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\controldates{16-DEC-2003,16-DEC-2003,16-DEC-2003,16-DEC-2003}
 
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  {American Mathematical Society}

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\begin{document}


\title[$3$-graded Lie  algebras]{On  
$\mathbf{3}$-graded Lie  algebras,  Jordan pairs and the 
canonical kernel function}

\author{M. P. de Oliveira}
\address {Department of Mathematics, University of Toronto, Canada}
\email{mpdeoliv@math.toronto.edu}
\thanks{The  author has been partially supported by FAPESP}

\keywords{Bergman kernel,  symmetric domain, $3$-graded Lie algebra}
\subjclass[2000]{Primary 32M15; Secondary 22E46, 46E22}
\date{October 11, 2001, and, in revised form, October 6, 2003}
\commby{Efim Zelmanov}
\begin{abstract}
We present several embedding results for  $3$-graded Lie
algebras and KKT algebras that are generated by two homogeneous 
elements of degrees $1$ and $-1$. We also propose the
canonical kernel function for
a ``universal Bergman  kernel'' which  extends  the usual Bergman
kernel on a bounded symmetric domain to a group-valued function or,
in terms of formal series, to an element in the formal completion
of the universal enveloping algebra of the free $3$-graded Lie
algebra in a pair of generators.
\end{abstract}
\maketitle

\section{Introduction}\label{S:Fgla}
  A $\mathbb Z$-graded Lie algebra ${\mathfrak  g}$
  of the form
\begin{equation}
  {\mathfrak  g }={\mathfrak  g_{-1} }\oplus{\mathfrak  g_0 }
  \oplus{\mathfrak  g_1 }
\end{equation}
  over a field $K$  is called a
  {\it $3$-graded Lie algebra.}
 They include the complexification of
 semisimple Lie algebras of
 Hermitian type with  gradation provided by
  the Harish-Chandra  decomposition,
 Heisenberg algebras, KKT (Kantor-Koecher-Tits) algebras, among
 other examples (see \cite{NE},
\cite{SA}).

    Several commutation expressions obtained for those Lie algebras
    can be generalized for $3$-graded Lie algebras
    over a field $K$ of \hbox{characteristic} zero,
     especially those
    between elements of the
    universal  \hbox{enveloping} algebras of  ${\mathfrak  g_{-1}} $ and
${\mathfrak  g_1 }$.
    In order to work out these  relations,
    we \hbox{introduce} the  free $3$-graded Lie algebra $\fla $
      \hbox{generated} by
   elements $ x $  of degree $ 1 $ and  $ y $
   of degree  $ -1 .$
Indeed, one can manipulate \hbox{formal} series in
   the elements of  $\fla$
    and exponentiate them to get a group, which can be handled
    by formal analytic  methods.
   A major step to prove these \hbox{relations} is
   to show that the center of $\fla$
   is zero if char$\, K = 0$. As a consequence,
  $\fla$ can be realized as a subalgebra of $\slpo$,
    where $t$ is an indeterminate.

   Recall that a {\it Kantor-Koecher-Tits algebra},
   or {\it KKT algebra} for short \cite{NE}, is a
  $3$-graded Lie algebra satisfying
\begin{equation}\label{E:KKT_def1}
    \; {(\rm i)} \; {\mathfrak  g_0 }=
   [{\mathfrak  g_{-1} }, \s {\mathfrak  g_1 }];
   \; {(\rm ii)} \; \{ x \in {\mathfrak  g_0 }\s |\;
   [x, {\mathfrak  g_{-1} }\s] = [x, {\mathfrak  g_1 }] = 0 \}
   = 0 \, .
\end{equation}
They
 correspond to the concept of Jordan pairs  written in terms of Lie
algebras for char$\, K \neq 2, 3$ (see \cite{LO},
\cite{NE}).
   One  still obtains  very interesting results
   if condition (\rm ii) above is waived.
   This situation can  sometimes be compared
   to the one existing between reductive and semisimple Lie algebras.


