L(\alpha)W^+(\alpha)= W^+(\alpha)\partial, &\ \\
M(\alpha)W^+(\alpha)=W^+(\alpha)x, &\ \\
N(\alpha,\beta)W^+(\beta)=W^+(\alpha),&\ \\
C^{(ij)}(\alpha)W^+(\alpha) = W^+(\alpha)e_{ij}, &\ \\
\displaystyle{\frac{\partial W^+(\alpha)}{\partial t^{(j)}_{k}}
= B^{(j)}_k(\alpha) W^+(\alpha),}
\endcases
\qquad
\cases L(\alpha)\Psi^+(\alpha,z)= z\Psi^+(\alpha,z),&\ \\
M(\alpha)\Psi^+(\alpha,z)={\partial\Psi^+(\alpha,z)\over\partial z},
&\ \\
N(\alpha,\beta)\Psi^+(\beta,z)=\Psi^+(\alpha,z),&\ \\
C^{(ij)}(\alpha)\Psi^+(\alpha,z) = \Psi^+(\alpha,z) e_{ij}, &\ \\
\displaystyle{\frac{\partial \Psi^+(\alpha,z)}{\partial t^{(j)}_{k}}
= B^{(j)}_k(\alpha) \Psi^+(\alpha,z).}
\endcases
\tag{4.24}$$

\subheading{\S5 The Adler--Shiota--van Moerbeke formula}
\vskip 10pt
\noindent
Fix $\alpha, \beta\in Q$ and recall
$$Y^{(ab,\ell+1)}_k(\alpha,\beta)\equiv
Y^{(ab,\ell+1)}_k(\tau_\alpha,\tau_\beta)
=W^+(\alpha)x^\ell\partial^{k+\ell}e_{ab}
W^+(\beta)^{-1},\tag{5.1}$$
then define
$$\aligned
Y^{(ab)}(\alpha,\beta,y,w)&\equiv Y^{(ab)}(\tau_\alpha,\tau_\beta,y,w)=
\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell !}\sum_{k\in \Bbb Z}
w^{-k-\ell-1}Y^{(ab,\ell+1)}_k(\alpha,\beta)\\
&=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell !}\sum_{k\in \Bbb Z}
w^{-k-\ell-1}M(\alpha)^\ell L(\alpha)^{k+\ell}C^{(ab)}(\alpha)N(\alpha,\beta).
\endaligned\tag{5.2}$$
We write $Y^{(ab,\ell+1)}_k(\alpha)$ and
$Y^{(ab)}(\alpha,y,w)\equiv Y^{(ab)}(\tau_\alpha,y,w)$ for respectively
$Y^{(ab,\ell+1)}_k(\alpha,\alpha)$,
$Y^{(ab)}(\alpha,\alpha,y,w)$,
then
$$Y^{(ab)}(\alpha,y,w)=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell !}\sum_{k\in
\Bbb Z}
w^{-k-\ell-1}M(\alpha)^\ell L(\alpha)^{k+\ell}C^{(ab)}(\alpha).$$
One deduces  the following
%
%
\proclaim{Proposition 5.1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
$$Y^{(ab)}(\alpha,\beta,y,w)_-=\Psi^+(\alpha,y)e_{ab}\partial^{-1}\,^t\Psi^-(\beta,
w)\tag{5.3}$$
\endproclaim
\demo{Proof}
First, notice that
%
$$\aligned
(W^+(\alpha)x^\ell\partial^{k+\ell}e_{ab}W^{+}(\beta)^{-1})_-
&=\sum_{j=1}^\infty\partial^{-j}\text{Res}_\partial\partial^{j-1}
W^+(\alpha)x^\ell\partial^{k+\ell}e_{ab}W^{+}(\beta)^{-1}\\
&=\sum_{j=1}^\infty\partial^{-j}\text{Res}_{z=0}dz (\partial^{j-1}
W^+(\alpha)x^\ell\partial^{k+\ell}e^{zx})e_{ab}\,^t(W^-(\beta)e^{-zx})\\
&=\text{Res}_{z=0}dz\sum_{j=1}^\infty\partial^{-j}[({\partial\over\partial
x})^{j-1}(z^{k+\ell}{\partial^\ell \Psi^+(\alpha,z)\over\partial
z^\ell})]e_{ab}\,^t\Psi^-(\beta,z)\\
&=\text{Res}_{z=0}dz z^{k+\ell}{\partial^\ell \Psi^+(\alpha,z)\over\partial
z^\ell} e_{ab}\partial^{-1}\,^t\Psi^-(\beta,z)
\endaligned
$$
%
Hence
%
$$\aligned
Y^{(ab)}(\alpha,\beta,y,w)_-&=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell
!}\sum_{k\in \Bbb Z}
w^{-k-\ell-1}\text{Res}_{z=0}dz z^{k+\ell}{\partial^\ell
\Psi^+(\alpha,z)\over\partial
z^\ell} e_{ab}\partial^{-1}\,^t\Psi^-(\beta,z)\\
&=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell !}
{\partial^\ell \Psi^+(\alpha,w)\over\partial
w^\ell} e_{ab}\partial^{-1}\,^t\Psi^-(\beta,w)\\
&=\Psi^+(\alpha,y) e_{ab}\partial^{-1}\,^t\Psi^-(\beta,w)\qquad\square
\endaligned
$$
\enddemo
\noindent Proposition 5.1 was obtained in the 1--component case by
Dickey [D].

