%Paper: hep-th/9405150
%From: PTVD@IBM.RZ.TU-CLAUSTHAL.DE
%Date: Tue, 24 May 94 09:39:38 MET



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%%%%%%%%%%% DEFINITIONS


\def\mt{\mapsto} \def\D{\Delta} \def\bbz{Z\!\!\!Z}
\def\bbc{C\kern-6.5pt I} \def\bbr{I\!\!R} \def\bbn{I\!\!N}
\def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta}
\def\k{\kappa} \def\hp{\hat{\varphi}} \def\l{\lambda}
\def\ca{{\cal A}} \def\L{\Lambda} \def\cc{{\cal C}} \def\ce{{\cal
E}} \def\ct{{\cal T}} \def\cu{{\cal U}} \def\cp{{\cal P}}
\def\cg{{\cal G}} \def\hd{{\hat{\cal D}}} \def\hm{{\hat M}}
\def\cz{{\cal Z}} \def\cd{D} \def\r{\rho} \def\eps{\epsilon}
\def\s{\sigma} \def\uq{U_q(sl(3))} \def\un{U_q(sl(n))}
\def\ag{{{\cal A}_g}} \def\an{SL_q(n)} \def\aq{SL_q(3)}
\def\x{\xi} \def\z{\zeta} \def\lg{\langle} \def\rg{\rangle}
\def\ve{\varepsilon} \def\vf{\varphi} \def\y{\eta} \def\th{{\hat
\ct}} \def\tv{{\tilde\vf}} \def\tc{{\hat \cc}} \def\hc{{\tilde
\cc}} \def\rra{\longrightarrow} \def\ra{\rightarrow}
\def\vr{\vert} \def\bl{{\bar \ell}} \def\bm{{\bar m}}
\def\bn{{\bar n}} \def\sa{{\bar\alpha}} \def\sb{{\bar\beta}}
\def\sg{{\bar\gamma}} \def\sd{{\bar\delta}} \def\se{{\bar e}}
\def\sf{{\bar f}} \def\sk{{\bar k}} \def\br{{\bar r}}
\def\sy{{\bar\y}} \def\bi{{\bar I}} \def\ci{{\cal I}}
\def\bci{{\bar \ci}} \def\be{{\bar \ce}} \def\he{{\tilde \ce}}
\def\dda{\downarrow} \def\pd{\partial} \def\sz{{\bar z}}
\def\bs{{\bar s}} \def\hr{{\hat r}} \def\bc{{\bar c}}
\def\xiL#1#2{\xi^{1\,\cdots\,#1}_{1\,\cdots\,#2}\,}

%%%%%%%% start of paper

\baselineskip=12pt

\line{\hfill ASI-TPA/10/93} \line{\hfill October 1993} \vskip
1.5cm


\baselineskip=16pt plus 2pt minus 1pt

\centerline{{\sfont q - Difference Intertwining Operators for
U$_{{\bf q}}$(sl(n))~: }} \centerline{{\sfont General Setting and
the Case ~n=3}}

\baselineskip=12pt

\vskip 1.5cm

\centerline{{\bf V.K. Dobrev$^{*}$} }\footnote{}{$^{*}$
{}~\male{Permanent address : Bulgarian Academy of Sciences,
Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko
Chaussee, 1784 Sofia, Bulgaria.}}

\vskip 0.5cm

\centerline{Arnold Sommerfeld Institute for Mathematical
Physics}\centerline{Technical University Clausthal}
\centerline{Leibnizstr. 10, 38678 Clausthal-Zellerfeld, Germany}




\baselineskip=16pt plus 2pt minus 1pt

\parskip=7pt plus 1pt

\vskip 2cm

\centerline{\bf Abstract}

We construct representations ~$\hat\pi_{\br}$~ of the quantum
algebra ~$\un$~ labelled by $n-1$ complex numbers ~$r_i$~ and
acting in the space of formal power series of ~$n(n-1)/2$~
non-commuting variables. These variables generate a flag manifold
of the matrix quantum group ~$\an$~ which is dual to ~$\un$~. The
conditions for reducibility of ~$\hat\pi_{\br}$~ and the
procedure for the construction of the ~$q$ - difference
intertwining operators are given. The representations and $q$ -
difference intertwining operators are given in the most explicit
form for $n=3$.

\vfill\eject

\baselineskip=16pt plus 2pt minus 1pt

\parskip=7pt plus 1pt

\null\bigskip

\noindent {\bf 1. ~~Introduction}

Invariant differential equations ~$\ci ~f ~=~ 0$~ play a very
important role in the description of physical symmetries -
recall, e.g., the examples of Dirac, Maxwell equations, (for more
examples cf., e.g., \ref\BR{A.O.Barut and R. R\c aczka, {\it
Theory of Group Representations and Applications}, \hfil\break II
edition, (Polish Sci. Publ., Warsaw, 1980).}). It is an
important and yet unsolved problem to find such equations for the
setting of quantum groups, where they are expected as
$q$-difference equations, especially, in the case of
non-commuting variables.

The approach to this problem used here relies on the following.
In the classical situation the invariant differential operators
{}~$\ci$~ giving the equations above may be described as operators
intertwining representations of complex and real semisimple Lie
groups \ref\KS{A.W. Knapp and E.M. Stein, Ann. Math. {\bf 93}
(1971) 489; ~Inv. Math. {\bf 60} (1980) 9.}, \ref\Ko{B. Kostant,
Lecture Notes in Math., Vol. 466 (Springer-Verlag, Berlin, 1975)
p. 101.}, \ref\Zh{D.P. Zhelobenko, Math. USSR Izv. {\bf 10}
(1976) 1003.}, \ref\Dob{V.K. Dobrev, Rep. Math. Phys. {\bf 25}
(1988) 159.}. There are many ways to find such operators, cf.,
e.g., \BR, however, most of these rely on constructions which are
not available for quantum groups. Here we shall apply a procedure
\Dob\ which is rather algebraic and can be generalized almost
straightforwardly to quantum groups. According to this procedure
one first needs to know these constructions for the complex
semisimple Lie groups since the consideration of a real
semisimple Lie group involves also its complexification. That is
why we start here with the case of ~$\un$ ~(we write $sl(n)$
instead of $sl(n,\bbc)$). For the procedure one needs
{}~$q$-difference realizations of the representations in terms of
functions of non-commuting variables. Until now such a
realization of the representations and of the intertwining
operators was found only for a Lorentz quantum algebra (dual to
the matrix Lorentz quantum group of \ref\WZ{S.L. Woronowicz and
S. Zakrzewski, Quantum Lorentz group having Gauss decomposition
property, ETH Z\"urich and TU Clausthal preprint, (July 1991).})
in \ref\DDF{L. D\c abrowski, V.K. Dobrev and R. Floreanini,
preprint Trieste Univ. UTS-DFT-17-93 (1993).}. The construction
in \DDF\ (also applying the procedure of \Dob) involves two
$q$-commuting variables $\y\bar\y = q\bar\y\y$ and uses the
complexification ~$U_q(sl(2)) \otimes U_q(sl(2))$~ of the Lorentz
quantum algebra.

In the present paper following the mentioned procedure we
construct representations ~$\hat\pi_{\br}$~ of ~$\un$~ labelled
by ~$n-1$~ complex numbers ~$\br ~=~ \{ r_1,\ldots, r_{n-1}\}$~
and acting in the spaces of formal power series of ~$n(n-1)/2$~
non-commuting (for $n>2$) variables ~$Y_{ij}$~, ~$1\leq j < i
\leq n$. These variables generate a flag manifold of the matrix
quantum group ~$\an$~ which is dual to ~$\un$~. ~For generic
$r_i\in\bbc$ the representations $\hat\pi_{\br}$ are irreducible.
We give the values of $r_i$ when the representations
$\hat\pi_{\br}$ are reducible. It is in the latter cases that
there arise various partial equivalences among these
representations. These partial equivalences are realized by ~$q$
- difference intertwining operators for which we give a canonical
derivation following \Dob. For $q=1$ these operators become the
invariant differential operators mentioned above. We should also
note that our considerations below are for general $n\geq 2$,
though the case $n=2$ is not interesting from the non-commutative
point of view since it involve functions of one variable, and
furthermore the representations and the only possible
$q$-difference intertwining operator are known for $U_q(sl(2))$,
(though derived by a different method), \ref\GP{A.Ch. Ganchev and
V.B. Petkova, Phys. Lett {\bf 233B} (1989) 374-382.}.


The paper is organized as follows. In Section 2 we recall the
matrix quantum group $GL_q(n)$ and its dual quantum algebra
$\cu_g$. In Section 3 we give the explicit construction of
representations of $\cu_g$ and its semisimple part $\un$. In
Section 4 we give the reducibility conditions for these
representations and the procedure for the construction of the $q$
- difference intertwining operators. In Section 5 we consider in
more detail the case $n=3$.


