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\topmatter
\title   
Schlesinger--B\"acklund transformations for the 
{\it n}-component KP 
\endtitle
\author
Johan van de Leur
\endauthor
\affil
Faculty of Applied Mathematics\\
University of Twente\\
P.O. Box 217\\
7500 AE Enschede\\
 The Netherlands\\
\endaffil
\thanks
The author is
supported by the ``Stichting Fundamenteel Onderzoek der
Materie (F.O.M.)''. E-mail: vdleur\@math.utwente.nl
\endthanks
\abstract
The $n$--component Kadomtsev--Petviashvili (KP) hierarchy was
introduced by Sato as the set of deformation 
equations, i.e. Lax equations,
of $n+1$ $n\times n$--matrix pseudodifferential operators.
On such a set of operators 
we define a 
Schlesinger--B\"acklund transformation, which produces from a set of
solutions of the KP hierachy 
a new set of solutions. Applying such 
transformations recursively we obtain a whole family of 
solutions, from which we can then destilate 
the KP tau-function. 
%
\vskip .3cm \noindent {\it Keywords}: 
KP hierarchy, 
B\"acklund transformation,
tau-function
\vskip .3cm \noindent {\it AMS Subject Classification (1991)}:
35Q58, 58F07, 58F37.
\vskip .3cm \noindent {\it PACS numbers} 02.30J, 02.20T,
02.90.

\endabstract
\endtopmatter

\break

\subheading{I. Introduction and formulation of the problem}
\vskip 10pt
\noindent 
The $n$--component Kadomtsev--Petviashvili (KP) hierarchy was
introduced by Sato in [1] (see also [2], [3]) as the set of deformation 
equations, also called Lax equations
($i,j=1,2,\ldots,n$, $k=1,2,\ldots$)
$$\aligned
{{\partial L}\over{\partial t_k^{(i)}}}&=[(L^kC^{(i)})_+,L],\\
{{\partial C^{(j)}}\over {\partial t_k^{(i)}}}
&=[(L^kC^{(i)})_+,C^{(j)}]\\
\endaligned\tag{1.1}
$$
of the set of $n\times n$--matrix pseudodifferential operator
$$\aligned 
L\equiv
L(x,t,\partial)=&
I_n\partial +U^{(1)}(x,t)
\partial^{-1}+U^{(2)}(x,t)\partial^{
-2}+\cdots,\\
C^{(j)}\equiv C^{(j)}(x,t,\partial)&=
E_{jj}+C^{(j1)}(x,t)\partial^{-1}
+C^{(j2)}(x,t)\partial^{-2}+\cdots\\
\endaligned
\tag{1.2}
$$
that satisfies the conditions
$$
LC^{(j)}=C^{(j)}L,\qquad C^{(i)}C^{(j)}=\delta_{ij}C^{(i)},
\qquad \sum_{i=1}^n C^{(i)}=I_n.\tag{1.3}
$$
Here $\partial={\partial\over\partial x}$, 
$t=\{t_k^{(i)}|1\le i\le n,\ k=1,2,\ldots\}$ and the 
coefficients $L^{(k)}$ and $C^{(jk)}$ are formal power series
over $\Bbb C$ in the indeterminates $x$ and $t$. $(A)_+$
stands for the differential part 
of a pseudodifferential operator $A$. The multiplication of
pseudodifferential operators is given by
$$A(x,t)\partial^k\circ B(x,t)\partial^{-\ell}=
\sum_{m=0}^\infty {k\choose m}A(x,t){{\partial^m B(x,t)}\over
{\partial x^m}}\partial^{k-m-\ell}
\qquad k\in {\Bbb Z}.
$$ 
The $n$--component KP hierarchy contains many of the 
well--known partial differential equations, e.g. the classical
Kadomtsev--Petviashvili ($n=1$), 
Davey--Stewartson ($n=2$) 
and the $n$--wave interaction equations ($n\ge 3$).

Starting from the level 1 representation theory of the 
infinite matrix group $GL_\infty$, V. Kac and the author
[4] and independently M. Bergvelt and
F. ten Kroode [5] showed that to every element $\tau$ of the
$GL_\infty$--group orbit of the highest weight vector there
exists a whole family of solutions $L(\alpha)$, 
$C^{(j)}(\alpha)$ of (1.1). These solutions are parametrized
by the root lattice $\frak L$ of $sl_n$, i.e. $\alpha\in \frak L$. 

Inspired by these articles, we want to consider the following
problem in this paper. Suppose one has one non-trivial set of
solutions  $L$, $C^{(j)}$ ($1\le j\le n$), of the KP 
hierarchy. Is it then possible to find the corresponding
tau--function? So, does one such solution contain enough 
information? One can solve this problem using the geometry, i.e.
Sato's infinite Grassmannian. However, this is in general not so
simple. In this paper we want to give
an algebraic and recursive approach. 
So, in order to solve this problem, one first has to
find a whole family of such pseudodifferential operators that
are all separate solutions of the KP hierarchy. 
To achieve such a result, we define a 
Schlesinger--B\"acklund transformation, which produces from a
solution of (1.1) a new solution of (1.1). Applying such
transformations recursively we obtain a whole family of 
solutions of (1.1), from which we can then destilate, with
some restrictions, the $\tau$--function. For this construction, 
it will be
convenient to formulate the KP hierarchy in terms of the
Sato equation.  

For more information on $n$-component KP and on some of the ideas used in
this paper we refer the reader to [4].

\subheading{II. The wave function and the Sato equation}
\vskip 10pt

The $n$--component KP hierarchy can be seen as the 
integrability or compatibility conditions for 
a linear problem. A formal oscillating function 
%
$$\aligned
W(x,t,z)&=\sum_{k=0}^\infty W^{(k)}(x,t)z^{-k}
e^{\Xi(t,z)}e^{xz},\quad\text{with}\\
\Xi(t,z)&=\sum_{j=1}^n 
\left( \sum_{k=1}^\infty t_k^{(j)}z^k E_{jj}\right)
\endaligned\tag{2.1}
$$
and $W^{(0)}(x,t)=I_n$, is called a wave function  
for the set of pseudodifferential operators $L$ and 
$C^{(j)}$ 
($1\le j\le n$) of the form (1.2) satisfying (1.3), 
if it satisfies the following set of linear equations:
$$LW=Wz,\quad C^{(j)}W=WE_{jj}, \quad 
{{\partial W}\over {\partial t_k^{(j)}}}=
(L^kC^{(j)})_+W.\tag{2.2}$$
We rewrite (2.1) as follows
$$W(x,t,z)=P(x,t,\partial)e^{\Xi(t,\partial)}e^{xz}$$
with 
$$P\equiv P(x,t,\partial)
=I_n+\sum_{k=1}^\infty W^{(k)}(x,t)\partial^{-k}.\tag{2.3}$$
Denote by $A_+$ (resp. $A_-=A-A_+$) the purely differential 
(resp. negative) part of a 
pseudodifferential operator $A$. Then clearly one has the following 
\proclaim{Proposition 2.1} One can find a wave function of the
form $(2.1)$ for the set of pseudodifferential operators $L$,
$C^{(j)}$, $1\le j\le n$ of the form $(1.2)$ which satisfy
$(2.3)$ if and only if we can find an operator
$P$ of the form $(2.3)$ which satisfies the following set of
equations
$$LP=P\partial, \quad C^{(j)}P=PE_{jj},\quad
{{\partial P}\over{\partial t_k^{(j)}}}=
-(L^kC^{(j)})_-P\tag{2.4}$$ 
\endproclaim
This proposition is intimately related to the Zakharov--Shabat
dressing method, viz. from (2.4) we deduce that 
$L=P\partial P^{-1}$, $C^{(j)}=PE_{jj}P^{-1}$ and hence the
last equation of (2.4) can be rewritten as
$${{\partial P}\over{\partial t_k^{(j)}}}=
-(P\partial^kE_{jj}P^{-1})_-P.\tag{2.5}$$
This equation is called the Sato equation.
\proclaim{Proposition 2.2} Let $L$ and $C^{(j)}$, 
$1\le j\le n$, be of the form $(1.2)$, satisfying $(1.3)$.
Then the  equations $(2.4)$ have a solution $P$ of the form
$(2.3)$ if and only if 
$L$ and $C^{(j)}$, $1\le j\le n$, satisfy $(1.1)$. Such a $P$
is unique upto multiplication on the right 
by a pseudodifferential operator of the form
$$D(\partial)=I_n+ \sum_{j=1}^\infty 
A_j\partial^{-j},\tag{2.6}$$
where the  $A_j$ are matrices over $\Bbb C$. 
\endproclaim
\demo{Proof} Given $L$ and $C^{(j)}$'s, the equations (2.4)
reduce to infinitely many ordinary differential equations of
the form
$$\aligned
{{\partial W^{(j)}}\over{\partial x}}E_{ii}&=
\text{linear combination of }
{{\partial^\ell W^{(k)}}\over{ \partial x^\ell}}\ \text{with }
k+\ell<j,\\
{{\partial W^{(j)}}\over{\partial t_m^{(i)}}}&=
\text{linear combination of }
{{\partial^\ell W^{(k)}}\over{\partial x^\ell}}\ \text{with }
k+\ell<j,\\
\endaligned
$$
with coefficient scalar multiples of the coefficients of the
$LC^{(i)}$. These equations can be solved recursiveley.
Uniqueness upto multiplication on the right by elements 
$D(\partial)$ of the form (2.6) is clear.
The converse is obvious, $L=P\partial P^{-1}$, 
$C^{(j)}=PE_{jj}P^{-1}$ and the Lax equation (1.1) follows
then immediately from the Sato equation.\hfill$\square$
\enddemo
Notice that from the first Lax equations (1.1) and the first
Sato equations (2.5), i.e. $k=1$,
we deduce that
$$\sum_{i=1}^n {{\partial L}\over {\partial t_1^{(i)}}}=
{{\partial L}\over {\partial x}},\quad
\sum_{i=1}^n {{\partial C^{(j)}}\over {\partial t_1^{(i)}}}=
{{\partial C^{(j)}}\over {\partial x}},\quad
\sum_{i=1}^n {{\partial P}\over {\partial t_1^{(i)}}}=
{{\partial P}\over {\partial x}}.\tag{2.7}
$$
Using the Sato equation we calculate explicitly   
$${{\partial W_{ij}^{(1)}}\over{\partial t_1^{(k)}}}=
W^{(1)}_{ik}W_{kj}^{(1)}-\delta_{jk}W_{ij}^{(2)}
+\delta_{ik}
\left({{\partial W_{ij}^{(1)}}\over {\partial x}}
+W_{ij}^{(2)}-\sum_{\ell=1}^n  W^{(1)}_{i\ell}
W_{\ell j}^{(1)}\right).
\tag{2.8}
$$