Next we calculate
$$\aligned
Y^{(ab)}(\alpha,\beta,y,w)\Psi^+(\beta,z)
&=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell !}\sum_{k\in \Bbb Z}
w^{-k-\ell-1}W^+(\alpha)x^\ell\partial^{k+\ell}e_{ab}e^{zx}\\
&=W^+(\alpha)\delta(w,z)e^{(y-w){\partial\over\partial
z}}e^{zx}e_{ab}\\
&=W^+(\alpha)\delta(w,z)e^{(z+y-w)x}e_{ab}\\
&=W^+(\alpha)\delta(w,z)e^{yx}e_{ab}\\
&=\delta(w,z)\Psi^+(\alpha,y)e_{ab},
\endaligned\tag{5.4}
$$
%
where $\delta(w,z)=\sum_{n\in\Bbb Z}w^{-n}z^{n-1}$.

Define $$\aligned
\Bbb X^{(ab)}(y,w)&=X^{(ab)}(y,w)e^{(y-w)x},\\
\Bbb W^{(ab)}(y,w)&=\sum_{\ell=0}^\infty {(y-w)^\ell\over \ell !}
\sum_{k\in \Bbb Z}\Bbb W_k^{(ab,\ell+1)} w^{-k-\ell-1}\\
&=(y-w)^{-\delta_{ab}}(\Bbb X^{(ab)}(y,w)-\delta_{ab}),\endaligned
\tag{5.5}$$
then
$$\Bbb W^{(ab)}(y,w)=:\psi^{+(a)}(y)\psi^{-(b)}(w):e^{(y-w)x}.$$
It is straightforward that the ${\Bbb W}_k^{(ab,\ell+1)}$ have the
same commutation relations as the $ W_k^{(ab,\ell+1)}$, since we have
only replaced  all $t^{(j)}_1$ by $t^{(j)}_1+x$ and kept
$\partial\over\partial t^{(j)}_1$ unchanged in the vertex operator
(3.11) of $W^{(ab)}(y,w)$ ($\partial\over\partial x$ does not appear
in this expression).