    From a $3$-graded Lie algebra ${\mathfrak  g}$ one obtains a KKT
   algebra ${\mathfrak  g}^{\#}$,   defining
\begin{equation}\label{E:3G-KKT}
   {\mathfrak  g}^{\#} ={\mathfrak  g'}\s /({\mathfrak  g'_0} \cap
Z_{\mathfrak  g'}),
\end{equation}
  where ${\mathfrak  g'}$ is the 3-graded Lie
  subalgebra of ${\mathfrak  g}$
 spanned by ${\mathfrak  g_{-1}}$ and ${\mathfrak  g_1}$, and
 $Z_{\mathfrak  g'}$ denotes its center.
  One has
\begin{equation}\label{E:3G-KKT2}
   {\mathfrak  g}^{\#}_i \cong {\mathfrak  g}_i,\;  i = -1,1\s , \quad
   {\mathfrak  g}^{\#}_0 \cong {\mathfrak  g'_0} \s
   /({\mathfrak  g'_0} \cap Z_{\mathfrak  g'})\s \cong
        \ad_{\mathfrak  g'} {\mathfrak  g'_0}\s , \quad
    {\mathfrak  g'_0 }= [{\mathfrak  g_{-1} }, \s {\mathfrak  g_1 }].
\end{equation}


   According to Corollary \ref{C:equiv} and Theorem
   \ref{T:com1}, $\fla$ is isomorphic to
   $\flaj$, the free KKT algebra in a pair of variables,
   if \hbox{{\rm char}$\, K = 0$.}
 From  the
  theory of Jordan pairs, it is known
  that, for \hbox{{\rm char}$ \, K \neq 2,3$,} $\flaj$ can
  be  embedded into $\slpo$
  \hbox{(cf Theorem \ref{T:KKT},} \hbox{item (a));}
    in contrast,  this is always false for $\fla$ if
     $K$ has nonzero characteristic greater than $3$ (cf.
    Remark \ref{R:charp}).



   In at least two important instances of the theory,
    the  canonical kernel function appears as
   a fundamental concept. In the setting of Lie groups of
   Hermitian type, it can be applied to the description of
   reproducing kernels
   for Hilbert spaces of holomorphic functions
      associated with the holomorphic discrete series
     representations (see \cite{DO1}). In terms of free
     $3$-graded Lie algebras, it provides the zero degree
     part in the formal Harish-Chandra decomposition of the product
      of certain exponentials. As a result,  one obtains
      commutation relations for the universal enveloping algebra of
      a $3$-graded Lie algebra, which generalize well-known
      expressions for $\mathfrak{sl}_2(K)$ (see~\cite{DO2}).

\section{The free case} \label{SS:UO}
      We say that a Lie algebra
   $\fla $ over $K$ is   the {\it free $3$-graded
   Lie algebra generated by
   variables $ x $  of degree $ 1 $  and  $ y $
   of degree  $-1$} ({\it or freely generated by the pair $(x,y)$})
    if, for any $3$-graded Lie algebra $ \mathfrak h$
  over $K$  and elements $ \overline{x} \in  \mathfrak h_{1}$
   and $ \overline{y} \in  \mathfrak h_{-1},$
     there is a unique homomorphism of graded Lie algebras
     $\psi :\fla \rightarrow \mathfrak h$
     such that $\psi(x)=  \overline{x}$ and $\psi(y)=  \overline{y}$.
    Clearly $\fla$    exists and it is unique up to an isomorphism
    of graded Lie algebras.

   Recall that  $ \slpo $
   is also a $3$-graded Lie algebra
     over $K$ with respect to the
      gradation
\begin{align*}
    &\slpo_{-1} = \biggl \{ \bl \left( \begin{array}{cc}
                          0 & 0 \\
                          p(t) & 0
                        \end{array} \right)  \bb |
                        \bb p(t) \in \s K [ t ] \bb  \biggr \},\\
            \quad \\
    &\slpo_{0} \bb  =  \biggl \{ \bl \left( \begin{array}{cc}
                          p(t) & 0 \\
                          0    & -p(t)\bl
                            \end{array} \right)  \bb |
                            \bb p(t) \in  \s K [ t ] \bb  \biggr \},\\
            \quad \\
    &\slpo_{1} \bb = \biggl \{ \bl \left( \begin{array}{cc}
                          0 & p(t) \\
                          0 & 0
                        \end{array} \right)  \bb | \bb p(t)
                         \in \s K [ t ] \bb  \biggr \},\\
             \quad \\
     &\slpo_{i} \bb = \bl \{ 0 \},
     \bb i \in {\mathbb Z} \s - \s  \{-1, 0, 1\}.
\end{align*}