\subheading{III. The Schlesinger--B\"acklund transformation}
\vskip 10pt

Let $P$ be of the form (2.3) satisfying the Sato equation
(2.5) and let $L=P\partial P^{-1}$, 
$C^{(j)}=PE_{jj}P^{-1}$, $1\le j\le n$, then the equations 
(1.1), (1.3) and (2.4) clearly hold.
Suppose that the coefficient 
$W_{ij}:=W^{(1)}_{ij}=C^{(j1)}_{ij}\ne 0$.
Inspired by Proposition 3.3 of [4] and the articles [5] and [6],
we define the following differential operator, which 
we call a Schlesinger--B\"acklund operator for $P$
or for the set $\{L,\ C^{(1)}, \dots,\ C^{(n)}\}$:
%
%
$$\aligned
Q(i,j;P)=\epsilon_{ij}&
\left( (\partial-{{\partial \log W_{ij}}
\over{\partial t^{(i)}_1}})E_{ii}
-{1\over W_{ij}}E_{ji}
-\sum_{k\ne i} W_{ik}E_{ik}\right)\\
&\hskip .5cm+\sum_{k\ne i,j} \epsilon_{ki}\epsilon_{kj}
 \left(E_{kk}-{W_{kj}\over W_{ij}}E_{ki}\right).
\endaligned
\tag{3.1}
$$
Here $\epsilon_{ij}$ is defined by 
$$\epsilon_{ij}=\cases
1&\text{if}\ i\le j,\\
-1&\text{if}\ i>j.\\
\endcases
$$
\proclaim{Lemma 3.1} The Schlesinger--B\"acklund 
operator $Q(i,j;P)$  is invertible, its inverse is
$$\aligned Q(i,j;P)^{-1}=
\epsilon_{ij}&
\left( (-\partial-{{\partial \log W_{ij}}
\over{\partial t^{(j)}_1}})E_{jj}
-{1\over W_{ij}}E_{ji}
-\sum_{k\ne j} W_{kj}E_{kj}\right)\\
&\hskip .5cm +\sum_{k\ne i,j} \epsilon_{ki}\epsilon_{kj}
 \left(E_{kk}-{W_{ik}\over W_{ij}}E_{jk}\right).
\endaligned
\tag{3.2}$$
\endproclaim
\demo{Proof} Multiply $Q(i,j;P)$ on the left (or on the right)
by the right--hand side of (3.2). A  straightforward 
calculation using the equations (2.8) then shows that this product is
the identity matrix.
\hfill$\square$
\enddemo
%
%
%
Define the constant pseudodifferential operator
$$R(ij)=\epsilon_{ij}(\partial E_{ii}-\partial^{-1}E_{jj})
+\sum_{k\ne i,j}\epsilon_{ki}\epsilon_{kj}E_{kk}
\tag{3.3}
$$ 
We now calculate the coefficients of
 $\tilde P:=Q(i,j;P)PR(i,j)^{-1}
=\sum_\ell \tilde W^{(\ell)}\partial^{-\ell}$ explicitly.
One obtains $1\le p,k\le n, k\ne i,j$:
$$\aligned
\tilde W^{(\ell)}_{ip}&=
\epsilon_{ij}\epsilon_{pi}\epsilon_{pj}
\Bigg( W^{(\ell+1+\delta_{pj}-\delta_{pi})}_{ip}
+{{\partial W^{(\ell+\delta_{pj}-\delta_{pi})}_{ip}}
\over{\partial x}}\\
&\hskip 1.9cm -W_{ip}^{(\ell+\delta_{pj}-\delta_{pi})}
{{\partial\log W^{(1)}_{ij}}
\over {\partial t_1^{(i)}}}
-\sum_{r\ne i}W_{ir}^{(1)}
W_{rp}^{(\ell+\delta_{pj}-\delta_{pi})}\Bigg),\\
\tilde W^{(\ell)}_{jp}&=
-\epsilon_{ij}\epsilon_{pi}\epsilon_{pj}
{{W_{ip}^{(\ell+\delta_{pj}-\delta_{pi})}}
\over {W^{(1)}_{ij}}},\\
\tilde W^{(\ell)}_{kp}&=
\epsilon_{ki}\epsilon_{kj}\epsilon_{pi}\epsilon_{pj}
\left( W_{kp}^{(\ell+\delta_{pj}-\delta_{pi})}
-{{W^{(1)}_{kj}W^{(\ell+\delta_{pj}-\delta_{pi})}_{ip}}
\over W^{(1)}_{ij}}\right).
\endaligned\tag{3.4}
$$
From this it is easy to deduce that $\tilde P$
is again of the form (2.3).
The proof of the following Lemma is less straightforward.
\proclaim{Lemma 3.2}If P satisfies the Sato equation then so
does $\tilde P$, i.e.
$${{\partial \tilde P}\over{\partial t_k^{(i)}}}=
-(\tilde P\partial^kE_{ii}\tilde P^{-1})_-\tilde P.$$
\endproclaim
\demo{Proof}We calculate
$$\aligned
{{\partial \tilde P}\over{\partial t_k^{(i)}}}
\tilde P^{-1}
&=
{{\partial Q(i,j; P)}\over{\partial t_k^{(i)}}}
Q(i,j;P)^{-1}
+
Q(i,j;P)
{{\partial P}\over{\partial t_k^{(i)}}}P^{-1}
Q(ij;P)^{-1}\\
&=
{{\partial Q(i,j; P)}\over{\partial t_k^{(i)}}}
Q(i,j;P)^{-1}
-
Q(i,j;P)(P\partial^kE_{ii}P^{-1})_- Q(i,j)^{-1}
\endaligned
$$
Now clearly the Sato equation holds if the following 
equation holds
$$\aligned
{{\partial Q(i,j; P)}\over{\partial t_k^{(i)}}}
Q(i,j;P)^{-1}
&=\\
-Q(i,j;P)
{{\partial P}\over{\partial t_k^{(i)}}}P^{-1}&
Q(ij;P)^{-1}-(Q(i,j;P)P\partial^kE_{ii}P^{-1}Q(i,j;P)^{-1})_-.
\endaligned
\tag{3.5}
$$
Denote for simplicity of notation
$$-{{\partial P}\over{\partial t_k^{(i)}}}P^{-1}=\sum_{k=1}^\infty
S^{(k)}\partial^{-k},$$ 
then 
$$\aligned
S^{(1)}&=-{{\partial W^{(1)}}\over
{\partial t_k^{(i)}}},\\
S^{(2)}&=-{{\partial W^{(2)}}\over
{\partial t_k^{(i)}}}
+{{\partial W^{(1)}}\over
{\partial t_k^{(i)}}}W^{(1)}.\\
\endaligned\tag{3.6}
$$
And split up 
$$\aligned
Q=Q(i,j;P)&=Q_1\partial^{1}+Q_0\quad \text{and }
Q(i,j;P)^{-1}=T_1\partial^{1}+T_0\,\quad\text{ with}\\
Q_1&=\epsilon_{ij}E_{ii},\qquad 
T_1=-\epsilon_{ij}E_{jj},\\
Q_0&=\epsilon_{ij}
\left( (-{{\partial \log W_{ij}}
\over{\partial t^{(i)}_1}})E_{ii}
-{1\over W_{ij}}E_{ji}
-\sum_{k\ne i} W_{ik}E_{ik}\right)\\
&\hskip .5cm+\sum_{k\ne i,j} \epsilon_{ki}\epsilon_{kj}
 \left(E_{kk}-{W_{kj}\over W_{ij}}E_{ki}\right),\\
T_0&=\epsilon_{ij}
\left( (-{{\partial \log W_{ij}}
\over{\partial t^{(j)}_1}})E_{jj}
-{1\over W_{ij}}E_{ji}
-\sum_{k\ne j} W_{kj}E_{kj}\right)\\
&\hskip .5cm +\sum_{k\ne i,j} \epsilon_{ki}\epsilon_{kj}
 \left(E_{kk}-{W_{ik}\over W_{ij}}E_{jk}\right).\\
\endaligned\tag{3.7}
$$
then it is straightforward to check that (3.5) is equivalent to
$$-{{\partial Q^{(0)}}\over
{\partial t_k^{(i)}}}(T_1\partial^{1}+T_0)+Q^{(0)}S^{(1)}T^{(1)}+
Q^{(1)}S^{(1)}T^{(0)}+
Q^{(1)}(S^{(1)}\partial+{{\partial S^{(1)}}\over
{\partial x}}+ S^{(2)})T^{(1)}=0.\tag{3.8}$$
Next substitute (3.6) and (3.7) into the left-hand side of 
(3.8) and calculate the coefficients of
$E_{pq}$. It is then straightforward to verify that indeed (3.8)
holds.\hfill$\square$
\enddemo
\noindent From all this we have the following Schlesinger--B\"acklund
transformation.
\proclaim{Proposition 3.3}Let $P$ be a pseudodifferential operator of
the form $(2.3)$ that satisfies the Sato equation $(2.5)$ and let
$\{ L,C^{(1)},C^{(2)},\ldots,C^{(n)}\}$ be a set of 
pseudodifferential operators of the form $(1.2)$ satisfying $(1.1)$ 
and $(1.3)$. If $W^{(1)}_{ij}=C^{(j1)}_{ij}\ne 0$ for $i\ne j$, then
$Q(i,j;P)PR(i,j)^{-1}$ also satisfies $(2.3)$ and $(2.5)$,
respectively
the set $\{ Q(i,j;P)LQ(i,j;P)^{-1}$, $Q(i,j;P)C^{(1)}Q(i,j;P)^{-1}$,
$\ldots$, $Q(i,j;P)C^{(n)}Q(i,j;P)^{-1}\}$ satisfies $(1.1-3)$.
\endproclaim
%
%
%