One has the following
\proclaim {Lemma 5.2}
$$\Bbb
X^{(ab)}(y,w)\psi^{+(j)}(z)e^{zx}=\delta_{bj}(y-w)^{\delta_{ab}}
\delta(w,z)\psi^{+(a)}(y)e^{yx}+\psi^{+(j)}(z)e^{zx}\Bbb
X^{(ab)}(y,w).$$
\endproclaim
\demo{Proof}
Let $\gamma\in Q$, we calculate
$$\aligned \Bbb
X^{(ab)}&(y,w)\psi^{+(j)}(z)e^{zx}\tau_\gamma(x,t)e^\gamma=\\
&\epsilon(\delta_a,\delta_b)\epsilon(\delta_a-\delta_b,\gamma+\delta_j)
\epsilon(\delta_j,\gamma)y^{(\delta_a|\gamma+\delta_j)}
w^{-(\delta_b|\gamma+\delta_j)}z^{(\delta_j|\gamma)}
(1-{z\over y})^{\delta_{aj}}(1-{z\over w})^{-\delta_{bj}}\\
&\qquad\times e^{\xi^{(a)}(x,t,y)-\xi^{(b)}(x,t,w)+\xi^{(j)}(x,t,z)}
e^{-\eta^{(a)}(t,y)+\eta^{(b)}(t,w)-\eta^{(j)}(t,z)}\tau_\gamma
e^{\gamma+\delta_j+\delta_a-\delta_b}
\endaligned$$
Now use the fact that
$(1-{z\over w})^{-1}=w\delta(w,z)-{w\over z}(1-{w\over z})^{-1}$,
$1-{z\over y}=-{z\over y}(1-{y\over z})$ and that
$\epsilon(\delta_m,\delta_n)\epsilon(\delta_n,\delta_m)=-(-1)^{\delta_{mn}}$,
then
$$\aligned \Bbb
X^{(ab)}&(y,w)\psi^{+(j)}(z)e^{zx}\tau_\gamma(x,t)e^\gamma\\
&=\delta_{bj}
\epsilon(\delta_a,\delta_b)\epsilon(\delta_a-\delta_b,\gamma+\delta_b)
\epsilon(\delta_b,\gamma)(y-z)^{\delta_{ab}}\delta(w,z)
y^{(\delta_a|\gamma)}
w^{-(\delta_b|\gamma)}z^{(\delta_b|\gamma)}\\
&\qquad\times e^{\xi^{(a)}(x,t,y)-\xi^{(b)}(x,t,w)+\xi^{(b)}(x,t,z)}
e^{-\eta^{(a)}(t,y)+\eta^{(b)}(t,w)-\eta^{(b)}(t,z)}\tau_\gamma
e^{\gamma+\delta_a}\\
&\quad+\epsilon(\delta_a,\delta_b)\epsilon(\delta_a-\delta_b,\gamma+\delta_j)
\epsilon(\delta_j,\gamma)(-)^{\delta_{aj}}(-)^{\delta_{bj}}
y^{(\delta_a|\gamma)}
w^{-(\delta_b|\gamma)}z^{(\delta_j|\gamma+\delta_a-\delta_b)}\\
&\qquad\times e^{\xi^{(j)}(x,t,z)}e^{\eta^{(j)}(t,z)}
e^{\xi^{(a)}(x,t,y)-\xi^{(b)}(x,t,w)}
e^{-\eta^{(a)}(t,y)+\eta^{(b)}(t,w)}\tau_\gamma
e^{\gamma+\delta_j+\delta_a-\delta_b}\\
&=\{ \delta_{bj}(y-w)^{\delta_{ab}}
\delta(w,z)\psi^{+(a)}(y)e^{yx}+
+\psi^{+(j)}(z)e^{zx}\Bbb
X^{(ab)}(y,w)\}\tau_\gamma e^\gamma.\qquad\square
\endaligned
$$\enddemo