\vskip 1cm

\noindent {\bf 2. ~~The matrix quantum group}

Let us consider an ~$n\times n$~ quantum matrix ~$M$~ with
non-commuting matrix elements ~$a_{ij}$, $1 \leq i,j \leq n$.
The matrix quantum group ~$\ag ~=~ GL_q(n)$, ~$q\in\bbc$, ~is
generated by the matrix elements ~$a_{ij}$~ with the following
commutation relations \ref\Ma{Yu.I. Manin, Quantum groups and
non-commutative geometry, Montreal University preprint, CRM-1561
(1988); ~ Comm. Math. Phys. {\bf 123} (1989) 163-175.}~ ($\l =
q-q^{-1}$)~: \eqna\1 $$\eqalignno{ a_{ij} a_{i\ell} ~&=~ q^{-1}
a_{i\ell} a_{ij} ~, ~{\rm for} ~ j < \ell ~, &\1 a\cr a_{ij}
a_{kj} ~&=~ q^{-1} a_{kj} a_{ij} ~, ~{\rm for} ~ i < k ~, &\1 b
\cr a_{i\ell} a_{kj} ~&=~ a_{kj} a_{i\ell} ~, ~{\rm for} ~ i < k
{}~, ~j < \ell ~, &\1 c \cr a_{k\ell} a_{ij} &- a_{ij} a_{k\ell}
{}~=~ \l a_{i\ell} a_{kj} ~, ~{\rm for} ~ i < k ~, ~j < \ell ~.
&\1 d \cr }$$ Considered as a bialgebra, it has the following
comultiplication ~$\d_\ca$~ and counit ~$\ve_\ca$~:
\eqn\2{\d_\ca(a_{ij}) ~=~ \sum_{k=1}^n a_{ik} \otimes a_{kj} ~,
{}~~~\ve_\ca(a_{ij}) ~=~ \d_{ij} ~. } This algebra has determinant
{}~$\cd$~ given by \Ma: \eqn\det{ \cd ~=~ \sum_{\r \in S_n} ~
\eps (\r) ~ a_{1,\r (1)} \ldots a_{n,\r (n)} ~=~ \sum_{\r \in
S_n} ~\eps (\r) ~ a_{\r (1),1} \ldots a_{\r (n),n} ~ ~, } where
summations are over all permutations $\r$ of $\{1, \ldots, n\}$
and the quantum signature is: \eqn\qs{ \eps (\r) ~=~ \prod_{j<k
\atop \r (j) > \r (k) } ~(- q^{-1} ) ~. } The determinant obeys
\Ma: \eqn\deta{ \d_\ca(\cd) ~=~ \cd \otimes \cd ~, ~~~\ve_\ca (\cd)
{}~=~ 1 ~. } The determinant is central, i.e., it commutes with the
elements $a_{ik}$~ \Ma: \eqn\detc{ a_{ik} ~\cd ~=~ \cd ~a_{ik} ~.
} Further, if ~$\cd \neq 0$~ one extends the algebra by an
element ~$\cd^{-1}$~ which obeys \Ma: \eqn\deti{\cd \cd^{-1} ~=~
\cd^{-1} \cd ~=~ 1_\ca ~. }

Next one defines the left and right quantum cofactor matrix
{}~$A_{ij}$~ \Ma: \eqn\cof{ \eqalign{ A_{ij} ~=&~ \sum_{\r (i) = j}
{}~ { \eps (\r \circ \s_i ) \over \eps ( \s_i) } ~ a_{1,\r (1)}
\ldots {\widehat a_{ij}} \ldots a_{n,\r (n)} ~= \cr =& ~ \sum_{\r
(j) = i} ~ { \eps (\r \circ \s'_j ) \over \eps ( \s'_j) } ~ a_{\r
(1),1} \ldots {\widehat a_{ij} } \ldots a_{\r (n),n} ~, \cr } }
where $\s_i$ and $\s'_j$ denote the cyclic permutations:
\eqn\perm{ \s_i ~=~ \{ i,\ldots,1\} ~, ~~~\s'_j ~=~ \{
j,\ldots,n\} ~, } and the notation $ {\hat x }$ indicates that
$x$ is to be omited. Now one can show that \Ma: \eqn\uni{ \sum_j
{}~a_{ij} ~A_{\ell j} ~~=~~ \sum_j ~A_{ji} ~a_{j\ell} ~~=~~
\d_{i\ell} ~\cd ~, } and obtain the left and right inverse \Ma:
\eqn\inv{ M^{-1} ~~=~~ \cd^{-1} ~ A ~~=~~ A ~ \cd^{-1} ~.}
Thus, one can introduce the antipode in ~$GL_q(n)$~ \Ma\ :
\eqn\ant{ \g_\ca (a_{ij}) ~~=~~ \cd^{-1} ~ A_{ji} ~~=~~ A_{ji} ~
\cd^{-1} ~.}

Next we introduce a basis of $GL_q(n)$ which consists of
monomials \eqn\mon{\eqalign{ f ~~=&~~ (a_{21})^{p_{21}} \ldots
(a_{n,n-1})^{p_{n,n-1}} (a_{11})^{\ell_{1}} \ldots
(a_{nn})^{\ell_{n}} (a_{n-1,n})^{n_{n-1,n}} \ldots
(a_{12})^{n_{12}} ~= \cr =&~~ f_{\bar \ell , \bar p, \bar n} ~,
\cr} } where $\bar \ell , \bar p, \bar n$ denote the sets $\{
\ell_{i}\}$, $\{p_{ij}\}$, $\{n_{ij}\}$, resp., $\ell_{i},
p_{ij}, n_{ij} \in \bbz_+$ and we have used the so-called normal
ordering of the elements $a_{ij}$. Namely, we first put the
elements $a_{ij}$ with $i>j$ in lexicographic order, i.e., if
$i<k$ then $a_{ij}$ ($i>j$) is before $a_{k\ell}$ ($k>\ell$) and
$a_{ti}$ ($t>i$) is before $a_{tk}$ ($t>k$); ~then we put the
elements $a_{ii}$; ~finally we put the elements $a_{ij}$ with
$i<j$ in antilexicographic order, i.e., if $i>k$ then $a_{ij}$
($i<j$) is before $a_{k\ell}$ ($k<\ell$) and $a_{ti}$ ($t<i$) is
before $a_{tk}$ ($t<k$). Note that the basis \mon\ icludes also
the unit element $1_\ag$ of $\ag$ when all $\{\ell_{i}\}$,
$\{p_{ij}\}$, $\{n_{ij}\}$ are equal to zero, i.e.: \eqn\monu{
f_{ \bar 0 , \bar 0, \bar 0} ~~=~~ 1_\ag ~. }


We need the dual algebra of $GL_q(n)$. This is the algebra
{}~$\cu_g = U_q(sl(n)) \otimes U_q(\cz)$, where $U_q(\cz)$ is
central in $\cu_g$ \ref\DP{ V.K. Dobrev and P. Parashar, J. Phys.
A: Math. Gen. {\bf 26} (1993) 6991.}. Let us denote the
Chevalley generators of $sl(n)$ by $H_i$, $X^\pm_i$, $i = 1,
\ldots, n-1$. Then we take for the 'Chevalley' generators of
{}~$\cu = U_q(sl(n))$~:~ $k_i = q^{H_i/2}$, $k_i^{-1} =
q^{-H_i/2}$, $X^\pm_i$, $i = 1, \ldots, n-1$, with the following
{}~{\it algebra}~ relations: \eqna\wb $$\eqalignno{ &k_i k_j ~=~
k_j k_i ~, \quad k_i k^{-1}_i ~=~ k^{-1}_i k_i ~=~ 1_{\cu_g} ~,
\quad k_i X^\pm_j ~=~ q^{\pm c_{ij}} X^\pm_j k_i &\wb a\cr
&[X^+_i , X^-_j ] ~=~ \d_{ij} \left( k^2_i - k^{-2}_i \right)/\l
{}~, &\wb b\cr &\left(X^\pm_i\right)^2 X^\pm_j ~-~ [2]_q X^\pm_i
X^\pm_j X^\pm_i ~+~ X^\pm_j \left(X^\pm_i\right)^2 ~~=~~ 0 ~,
{}~~~~\vr i- j\vr =1 ~, &\wb c\cr &[X^\pm_i , X^\pm_j ] ~=~ 0 ~,
{}~~\vr i- j\vr \neq 1 ~, &\wb d\cr } $$ where ~$c_{ij}$~ is the
Cartan matrix of ~$sl(n)$, ~and ~{\it coalgebra}~ relations:
\eqna\wg $$\eqalignno{ \d_\cu (k^\pm_i) ~&=~ k^\pm_i \otimes
k^\pm_i ~, &\wg a \cr \d_\cu (X^\pm_i) ~&=~ X^\pm_i \otimes k_i
{}~+~ k^{-1}_i \otimes X^\pm_i ~, &\wg b\cr \ve_\cu (k^\pm_i) ~&=~
1 ~, \quad \ve_\cu (X^\pm_i) ~=~ 0 ~, &\wg c\cr \g_\cu (k_i) ~&=~
k_i^{-1} ~, \quad \g_\cu (X^\pm_i) ~=~ - q^{\pm 1} ~X^\pm_i ~,
&\wg d\cr}$$ where $k^+_i = k_i$, $k^-_i = k^{-1}_i$. Further,
we denote the generator of $\cz$ by $H$ and the generators of
$U_q(\cz)$ by $k = q^{H/2}$, $k^{-1} = q^{-H/2}$, $k k^{-1} ~=~
k^{-1} k ~=~ 1_{\cu_g}$. The generators $k, k^{-1}$ commute with
the generators of $\cu$, and their coalgebra relations are as
those of any $k_i$. From now on we shall give most formulae only
for the generators $k_i, ~X^\pm_i, ~k$, since the analogous
formulae for $k_i^{-1}, ~k^{-1}$ follow trivially from those for
$k_i, ~k$, resp.


The bilinear form giving the duality between ~$\cu_g$~ and
{}~$\ag$~ is given by \DP: \eqna\pai $$\eqalignno{\lg ~k_i ~,
{}~a_{j\ell} ~\rg ~~&= ~~ \d_{j\ell} ~q^{(\d_{ij} - \d_{i,j+1})/2}
{}~, &\pai a\cr \lg ~X^+_i ~, ~a_{j\ell} ~\rg ~~&= ~~ \d_{j+1,\ell}
\d_{ij} ~, &\pai b\cr \lg ~X^-_i ~, ~a_{j\ell} ~\rg ~~&= ~~
\d_{j-1,\ell} \d_{i\ell} ~, &\pai c\cr \lg ~k ~, ~a_{j\ell} ~\rg
{}~~&= ~~ \d_{j\ell} ~q^{1/2} ~. &\pai d\cr}$$ The pairing between
arbitrary elements of ~$\cu_g$~ and $f$ follows then from the
properties of the duality pairing. All this is given in \DP\ and
is not reproduced here since we shall not need these formulae.
The pairing \pai{} is standardly supplemented with \eqn\du{ \lg
{}~y ~, ~1_{\ag} ~\rg ~=~ \ve_{\cu_g} (y) ~. }

It is well know that the pairing provides the fundamental
representation of ~$\cu_g$~: \eqn\fun{ F (y)_{j\ell} ~~=~~ \lg ~y
{}~, ~a_{j\ell} ~\rg ~, \quad y ~=~ k_i, X^\pm_i , k ~. } Of
course, ~$F(k) = q^{1/2} I_n$~, ~where ~$I_n$ ~is the unit
{}~$n\times n$ ~matrix.