\subheading{IV. A family of solutions}
\vskip 10pt

Let $P$ be a pseudodifferential operator satisfying (2.3) and (2.5). 
If $W^{(1)}_{ij}\ne 0$ for $i\ne j$, then we can use the Schlesinger--B\"acklund
operator $Q(i,j;P)$ to obtain new solutions of (2.3) and (2.5). Doing this
recursively, we obtain a whole family of solutions. To describe this
family better we introduce the root lattice of $sl_n$.

Let $\tilde {\frak L}$ be a lattice over $\Bbb Z$ with basis 
$\delta_1,\delta_2,\ldots,\delta_n$ and let $(\ |\ )$ be a symmetric 
bilinear form on $\tilde {\frak L}$, defined by $(\delta_i |\delta_j)=
\delta_{ij}$. Let $\delta =\delta_1 +\delta_2+\cdots\delta_n$,
denote by ${\frak L}$ the sublattice of $\tilde {\frak L}$ defined by
${\frak L}=\{ \gamma \in\tilde {\frak L}| (\delta |\gamma)=0\}$. This
lattice ${\frak L}$ is the root lattice of $sl_n$. Using the 
Schlessinger--B\"acklund transformation we can define new solutions of
(2.5) parametrized by ${\frak L}$. Define $P(0)=P$, suppose we have
defined $P(\alpha)=\sum_\ell W^{(\ell)}\partial^{-\ell}$ 
for some $\alpha \in{\frak L}$ If $W^{(1)}_{ij}(\alpha)\ne 0$,
we define
$$P(\alpha+\delta_i-\delta_j)=Q(i,j;P(\alpha))P(\alpha)R(i,j)^{-1}.\tag{4.1}$$
However, there is a slight problem, it is not clear that in this way
these solutions are well--defined, e.g. does the following equation
hold?
$$\aligned
Q(i,j&;Q(k,\ell;P(0))P(0)R(k,\ell)^{-1})Q(k,\ell;P(0))P(0)R(k,\ell)^{-1}
R(i,j)^{-1}=\\
&Q(k,\ell;Q(i,j;P(0))P(0)R(i,j)^{-1})Q(i,j;P(0))P(0)R(i,j)^{-1}R(k,\ell)^{-1}.
\endaligned\tag{4.2}$$
Clearly,
$$R(k,\ell)^{-1}
R(i,j)^{-1}=R(i,j)^{-1}R(k,\ell)^{-1}$$
and for $\alpha\in {\frak L}$ it makes sense to define $R(\alpha)$,
let $\alpha=\sum_{i=1}^n
k_i\delta_i\in{\frak L}$, then we define
$$R(\alpha)=R(1,2)^{k_1}R(2,3)^{k_1+k_2}\cdots
R(n-1,n)^{k_1+k_2+\cdots k_{n-1}}.\tag{4.3}$$
Notice that this is well defined.
Following [4], we define a bimultiplicative function 
$\epsilon:{\frak L}\times {\frak L}\to \{ \pm 1\}$
by letting 
$$\epsilon(\delta_i,\delta_j)=\epsilon_{ij}\tag{4.4}$$
then 
$$R(\alpha )=\sum_{i=1}^n \epsilon(\delta_i,\alpha)
E_{ii}\partial^{(\delta_i|\alpha)}.\tag{4.5}$$