Recall the bilinear identity (3.14) with $\alpha$ replaced by
$\alpha+\delta_b-\delta_a$ in a slightly different version, viz.,
%
$$\aligned \text{Res}_{z=0}dz&\sum_{j=1}^s\psi^{+(j)}(z)e^{zx}
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)
e^{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}\\
&\psi^{-(j)'}(z)e^{-zx'}
\tau_{\beta+\delta_j-\delta_k}(x',t')(e^{\beta+\delta_j-\delta_k})' =0.
\endaligned
$$
Let $\Bbb X^{(ab)}(y,w)$ act on this identity, then using Lemma 5.2
one obtains:
$$\aligned
&\text{Res}_{z=0}dz\{ (y-w)^{\delta_{ab}}\delta(w,z)
\psi^{+(a)}(y)e^{zx}
\tau_{\alpha+\delta_i-\delta_a}(x,t)
e^{\alpha+\delta_i-\delta_a}\psi^{-(a)'}(z)\\
&+\sum_{j=1}^s\psi^{+(j)}(z)e^{zx}\Bbb X^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)
e^{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}\psi^{-(j)'}(z)\} \\
&\times e^{-zx'}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\tau_{\beta+\delta_j-\delta_k}(x',t')(e^{\beta+\delta_j-\delta_k})'=0.\endaligned
\tag{5.6}$$
Now divide by $\tau_\alpha(x,t)\tau_\beta(x',t')$ and remove the factors
$e^{\alpha+\delta_i}$ and $(e^{\beta-\delta_k})'$. Notice that by
doing this, the action of $\Bbb X^{(ab)}(y,w)$ is no longer
well--defined, for that reason we introduce  ${\hat X}^{(ab)}(y,w)$
by
$$\aligned
\hat X^{(ab)}&(y,w)\tau_\gamma(x,t)\\
&=\epsilon(\delta_a,\delta_b)
\epsilon(\delta_a-\delta_b,\gamma)y^{(\delta_a|\gamma)}w^{-(\delta_b|\gamma)}
e^{\xi^{(a)}(x,t,y)-\xi^{(b)}(x,t,w)}e^{-\eta^{(a)}(t,y)+\eta^{(b)}(t,w)}
\tau_\gamma(x,t).\endaligned$$ and
$$\aligned
{\hat W}^{(ab)}(y,w)&=\sum_{\ell=0}^\infty {(y-w)^\ell\over\ell
!}\sum_{k\in\Bbb Z} {\hat W}_k^{(ab,\ell+1)}w^{-k-\ell-1}\\
&=(y-w)^{-\delta_{ab}}(\hat X^{(ab)}(y,w)-\delta_{ab}).\endaligned
\tag{5.7}$$
Then (5.6) turns into
$$\aligned
\text{Res}_{z=0}dz&\{ (y-w)^{\delta_{ab}}\delta(w,z)
\Psi^{+}_{ia}(\alpha,y)\Psi^{-}_{kb}(\beta,z)'\\
&+\sum_{j=1}^s
e^{-\eta^{(j)}(t,z)}\left ({{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )\Psi^{+}_{ij}(\alpha,z)
\Psi^{-}_{kj}(\beta,z)'\}=0.\endaligned$$
Using (5.4) one obtains
$$
\aligned
\text{Res}_{z=0}dz&\{ e_{ii}  ( (y-w)^{\delta_{ab}}Y^{(ab)}(\alpha,y,w)
\Psi^+(\alpha,z)\\
&+\sum_{j=1}^s
e^{-\eta^{(j)}(t,z)}\left ({{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right ))\Psi^{+}(\alpha,z)e_{jj}
\}^t\Psi^{-}(\beta,z)'=0.\endaligned$$
Now notice that
$$\aligned
e^{-\eta^{(j)}(t,z)}&\left ({{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right
)\Psi^{+}(\alpha,z)e_{jj}=
\sum_{k=0}^\infty
c_{jk}L(\alpha)^{-k}C^{(j)}(\alpha)\Psi^{+}(\alpha,z)\\
&=\{ (\sum_{k=1}^\infty
c_{jk}L(\alpha)^{-k}C^{(j)}(\alpha))_-
+{{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)} e_{jj}\}\Psi^{+}(\alpha,z),
\endaligned
$$
hence
$$\text{Res}_{z=0}dz e_{ii} \left (
(y-w)^{\delta_{ab}}Y^{(ab)}(\alpha,y,w)+\sum_{j=1}^s\sum_{k=1}^\infty
c_{jk}L(\alpha)^{-k}C^{(j)}(\alpha)\right
)\Psi^+(\alpha,z)^t\Psi^{-}(\beta,z)'=0.$$
Now using Proposition 4.2 for  $\beta=\alpha$, one obtains
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
$$e_{ii}((y-w)^{\delta_{ab}}Y^{(ab)}(\alpha,y,w)_-+\sum_{j=1}^s(\sum_{k=0}^\infty
c_{jk}L(\alpha)^{-k}C^{(j)}(\alpha))_-)=0.$$
Hence
$$\aligned
-e_{ii}(y-w)^{\delta_{ab}}&Y^{(ab)}(\alpha,y,w)_-\Psi^+(\alpha,z)=\\
&e_{ii}\sum_{j=1}^s  e^{-\eta^{(j)}(t,z)}\left ({{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )\Psi^+(\alpha,z)e_{jj}\\
&\qquad -
{{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_b-\delta_a}(x,t)\over
\tau_{\alpha}(x,t)} e_{jj}\Psi^{+}(\alpha,z).
\endaligned
$$
So we obtain the following generalization of the Adler--Shiota--van
Moerbeke formula
\proclaim{Theorem 5.3}
%
$$\aligned
&(y-w)^{\delta_{ab}}(-Y^{(ab)}(\alpha,y,w)_-\Psi^+(\alpha,z))_{ij}=\\
&\{ e^{-\eta^{(j)}(t,z)}\left ({{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )-
{{\hat X}^{(ab)}(y,w)
\tau_{\alpha+\delta_b-\delta_a}(x,t)\over
\tau_{\alpha}(x,t)}\} \Psi^{+}_{ij}(\alpha,z).
\endaligned\tag {5.8}
$$
\endproclaim
In a similar way as in the introduction the operator
$e^{\lambda \Bbb X^{(ab)}(y,w)}=1+\lambda \Bbb X^{(ab)}(y,w)$ is an
auto--B\"acklund transformation of the $s$--component KP hierarchy (see
also [KV]). Now let
$$\sigma (x,t)=\sum_{\gamma\in
Q}\sigma_\gamma(x,t)e^\gamma=e^{\lambda \Bbb X^{(ab)}(y,w)}\sum_{\gamma\in
Q}\tau_\gamma(x,t)e^\gamma=
\sum_{\gamma\in Q}(\tau_\gamma(x,t)
+\lambda{\hat X}^{(ab)}(y,w)\tau_{\gamma+\delta_b-\delta_a})e^\gamma,$$
then (5.8) is equal to
$$\aligned
-\lambda(y-w)^{\delta_{ab}}(
Y^{(ab)}(\tau_\alpha,y,w)_-&\Psi^+(\tau_\alpha,x,t,z))_{ij}=\\
&\{ e^{-\eta^{(j)}(t,z)}\left ({\sigma_{\alpha+\delta_i-\delta_j}(x,t)
\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )
-
{\sigma_\alpha(x,t)\over
\tau_{\alpha}(x,t)}\} \Psi^{+}_{ij}(\tau_\alpha,x,t,z).
\endaligned
$$
So
$$\aligned
\Psi_{ij}^+(\sigma_\alpha,x,t,z)&=
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{e^{-\eta^{(j)}(t,z)}\sigma_{\alpha+\delta_i-\delta_j}(x,t)\over\sigma_{\alpha}(x,t)}
e^{\xi^{(j)}(x,t,z)}\\
&={\tau_\alpha(x,t)\over\sigma_{\alpha}(x,t)}e^{-\eta^{(j)}(t,z)}\left(
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{\sigma_{\alpha+\delta_i-\delta_j}(x,t)\over\tau_{\alpha+\delta_i-\delta_j}(x,t)}
\right)\Psi_{ij}^+(\tau_\alpha,x,t,z)\\
&=\Psi_{ij}^+(\sigma_\alpha,x,t,z)+{\tau_\alpha(x,t)\over\sigma_{\alpha}(x,t)}
(e^{-\eta^{(j)}(t,z)}-1)\left(
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{\sigma_{\alpha+\delta_i-\delta_j}(x,t)\over\tau_{\alpha+\delta_i-\delta_j}(x,t)}
\right)\Psi_{ij}^+(\tau_\alpha,x,t,z)\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&=\Psi_{ij}^+(\sigma_\alpha,x,t,z)-\lambda{\tau_\alpha(x,t)\over\sigma_{\alpha}(x,t)}
(y-w)^{\delta_{ab}}\left
(Y^{(ab)}(\tau_\alpha,y,w)_-\Psi^+(\tau_\alpha,x,t,z)\right )_{ij}\\
&=\left (\left(1-\lambda{\tau_\alpha(x,t)\over\sigma_{\alpha}(x,t)}
(y-w)^{\delta_{ab}}Y^{(ab)}(\tau_\alpha,y,w)_-\right
)\Psi^+(\tau_\alpha,x,t,z)\right )_{ij}
\endaligned $$
and we obtain the following consequence of Theorem 5.3:
\proclaim{Corollary 5.4}
Let $\tau(x,t)$ be a solution of the $s$--component KP hierarchy, then
$\sigma(x,t)=e^{\lambda \Bbb X^{(ab)}(y,w)}\tau(x,t)$ is a new
solution of this hierarchy and the wave functions are related by
$$\Psi_{ij}^+(\sigma_\alpha,x,t,z)=
\left (\left(1-\lambda{\tau_\alpha(x,t)\over\sigma_{\alpha}(x,t)}
(y-w)^{\delta_{ab}}Y^{(ab)}(\tau_\alpha,y,w)_-\right
)\Psi^+(\tau_\alpha,x,t,z)\right )_{ij}$$
\endproclaim
\noindent Hence (5.8) relates this B\"acklund transformation of the
$s$--component KP hierarchy acting on the $\tau$--function to a
``B\"acklund transformation''
on the wave function.