\vskip 1cm

\noindent {\bf 3. ~~Representations of ~$\cu_g$~ and ~$\cu$ }

We begin by defining ~{\it two actions}~ of the dual algebra
{}~$\cu_g$~ on the basis \mon\ of ~$\ag$.

First we introduce the ~{\it left regular representation}~ of
{}~$\cu_g$~ which in the $q=1$ case is the infinitesimal version
of~: \eqn\lrcl{ \pi (Y) \ M ~~=~~ Y^{-1}\ M ~, \quad Y,M ~\in
{}~GL(n) ~. } Explicitly, we define the action of ~$\cu_g$~ as
follows (cf. \fun): \eqn\lrm{ \pi (y)~ a_{i\ell} ~\doteq ~
\left( F \left(y^{-1}\right) M \right)_{i\ell} ~=~ \sum_j ~F
\left(y^{-1}\right)_{ij} ~a_{j\ell} ~=~ \sum_j ~\lg ~y^{-1}~ ,
{}~a_{ij}~ \rg ~a_{j\ell} ~, } where ~$y$~ denotes the generators
of ~$\cu_g$~ and ~$y^{-1}$~ is symbolic notation, the possible
pairs being given explicitly by: \eqn\lrma{ (y, y^{-1}) ~~~=~~~
(k_i , k_i^{-1}), ~~ (X^\pm_i , ~-X^\pm_i), ~~(k, k^{-1}) ~. }
{}From \lrm\ we find the explicit action of the generators of
{}~$\cu_g$~: \eqna\lr $$\eqalignno{ \pi (k_i)~ a_{j\ell} ~&=~
q^{(\d_{i+1,j} - \d_{ij})/2} ~a_{j\ell} ~, &\lr a\cr \pi
(X^+_i)~ a_{j\ell} ~&=~ - \d_{ij} ~a_{j+1\ell} ~, &\lr b\cr \pi
(X^-_i)~ a_{j\ell} ~&=~ - \d_{i+1,j} ~a_{j-1\ell} ~, &\lr c\cr
\pi (k)~ a_{j\ell} ~&=~ q^{-1/2} ~a_{j\ell} ~. &\lr d\cr}$$ The
above is supplemented with the following action on the unit
element of $\ag$: \eqn\lru{ \pi (k_i) ~1_\ag ~~=~~ 1_\ag ~, \quad
\pi (X^\pm_i) ~1_\ag ~~=~~ 0 ~, \quad \pi (k) ~1_\ag ~~=~~ 1_\ag
{}~. }

In order to derive the action of $\pi(y)$ on arbitrary elements
of the basis \mon, we use the twisted derivation rule consistent
with the coproduct and the representation structure, namely, we
take: ~$\pi (y)\vf\psi ~=~ \pi (\d'_{\cu_g}(y))(\vf\otimes
\psi)$, ~where ~$\d'_{\cu_g} ~= ~\sigma \circ \d_{\cu_g}$ ~is the
opposite coproduct, ($\sigma$ is the permutation operator). Thus,
we have: \eqna\tw $$\eqalignno{ \pi (k_i)\vf\psi ~~&=~~ \pi
(k_i)\vf\cdot \pi (k_i)\psi\ , &\tw a\cr \pi (X^\pm_i)\vf\psi
{}~~&=~~ \pi (X^\pm_i)\vf\cdot \pi (k^{-1}_i)\psi ~+~ \pi
(k_i)\vf\cdot \pi (X^\pm_i)\psi\ , &\tw b\cr \pi (k)\vf\psi
{}~~&=~~ \pi (k)\vf\cdot \pi (k)\psi\ . &\tw c\cr}$$

{}From now on we suppose that ~$q$~ is not a nontrivial root of
unity. Applying the above rules one obtains: \eqna\act
$$\eqalignno{ \pi (k_i)~ (a_{j\ell})^n ~&=~ q^{n(\d_{i+1,j} -
\d_{ij})/2} ~(a_{j\ell})^n ~, &\act a\cr \pi (X^+_i)~
(a_{j\ell})^n ~&=~ - \d_{ij} ~c_n ~ (a_{j\ell})^{n-1}
{}~a_{j+1\ell} ~, &\act b\cr \pi (X^-_i)~ (a_{j\ell})^n ~&=~ -
\d_{i+1,j} ~c_n ~a_{j-1\ell} ~ (a_{j\ell})^{n-1} ~, &\act c\cr
\pi (k)~ (a_{j\ell})^n ~&=~ q^{-n/2} ~(a_{j\ell})^n ~, &\act
d\cr}$$ where \eqn\cof{ c_n ~=~ q^{(n-1)/2} ~[n]_q ~, ~~~[n]_q =
(q^n - q^{-n})/\l \ . } Note that \lru\ and \lr{} are partial
cases of \act{} for $n=0$ and $n=1$ resp. (cf. \monu).


\vskip 5mm

Analogously, we introduce the ~{\it right action}~ (see also
\ref\MNS{T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Sabuti
and K. Ueno, Lett. Math. Phys. {\bf 19} (1990) 187-194; Lett.
Math. Phys. {\bf 19} (1990) 195-204.}) which in the classical
case is the infinitesimal counterpart of~: \eqn\lrcr{ \pi_R (Y) ~
M ~~=~~ M ~ Y ~, \quad Y,M ~\in ~GL(n) ~. } Thus, we define the
right action of ~$\cu_g$~ as follows (cf. \fun): \eqn\lrr{ \pi_R
(y)~ a_{i\ell} ~=~ \left( M F (y) \right)_{i\ell} ~=~ \sum_j
{}~a_{ij} ~F (y)_{j\ell} ~ ~=~ \sum_j ~a_{ij} ~ \lg ~y~ ,
{}~a_{j\ell}~ \rg ~, } where ~$y$~ denotes the generators of
{}~$\cu_g$~.

{}From \lrr\ we find the explicit right action of the generators of
{}~$\cu_g$~: \eqna\lrrr $$\eqalignno{ \pi_R (k_i)~ a_{j\ell} ~&=~
q^{(\d_{i\ell} - \d_{i+1,\ell})/2} ~a_{j\ell} ~, &\lrrr a\cr
\pi_R (X^+_i)~ a_{j\ell} ~&=~ \d_{i+1,\ell} ~a_{j,\ell -1} ~,
&\lrrr b\cr \pi_R (X^-_i)~ a_{j\ell} ~&=~ \d_{i\ell} ~a_{j,\ell
+1} ~, &\lr c\cr \pi_R (k)~ a_{j\ell} ~&=~ q^{1/2} ~a_{j\ell} ~,
&\lrrr d\cr}$$ supplemented by the right action on the unit
element: \eqn\lrur{\pi_R (k_i) ~1_\ag ~=~ 1_\ag ~, \quad \pi_R
(X^\pm_i) ~1_\ag ~~=~~ 0 ~, \quad \pi_R (k) ~1_\ag ~~=~~ 1_\ag ~.
}





The twisted derivation rule is now given by ~$\pi_R (y)\vf\psi
{}~=~ \pi_R (\d_{\cu_g}(y))(\vf\otimes \psi)$, i.e., \eqna\twr
$$\eqalignno{ \pi_R (k_i)\vf\psi ~&=~ \pi_R (k_i)\vf\cdot \pi_R
(k_i)\psi\ ,&\twr a\cr \pi_R (X^\pm_i)\vf\psi ~&=~ \pi_R
(X^\pm_i)\vf\cdot \pi_R (k_i)\psi +\pi_R (k_i^{-1})\vf\cdot \pi_R
(X^\pm_i)\psi\ ,&\twr b\cr \pi_R (k)\vf\psi ~&=~ \pi_R
(k)\vf\cdot \pi_R (k)\psi\ ,&\twr c\cr }$$ Using this, we find:
\eqna\ract $$\eqalignno{ \pi_R (k_i)~ (a_{j\ell})^n ~&=~
q^{n(\d_{i\ell} - \d_{i+1,\ell})/2} ~(a_{j\ell})^n ~, &\ract a\cr
\pi_R (X^+_i)~ (a_{j\ell})^n ~&=~ \d_{i+1,\ell} ~c_n ~a_{j,\ell
-1} ~ (a_{j\ell})^{n-1} ~, &\ract b\cr \pi_R (X^-_i)~
(a_{j\ell})^n ~&=~ \d_{i\ell} ~c_n ~ (a_{j\ell})^{n-1}
{}~a_{j,\ell+1} ~, &\ract c\cr \pi_R (k)~ (a_{j\ell})^n ~&=~
q^{n/2} ~(a_{j\ell})^n ~. &\ract d\cr}$$


Let us now introduce the elements ~$\vf$~ as formal power series
of the basis \mon: \eqn\ser{\eqalign{ \vf ~=&~ \sum_{{\bar \ell
,\bar m,\bar n \in\bbz_+}} \mu_{{\bar \ell , \bar m, \bar n }} ~
(a_{21})^{m_{21}} \ldots (a_{n,n-1})^{m_{n,n-1}}
(a_{11})^{\ell_{1}} \ldots (a_{nn})^{\ell_{n}} \times \cr &\times
(a_{n-1,n})^{n_{n-1,n}} \ldots (a_{12})^{n_{12}} ~. \cr}}

By \act{} and \ract{} we have defined left and right action of
{}~$\cu_g$~ on ~$\vf$. As in the classical case the left and right
actions commute, and as in \Dob\ we shall use the right action to
reduce the left regular representation (which is highly
reducible). In particular, we would like the right action to
mimic some properties of a highest weight module, i.e.,
annihilation by the raising generators $X^+_i$ and scalar action
by the (exponents of the) Cartan operators $k_i,k$. In the
classical case these properties are also called right covariance
\Dob. However, first we have to make a change of basis using the
$q$-analogue of the classical Gauss decomposition. For this we
have to suppose that the principal minor determinants of $M$~:
\eqn\prm{ \eqalign{ \cd_m ~&=~ \sum_{\r \in S_m} ~\eps (\r) ~
a_{1,\r (1)} \ldots a_{m,\r (m)} ~=\cr &=~ \sum_{\r \in S_m}
{}~\eps (\r) ~ a_{\r (1),1} \ldots a_{\r (m),m} ~ ~, \quad m\leq n
{}~, \cr } } are invertible; note that ~$\cd_n = \cd$, ~$\cd_{n-1}
= A_{nn}$. Thus, using \uni\ for $i=\ell=n$ we can express,
e.g., $a_{nn}$ in terms of other elements: \eqn\ann{ a_{nn} ~=~
\left( \cd ~-~ \sum_{j<n} ~a_{nj} ~A_{nj} \right) ~
\cd_{n-1}^{-1} ~=~ \cd_{n-1}^{-1} ~\left( \cd ~-~
\sum_{j<n} ~A_{jn} ~a_{jn} \right) ~. }