One could try to prove (4.2) by a straightforward calculation.
This is tedious but possible. One has to use the equations
(2.8), i.e. the first Sato equations to show that (4.2) holds.
So one can say that the equations (2.8) are the Bianchi 
equations for the Schlesinger--B\"acklund transformations. 
However, such a computation does not exclude the fact 
that $P(\alpha)$ might be not uniquely defined, that it might
depend on the path. However, this is not the case as the
following proposition shows.
%
%
%
\proclaim{Proposition 4.1}Suppose we have two paths
$$\aligned 0=\alpha_0\mapsto \alpha_1\mapsto \alpha_2
\mapsto\cdots\mapsto\alpha_p=\alpha&\qquad \text{with }
\alpha_k-\alpha_{k-1}=\delta_{i_k}-\delta_{j_k}\\
0=\beta_0\mapsto \beta_1\mapsto \beta_2
\mapsto\cdots\mapsto\beta_q=\beta&\qquad \text{with }
\beta_k-\beta_{k-1}=\delta_{\ell_k}-\delta_{m_k}
\endaligned\tag{4.6}$$ 
such that
$$\aligned
P(\alpha_k)&=Q(i_k,j_k;P(\alpha_{k-1}))P(\alpha_{k-1})
R(i_k,j_k)^{-1}\\
P(\beta_k)&=Q(\ell_k,m_k;P(\beta_{k-1}))P(\beta_{k-1})
R(\ell_k,m_k)^{-1}.
\endaligned\tag{4.7}
$$
Then 
$$(P(\alpha)R(\alpha-\beta)P(\beta)^{-1})_-=0\tag{4.8}$$
and if $\alpha=\beta$, then  $P(\alpha)=P(\beta)$
\endproclaim
\demo{Proof}Write $Q_k=Q(i_k,j_k;P(\alpha_{k-1}))$ and
$\tilde Q_k=Q(\ell_k,m_k;P(\beta_{k-1}))$ and notice that 
$Q_k,\tilde Q_k, Q_k^{-1}$ and $\tilde Q_k^{-1}$ are all 
first order differential operators. Hence,
$$\aligned
P(\alpha)R(\alpha-\beta)P(\beta)^{-1}&=
Q_pQ_{p-1}\cdots Q_1P(0)R(\alpha)^{-1}
R(\alpha-\beta)R(\beta)P(0)^{-1}\tilde Q_1^{-1}
\tilde Q_2^{-1}\cdots \tilde Q_q^{-1}\\
&=Q_pQ_{p-1}\cdots Q_1\tilde Q_1^{-1}
\tilde Q_2^{-1}\cdots \tilde Q_q^{-1}
\endaligned$$
is a differential operator. Now if $\alpha=\beta$
then since $R(0)=I_n$ and both $P(\alpha)$ and 
$P(\beta)^{-1}$ are of the form (2.3), we deduce that
$P(\alpha)P(\beta)^{-1}=I_n$.\hfill$\square$
\enddemo
\noindent
We conclude from this proposition that there is no ambiguity 
in the definition of $P(\alpha)$. 

Now suppose $P(\alpha)$ and $P(\beta)$ exist and $\alpha-
\beta=\delta_i-\delta_j$, then
$$P(\alpha)R(\delta_i-\delta_j)P(\beta)^{-1}=
\epsilon_{ij}\partial E_{ii}+\sum_{k\ne i,j}\epsilon_{ki}
\epsilon_{kj}E_{kk}+\epsilon_{ij}
(W^{(1)}(\alpha)E_{ii}
-E_{ii}W^{(1)}(\beta)),\tag{4.9}$$ 
where $P(\gamma)=I_n+\sum_{k>0}W^{(k)}(\gamma)\partial^{-k}$.
Now expressing $P(\alpha)$ in $P(\beta)$ one obtains the
following expression:
$$\aligned
\epsilon_{ij}&W^{(m+1)}(\alpha)E_{ii}+
\sum_{k\ne i,j}\epsilon_{ki}
\epsilon_{kj}W^{(m)}(\alpha)E_{kk}
-\epsilon_{ij}W^{(m-1)}(\alpha)E_{jj}=\\
&\epsilon_{ij}E_{ii}\left(
{{\partial W^{(m)}({\beta})}\over{\partial x}}+
W^{(m+1)}(\beta)-W^{(1)}(\beta)W^{(m)}(\beta)\right)+\\
&+\sum_{k\ne i,j}\epsilon_{ki}
\epsilon_{kj}E_{kk}W^{(m)}(\beta)+
\epsilon_{ij}W^{(1)}(\alpha)E_{ii}W^{(m)}(\beta).
\endaligned\tag{4.10}$$
Now take $m=1$ and take the $(\ell,j)$th coefficient of this 
expression, which gives 
$$\aligned
\epsilon_{ij}\delta_{\ell i}\left(
{{\partial W^{(1)}_{ij}({\beta})}\over{\partial x}}+
W^{(2)}_{ij}(\beta)-\sum_{k=1}^n
W^{(1)}_{ik}(\beta)W^{(1)}_{kj}(\beta)\right)+&\\
+(1-\delta_{i\ell}-\delta_{j\ell})\epsilon_{\ell i}
\epsilon_{\ell j}W^{(1)}_{\ell j}(\beta)+\epsilon_{ij}W^{(1)}_{\ell
i}
(\alpha)E_{ii}W^{(1)}_{ij}(\beta)&=
-\epsilon_{ij}\delta_{\ell j}.
\endaligned\tag{4.11}
$$
Taking $\ell=j$ we see that 
$W^{(1)}_{ij}(\beta)$ is invertible and we can 
determine $W^{(1)}_{ji}(\alpha)$. If $\ell\ne i,j$
Then one easily determines $W^{(1)}_{\ell i}(\alpha)$.
If $\ell=i$ one can use the equations (2.8) to express
$W^{(1)}_{ii}(\alpha)$ in $W^{(1)}(\beta)$. 
In fact one obtains that
$P(\alpha)R(\delta_i-\delta_j)P(\beta)^{-1}=Q(i,j;P(\beta))$.
Using (3.4) we see that since  $W^{(1)}_{ij}(\beta)\ne 0$ then
also $W^{(1)}_{ji}(\alpha)\ne 0$ hence one can define the
inverse Schlesinger--B\"acklund operator
$Q(j,i;P(\alpha))$.

Define the support $supp(P)\subset \frak L$ of $P$. We say
that  
$\alpha\in\frak L$ is in $supp(P)$ if there exists a path of the form
(4.6) from $0$ to $\alpha$ such that (4.7) holds. Above we proved the
following proposition:
%
%
\proclaim{Proposition 4.2}Let $\alpha,\beta\in supp(P)$ be
such that $\alpha-\beta=\delta_i-\delta_j$, then
$$P(\alpha)=Q(i,j;P(\beta))P(\beta)R(i,j)^{-1}\quad
\text{and }P(\beta)=Q(j,i;P(\alpha))P(\alpha)R(j,i)^{-1}.$$
\endproclaim

We will now introduce a wave function for every
$\alpha\in supp(P)$. Define
$$W(\alpha)\equiv W(\alpha,x,t,z)=P(\alpha)R(\alpha)
e^{\Xi(t,z)}e^{xz}\tag{4.12}$$
and the corresponding operators
$$\aligned
L(\alpha)&\equiv L(\alpha,x,t,\partial)=P(\alpha)\partial
P(\alpha)^{-1},\\
C^{(j)}(\alpha)&\equiv C^{(j)}(\alpha,x,t,\partial)=
P(\alpha)E_{jj}P(\alpha)^{-1},\\
N(\beta,\alpha)&\equiv N(\beta,\alpha,x,t,\partial)=
P(\beta)R(\beta-\alpha)P(\alpha)^{-1}.
\endaligned\tag{4.13}$$
Notice that by Proposition 4.1 $N(\beta,\alpha)$ is a differential operator.
Since $P(\alpha)$ satisfies the Sato equation
$${{\partial P(\alpha)}\over{\partial t_k^{(j)}}}=
-(P(\alpha)\partial^kE_{jj}P(\alpha)^{-1})_-P(\alpha),\tag{4.14}$$
the set of wave functions $W(\alpha)$ for $\alpha\in supp(P)$
satisfy the following equations:
$$\aligned
L(\alpha)W(\alpha)&=W(\alpha)z,\\
C^{(j)}(\alpha)W(\alpha)&=W(\alpha)E_{jj},\\
N(\beta,\alpha)W(\alpha)&=W(\beta),\\
{{\partial W(\alpha)}\over{\partial t_k^{(i)}}}
&=((L(\alpha)^kC^{(i)}(\alpha))_+W(\alpha)
\endaligned\tag{4.15}$$
and the operators $L(\alpha),C^{(j)}(\alpha),N(\beta,\alpha)$
satisfy (1.3) and the Lax equations:
$$\aligned
{{\partial L(\alpha)}\over{\partial t_k^{(i)}}}
&=[(L(\alpha)^kC^{(i)}(\alpha))_+,L(\alpha)],\\
{{\partial C^{(j)}(\alpha)}\over {\partial t_k^{(i)}}}
&=[((L(\alpha)^kC^{(i)}(\alpha))_+,C^{(j)}(\alpha)],
\\
{{\partial N(\beta,\alpha)}\over{\partial t_k^{(i)}}}
&=(L(\beta)^kC^{(i)}(\beta))_+N(\beta,\alpha)-
N(\beta,\alpha)(L(\alpha)^kC^{(i)}(\alpha))_+.
\endaligned
\tag{4.16}
$$