     We have the following relations between $\fla$ and $\flaj$,
     the free KKT algebra in a pair of generators.


    \begin{Th}[\cite{DO2}] \label{T:KKT}
  Suppose {\rm char}$\, K \neq 2,3$.
  Let $\pi: \fla \rightarrow \flaj $ be
  the natural projection, and let
 $\mathfrak i: \fla \rightarrow \slpos   $ be
  the homomorphism  of  graded Lie algebras defined  by
\[ \mathfrak i(x) = tE =
                        \left( \begin{array}{ll}
                          0 & t   \\
                          0 & 0
                        \end{array} \right),
  \qquad \qquad  {\mathfrak i} (y) = tF =
                       \left( \begin{array}{ll}
                          0 & 0 \\
                          t &  0
                        \end{array} \right). \]

  {\rm(a)} There exists a monomorphism
\[ \mathfrak i^{\#}: \flaj \rightarrow \slpos \]
    of  graded Lie algebras such that
\[ {\mathfrak i} = {\mathfrak i^{\#}}\circ \pi .\]

   {\rm(b)}
    $ Z_{\fla} = \ker {\mathfrak i} = \ker \pi
      = \text{\rm linear span}
    \{\, [ \; [ x, y ],  (\sdec \ad x \ad y)^{i}\, [ x,y  ]\; ] \,| \,
    i \geqslant 1 \}.$
  \end{Th}



  \begin{Cor}[\cite{DO2}] \label{C:equiv}
 If {\rm char\,}$K \neq 2,3$, the
following  statements are equivalent:
 \begin{itemize}
 \item[{\rm(a)}] $\fla = \flaj $ {\rm(}$\fla$ is a KKT algebra{\rm)}.

 \item[{\rm (b)}] The center of $ \fla $ is zero.

 \item[{\rm (c)}] The homomorphism  of  graded Lie algebras
                     $\mathfrak i: \fla \rightarrow \slpos   $
                    defined  by
                   $ {\mathfrak i}(x) = t\, E ,\;
                   {\mathfrak i} (y) =  t \, F $
                    is a monomorphism.
\end{itemize}
 \end{Cor}


  When the field has characteristic zero,  the above statements are
  satisfied, as shown next:

  \begin{Th}[\cite{DO2}] \label{T:com1}
   If {\rm char\,}$K = 0$, the center of $\fla$ is zero.\end{Th}
  \begin{proof}
     Let $ I : \fla \rightarrow \fla $  be the identity map.
     We define
\[ A_i : \fla \rightarrow \fla, \;
  i \in {\mathbb N} , \] 
as
   $A_0 = I$  on  ${\fla_1} $,
    $ -I $ on  $ \fla_{-1}$
    and zero   on $\fla_0$,
\[ A_i = \ad ((\ad x \ad y)^{i-1}[ x, y]),\;
  i \geqslant 1 ,\]
   and write  $A = A_1$ for short. It is enough  to show that
\[[\bl [ x,y \es ],\bl (\ad x \ad y)^{m-1}[x,y \es ] \bl ] = 0\]
  for any positive integer $m,$
   in view of  Theorem (\ref{T:KKT}), item (b).


   We prove it by induction. For $ m=1$ it is trivial.
   Suppose it is valid for $m\leqslant p \bl $. Then the following
   identities hold:


   \begin{itemize}

    \item[{(\rm i)}] $A_{i+1}x = AA_i x$, $p \geqslant i \geqslant 0$.