Further, for the ordered sets $I=\{ i_1 < \cdots < i_r \}$ and
$J=\{ j_1 < \cdots < j_r \}$, let $\xi^I_J$ be the $r$-minor
determinant with respect to rows $I$ and columns $J$ such that
\eqn\mnr{\xi^I_J ~=~ \sum_{\r\in S_r}
\eps(\r)~a_{i_{\r(1)}j_1}\cdots a_{i_{\r(r)}j_r} ~.} Note that
$\xiL{i }{i } = \cd_i$~. ~ Then one has \ref\ANO{H. Awata, M.
Noumi and S. Odake, preprint YITP/K-1016 (1993).} ($i,j,\ell =
1,\ldots,n$)~: \eqn\gau{ a_{i\ell} ~=~ \sum_j B_{ij}Z_{j\ell}
{}~,\quad B_{i\ell} ~=~ \xiL{\ell-1\,i}{\ell } \cd_{\ell-1}^{-1}
{}~,\quad Z_{i\ell} ~=~ \cd_i^{-1} \xiL{i }{i-1\,\ell} ~,}
$B_{i\ell}=0$ for $i<\ell$, ~$Z_{i\ell}=0$ for $i>\ell$,
{}~$\{1\cdots 0\} = \emptyset$, ~and~ $Z_{ij}$, $i<j$, ~may be
regarded as a $q$-analogue of local coordinates of the flag
manifold ~$B\backslash GL(n)$.

For our purposes we need a refinement of this decomposition:
\eqn\gua{B_{i\ell} ~~=~~ Y_{i\ell} D_{\ell\ell} ~, \quad
Y_{i\ell} ~=~ \xiL{\ell-1\,i}{\ell } \cd_{\ell}^{-1} ~,\quad
D_{\ell\ell} ~=~ \cd_{\ell} \cd_{\ell -1}^{-1} ~, \quad (\cd_0
\equiv 1_\ag) ~,} where ~$Y_{j\ell}$, $j>\ell$, ~may be regarded
as a $q$-analogue of local coordinates of the flag manifold
{}~$GL(n)/DZ$.

Clearly, we can replace the basis \mon\ of $\ag$ with a basis in
terms of $Y_{i\ell}$, $i>\ell$, $\cd_{\ell}$, $Z_{i\ell}$,
$i<\ell$. (Note that $Y_{ii} = Z_{ii} = 1_\ag$.) We could have
used also $D_{\ell\ell}$ instead of $\cd_\ell$, but this choice
is more convenient since below we shall impose $\cd_n = \cd =
1_\ag$. Thus, we consider formal power series:
\eqn\sera{\eqalign{ \vf ~=&~ \sum_{{ \bar m,\bar n \in\bbz_+
\atop \bar \ell \in\bbz}} \mu'_{{\bar \ell , \bar m, \bar n }} ~
(Y_{21})^{m_{21}} \ldots (Y_{n,n-1})^{m_{n,n-1}}
(\cd_{1})^{\ell_{1}} \ldots (\cd_{n})^{\ell_{n}} \times \cr
&\times (Z_{n-1,n})^{n_{n-1,n}} \ldots (Z_{12})^{n_{12}} ~. \cr}}



Now, let us impose right covariance \Dob\ with respect to
{}~$X^+_i$~, i.e., we require: \eqn\rcf{ \pi_R (X^+_i) ~\vf ~~=~~ 0
{}~. } First we notice that: \eqn\rcfa{ \pi_R (X^+_i) ~ \xi^I_J
{}~~=~~ 0 ~, \quad {\rm for}~~ J ~=~ \{1,\ldots,j\} ~, ~\forall ~I
{}~, } from which follow: \eqn\rcff{ \pi_R (X^+_i) ~\cd_j ~~=~~ 0
{}~, \quad \pi_R (X^+_i) ~Y_{j\ell} ~~=~~ 0 ~.} On the other hand
{}~$\pi_R (X^+_i)$~ acts nontrivially on ~$Z_{j\ell}$~. ~ Thus,
\rcf\ simply means that our functions ~$\vf$~ do not depend on
{}~$Z_{j\ell}$~. ~Thus, the functions obeying \rcf\ are:
\eqn\seras{ \vf ~=~ \sum_{{\bar \ell \in\bbz ~, ~ \bar m
\in\bbz_+ }} \mu_{{\bar \ell , \bar m}} ~ (Y_{21})^{m_{21}}
\ldots (Y_{n,n-1})^{m_{n,n-1}} (\cd_{1})^{\ell_{1}} \ldots
(\cd_{n})^{\ell_{n}} ~. }

Next, we impose right covariance with respect to ~$k_i,k$~:
\eqna\rck $$\eqalignno{ &\pi_R (k_i) ~\vf ~~=~~ q^{r_i/2} ~\vf ~,
&\rck a\cr &\pi_R (k) ~\vf ~~=~~ q^{\hr/2} ~\vf ~, &\rck b\cr}$$
where $r_i,\hr$ are parameters to be specified below. On the
other hand using \twr{a,c}, \ract{a,c} we have: \eqn\rcfka{\pi_R
(k_i) ~\xi^I_J ~~=~~ q^{\d_{ij}/2}~\xi^I_J ~, \quad \pi_R (k)
{}~\xi^I_J ~~=~~ q^{j/2}~\xi^I_J ~, \quad {\rm for}~~ J ~=~
\{1,\ldots,j\} ~, ~\forall ~I ~, } from which follows:
\eqna\rcfk $$\eqalignno{ &\pi_R (k_i) ~\cd_j ~~=~~ q^{\d_{ij}/2}~
\cd_j ~, \quad \pi_R (k) ~\cd_j ~~=~~ q^{j/2}~ \cd_j ~, &\rcfk
a\cr &\pi_R (k_i) ~Y_{j\ell} ~~=~~ Y_{j\ell} ~, \quad
\pi_R (k) ~Y_{j\ell} ~~=~~ Y_{j\ell} ~, &\rcfk b\cr}$$
and thus we have: \eqna\rk $$\eqalignno{&\pi_R(k_i) ~\vf ~=~
q^{\ell_i/2}~ \vf ~, &\rk a\cr &\pi_R(k) ~\vf ~=~ q^{\sum_{j=1}^n
j\ell_j/2}~ \vf ~. &\rk b\cr}$$

Comparing right covariance conditions \rck{} with the direct
calculations \rk{} we obtain $\ell_i = r_i$, for $i<n$,
$\sum_{j=1}^n j\ell_j = \hr$. This means that ~$r_i,\hr\in\bbz$~
and that there is no summation in $\ell_i$, also $\ell_n = (\hr -
\sum_{i=1}^{n-1} ir_i)/n$.

Thus, the reduced functions obeying \rcf\ and \rck\ are:
\eqn\seras{ \vf ~=~ \sum_{{ \bar m \in\bbz_+}} \mu_{\bar m}
{}~(Y_{21})^{m_{21}} \ldots (Y_{n,n-1})^{m_{n,n-1}}
(\cd_{1})^{r_{1}} \ldots (\cd_{n-1})^{r_{n-1}} (\cd_{n})^{{\hat
\ell}} ~, } where $ {\hat \ell} ~=~ (\hr - \sum_{i=1}^{n-1}
ir_i)/n $.

Next we would like to derive the ~$\cu_g$~ - action ~$\pi$~ on
{}~$\vf$~. ~First, we notice that ~$\cu$~ acts trivially on $\cd_n
= \cd$~: \eqn\actd{ \pi (X^\pm_i) ~\cd ~~=~~ 0 ~, \quad \pi(k_i)
{}~ \cd ~~=~~ \cd ~.} Then we note: \eqn\acad{ \pi (k) ~\cd_j
{}~~=~~ q^{-j/2} ~\cd_j ~, \quad \pi (k) ~Y_{j\ell} ~~=~~ Y_{j\ell}
{}~, } from which follows: \eqn\acbd{ \pi (k) ~\vf ~~=~~ q^{-\hr/2}
{}~\vf ~.} Thus, the action of ~$\cu$~ involves only the parameters
{}~$r_i$, $i<n$, while the action of ~$U_q(\cz)$~ involves only the
parameter ~$\hr$. Thus we can consistently also from the
representation theory point of view restrict to the matrix
quantum group $SL_q(n)$, i.e., we set: \eqn\sl{\cd ~ =~ \cd^{-1}
{}~=~ 1_\ag ~. } Then the dual algebra is $\cu = U_q(sl(n))$. This
is justified as in the $q=1$ case \Dob\ since for our
considerations only the semisimple part of the algebra is
important. (This would not be possible for the multiparameter
deformation of $GL(n)$ \ref\Sua{A. Sudbery, J. Phys. A : Math.
Gen. {\bf 23} (1990) L697.}, \ref\Schi{A. Schirrmacher, Zeit. f.
Physik {\bf C50} (1991) 321.}, since there $\cd$ is not central.
Nevertheless, we expect most of the essential features of our
approach to be preserved since the dual algebra can be
transformed as a commutation algebra to the one-parameter
$\cu_g$, with the extra parameters entering only the co-algebra
structure \DP.)