\subheading{V. The adjoint wave functions and the bilinear
identity}
\vskip 10pt
On the algebra of formal $n\times n$--matrix pseudodifferential
operators exists  a natural linear anti--involution $*$, defined by
$$(A(x,t)\partial^k)^*=(-\partial)^k\circ^t\!\!A(x,t),\tag{5.1}$$
here$\ ^tA$ stands for the transposed of the matrix $A$. The 
anti--involution preserves the decomposition 
$A=A_++A_-$, i.e., $(A^*)_+=(A_+)^*$ and $(A^*)_-=(A_-)^*$.
Let $W(\alpha)$ be defined by (4.12), we introduce the corresponding 
adjoint wave function by
$$W^*(\alpha)\equiv W^*(\alpha,x,t,z)=P(\alpha)^{*-1}R(\alpha)^{*-1}
e^{-\Xi(t,z)}e^{-xz}.\tag{5.2}$$


Since $P(\alpha)^{*-1}$ satisfies the adjoint Sato equation:
$${{\partial P(\alpha)^{*-1}}\over{\partial t_k^{(i)}}}=
(P(\alpha)^{*-1}(-\partial)^k E_{ii}P(\alpha)^{*})_-P(\alpha)^{*-1},
\tag{5.3}$$
we have the following equations for the adjoint wave function:
$$\aligned
L(\alpha)^*W^*(\alpha)&=W^*(\alpha)z,\\
C^{(j)}(\alpha)^*W^*(\alpha)&=W^*(\alpha)E_{jj},\\
N(\beta,\alpha)^*W^*(\beta)&=W^*(\alpha),\\
{{\partial W^*(\alpha)}\over{\partial t_k^{(i)}}}
&=-((L(\alpha)^{*k}C^{(i)}(\alpha)^*)_+W^*(\alpha)
\endaligned\tag{5.4}$$
and the operators $L(\alpha)^*,C^{(j)}(\alpha)^*,N(\beta,\alpha)^*$
satisfy (1.3) and the Lax equations:
$$\aligned
{{\partial L(\alpha)^*}\over{\partial t_k^{(i)}}}
&=-[(L(\alpha)^{*k}C^{(i)}(\alpha)^*)_+,L(\alpha)^*],\\
{{\partial C^{(j)}(\alpha)^*}\over {\partial t_k^{(i)}}}
&=-[((L(\alpha)^{*k}C^{(i)}(\alpha)^*)_+,C^{(j)}(\alpha)^*],
\\
{{\partial N(\beta,\alpha)^*}\over{\partial t_k^{(i)}}}
&=-(L(\beta)^{*k}C^{(i)}(\beta)^*)_+N(\beta,\alpha)^*+
N(\beta,\alpha)^*(L(\alpha)^{*k}C^{(i)}(\alpha)^*)_+.
\endaligned
\tag{5.5}
$$


The set of wave functions and their corresponding adjoint wave
functions satisfy the following:
\proclaim {Theorem 5.1}Let $\alpha,\beta\in supp(P)$, then one has
the following bilinear identity:
$$Res_{z=0}W(\alpha,x,t,z)\,^tW^*(\beta,x',t',z)=0.$$
\endproclaim
\demo{Proof}Let $P(\alpha,x,t,z)R(\alpha)=\sum_i P_i\partial^{-i}$ and
$P(\beta,x',t,z)^{*-1}R(\beta)^{*-1}
=\sum_jQ_j\partial^{-j}$, then (4.8) is equivalent to
$$\aligned
0&=(\sum_i P_i\partial^{-i}\sum_j(-\partial)^{-j}\,^tQ_j)_-\\
&=(\sum_{{i,j}\atop{\ell\ge 0}}(-)^j
{{-i-j}\choose{\ell}}P_i{{\partial^\ell\,^tQ_j}\over {\partial x^\ell}}
\partial^{-i-j-\ell})_-.\\
\endaligned$$
Hence each coefficient of $\partial^{-m-1}$ for $m\ge 0$ of this 
last expression, which is 
$$\sum_{{0\le \ell}\atop{i+j+\ell=m+1}}(-)^j{{-i-j}\choose{\ell}}
P_i{{\partial^\ell\,^tQ_j}\over {\partial x^\ell}},$$
must be equal to $0$. We rewrite this as 
$$\aligned
0&=\sum_{{0\le \ell\le m}\atop{i+j+\ell=m+1}}(-)^{j+\ell}
{m\choose \ell}P_i{{\partial^\ell\,^tQ_j}\over {\partial x^\ell}}\\
&=Res_{z=0}\sum_{i,j}\sum_{\ell=0}^m(-)^{j+m-\ell}
{m\choose \ell}P_i{{\partial^\ell\,^tQ_j}\over {\partial x^\ell}}z^{m-i-j-\ell}\\
&=Res_{z=0}\sum_{i,j}P_iz^{-i}e^{\Xi(t,z)}e^{xz}
{{\partial^m}\over {\partial
x^m}}\,^t(Q_j(-z)^je^{-\Xi(t,z)}e^{-xz}).
\endaligned
$$
The Lemma then follows by using the equations  (5.4) and Taylor's
formula.
\hfill $\square$
\enddemo
%
%
%
%
\subheading{VI. The tau-function}
\vskip 10pt
In this section we want to introduce tau-functions. For every
$\alpha\in
supp(P)$ we introduce a tau-function $\tau_\alpha$. We give an
algebraic proof for the existence of these tau-functions. This proof
was inspired by the work of Ueno and Takasaki [7] on the Toda lattice
hierarchy.

Before we give the main Theorem, we first introduce some notations.
We write 
$$\aligned
P^{\vee}_{ij}(\alpha,x,t,z)&=(P(\alpha,x,t,-z)^{*-1})_{ij}\\
R^\vee(\alpha,z)&=R(\alpha,-z)^{*-1},
\endaligned$$
then
$$W^*(\alpha,x,t,z)=P^\vee(\alpha,x,t,z)R^\vee(\alpha,x,z)e^{-\Xi(t,z)}e^{-xz}.$$
\proclaim{Theorem 6.1}For all $\alpha\in supp(P)$ we find 
$\tau_{\alpha+\delta_i-\delta_j}$, uniquely determined upto a constant
multiple, such that 
$$\aligned
P_{ij}(\alpha,x,t,z)&=\epsilon_{ji}{{\tau_{\alpha+\delta_i-\delta_j}
(t^{(p)}_k+\delta_{k,1}x-\delta_{pj}{z^{-k}\over k})}\over
{\tau_\alpha(t^{(p)}_k+\delta_{k,1}x)}},\\
P^\vee_{ij}
(\alpha,x,t,z)&=\epsilon_{ji}{{\tau_{\alpha+\delta_j-\delta_i}
(t^{(p)}_k+\delta_{k,1}x+\delta_{pj}{z^{-k}\over k})}\over
{\tau_\alpha(t^{(p)}_k+\delta_{k,1}x)}}.
\endaligned\tag{6.1}$$
\endproclaim
\noindent
The proof of this theorem is based on several identities, which
all follow from the bilinear identity for the wave function (Theorem
5.1). Notice that the $x$-dependence is given by formula (2.7) and a
similar formula for the adjoint wave function (which follows from
(5.3)). So from now on in the proof of the above Theorem we will
concentrate on the $t$-dependence.