   \item[{(\rm ii)}] $A_{i+1}y = - AA_i y$, $i\geqslant 0 .$

   \item[{(\rm  iii)}] If $i+j=p$ \s  for integers $ i,j\geqslant 0 $, then
   \[[A_{i+1}x,A_jy \es ]=A[x,A_py \es ] + [A_ix,A_{j+1}y \es ].\]
\end{itemize}
We prove $ {(\rm  iii)}.$ This is
   obviously valid if $p =1 .$ Now suppose $p \geqslant 2$. One has
\[[A_{i+1}x,A_jy \es ]= [AA_ix,A_jy \es ] = A[A_ix,A_jy \es ] +
[A_ix,A_{j+1}y \es ] .\]
But
\begin{align*} A[A_ix,A_jy \es ] &= A[A_iA_jy,x \es ] + AA_i[x,A_jy \es ] =
A[A_iA_jy,x \es]\\
      &= (-1)^{j-1}A[A_iA^jy,x \es] = (-1)^{j-1}A[A^jA_iy,x \es ] \\
      &= (-1)^{i+j}A[A^jA^iy,x \es ]
     = -A[A_py,x \es ]= A[x,A_py \es ].
     \end{align*}
 Hence
\[[A_{i+1}x,A_jy \es ]= A[x,A_py \es ]+[A_ix,A_{j+1}y \es ].\]

    Back to the proof of the theorem, one has for $m=p+1$:
\begin{align*}
A_{p+1}[x,y \es ] &=  [A_{p+1}x,y \es ] + [x,A_{p+1}y \es ] =
     - [A_{p+1}x,A_0y \es ] + [x, A_{p+1}y \es ]\\
    &= -(p+1)A[x,A_py \es ] - [A_0x, A_{p+1}y \es ] + [x, A_{p+1}y \es ]
   \;\;(\text{by } {(\rm iii)}).
  \end{align*} 
    Therefore
\[ (p+2) A_{p+1}[x,y \es ] = -[x,A_{p+1}y \es ]
     + [x,A_{p+1}y \es ] =0\]
     and the theorem is proved.
    \end{proof}

    \begin{Rem}\label{R:charp} The following   construction of $\fla$ over
$K,$
    {\rm char}\,$K =p  > 3, $ which also works in characteristic zero,
    has been pointed out to us
   by E. Zelmanov:
   Let
\[\slpo \oplus K[t]   \]
be the $3$-graded Lie algebra such that
       $\slpo$ is graded as before,  the elements of $K[t]$ are
       central and  homogeneous  of degree $0$ and
\[ [t^i a, t^j b] = t^{i+j} [a,b]+ i \, \delta_{i+j,\,p}\,
(a,b)\, t^{i+j},\]
        where $a,b\!\in \sltwo$, $(\, ,\,)$ denotes the
         Killing form of $\sltwo$, and
         $\delta_{k,\, p}=1$ if \hbox{$k=0 \modu p$},
         $\delta_{k,\, p}=0$ otherwise. (Notice its resemblance to
         an affine Lie algebra.) In view of the previous
         development, it is clear that the $3$-graded subalgebra
         spanned by
\[ x = t\,E, \quad y = t\,F, \]
            gives  a realization of $\fla.$ In particular, for
    {\rm char}\,$K > 3,$ $\fla$ has a nontrivial center and does not satisfy
    Corollary \ref{C:equiv}.
  \end{Rem}
  In what follows, we  restrict our attention to
  the theory in characteristic zero.

  We notice that $\fla$ and its universal enveloping algebra
  $ {\mathscr U} (\fla)$
  admit an ${\mathbb N}$-gradation given by the sum of  occurrences
  of $x$ and $y$ in each monomial, which is called {\it total
  gradation} in either case. We  consider here
  their formal completions $  \hfla$ and
  $ \widehat {\mathscr U} ( \fla)$, which consist of formal series
  with finitely many terms for each degree. Similarly the
   {\it total gradation} or
     ${\mathbb N}$-gradation of $\slpo $ is the one
   whose homogeneous elements of degree $n$ are the matrices over
     scalar multiples of $t^n$ having  trace zero, and
     so forth.
  We have the following useful characterization of $\fla$ and its
  completion $  \hfla:$

  \begin{Cor}[\cite{DO2}] \label{T:embed}
  Let $\fla $ be the free $3$-graded Lie algebra over $K$,
${\rm char}\, K = 0$,
   generated by
   variables $ x $ of degree $ 1 $ and  $ y $ of degree $ -1 $,
   endowed with the total gradation defined by
   the sum of occurrences of x and y
  in each Lie monomial.
  
  \begin{itemize}



   \item[{\rm (a)}] The homomorphism
   $\mathfrak i: \fla \rightarrow \slpos $ given by
\[ \mathfrak i(x) =  \left( \begin{array}{ll}
                          0 & t   \\
                          0 & 0
                        \end{array} \right),
  \qquad \qquad  {\mathfrak i} (y) =  \left( \begin{array}{ll}
                          0 & 0 \\
                          t &  0
                        \end{array} \right) \]
 is a monomorphism of  ${\mathbb N}$-graded Lie algebras.