Thus, the reduced functions for the ~$\cu$~ action are:
\eqna\serasa $$\eqalignno{ \tv ({\bar Y}, {\bar \cd}) ~=&~
\sum_{{ \bar m \in\bbz_+}} \mu_{\bar m} ~(Y_{21})^{m_{21}} \ldots
(Y_{n,n-1})^{m_{n,n-1}} (\cd_{1})^{r_{1}} \ldots
(\cd_{n-1})^{r_{n-1}} ~= &\serasa a\cr ~=& ~~\hp ({\bar Y})~
(\cd_{1})^{r_{1}} \ldots (\cd_{n-1})^{r_{n-1}} ~, &\serasa
b\cr}$$ where ${\bar Y}, {\bar \cd}$ denote the variables
$Y_{il}$, $i>\ell$, $\cd_i$, $i<n$. Next we calculate:
\eqna\accd \eqna\acdd $$\eqalignno{ &\pi (k_i) ~\cd_j ~~=~~
q^{-\d_{ij}/2} ~\cd_j ~, &\accd a\cr &\pi (X^+_i) ~\cd_j ~~=~~
-\d_{ij} ~Y_{j+1,j} ~\cd_j ~, &\accd b\cr &\pi (X^-_i) ~\cd_j
{}~~=~~ 0 ~, &\accd c \cr &\pi (k_i) ~Y_{j\ell} ~~=~~ q^{{1\over
2}(\d_{i+1,j} - \d_{ij} - \d_{i+1,\ell} + \d_{i\ell})} ~Y_{j\ell}
&\acdd a\cr &\eqalign{\pi (X^+_i) ~Y_{j\ell} ~~=&~~ -\d_{ij}
{}~Y_{j+1,\ell} ~+~\d_{i\ell} ~ q^{1 - \d_{j,\ell +1}/2} ~Y_{\ell
+1, \ell} ~Y_{j\ell} ~+\cr ~&+~ ~\d_{i+1,\ell} ~ \left( q^{-1}
{}~Y_{j,\ell -1} ~-~ Y_{\ell,\ell -1} ~ Y_{j\ell} \right) ~, \cr}
&\acdd b\cr &\pi (X^-_i) ~Y_{j\ell} ~~=~~ - \d_{i+1,j} ~
q^{-\d_{i\ell}/2} ~ Y_{j-1,\ell} ~. &\acdd c \cr } $$

These results have the important consequence that the degrees of
the variables ~$\cd_j$~ are not changed by the action of ~$\cu$.
Thus, the parameters ~$r_i$~ indeed characterize the action of
{}~$\cu$~, ~i.e., we have obtained representations of ~$\cu$. We
shall denote by ~$\cc_{\br}$~ the representation space of
functions in \serasa{} which have covariance properties \rcf,
\rck{a}, and the representation acting in ~$\cc_{\br}$~ we denote
by ~$\tilde\pi_{\br}$ - here a renormalization of the explicit
formulae may be done to simplify things. To obtain this
representation more explicitly one just applies \accd{}, \acdd{}
to the basis in \serasa{} using \tw{}. In particular, we have:
\eqna\abcd $$\eqalignno{ \pi (k_i) ~(\cd_j)^n ~~=&~~
q^{-n\d_{ij}/2} ~(\cd_j)^n ~, \quad n\in\bbz ~, &\abcd a\cr \pi
(X^+_i) ~(\cd_j)^n ~~=&~~ -\d_{ij} ~\bc_n ~ ~Y_{j+1,j} ~(\cd_j)^n
{}~, \quad n\in\bbz ~, &\abcd b\cr \pi (X^-_i) ~(\cd_j)^n ~~=&~~ 0
{}~, \quad n\in\bbz ~, &\abcd c \cr \eqna\abdd \pi (k_i)
{}~(Y_{j\ell})^n ~~=&~~ q^{{n\over 2}(\d_{i+1,j} - \d_{ij} -
\d_{i+1,\ell} + \d_{i\ell})} ~(Y_{j\ell})^n ~, \quad n\in\bbz_+
{}~, &\abdd a\cr \pi (X^+_i) ~(Y_{j\ell})^n ~~=&~~ -\d_{ij} ~\bc_n
{}~(Y_{j\ell})^{n-1} ~Y_{j+1,\ell} ~+&\cr &~~+~ \d_{i\ell} ~ q^{1
-n\d_{j,\ell +1}/2} ~c_n ~Y_{\ell +1, \ell} ~(Y_{j\ell})^n ~+&\cr
&~~+ ~\d_{i+1,\ell} ~ \bc_n ~\left( q^{-1} ~Y_{j,\ell -1}
{}~(Y_{j\ell})^{n-1} ~-~ Y_{\ell,\ell -1} ~ Y_{j\ell}^n \right),
\quad n\in\bbz_+ &\abdd b\cr \pi (X^-_i) ~(Y_{j\ell})^n ~~=&~~ -
\d_{i+1,j} ~ q^{-\d_{j,\ell +1}n/2} ~c_n ~ Y_{j-1,\ell}
{}~(Y_{j\ell})^{n-1} ~, \quad n\in\bbz_+ ~, &\abdd c \cr } $$ where
\eqn\coe{\bc_n ~~=~~ q^{(1-n)/2}~ [n]_q ~. }

Further, since the action of ~$\cu$~ is not affecting the degrees
of $\cd_i$, we introduce (as in \Dob) the restricted functions
{}~$\hp ({\bar Y})$~ by the formula which is prompted in
\serasa{b}~: \eqn\res{ \hp({\bar Y}) ~\equiv~ \bigl( \ca\tv )
({\bar Y}) ~\doteq ~\tv ({\bar Y}, \cd_1 = \cdots = \cd_{n-1} =
1_\ag) ~. } We denote the representation space of ~$\hp({\bar
Y})$~ by ~$\tc_{\br}$~ and the representation acting in
{}~$\tc_{\br}$~ by ~$\hat\pi_{\br}$~. ~Thus, the operator ~$\ca$~
acts from ~$\cc_{\br}$~ to ~$\tc_{\br}$~. ~The properties of
{}~$\tc_{\br}$~ follow from the intertwining requirement for
{}~$\ca$~\Dob: \eqn\int{ \hat\pi_{\br} ~\ca ~~=~~ \ca ~ \tilde
\pi_{\br} ~.}

\vskip 1cm

\noindent {\bf 4. ~~Reducibilty and ~$q$ - difference
intertwining operators}

We have defined the representations ~$\hat\pi_{\br}$~ for
{}~$r_i\in\bbz$. However, notice that we can consider the
restricted functions $\hp({\bar Y})$ for arbitrary complex $r_i$.
We shall make these extension from now on, since this gives the
same set of representations for ~$\un$~ as in the case $q=1$.

Now we make some statements which are true in the classical case
\Dob, and will be illustrated below. For any ~$i,j$, such that
{}~$1\leq i \leq j \leq n-1$, define: \eqn\rij{m_{ij} ~\equiv ~ r_i
+ \cdots + r_j + j-i+1 ~, } note $m_i = m_{ii} = r_i +1$, $m_{ij}
{}~=~ m_i + \cdots + m_j$. Note that the possible choices of $i,j$
are in 1-to-1 correspondence with the positive roots ~$\a =
\a_{ij} = \a_i + \cdots + \a_j$ of the root system of $sl(n)$,
the cases $i=j = 1 \ldots, n-1$ enumerating the simple roots
$\a_i = \a_{ii}$. In general, ~$m_{ij}\in\bbc$~ for the
representations ~$\hat\pi_{\br}$, while ~$m_{ij}\in\bbz$~ for the
representations ~$\pi_{\br}$. If ~$m_{ij} \notin \bbn$~ for all
possible $i,j$ the representations ~$\hat\pi_{\br}$, $\pi_{\br}$
{}~are irreducible. If ~$m_{ij} \in \bbn$~ for some $i,j$ the
representations $\hat\pi_{\br}$, $\pi_{\br}$ ~are reducible. The
corresponding irreducible subrepresentations are still
infinite-dimensional unless ~$m_i\in\bbn$~ for all
$i=1,\ldots,n-1$. The representation spaces of the irreducible
subrepresentations are invariant irreducible subspaces of our
representation spaces. These invariant subspaces are spanned by
functions depending on all variables ~$Y_{j\ell}$~, ~except when
for some ~$s\in\bbn$, $1\leq s \leq n-1$, we have ~$m_s = m_{s+1}
= \cdots = m_{n-1} =1$. In the latter case these functions depend
only on the ~$(s-1)(2n-s)/2$~ variables ~$Y_{j\ell}$~ with
{}~$\ell<s$, ~ (the unrestricted subrepresentation functions depend
still on $D_\ell$ with $\ell<s$). ~In particular, for ~$s=2$~ the
restricted subrepresentation functions depend only on the $n-1$
variables ~$Y_{j1}$. The latter situation is relatively simple
also in the $q$ case since these variables are $q$-commuting~:
{}~$Y_{j1}Y_{k1} ~=~ qY_{k1}Y_{j1}$~, ~$j>k$. (For $s=1$ the
irreducible subrepresentation is one dimensional, hence no
dependence on any variables.)


Furthermore, for ~$m_{ij} \in \bbn$~ the representation
$\hat\pi_{\br}$, $\pi_{\br}$, resp., is partially equivalent to
the representation $\hat\pi_{\br'}$, $\pi_{\br'}$, resp., with
$m'_\ell = r'_\ell +1$ being explicitly given as follows \Dob:
\eqn\redd{ m'_\ell ~~=~~ \cases{ m_\ell ~, &\quad for~ $\ell \neq
i-1,i,j,j+1$ ~, \cr m_{\ell j} ~, & \quad for~ $\ell =i-1$ ~, \cr
-m_{\ell +1,j} ~, & \quad for~ $\ell =i< j$ ~, \cr -m_{i,\ell -1}
{}~, & \quad for~ $\ell =j> i$ ~, \cr -m_\ell ~, & \quad for~ $\ell
=i=j$ ~, \cr m_{i\ell} ~, & \quad for~ $\ell =j+1$ ~. \cr}}

These partial equivalences are realized by intertwining
operators: \eqna\op $$\eqalignno{ &\ci_{ij} ~: ~\cc_{\br}
{}~\longrightarrow ~\cc_{\br'} ~, \quad m_{ij}\in\bbn ~, &\op a\cr
&I_{ij} ~: ~\tc_{\br} ~\longrightarrow ~\tc_{\br'} ~, \quad
m_{ij}\in\bbn ~, &\op b\cr }$$ i.e., one has: \eqna\inta
$$\eqalignno{ & \ci_{ij} \circ \pi_{\br} ~=~ \pi_{\br'}
\circ \ci_{ij} ~, \quad m_{ij}\in\bbn ~, &\inta a\cr &I_{ij}
\circ \hat\pi_{\br} ~=~ \hat\pi_{\br'} \circ I_{ij} ~, \quad
m_{ij}\in\bbn ~. &\inta b\cr }$$ The invariant irreducible
subspace of ~$\hat\pi_{\br}$~ (resp. $\pi_{\br}$) discussed above
is the intersection of the kernels of all intertwining operators
acting from ~$\hat\pi_{\br}$ ~(resp. $\pi_{\br}$). When all
$m_i\in\bbn$ the invariant subspace is finite-dimensional with
dimension ~$\prod_{ 1\leq i \leq j \leq n - i } ~ m_{ij} ~/ ~
\prod_{t = 1}^{n-1} t!$~, and all finite-dimensional irreps
of $U_q(sl(n))$ can be obtained in this way.