Since the left--hand--side of (5.8) is also equal to
$$\aligned
 \{ e^{-\eta^{(j)}(t,z)}\left ({({\hat X}^{(ab)}(y,w)-\delta_{ab})
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )&\ \\
-
{({\hat X}^{(ab)}(y,w)-\delta_{ab})
\tau_{\alpha+\delta_b-\delta_a}(x,t)\over
\tau_{\alpha}(x,t)}& \} \Psi^{+}_{ij}(\alpha,z),\endaligned$$
we have the following
\proclaim{Corollary 5.5}
$$\aligned (-(M&(\alpha)^\ell
L(\alpha)^{k+\ell}C^{(ab)}(\alpha))_-\Psi^+(\alpha,z))_{ij}=\\
&\{ e^{-\eta^{(j)}(t,z)}\left ({{\hat W}^{(ab,\ell+1)}_k
\tau_{\alpha+\delta_i+\delta_b-\delta_j-\delta_a}(x,t)\over
\tau_{\alpha+\delta_i-\delta_j}(x,t)}\right )-
{{\hat W}^{(ab,\ell+1)}_k
\tau_{\alpha+\delta_b-\delta_a}(x,t)\over
\tau_{\alpha}(x,t)}\} \Psi^{+}_{ij}(\alpha,z).\endaligned$$
\endproclaim
\demo{Proof} Compare in (5.8) the expansions for the vertex operators
$Y^{(ab)}(\alpha,y,w)$
as in (5.2) and for ${\hat W}^{(ab)}(y,w)$ as in (5.7).\qquad $\square$
\enddemo