  \item[{\rm (b)}]  This monomorphism extends  to a monomorphism
\[  \widehat {\mathfrak i} :  \hfla \rightarrow {\slses}\]
    of   Lie algebras, where $\hfla$ denotes the completion
    of $\fla$ with respect to the   total gradation. 

  \item[{\rm (c)}] Let $  \exla$  and $\exses $ be  the groups obtained
   formally exponentiating  $  \hfla$ \hbox{and} $\slses$
    inside  $ \widehat {\mathscr U} ( \fla)$ and
    $ M_{2} (\ser)$, respectively. Then the  diagram
\[  \begin{CD}
         \hfla @>\widehat {\mathfrak i}>>       \slses  \\
          @V{\exp}VV                           @VV{\exp}V\\
         \exla            @>>{\mathfrak I}>          \exses
\end{CD} \]     
is commutative, the  vertical maps are bijective, and the
     map $\mathfrak I$ is a mono\-morphism of groups.
     \end{itemize}
 \end{Cor}


 \section{The canonical kernel function} \label{S:ckfction}



The canonical kernel function is a generalization of the usual
Bergman kernel function of a bounded symmetric domain $\myD$
    but it is group-valued instead. It has been introduced  by
    I.  Satake
      in  \cite{SA0},
\cite{SA1},
\cite{SA} through  several distinct contexts.
       It is  related to   different  topics such as
    the geometry of $\myD$, the contravariant
    form on highest weight modules, the Harish-Chandra decomposition for
   groups of Hermitian type  and also reproducing kernels for families
    of holomorphic functions
     in the ``holomorphic discrete series"
     (which includes the Hilbert space
    of square-integrable
    holomorphic   functions on $\myD$ as a special case).



   Let $ \g $ be a  real semisimple Lie algebra
  of Hermitian  type. By Hermitian type we
  mean that for  any Cartan decomposition
   $ {\mathfrak g = \mathfrak k + \mathfrak p} $
  there is an element  $H_0$  in the center $ \mathfrak c$
  of $ \mathfrak k$ such that
  $  \ad  H_0$ is a complex structure on $ \mathfrak p $.
 Fix one of them and let $\theta$ denote
  the corresponding Cartan involution.
    Choose  a Cartan  subalgebra $ \mathfrak h$ of
    $\mathfrak g$ contained in $\mathfrak k$. Let $\mathfrak g_{c}$,
   $\mathfrak k_{c}$, $\mathfrak c_{c}$,  $\mathfrak h_{c}$,
   $\mathfrak p_{c}$ be the complexifications of
  $\mathfrak g$,  $\mathfrak k$,
    $\mathfrak c$,  $\mathfrak h$, $\mathfrak p$,
    and $\mathfrak p_+$, $\mathfrak p_-$ the eigenspaces
    of $\ad_{\mathfrak p_c} H_0 $ with eigenvalues
    $ i $ and $ -i $, respectively.
   Let $G_c$ be the connected simply connected Lie group with Lie algebra
    $\mathfrak g_{c}$ and $ P_+ $, $P_- \s$, ${\mathcal K_c} \, ,$
    $  { \mathstrut  G},$
     $ { \mathcal   K}$ the
   analytic subgroups of  $G_c$ with  Lie algebras $\mathfrak p_+$,
$\mathfrak p_-$,
    $\mathfrak k_c$, $\mathfrak g$ and $\mathfrak k$, respectively.
  The exponential map of $G_c$ induces a holomorphic diffeomorphism
  of $\mathfrak p_+$ ($\mathfrak p_-$) onto $ P_+ $ ($ P_-$)
  (see \cite{HC3}).