We present now a canonical procedure for the derivation of these
intertwining operators following the $q=1$ procedure of \Dob. By
this procedure one should take as intertwiners (up to nonzero
multiplicative constants): \eqna\intb $$\eqalignno{ &\ci_{ij}
{}~~=~~ \cp_{ij} \left( \pi_R (X^-_i), \ldots, \pi_R (X^-_j)
\right), ~~~~m_{ij} \in\bbn ~, &\intb a\cr &I_{ij} ~~=~~ \cp_{ij}
\left( \hat\pi_R (X^-_i), \ldots, \hat\pi_R (X^-_j) \right),
{}~~~~m_{ij} \in\bbn ~, &\intb b\cr }$$ where ~$\cp_{ij}$~ is a
homogeneous polynomial in each of its ~$(j-i+1)$~ variables of
degree ~$m_{ij}$. This polynomial gives a singular vector
{}~$v_{ij}$~ in a Verma module $V^{\L(\br)}$ with highest weight
$\L(\br)$ determined by $\br$, (cf. \Dob), i.e.:
\eqn\sing{ v_{ij} ~~=~~ \cp_{ij} \left( X^-_i, \ldots, X^-_j
\right) ~\otimes ~v_0 ~, } where $v_0$ is the highest weight
vector of $V^{\L(\br)}$. In particular, in the case of the
simple roots, i.e., when ~$m_i = m_{ii} = r_i +1 \in\bbn$, we
have \eqna\intc $$\eqalignno{ &\ci_{i} ~~=~~ \left( \pi_R (X^-_i)
\right)^{m_i}, ~~~~m_{i} \in\bbn ~, &\intc a\cr &I_{i} ~~=~~
\left( \hat\pi_R (X^-_i) \right)^{m_i}, ~~~~m_{i} \in\bbn ~.
&\intc b\cr }$$ For the nonsimple roots one should use the
explicit expressions for the singular vectors of the Verma
modules over ~$\un$~ given in \ref\Doc{V.K. Dobrev, J. Phys. A:
Math. Gen. {\bf 25} (1992) 149.}. Implementing the above one
should be careful since $\hat\pi_R (X^-_i)$ is not preserving the
reduced spaces ~$\cc_{\br}$, ~$\tc_{\br}$, which is of course a
prerequisite for \inta{}, \intb{}, \intc{}.

\vfil\eject


\noindent {\bf 5. ~~The case of ~$\uq$}

In this Section we consider in more detail the case $n=3$. We
could have started also with the case $n=2$ involving functions
of one variable. However, though by a different method, this case
was obtained in \GP. It can also be obtained by restricting the
construction for the (complexification of the) Lorentz quantum
algebra of \DDF\ to one of its $U_q(sl(2))$ subalgebras.

Let us now for $n=3$ denote the coordinates on the flag manifold
by: ~$\x ~=~ Y_{21}$, ~$\y ~=~ Y_{32}$, ~$\z ~=~ Y_{31}$. We note
for future use the commutation relations between these
coordinates: \eqn\coo{ \x\y ~=~ q\y\x - \l\z ~, \quad \y\z ~=~
q\z\y ~, \quad \z\x ~=~ q\x\z ~.}

The reduced functions for the ~$\cu$~ action are (cf. \serasa{}):
\eqna\sersa $$\eqalignno{ \tv ({\bar Y}, {\bar \cd}) ~~=&~~
\sum_{{ j,n,\ell \in\bbz_+}} \mu_{j,n,\ell}~ \x^j ~ \z^n ~
\y^\ell (\cd_{1})^{r_{1}} ~ (\cd_{2})^{r_{2}} ~= &\sersa a\cr =&
{}~~\sum_{{ j,n,\ell \in\bbz_+}} \mu_{j,n,\ell}~ \tv_{jn\ell} ~,
&\sersa b\cr \tv_{jn\ell} ~~=& ~~ \x^j ~ \z^n ~ \y^\ell
{}~(\cd_{1})^{r_{1}} ~ (\cd_{2})^{r_{2}} ~. &\sersa c\cr }$$

Now the action of $\uq$ on \sersa{} is given explicitly by:
\eqna\aba $$\eqalignno{ \pi (k_1) ~ \tv_{jn\ell} ~~=&~~ q^{j +
(n-\ell-r_1)/2} ~\tv_{jn\ell} ~, &\aba a\cr \pi (k_2) ~
{}~\tv_{jn\ell} ~~=&~~ q^{\ell + (n-j-r_2)/2} ~\tv_{jn\ell} ~,
&\aba b\cr \pi (X^+_1) ~\tv_{jn\ell} ~~=&~~ q^{(1 + n - \ell -
r_1)/2} ~[n+j-\ell -r_1]_q ~\tv_{j+1,n\ell} ~+&\cr &~~+~ q^{j +
(n - \ell - 3r_1 -1)/2} ~[\ell]_q ~ ~\tv_{j,n+1,\ell -1} ~, &\aba
c\cr \pi (X^+_2) ~\tv_{jn\ell} ~~=&~~ q^{(1 + n - j - r_2)/2}
{}~[\ell -r_2]_q ~\tv_{jn,\ell +1} ~-&\cr &~~-~ q^{-\ell + (j - n +
r_2 -1)/2} ~[j]_q ~ ~\tv_{j-1,n+1,\ell} ~, &\aba d\cr
\pi (X^-_1) ~\tv_{jn\ell} ~~=&~~ q^{(\ell - n + r_1 -1)/2} ~[j]_q
{}~\tv_{j-1,n\ell} ~, &\aba e\cr \pi (X^-_2) ~\tv_{jn\ell} ~~=&~~
-~q^{(n - j + r_2 -1)/2} ~[\ell]_q ~\tv_{jn,\ell -1} ~-&\cr &~~-~
q^{-\ell + (n - j + r_2 -1)/2} ~[n]_q ~\tv_{j+1,n-1,\ell} ~.
&\aba f\cr } $$


It is easy to check that ~$\pi(k_i)$, $\pi(X^\pm_i)$ satisfy
\wb{}. It is also clear that we can remove the inessential
phases by setting: \eqn\red{\tilde\pi_{r_1,r_2} (k_i) ~~=~~ \pi
(k_i) ~, \quad \tilde\pi_{r_1,r_2} (X^\pm_i) ~~=~~
q^{\mp (r_i-1)/2} ~\pi (X^\pm_i) ~. } Then ~$\tilde\pi_{r_1,r_2}$~
also satisfy \wb{}.

Then we consider the restricted functions (cf. \res): \eqna\sersb
$$\eqalignno{ \hp({\bar Y}) ~~=&~~ \sum_{{ j,n,\ell \in\bbz_+}}
\mu_{j,n,\ell}~ \x^j ~ \z^n ~ \y^\ell ~= &\sersb a\cr =&
{}~~\sum_{{ j,n,\ell \in\bbz_+}} \mu_{j,n,\ell}~ \hp_{jn\ell} ~,
&\sersb b\cr \hp_{jn\ell} ~~=& ~~ \x^j ~ \z^n ~ \y^\ell ~~.
&\sersb c\cr}$$

As a consequence of the intertwining property \int\ we obtain
that $\hp_{jn\ell}$ obey the same transformation rules \aba{} as
$\tv_{jn\ell}$, i.e., (cf. also \red) we have: \eqna\abb
$$\eqalignno{ \hat\pi_{r_1,r_2} (k_1) ~ \hp_{jn\ell} ~~=&~~ q^{j
+ (n-\ell-r_1)/2} ~\hp_{jn\ell} ~, &\abb a\cr \hat\pi_{r_1,r_2}
(k_2) ~ ~\hp_{jn\ell} ~~=&~~ q^{\ell + (n-j-r_2)/2} ~\hp_{jn\ell}
{}~, &\abb b\cr \hat\pi_{r_1,r_2} (X^+_1) ~\hp_{jn\ell} ~~=&~~
q^{(n - \ell )/2} ~[n+j-\ell -r_1]_q ~\hp_{j+1,n\ell} ~+&\cr
&~~+~ q^{j - r_1 -1 + (n - \ell )/2} ~[\ell]_q ~ ~\hp_{j,n+1,\ell
-1} ~, &\abb c\cr \hat\pi_{r_1,r_2} (X^+_2) ~\hp_{jn\ell} ~~=&~~
q^{(n - j)/2} ~[\ell -r_2]_q ~\hp_{jn,\ell +1} ~-&\cr &~~-~ q^{
r_2 - 1 - \ell + (j - n )/2} ~[j]_q ~ ~\hp_{j-1,n+1,\ell} ~,
&\abb d\cr \hat\pi_{r_1,r_2} (X^-_1) ~\hp_{jn\ell} ~~=&~~
q^{(\ell - n)/2} ~[j]_q ~\hp_{j-1,n\ell} ~, &\abb e\cr
\hat\pi_{r_1,r_2} (X^-_2) ~\hp_{jn\ell} ~~=&~~ -~q^{(n - j)/2}
{}~[\ell]_q ~\hp_{jn,\ell -1} ~-&\cr &~~-~ q^{-\ell + (n - j)/2}
{}~[n]_q ~\hp_{j+1,n-1,\ell} ~. &\abb f\cr } $$