\proclaim{Lemma 6.2}Let $\alpha$ and $\alpha+\delta_a-\delta_c\in
supp(P)$ ($0\le a\ne c\le n$), then the following identities hold 
($0\le b\le n$):
$$\align
\nu P_{ab}(\alpha,x,t_r^{(p)},\nu)&
P^\vee_{bb}(\alpha,x,t_r^{(p)}-\delta_{pb}{\nu^{-r}\over r},\nu)=\\
&\delta_{ab}\nu
-(1-\delta_{ab})P_{ba}^{\vee
(1)}(\alpha,x,t_r^{(p)}-\delta_{pb}{\nu^{-r}\over r}),\tag{6.2}\\ 
\nu P_{ab}^\vee(\alpha,x,t_r^{(p)},\nu)&
 P_{aa}(\alpha,x,t_r^{(p)}+\delta_{pa}{\nu^{-r}\over r},\nu)
=\\
&\delta_{ab}\nu
-(1-\delta_{ab})P_{ba}^{(1)}
(\alpha,x,t_r^{(p)}+\delta_{pa}{\nu^{-r}\over r}),\tag{6.3}\\
\nu P_{ac}(\alpha,x,t_r^{(p)},\nu)&P_{bc}^\vee 
(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over
r}-\delta_{pa}{\rho^{-r}\over r},\nu)=\\  
&\delta_{ab}\rho
-(1-\delta_{ab}-\delta_{bc})P_{ab}^{(1)}(\alpha,x,t_r^{(p)})\\
&-\rho P_{aa}(\alpha,x,t_r^{(p)},\rho)P_{ba}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over
r}-\delta_{pa}{\rho^{-r}\over r},\rho),\tag{6.4}\\
\epsilon_{bc}\nu^{1-\delta_{bc}}&P_{bb}(\alpha+\delta_a-\delta_c,x,t_r^{(p)},\nu)
P_{cb}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pb}{\nu^{-r}\over r},\nu)=\\
&-\epsilon_{ca}\epsilon_{ab}
P_{ba}^{(1-\delta_{ab})}(\alpha+\delta_a-\delta_c,x,t_r^{(p)})
P_{ca}^{\vee (1)}
(\alpha,x,t_r^{(p)}+\delta_{pb}{\nu^{-r}\over r}), \tag{6.5}\\
P_{aa}(\alpha,x,t_r^{(p)}+\delta_{pa}&{\nu^{-r}\over r},\nu)
P_{aa}^\vee (\alpha,x,t_r^{(p)}-\delta_{pa}{\rho^{-r}\over r},\nu)=\\
&P_{aa}(\alpha,x,t_r^{(p)}+\delta_{pa}{\nu^{-r}\over r},\rho)
P_{aa}^\vee (\alpha,x,t_r^{(p)}-\delta_{pa}{\rho^{-r}\over r},\rho).\tag{6.6}\\
\endalign
$$
\endproclaim
\demo{Proof}We will give the proof of the identity (6.4), the proof of
all other identities are analogous. We first introduce the formal 
delta-function $\delta(y-z)=z^{-1}\sum_{k\in \Bbb Z}(z/y)^k$, for which
one has $\sum_{k=0}^\infty (z/y)^k=z\delta(y-z)-\sum_{k=1}^\infty
(y/z)^k$
and $Res_{z=0}f(y,z)\delta(y-z)=f(y,y)$. Using Theorem 5.1, we deduce
$$\aligned
0=&Res_{z=0}\sum_{\ell=1}^n W_{a\ell}(\alpha,x,t_r^{(p)},z)
W_{b\ell}^*(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},z)\\
=&Res_{z=0}\{ P_{ac}(\alpha,x,t_r^{(p)},z)P^\vee_{bc}
(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},z)\sum_{k=0}^\infty\left( z\over
\nu\right)^k\\
&\ +P_{aa}(\alpha,x,t_r^{(p)},z)
P_{bc}^\vee(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},z)\sum_{k=0}^\infty\left( z\over
\rho\right)^k\}\\
&\ +(1-\delta_{ab}-\delta_{ac})P_{ab}^{(1)}(\alpha,x,t_r^{(p)})\\
=&Res_{z=0}\{ P_{ac}(\alpha,x,t_r^{(p)},z)P^\vee_{bc}
(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},z)(z\delta(\nu-z)-\sum_{k=1}^\infty
\left( \nu\over z\right)^k)\\
&\ +P_{aa}(\alpha,x,t_r^{(p)},z)
P_{bc}^\vee(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},z)(z\delta(\rho-z)-\sum_{k=1}^\infty
\left( \rho\over z\right)^k)\}\\
&\ +(1-\delta_{ab}-\delta_{ac})P_{ab}^{(1)}(\alpha,x,t_r^{(p)})\\
=&\nu P_{ac}(\alpha,x,t_r^{(p)},\nu)P^\vee_{bc}
(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over
r}-\delta_{pa}{\rho^{-r}\over r},\nu)\\
&\ +\rho P_{aa}(\alpha,x,t_r^{(p)},\rho)
P_{bc}^\vee(\alpha,x,t_r^{(p)}-\delta_{pc}{\nu^{-r}\over r}
-\delta_{pa}{\rho^{-r}\over r},\rho)-\rho\delta_{ab}\\
&\ +(1-\delta_{ab}-\delta_{ac})P_{ab}^{(1)}(\alpha,x,t_r^{(p)}).
\hskip 7.9cm \hfill\square
\endaligned
$$
\enddemo
%
%
%
%
We are now able to proof the following proposition.
\proclaim{Proposition 6.3}For $\alpha\in supp(P)$ one has the following
identities ($1\le i\ne j\le n$):
$$\align
P_{jj}(\alpha,x,t_r^{(p)},\lambda)&=
P_{jj}(\alpha,x,t_r^{(p)}-\delta_{pj}{\lambda^{-r}\over r},\lambda)^{-1},\tag{6.7}\\
P_{ij}^{(1)}(\alpha,x,t_r^{(p)})&=
-P_{ji}^{(1)}(\alpha,x,t_r^{(p)}),\tag{6.8}\\
\lambda P_{ij}(\alpha,x,t_r^{(p)},\lambda)&=P_{ij}^{(1)}
(\alpha,x,t_r^{(p)}-\delta_{pj}{\lambda^{-r}\over r})
P_{jj}(\alpha,x,t_r^{(p)},\lambda),\tag{6.9}\\
\lambda P_{ij}^\vee(\alpha,x,t_r^{(p)},\lambda)&=P_{ij}^{\vee(1)}
(\alpha,x,t_r^{(p)}-\delta_{pj}{\lambda^{-r}\over r})
P_{jj}^\vee(\alpha,x,t_r^{(p)},\lambda).\tag{6.10}
\endalign
$$
\endproclaim
\demo{Proof}The identity (6.7) is a consequence of (6.2), where one
choses
$a=b=j$. To prove (6.8), we take the constant coefficient of (6.2) for
$i=a\ne b=j$. The other identities follow immediately using (6.2),
(6.3) and (6.8).\hfill$\square$
\enddemo
Consider now the following equations ($\alpha\in supp (P),\ 1\le i\ne
j\le n$):
$$\aligned
\log P_{jj}(\alpha,0,t,z)&=(e^{-\eta(t^{(j)},z)}-1)
\log \tau_\alpha(t),\\
P_{ij}^{(1)}(\alpha,0,t)&=\epsilon_{ji}
{{\tau_{\alpha+\delta_i-\delta_j}(t)}\over
{\tau_\alpha(t)}},
\endaligned\tag{6.11}$$
where
$$\eta(t,z):=\sum_{k=1}^\infty {\partial\over {\partial
t_k}}{z^{-k}\over k}.\tag{6.12}$$
Define $p_j(\partial^{(i)})$ for $\partial^{(i)}=({\partial\over {\partial
t_1^{(i)}}},{\partial\over {\partial t_2^{(i)}}},\ldots)$ as follows
$$\sum_{j=0}^\infty p_j(\partial^{(i)})z^{-j}=e^{-\eta(t^{(i)},z)}$$
and set 
$$\aligned 
\log P_{jj}(\alpha,0,t,z)&=\sum_{k=1}^\infty
T_j^{(k)}(\alpha)z^{-k},\\
\log\epsilon_{ji} P_{ij}^{(1)}(\alpha,0,t)&=T_{ij}(\alpha).
\endaligned\tag{6.13}
$$
Of course (6.13) is not well-defined if
$P_{ij}^{(1)}(\alpha,0,t,z)=0$,
however this is only the case if $\alpha+\delta_i-\delta_j\not\in
supp(P)$. In that case one simply defines
$\tau_{\alpha+\delta_i-\delta_j}(t)=0$.