As an application of Corollary 5.5 we see that if
$$\sum_{a=1}^s\sum_{\ell=0}^\infty\sum_{k\in\Bbb Z} c_{a\ell k}(M(\alpha)^\ell
L(\alpha)^{k+\ell}C^{(a)}(\alpha))_-=0,$$   one
finds that  that for any $1\le j\le s$
$$(e^{-\eta^{(j)}(t,z)}-1)\left
({\sum_{a=1}^s\sum_{\ell=0}^\infty\sum_{k\in\Bbb
Z} c_{a\ell k}
{\hat W}^{(a,\ell+1)}_k
\tau_{\alpha}(x,t)\over
\tau_{\alpha}(x,t)}\right )=0,$$
hence
$$\sum_{a=1}^s\sum_{\ell=0}^\infty\sum_{k\in\Bbb
Z} c_{a\ell k}
\Bbb W^{(a,\ell+1)}_k
\tau_{\alpha}(x,t)e^{\alpha}
=\text{constant}\,\tau_\alpha(x,t)e^\alpha.$$
Thus Corollary 5.5 provides an alternative proof of Theorem 6.5 of [V].



\vskip 1cm
\Refs
\widestnumber\key{\bf DJKM3}

\ref \key{\bf ASV1} \by M. Adler, T. Shiota and P. van Moerbeke\paper
{}From the $w_\infty$-algenra to its central extension: a
$\tau$-function approach\jour to appear in Physics Letters A\endref

\ref \key{\bf ASV2} \by M. Adler, T. Shiota and P. van Moerbeke\paper
A Lax representation of the vertex operator and the central extension
\jour
to appear in Comm. Math. Phys.\endref