Let us introduce the following operators acting on our functions:
\eqna\opes $$\eqalignno{ \hm^\pm_\k ~\hp({\bar Y}) ~~=&~~ \sum_{{
j,n,\ell \in\bbz_+}} \mu_{j,n,\ell}~ \hm^\pm_\k ~\hp_{jn\ell} ~,
&\opes a\cr T_\k ~\hp({\bar Y}) ~~=&~~ \sum_{{ j,n,\ell
\in\bbz_+}} \mu_{j,n,\ell}~ T_\k ~\hp_{jn\ell} ~, &\opes b\cr }$$
where ~$\k ~=~ \x,\y,\z$, ~and the explicit action on
$\hp_{jn\ell}$ is defined by: \eqna\ops $$\eqalignno{ \hm^\pm_\x
{}~\hp_{jn\ell} ~=&~ \hp_{j\pm 1,n\ell} ~, &\ops a\cr \hm^\pm_\y
{}~\hp_{jn\ell} ~=&~ \hp_{jn,\ell\pm 1} ~, &\ops b\cr \hm^\pm_\z
{}~\hp_{jn\ell} ~=&~ \hp_{j,n\pm 1,\ell} ~, &\ops c\cr T_\x
{}~\hp_{jn\ell} ~=&~ q^{ j} ~\hp_{jn\ell} ~, &\ops d\cr T_\y
{}~\hp_{jn\ell} ~=&~ q^{\ell} ~\hp_{jn\ell} ~, &\ops e\cr T_\z
{}~\hp_{jn\ell} ~=&~ q^{ n} ~\hp_{jn\ell} ~. &\ops f\cr }$$

Now we define the $q$-difference operators by: \eqn\qdif{ \hd_\k
{}~\hp({\bar Y}) ~~=~~ {1\over \l} ~\hm^-_\k ~\left( T_\k -
T^{-1}_\k \right) ~ \hp({\bar Y}) ~, \quad \k ~=~ \x,\y,\z ~. }
Thus, we have: \eqna\opr $$\eqalignno{ \hd_\x ~\hp_{jn\ell} ~=&~
[j] ~\hp_{j-1,n\ell} ~, &\opr a\cr \hd_\y ~\hp_{jn\ell} ~=&~
[\ell] ~\hp_{jn,\ell-1} ~, &\opr b\cr \hd_\z ~\hp_{jn\ell} ~=&~
[n] ~\hp_{j,n-1,\ell} ~. &\opr c\cr }$$ Of course, for ~$q\to
1$~ we have ~$\hd_\k ~\to ~\pd_\k ~\equiv ~\pd/\pd \k$.



In terms of the above operators the transformation rules \abb{}
are written as follows: \eqna\abc $$\eqalignno{ \hat\pi_{r_1,r_2}
(k_1) ~ \hp ({\bar Y}) ~~=&~~ q^{-r_1/2} ~T_\x ~T_\z^{1/2}
{}~T_\y^{-1/2} ~\hp ({\bar Y}) ~, &\abc a\cr \hat\pi_{r_1,r_2}
(k_2) ~ ~\hp ({\bar Y}) ~~=&~~ q^{-r_2/2} ~T_\y ~T_\z^{1/2}
{}~T_\x^{-1/2} ~\hp ({\bar Y}) ~, &\abc b\cr \hat\pi_{r_1,r_2}
(X^+_1) ~\hp ({\bar Y}) ~~=&~~ (1/\l) ~\hm_\x ~T_\z^{1/2}
{}~T_\y^{-1/2} ~\left( q^{-r_1} T_\x T_\z T_\y^{-1} - q^{r_1}
T_\x^{-1} T_\z^{-1} T_\y \right) ~\hp ({\bar Y}) ~+&\cr &~~+~
q^{-r_1 -1} ~\hm_\z ~\hd_\y ~T_\x ~T_\z^{1/2} ~T_\y^{-1/2} ~\hp
({\bar Y}) ~, &\abc c\cr \hat\pi_{r_1,r_2} (X^+_2) ~\hp ({\bar
Y}) ~~=&~~(1/\l) ~\hm_\y ~T_\z^{1/2} ~T_\x^{-1/2} ~\left(
q^{-r_2} T_\y - q^{r_2} T_\y^{-1} \right) ~\hp ({\bar Y}) ~-&\cr
&~~-~ q^{r_2 -1} ~\hm_\z ~\hd_\x ~T_\x^{1/2} ~T_\z^{-1/2}
{}~T_\y^{-1} ~\hp ({\bar Y}) ~, &\abc d\cr \hat\pi_{r_1,r_2}
(X^-_1) ~\hp ({\bar Y}) ~~=&~~ \hd_\x ~T_\z^{-1/2} ~T_\y^{1/2}
{}~\hp ({\bar Y}) ~, &\abc e\cr \hat\pi_{r_1,r_2} (X^-_2) ~\hp
({\bar Y}) ~~=&~~ -~ \hd_\y ~T_\z^{1/2} ~T_\x^{-1/2} ~\hp ({\bar
Y}) ~-&\cr &~~-~ \hm_\x ~\hd_\z ~T_\x^{-1/2} ~T_\z^{1/2}
{}~T_\y^{-1} ~\hp ({\bar Y}) ~, &\abc f\cr } $$ where ~$\hm_\k ~=~
\hm^+_\k$~.




Notice that it is possible to obtain a realization of the
representation ~$\hat\pi_{r_1,r_2}$~ on monomials in three
commuting variables $x,y,z$. Indeed, one can relate the
non-commuting algebra ~$\bbc ~[\x,\y,\z]$~ with the commuting one
{}~$\bbc ~[x,y,z]$~ by fixing an ordering prescription. However,
such realization in commuting variables may be obtained much more
directly as is done by other methods and for other purposes in
\ref\BDT{L.C. Biedenharn, V.K. Dobrev and P. Truini, to appear.}.
In the present paper we are interested in the non-commutative
case and we continue to work with the non-commuting variables
{}~$\x,\y,\z$.



\vskip 5mm

Now we can illustrate some of the general statements of the
previous Section. Let ~$m_2 = r_2 +1 \in\bbn$. Then it is clear
that functions $\hp$ from \sersb{} with ~$\mu_{j,n,\ell} =0$~ if
$\ell \geq m_2$ form an invariant subspace since:
\eqn\sbs{\hat\pi_{r_1,r_2} (X^+_2) ~\hp_{jnr_2} ~~=~~ -~ q^{- 1 +
(j - n )/2} ~[j]_q ~ ~\hp_{j-1,n+1,r_2} ~, } and all other
operators in \abb{} either preserve or lower the index $\ell$.
The same is true for the functions $\tv$. In particular, for
{}~$m_2=1$~ the functions in the invariant subspace do not depend
on the variable ~$\y$. In this case we have functions of two
$q$-commuting variables ~$\z\x =q\x\z$~ which are much easier to
handle that the general non-commutative case \coo.

The intertwining operator \intc{} for ~$m_2 \in\bbn$~ is given as
follows. First we calculate: \eqn\intd{ \eqalign{
\left( \pi_R(X^-_2)\right)^s ~\tv_{jn\ell} ~=&~ \left(
\pi_R(X^-_2)\right)^s ~ \x^j ~\z^n ~\y^\ell ~\cd^{r_1}_1
\cd^{r_2}_2 ~= \cr =&~ \x^j ~\z^n ~\sum_{t=0}^s ~a_{st} ~\y^{\ell
-t} ~\cd^{r_1 +t}_1 \cd^{r_2 -s-t }_2 ~(\x^{12}_{13})^{s-t} ~,
\cr a_{st} ~=&~ q^{t\ell +r_2s/2 - (s+t)(s+t+1)/4} ~{s \choose
t}_q ~ { [r_2 -t]_q! [\ell]_q! \over [r_2 -s]_q! [\ell -t]_q! }
{}~.\cr}} Thus, indeed $ \pi_R(X^-_2)$ is not preserving the
reduced space $\cc_{r_1,r_2}$, and furthermore there is the
additional variable $\x^{12}_{13}$. Since we would like $
\pi_R(X^-_2)$ to some power to map to another reduced space this
is only possible if the coefficients $a_{st}$ vanish for $s\neq
t$. This happens iff ~$s ~=~ r_2 + 1 ~=~ m_2$. Thus we have (in
terms of the representation parameters $m_i = r_i +1$):
\eqn\inte{ \eqalign{ \left( \pi_R(X^-_2)\right)^{m_2} ~ \x^j
{}~\z^n ~\y^\ell ~&\cd^{m_1-1}_1 \cd^{m_2-1}_2 ~=\cr =&~ q^{ m_2
(\ell - 1- m_2/2)} ~ { [\ell]_q! \over [\ell -m_2]_q! } ~\x^j
{}~\z^n ~\y^{\ell -m_2} ~\cd_1^{m_{12} -1} ~ \cd_2^{-m_2 -1} ~.
\cr} } Comparing the powers of $\cd_i$ we recover at once \redd\
for our situation, namely, $m'_1 = m_{12}$, $m'_2 = -m_2$. Thus,
we have shown \op{a} and \inta{a}. Then \op{b} and \inta{b}
follow using \int. This intertwining operator has a kernel which
is just the invariant subspace discussed above - from the factor
{}~$1/ [\ell -m_2]_q!$~ in \inte\ it is obvious that all monomials
with $\ell < m_2$ are mapped to zero.

For the restricted functions we have: \eqn\intee{ \eqalign{
\left( \pi_R(X^-_2)\right)^{m_2} ~ \hp_{jn\ell} ~~=&~~ q^{ m_2
(\ell - 1- m_2/2)} ~ { [\ell]_q! \over [\ell -m_2]_q! } ~\hp_{jn,
\ell -m_2} ~= \cr =&~~ q^{-3 m_2/2)} ~\left( \hd_\y ~T_\y
\right)^{m_2} ~ \hp_{jn\ell} ~.\cr }} Thus, renormalizing
\intc{b} by ~$q^{-3 m_2/2}$~ we finally have: \eqn\intef{
I_2^{m_2} ~~=~~ \left( \hd_\y ~T_\y \right)^{m_2} ~. } For $q=1$
this operator reduces to the known result: ~$I_2^{m_2} ~=~
(\pd_\y)^{m_2}$ ~\Dob.