     By a theorem of Harish-Chandra, the map
\[ \psi: P_+ \times {\mathcal K_c} \times P_- \rightarrow G_c \, ,\qquad
      \psi(p,k,q\s )=p k q\]
   is a holomorphic diffeomorphism onto an open dense   subset of
     $ G_c $ and
     \hbox{$  { \mathstrut   G }\subset P_+  { \mathcal K_c } P_- $.}
    In general, given $g \in  {\rm Im\s} \psi $,
    we write $ g = g_+ \,  g_0 \, g_- $ for the
      $ `` P_+ {\mathcal K_c} P_-  "$ decomposition of $g$,
      where $g_+ \in P_+$, $g_0
     \in { \mathcal K_c}$ and $g_- \in P_-$.



      We have the following
     open holomorphic embeddings:
\begin{equation}
     \mathcal D =   { \mathstrut   G} / { \mathstrut  {\mathcal  K} }
     \simeq  { \mathstrut   G} \, {\mathcal  K_c}  P_- /
     {\mathcal K_c}  P_-
     \hookrightarrow   P_+  {\mathcal K_c}  P_-/ { \mathcal K_c } P_-
     \simeq P_+ \simeq  \mathfrak p_+ \,.
\end{equation}

     One can prove  that the image of
     the composition of these
     embeddings is  bounded with respect to the
     metric on $\pp$ induced by the Killing form of $\g_c$ and
    a holomorphically symmetric connected open subset
    of $\mathfrak p_+,$
      which is known as the
    Harish-Chandra realization
     of $\mathcal D$ as a bounded symmetric domain (see \cite{SA}).


    The  \textit{canonical kernel function}
    of a bounded symmetric domain in its
       Harish-Chandra realization (\cite{SA}) is given by
\begin{equation}
     \kappa : \myD \times  \myD \rightarrow  {\mathcal K_{c}}, \quad
        \kappa (z,w\s ) =
    ((\exp -\overline{w} \,\exp z\s )_{0}\s )^{-1} \in
      {\mathcal K_{c}},
\end{equation}
   which is  well
  defined on ${\myD}\times{\myD}$ because, for $z, w \in \mathcal D$,
\begin{equation}
  \exp -\overline{w} \,\exp z \in
  (\overline{G {\mathcal K_c}  P_-})^{-1} G {\mathcal K_c}  P_- =
   P_+ {\mathcal K_c} G {\mathcal K_c} P_- = P_+  {\mathcal K_c} P_-.
\notag \end{equation}
In other words, $\kappa (z,w\s )$ is
   the inverse of the ${\mathcal K_c}$-part of
\begin{equation}\label{E:P1}
 \exp -\overline{w} \, \exp z
\end{equation}
 in the $ `` P_+ {\mathcal K_c} P_-  "$ (also called Harish-Chandra)
decomposition.

   It is  straightforward to verify that:
\begin{itemize}
   \item[(i)] $ \kappa(0,0\s )= e, $

    \item[(ii)] $\kappa(z,w\s )$ is holomorphic in $z,$


    \item[(iii)]  $\kappa(z,w\s ) =\overline {\kappa(w,z\s )^{-1}},$

    \item[(iv)]   $ \kappa(gz,gw\s ) =  J(g,z\s ) \kappa(z,w\s )
    \overline {J(g,w\s )^{-1}}, $
               $\;    g \in G,\; z,w \in {  \mathcal  D}$, where $J$ is called the
    {\it canonical automorphy factor} of
    $G$, given by
\[ J: G \times \myD \rightarrow  {\mathcal K_{c}}, \;
      {\mathstrut  J}(g,z\s ) = (g\s  \exp z\s )_0.\]
      \end{itemize}
   We recall that the canonical kernel function can, alternatively, be
defined by  properties (i)--(iv).

  \begin{Th}[\cite{DO1}] \label{T:maintheo}
   Let $\mathcal D$ be a bounded symmetric domain in its
  Harish-Chandra  rea\-lization, with notation as above. Then
\begin{equation*}
   \kappa (z, w\s ) = \exp
   \displaystyle\frac {\log (1-\ad z \ad\overline w /2\s )}
   {\ad z \ad\overline w/2}[z,\overline w \s ],
    \; \quad z , w \in \myD. \notag
\end{equation*}
    The series under the  exponential sign  converges uniformly inside
   compact subsets
   of ${\mathcal D}\times{\mathcal D}$ .
   \end{Th}
\noindent The series above should be understood as the formal development of
 $ ({\log(1-x\s )})/{x},$ where $x$ is replaced by the linear
 operator $\ad z \ad\overline w /2$ and the output is   evaluated at
  $[z,\overline w \s \s].$ We shall make use of this notation again
 (see Corollary \ref{C:AHCD}).