\ref \key{\bf DJKM1} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Operator approach to the Kadomtsev-Petviashvili equation.
Transformation groups for soliton equations. III \jour J. Phys. Soc. Japan
\vol 50 \yr 1981 \pages 3806--3812\endref

\ref \key{\bf DJKM2} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Transformation groups for soliton equations.  Euclidean Lie
algebras and reduction of the KP hierarchy \jour Publ. Res. Inst.
Math. Sci. \vol 18 \yr 1982 \pages 1077--1110\endref

\ref \key{\bf DJKM3} \by E. Date, M. Jimbo, M. Kashiwara and T. Miwa
\paper Transformation groups for soliton equations \jour in:
Nonlinear integrable systems---classical theory and quantum theory
eds M. Jimbo and T. Miwa, World Scientific, 1983\pages 39--120\endref

\ref \key{\bf D} \by  L.A. Dickey \paper On additional symmetries of
the KP hierarchy and Sato's B\"acklund transformation
 \jour preprint University of Oklahoma \yr 1992\endref

\ref \key{\bf FK} \by I.B. Frenkel and V.G. Kac \paper Basic
representations of affine Lie algebras and dual resonance models
\jour Invent. Math. \vol 62 \yr 1980 \pages 23--66\endref


\ref \key{\bf K} \by V.G. Kac \paper Infinite dimensional Lie algebras
\jour Progress in Math., vol. 44, Brikh\"{a}user, Boston, 1983; 2nd
ed., Cambridge Univ. Press, 1985; 3d ed., Cambridge Univ. Press,
1990\endref

\ref \key{\bf KP1} \by V.G. Kac and D.H. Peterson \paper Spin and
wedge representations of infinite dimensional lie algebras and
groups\jour Proc. Nat. Acad. Sci. U.S.A.\yr 1981\pages 3308-3312\endref

\ref \key{\bf KP2} \by V.G. Kac and D.H. Peterson \paper Lectures on the
infinite wedge representation and the MKP hierarchy \jour Sem. Math.
Sup., vol. 102, Presses Univ. Montreal, Montreal, 1986, pp.
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\ref\key{\bf KRa}\by V. Kac and A. Radul \paper Quasifinite highest weight
modules over the Lie algebra of differential operators on the circle
\jour Comm. Math. Phys.\vol 157 \yr 1993 \pages 429-457
\endref


\ref\key{\bf KV}\by V. Kac and J. van de Leur\paper The $n$--Component KP
hierarchy and Representation Theory.\jour in Important Developments in Soliton
Theory, eds. A.S. Fokas and V.E. Zakharov. Springer Series in Nonlinear
Dynamics,\yr 1993\pages 302--343\endref

\ref\key{\bf Ra} \by A.O. Radul\paper Lie algebras of differential operators,
their central extensions, and W--algebras \jour Funct. Anal. and its Appl.\vol
25
\yr 1991 \pages 33--49\endref


\ref\key{\bf OS}\by A.Yu. Orlov and E.I. Schulman\paper Additional
symmetries for integrable and conformal algebra representations\jour
Lett. Math. Phys.\vol 12\yr 1986 \pages 171\endref

\ref\key{\bf S}\by G.Segal\paper Unitary representations of some
infinite dimensional groups
\jour Commun. Math. Phys. \vol 80 \yr
1981\pages 301--342\endref

\ref \key{\bf TV} \by F. ten Kroode and J. van de Leur \paper Bosonic and
fermionic realizations of the affine algebra $\hat{gl}_{n}$ \jour
Comm. Math. Phys.\ \vol 137 \yr 1991 \pages 67--107\endref

\ref \key{\bf UT} \by K. Ueno and K. Takasaki \paper Toda lattice hierarchy
\jour Adv. stud. Pure Math., vol. 4, North-Holland, 1984 \pages
1--95\endref


\ref \key{\bf V}\by J. van de Leur \paper KdV--type hierarchies, the string
equation and $W_{1+\infty}$ constraint \jour Utrecht University preprint and
hep-th 9403080 to appear in Jour. Geom. and Phys\endref

\endRefs
\vskip 20pt
\noindent J.W. van de Leur

\noindent Faculty of Applied Mathematics

\noindent University of Twente

\noindent P.O. Box 217

\noindent 7500 AE Enschede

\noindent The Netherlands






\enddocument