\vskip 5mm


Let now ~$m_1 \in\bbn$. In a similar way, though the
calculations are more complicated, we find: \eqn\intf{ \eqalign{
\left( \pi_R(X^-_1)\right)^{m_1} ~& \x^j ~\z^n ~\y^\ell
{}~\cd^{m_1-1}_1 \cd^{m_2-1}_2 ~=\cr =&~~ q^{ m_1 (j+n-\ell - 1-
m_1/2) }~\sum_{t=0}^{m_1} ~q^{-t(t+3+2j)/2} ~\times \cr \times &
{}~{m_1 \choose t}_q ~ { [j]_q! [n]_q! \over [j -m_1 +t]_q!
[n-t]_q! } ~\x^{j+t-m_1} ~\z^{n-t} ~\y^{\ell +t} ~\cd_1^{-m_1 -1}
{}~\cd_2^{m_{12} -1} ~. \cr} } Comparing the powers of $\cd_i$ we
recover \redd\ for our situation, namely, $m'_1 = -m_1$, $m'_2 =
m_{12}$. Thus, we have shown \op{} and \inta{}.


For the restricted functions we have: \eqn\intfe{ \eqalign{
\left( \pi_R(X^-_1)\right)^{m_1} ~ \hp_{jn\ell} ~~=&~~ q^{ m_1
(j+n-\ell - 1- m_1/2) }~\sum_{t=0}^{m_1} ~q^{t(t+3+2j)/2} ~\times
\cr & \times ~{m_1 \choose t}_q ~ { [j]_q! [n]_q! \over [j -m_1
+t]_q! [n-t]_q! } ~\hp_{j+t-m_1,n-t, \ell +t} ~= \cr =&~~ q^{
-m_1 (3/2 + m_1) }~ T_\z^{m_1} ~ \sum_{t=0}^{m_1} ~\hm_\y^t
{}~\hd_\z^t ~(q\hd_\x T_x)^{m_1-t} ~ T_\y^{-m_1} ~\hp_{jn\ell} ~.
\cr }} Then, renormalizing \intc{b} we finally have: \eqn\intff{
I_1^{m_1} ~~=~~ T_\z^{m_1} ~ \sum_{t=0}^{m_1} ~\hm_\y^t ~\hd_\z^t
{}~(q\hd_\x T_x)^{m_1-t} ~ T_\y^{-m_1} ~.} For $q=1$ this operator
reduces to the known result: ~$I_1^{m_1} ~=~ ( \pd_\x + \y \pd_\z
)^{m_1}$ ~\Dob.



\vskip 5mm

Finally, let us consider the case ~$m =m_{12} ~=~ m_1 + m_2
\in\bbn$, first with $m_1, m_2 \notin\bbn$. In this case the
intertwining operator is given by \intb, \sing\ with
\ref\Doa{V.K. Dobrev, Talk at the International Group Theory
Conference (St. Andrews, 1989), Proceedings, Eds. C.M.  Campbell
and E.F. Robertson, Vol. 1, London Math. Soc. Lecture Note Series
159 (Cambridge University Press, 1991) pp. 87-104 \& ICTP Trieste
internal report IC/89/142 (June 1989).}, formula (27), (cf. also
\Doc): \eqn\sng{ \eqalign{ \cp^m_{12} \left( X^-_1, X^-_2 \right)
{}~~=&~~ \sum_{s=0}^m ~a_s ~(X^-_1)^{m-s} ~(X^-_2)^m ~(X^-_1)^s ~,
\cr a_s ~~=&~~ (-1)^s ~a ~{[m_1]_q\over [m_1 -s]_q} ~\left({m
\atop s}\right)_q ~~, ~~s = 0,\dots,m , ~~a\neq 0 ~. \cr} }

Let us illustrate the resulting intertwining operator in the case
$m=1$. Then, we have, setting in \sng\ ~$a = [1-m_1]_q$~:
\eqn\snga{ \ci_{12} ~~=~~ [1-m_1]_q ~\pi_R (X^-_1)~ \pi_R (X^-_2)
{}~+~ [m_1]_q ~\pi_R (X^-_2)~ \pi_R (X^-_1) ~.} Then we can see at
once the intertwining properties of $\ci_{12}$ by calculating:
\eqn\sngb{ \eqalign{ \ci_{12} ~\x^j ~\z^n ~\y^\ell ~\cd^{m_1-1}_1
\cd^{m_2-1}_2 ~~=&~~ q^{j+n-2-m_1} ~[j]_q ~[\ell]_q ~\x^{j-1}
{}~\z^n ~\y^{\ell -1} ~\cd^{m_1-2}_1 \cd^{m_2-2}_2 ~+\cr &+~
q^{n-2} ~[n]_q ~[\ell + m_1]_q ~\x^j ~\z^{n-1} ~\y^\ell
{}~\cd^{m_1-2}_1 \cd^{m_2-2}_2 ~. \cr } } Comparing the powers of
$\cd_i$ we recover \redd\ for our situation, namely, $m'_1 = -m_2
= m_1 -1$, $m'_2 = -m_1 =m_2 -1$.

For the restricted functions we have: \eqn\sngc{ \eqalign{ \bigl(
[1-m_1]_q& ~\pi_R (X^-_1)~ \pi_R (X^-_2) ~+~ [m_1]_q ~\pi_R
(X^-_2)~ \pi_R (X^-_1) \bigr) ~ \hp_{jn\ell} ~~=\cr =&~~ q^{n-2
+j-m_1} ~[j]_q ~[\ell]_q ~ \hp_{j-1,n,\ell-1} ~+~ q^{n-2} ~[n]_q
{}~[\ell + m_1]_q ~ \hp_{j,n-1,\ell} ~=\cr =&~~ q^{-2} ~\left(
q^{-m_1}~ \hd_\x ~T_\x ~ \hd_\y ~+ ~ (1/\l) ~\hd_\z ~( q^{m_1}
T_\y - q^{-m_1} T_\y^{-1}) \right) ~T_\z ~ \hp_{jn\ell} ~. \cr }
} Rescaling \intb{b} we finally have: \eqn\sngd{ I_{12}^{m=1}
{}~~=~~ \left( q^{-m_1}~ \hd_\x ~T_\x ~ \hd_\y ~+ ~ (1/\l) ~\hd_\z
{}~( q^{m_1} T_\y - q^{-m_1} T_\y^{-1}) \right) ~T_\z ~. } For
$q=1$ this operator is: ~$I^{m=1}_{12} ~=~ \pd_\x \pd_\y + (m_1+
\y\pd_\y) \pd_\z$ ~\Dob.

Above we have supposed that $m_1, m_2\notin\bbn$. However, after
the proper choice of $a$ in \sng, (e.g., as made above in \snga)
we can consider the singular vector \sng\ and the resulting
intertwining operator also when $m_1$ and/or $m_2$ are positive
integers. Of particular interest are the cases
$m_1,m_2\in\bbz_+$. In these cases the singular vector is reduced
in four different ways (cf. \Doa, \Doc\ formulae (33a-d)).
Accordingly, the intertwining operator becomes composite, i.e.,
it can be expressed as the composition of the intertwiners
introduced so far as follows: \eqna\snge $$\eqalignno{ I^m_{12}
{}~&=~ c_1~ I^{m_2}_1  ~I^{m}_2  ~I^{m_1}_1 ~= &\snge a\cr &=~ c_2~
I^{m_1}_2  ~I^{m}_1  ~I^{m_2}_2   ~= &\snge b\cr &=~ c_3 ~
I^{m_1}_2  ~I^{m_2}_{12}  ~I^{m_1}_1   ~= &\snge c\cr &=~ c_4 ~
I^{m_2}_1  ~I^{m_1}_{12}  ~I^{m_2}_2   ~.  &\snge d\cr }$$ The
four expressions were used to prove commutativity of the hexagon
diagram of $U_q(sl(3,\bbc))$) \Doa.  This diagram involves six
representations which are denoted by $V_{00}$, $V^1_{00}$,
$V^2_{00}$, $V^{12}_{00}$, $V^{21}_{00}$,  $V^3_{00}$, in (29) of
\Doa\ and which in our notation are connected by the intertwiners
in \snge{} as follows: \eqna\sngf $$\eqalignno{ &\tc_{m_1,m_2}
\quad {I^{m_1}_1 \atop \rra} \quad  \tc_{-m_1,m} \quad {I^{m}_2
\atop \rra} \quad  \tc_{m_2,-m} \quad {I^{m_2}_1 \atop \rra}
\quad \tc_{-m_2,-m_1} ~, &\sngf a\cr &&\cr &\tc_{m_1,m_2} \quad
{I^{m_2}_2 \atop \rra} \quad  \tc_{m,-m_2} \quad {I^{m}_1 \atop
\rra} \quad  \tc_{-m,m_1} \quad {I^{m_1}_2 \atop \rra} \quad
\tc_{-m_2,-m_1} ~, &\sngf b\cr &&\cr &\tc_{m_1,m_2} \quad
{I^{m_1}_1 \atop \rra}  \quad  \tc_{-m_1,m} \quad {I^{m_2}_{12}
\atop \rra} \quad  \tc_{-m,m_1} \quad {I^{m_1}_2 \atop \rra}
\quad \tc_{-m_2,-m_1} ~, &\sngf c\cr &&\cr &\tc_{m_1,m_2} \quad
{I^{m_2}_2 \atop \rra} \quad  \tc_{m,-m_2} \quad {I^{m_1}_{12}
\atop \rra} \quad  \tc_{m_2,-m} \quad {I^{m_2}_1 \atop \rra}
\quad \tc_{-m_2,-m_1} ~. &\sngf d\cr }$$ Of these six
representations only $\tc_{m_1,m_2}$ has a finite dimensional
irreducible subspace iff $m_1m_2>0$, the dimension being $m_1 m_2
m/2$ \Doa.  If $m_1=0$ the intertwinig operators with superscript
$m_1$ become the identity (since in these cases the intertwined
spaces coincide) and the compositions in \snge{}, \sngf{} are
shortened to two terms in cases (a,b,d) and one term in case (c),
(resp. for $m_2=0$, two terms in cases (a,b,c), one term in (d)).
Such considerations are part of the multiplet classification
given in \Doa.

\vfil\eject

\listrefs

\end