The equations (6.11) are solvable if the following compatibility
conditions hold ($1\le i,j,k\le n$, $\ell,m=1,2,\ldots$):
$$\aligned
p_\ell(\partial^{(i)})(T_j^{(m)}(\alpha))&
=p_m(\partial^{(j)})(T_i^{(\ell)}(\alpha))\\
p_\ell(\partial^{(i)})(T_{kj}(\alpha))&
=T_i^{(\ell)}(\alpha+\delta_k-\delta_j)-T_i^{(\ell)}(\alpha).
\endaligned\tag{6.14}$$
The ring of differential operators over $\Bbb C$ with generators 
${\partial\over {\partial t_\ell^{(i)}}}$, $1\le i\le n$,
$\ell=1,2,\ldots$,
also have the $p_\ell(\partial^{(i)})$ as generators. 
In fact, from the first formula of (6.11) one can deduce the more 
explicit formula [4]:
$${{\partial \log \tau_\alpha(t)}\over{\partial t_k^{(j)}}}=
Res_{z=0}z^k\left ( {\partial\over{\partial z}}-\sum_{\ell >1}z^{-\ell-1}
{\partial\over{\partial t_\ell^{(j)}}}\right )
\log P_{jj}(\alpha,0,t,z)$$
from which one can determine the tau-function $\tau_\alpha(t)$  upto a constant
factor.
Hence if the compatibility conditions (6.14)
hold, then the $\tau_{\alpha+\delta_i-\delta_j}(t)$ are uniquely
defined upto a constant multiple factor. These compatibility
conditions follow from the next proposition.
\proclaim{Proposition 6.4}Let $\alpha$ and
$\alpha+\delta_k-\delta_j\in supp(P)$, $1\le j\ne k\le n$,
then one has the following identities ($1\le i\le n$):
$$\align
P_{ii}(\alpha,x,t_r^{(p)},\mu)
P_{jj}(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over r},\lambda)&=\\
P_{jj}(\alpha,x,t_r^{(p)},\lambda)&
P_{ii}(\alpha,x,t_r^{(p)}-\delta_{pi}{\lambda^{-r}\over r},\mu),\tag{6.15}\\
\epsilon_{ji}P_{ii}(\alpha+\delta_k-\delta_j,x,t_r^{(p)},\mu)
P_{ij}^{(1-\delta_{ij})}(\alpha,x,t_r^{(p)})&=\\
\epsilon_{jk}\epsilon_{ki}P_{ii}(\alpha,x,t_r^{(p)},\mu)
P_{ik}^{(1-\delta_{ik})}(\alpha+&\delta_k-\delta_j,x,t_r^{(p)})
P_{kj}^{(1)}(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over
r}).\tag{6.16}
\endalign$$
\endproclaim
\demo{Proof}The identity (6.15) for $i=j$ follows immediately from
(6.6) and (6.7). If $i=j$ one uses (6.4) with $a=i$, $b=c=j$,
$\nu=\lambda$, $\rho=\mu$ and with $a=j$, $b=c=i$, $\nu=\mu$,
$\rho=\lambda$, respectively. On thus obtains:
$$\align
\mu^2 P_{ii}(\alpha,x,t_r^{(p)},\mu)&P_{ii}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over r}
-\delta_{jp}{\lambda^{-r}\over r},\mu)\times\\
&P_{ji}(\alpha,x,t_r^{(p)},\mu)P_{ji}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over r}
-\delta_{jp}{\lambda^{-r}\over r},\mu)=\\
\lambda^2 P_{jj}(\alpha,x,t_r^{(p)},\lambda)&P_{ii}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over r}
-\delta_{jp}{\lambda^{-r}\over r},\lambda)\times\\
&P_{ij}(\alpha,x,t_r^{(p)},\lambda)P_{ji}^\vee
(\alpha,x,t_r^{(p)}-\delta_{pi}{\mu^{-r}\over r}
-\delta_{jp}{\lambda^{-r}\over r},\lambda)
\endalign$$
Again use (6.4), but now with $a=b=i$, $c=j$, $\nu=\lambda$,
$\rho=\mu$ and $a=b=i$, $c=j$, $\nu=\mu$, $\rho=\lambda$,
respectively.
Then (6.15) follows immediately.

The identity (6.16) follows from (6.5), where one uses (6.7), (6.8)
and (6.10).\hfill$\square$
\enddemo
\proclaim{Corollary 6.5}Let $\alpha$ and $\alpha+\delta_i-\delta_k\in
supp(P)$, then one has 
($1\le i\le n$ $\ell,m=1,2,\ldots$):
$$\aligned
p_\ell(\partial^{(i)})(T_j^{(m)}(\alpha))&
=p_m(\partial^{(j)})(T_i^{(\ell)}(\alpha))\\
T_{kj}(\alpha)&=(1-\delta_{ij})T_{ij}(\alpha)
-(1-\delta_{ik})T_{ik}(\alpha+\delta_k-\delta_j)\\
p_\ell(\partial^{(i)})(T_{kj}(\alpha))&
=T_i^{(\ell)}(\alpha+\delta_k-\delta_j)-T_i^{(\ell)}(\alpha).
\endaligned$$
\endproclaim
Hence the compatibility conditions hold. Finally, combining Proposition
6.3, the formula (6.11) 
and the fact that
the $x$-dependence of the
tau-functions follow from the last formula of (2.7), this finishes the
proof of Theorem 6.1.

We next extend the definition of the tau-function (following [4]). We define
$\tau_\alpha=0$
for $\alpha\not\in supp(P)$. Let $\Bbb C[\tilde{\frak L}]$ be the twisted
group algebra of $\tilde{\frak  L}$, with basis $e^\gamma$, $\gamma\in
\tilde{\frak 
L}$ and algebra product
$$e^\alpha
e^\beta=\epsilon(\alpha,\beta)e^{\alpha+\beta}.\tag{6.17}$$
Define the operator $\gamma$ for $\gamma\in \tilde{\frak
L}$ on $\Bbb C[\tilde{\frak L}]$ by 
$$\gamma (e^\beta)= (\gamma|\beta)e^\beta.$$ 
Notice that $\Bbb
C[{\frak L}]$ is a subalgebra of $\Bbb C[\tilde{\frak L}]$. Using
this group algebra we define a generating series of tau-functions as follows: 
$$\tau\equiv\tau(P,x,t)=\sum_{\gamma\in{\frak
L}}\tau_\gamma(x,t)e^\gamma,\tag{6.18}$$
Then using Theorem 5.1 and Theorem 6.1 we deduce the following
important result:
\proclaim{Theorem 6.6}The following bilinear identity holds for the
tau-function $\tau(P,x,t)$:
$$\align
Res_{z=0}\sum_{j=1}^n\{ e^{\delta_j}z^{\delta_j}e^{\eta
(t^{(j)},z)} \tau(P,x,t)\}&
\{ e^{-\delta_j}z^{-\delta_j}e^{-\eta
(s^{(j)},z)} \tau(P,y,s)\}\times\\ 
&e^{\sum_{k=1}^\infty (t^{(j)}-s^{(j)})z^j} e^{(x-y)z}=0.
\endalign$$
\endproclaim

\proclaim{Remark 6.7}Notice that $\tau_0$ is given by the choice 
$P(0,x,t)=P(x,t)$. If one chooses a $P(\alpha,x,t)=P(x,t)$ for a
specific $\alpha\in \frak L$ one also gets a tau-function. So we may
conclude that the tau-function belonging to $P$ is uniquely defined
upto a constant multiple factor and upto a translation over the root
lattice $\frak L$. Such a translation $\alpha\mapsto \alpha+\beta$ is
called a Schlesinger transformation.
\endproclaim
\proclaim{Remark 6.8}Notice that with this construction it is only
possible to obtain tau functions that have a connected support. 
It is unclear if there exist tau-functions that do not have a connected
support. If they exist, our construction would clearly
only give a connected component.
It was shown in [4] that in the polynomial case the support is always convex. 
\endproclaim
%
%
%
%
%
\subheading{VII. The Davey--Stewartson and AKNS hierarchies}
\vskip 10pt
In this section we consider the $2$--component KP hierarchy, which we call the
Davey--Stewartson hierarchy and its reduction, the well-known AKNS
hierarchy
(see e.g. [4] for more details). 

From now on let $n=2$. Notice that $\alpha\in {\frak L}$ is of the
form
$\alpha=k\delta_1-k\delta_2$ with $k\in{\Bbb Z}$. So from now on it
will be convenient to write $W(k), P(k), Q(i,j,k),\ldots$ instead of
$W(\alpha), P(\alpha), Q(i,j;P(\alpha)),\ldots$.
Define 
$$\aligned
\overline W(k)&= P(k)R(k)=(I_2+W^{(1)}(k)\partial^{-1}+\cdots)
(E_{11}\partial^k + E_{22}(-\partial)^{-k}),\\
\overline W^*(k)&= P(k)^{*-1}R(k)^{*-1}.
\endaligned\tag{7.1}$$
We will also write
$$\overline W_{ij}(k)=(\delta_{ij}+W_{ij}^{(1)}(k)\partial^{-1}+\cdots)
(\delta_{j1}\partial^k +\delta_{j2}(-\partial)^{-k})$$
and a similar expression for $\overline W_{ij}^*(k)$. Then one has
$$
\aligned
\overline W(k+1)&=Q(i,j,k)\overline W(k),\\
\overline W^*(k+1)&=Q(i,j,k)^{*-1}\overline W^*(k),
\endaligned
$$
which gives for $a=1,2$:
$$
\align
\overline W_{1a}(k+1)&={{\partial \overline W_{1a}(k)}\over {\partial x}}
+\overline W_{1a}(k)\left ( \partial-{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t^{(1)}_1}}\right )
-W_{12}^{(1)}(k)\overline W_{2a}(k),\tag{7.2}\\
\overline W_{2a}(k+1)&=-{{\overline W_{1a}(k)}\over
{W_{12}^{(1)}(k)}},\tag{7.3}\\
\overline W^*_{1a}(k+1)&=-{{\overline W^*_{2a}(k)}\over
{W_{12}^{(1)}(k)}},\tag{7.4}\\
\overline W^*_{2a}(k+1)&={{\partial \overline W^*_{2a}(k)}\over {\partial x}}
+\overline W^*_{2a}(k)\left ( \partial-{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t^{(2)}_1}}\right )
-W_{12}^{(1)}(k)\overline W^*_{1a}(k).\tag{7.5}
\endalign
$$
Now substitute (7.3) and (7.4) in (7.2), respectively (7.5), one thus
obtains the following recursion relations:
$$
\aligned
\overline W_{1a}(k+1)&={{\partial \overline W_{1a}(k)}\over {\partial x}}
+\overline W_{1a}(k)\left ( \partial-{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t^{(1)}_1}}\right )
+{{W_{12}^{(1)}(k)}\over {W_{12}^{(1)}(k-1)}}\overline W_{1a}(k-1),\\
\overline W^*_{2a}(k+1)&={{\partial \overline W^*_{2a}(k)}\over {\partial x}}
+\overline W^*_{2a}(k)\left ( \partial-{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t^{(2)}_1}}\right )
+{{W_{12}^{(1)}(k)}\over {W_{12}^{(1)}(k-1)}}\overline W^*_{2a}(k-1).
\endaligned 
\tag{7.6}
$$
The recursion formulas (7.6), when corresponding to very special
tau-functions form  recursion relations for string-ortogonal
polynomials as considered by Adler and van Moerbeke in [8].

Now consider the $[1,1]$--reduction of the Davey--Stewartson hierarchy
(for more details see [4]), this is the famous AKNS hierarchy.
This means that $L(k)=\partial$ and $\partial W(k)/\partial x=zW(k)$,
so 
$${{\partial \overline W_{1a}(k)}\over {\partial x}}=
{{\partial \overline W_{2a}(k)}\over {\partial x}}=
{{\partial \overline W^*_{1a}(k)}\over {\partial x}}=
{{\partial \overline W^*_{2a}(k)}\over {\partial x}}=0.$$
using (2.7) one has $\partial/\partial t_1^{(1)}=-\partial/\partial
t_1^{(2)}$, hence the eqations (6.7) turn into ($t_1=t_1^{(1)}$):
$$
\aligned
\overline W_{1a}(k+1)&=
\overline W_{1a}(k)\left ( \partial-{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t_1}}\right )
+{{W_{12}^{(1)}(k)}\over {W_{12}^{(1)}(k-1)}}\overline W_{1a}(k-1),\\
\overline W^*_{2a}(k+1)&=
\overline W^*_{2a}(k)\left ( \partial+{{\partial\log W_{12}^{(1)}(k)}\over
{\partial t_1}}\right )
+{{W_{12}^{(1)}(k)}\over {W_{12}^{(1)}(k-1)}}\overline W^*_{2a}(k-1).
\endaligned 
\tag{7.7}
$$
These type of recursion relations also appear in the description of
orthogonal polynomials related to the Toda lattice hierarchy as
described in [9].

We finally investigate the effect of the Schlesinger--B\"acklund
transformation on the Lax formulation of the AKNS hierarchy.
Consider again the Davey--Stewartson hierarchy and make the following
change of variables.
$$x_i=t_i^{(1)}+t_i^{(2)},\quad y_i=t_i^{(1)}-t_i^{(2)}.$$
Again consider the reduction, so $L(k)=\partial$ and
$C^{(1)}(k)$ and $C^{(2)}(k)$ are independent of the $x_i$'s and of
$x$.
Denote by $S(k)=C^{(1)}(k)-C^{(2)}(k)$, then since $S(k)^2=I_2$ and
$\partial S(k)/\partial x=0$, the AKNS hierarchy is the following set
of Lax equations
$${{\partial S(k)}\over{\partial
y_i}}=[(S(k)\partial^i)_+,S(k)].\tag{7.8}$$
Let $y=y_1,\ t=-2iy_2,\ q_k=W^{(1)}_{12}(k)$ and
$r_k=W^{(1)}_{21}(k)$, then it is straightforward to check that 
$$\aligned
S(k)=&
\pmatrix
1&0\\
0&-1
\endpmatrix
+
\pmatrix
0&-2q_k\\
2r_k&0
\endpmatrix
\partial^{-1}+
\pmatrix
2q_kr_k&-2{{\partial q_k}/{\partial y}}\\
-2{{\partial r_k}/{\partial y}}&-2q_kr_k
\endpmatrix
\partial^{-2}+\\
&\quad +\pmatrix
*&-2{{\partial^2 q_k}/{\partial y^2}}-4q_k^2r_k\\
2{{\partial^2 r_k}/{\partial y^2}}+4q_kr_k^2&*
\endpmatrix
\partial^{-3}+\cdots\endaligned
$$
and that the $q_k$ and $r_k$ satisfy the Non-Linear Schr\"odinger
system
$$\aligned
i{{\partial q_k}\over{\partial t}}
&=-{1\over 2}{{\partial^2 q_k}\over{\partial y^2}}
-q_k^2r_k,\\
i{{\partial r_k}\over{\partial t}}
&={1\over 2}{{\partial^2 r_k}\over{\partial y^2}}+q_kr_k^2.
\endaligned$$
Now 
$$Q(1,2,k)=\pmatrix
\partial-{{\partial \log q_k}/ {\partial y}}&-q_k\\
-{{1}/ {q_k}}&0
\endpmatrix ,$$
so
$S(k+1)=Q(1,2,k)S(k)Q(1,2,k)^{-1}$ and this gives the following
recursion relation for the $q_k$'s and $r_k$'s:
$$\aligned 
q_{k+1}=&-{{\partial^2 \log q_k}\over {\partial y^2}}-q_k^2r_k\\
r_{k+1}=&-{1\over {q_k}}\endaligned
$$
These are the $1$-dimensional Toda lattice equations. Similar equations
were also obtained in [6] and [10].

\vskip10pt
\subheading{Acknowledgements} 
\vskip 10pt
\noindent
The author is very grateful to
Gerard Helminck for stimulating discussions on the Toda lattice
hierarchy and to the referee for correcting some errors.

\vskip 10pt
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\endRefs

 



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