  The relation between the canonical kernel function and the
  Bergman kernel of $\myD$ is the following (\cite{SA}):

\begin{Prop}\label{P:berker}
   The  Bergman  kernel $K(z,w\s )$ of $\myD $
     is given by
\begin{equation*}\label{E:berg}
      K(z,w\s ) = \dfrac{1\s }{ {\rm vol}\, \myD }
        \bl  \det \Ad^{-1}_{\pp}\Ke (z,w\s ).
 \end{equation*}
\end{Prop}

    The volume ${ {\rm vol}\, \myD }$ above is the one relative to the
    Euclidean metric on $\pp$ induced by the Killing form of $\g_c.$
     For the sake of completeness, we notice that (\cite{DO1},
\cite{SA}):
\begin{align*}
    &\Ad_{\pp} \Ke (z,w\s ) =
   1- \ad z \ad \overline w +
   \dfrac {1}{4}(\ad z\s )^2 (\ad \overline w\s)^2
     \\
    &\qquad = 1- \ad [z,\overline w\,] +
    \dfrac{1}{2}(\ad [z,\overline w\,]\s )^2
     - \dfrac{1}{4} \ad [z,[\overline w,[z,\overline w \s ] \s ]]
     \quad {\rm on}\;\; \pp.
\end{align*}

  We have considered here the representation
  $\pi = \det^{-1}  \Ad_{\mathfrak p_+}$ of
  ${\mathcal K}.$
  In their celebrated paper (\cite{FK}),
   A. Kor{\'a}nyi and  J. Faraut have studied reproducing kernels for
    holomorphic discrete series representations
   as they are induced by characters of ${\mathcal K}$ (the scalar case).
  As A. Kor{\'a}nyi remarked, {\it `` The analytic problems \ldots are
  meaningful and interesting for the vector-valued case too,
  but there is very little known
   about them at present \ldots~"}
  (see \cite{FA}, page 259).
   In  (\cite{DF}) and later, independently in (\cite{DO1}),
     the reproducing kernel of a
   general vector-valued
   holomorphic discrete series representation
   is shown to be equal to
    the representation   calculated at the
   canonical kernel function,
   up to a multiplicative factor.
   This  description  also appears,  in
    implicit form, earlier in \cite{LA}, 
\cite{W}.
    The
   multiplicative factor
    has been  calculated explicitly in \cite{DO1} in terms of
     the formal degree of the representation
      in the  holomorphic discrete series, the degree of  its
      irreducible
       $ { \mathcal   K}$-type
    having the same highest weight
       and the Euclidean
      geometry of the domain.


   The above results motivate us to regard
   the canonical kernel function
   as a natural candidate for a ``universal Bergman kernel''
   in the setting of Lie groups of Hermitian type.
    Next we go over this concept  in the context of
   free $3$-graded Lie algebras.




   \section{The formal canonical kernel function} \label{S:cr-fckf}
 Here  $K$ denotes a field of characteristic $0.$
 From Corollary \ref{T:embed}(c) it follows
\begin{Cor}[\cite{DO2}] \label{C:AHCD}
   In $\exla $ one has
\begin{equation*}\label{E: AHCD}
   \exp -x  \s  \exp y  =
    \exp \s (\s ({1  - \ad y \ad x /2})^{-1} y \s )
    \s    \s
    \kappa (x \s, y \s ) \s  \s
    \exp \s ( \s - (1 -  \ad x\s \ad y /2 )^{-1} x \s ),\notag
\end{equation*}
    where
\begin{equation}
   \kappa (x, y\s ) = \exp \s
   \bigr( \s \displaystyle\frac  {\log (1 - \ad x \ad y /2\s )}
   {\ad x \ad  y/2}[x,  y \s ] \s \bigl), \tag{i}
\end{equation}
which can be expanded as 
\begin{equation}\label{E:EXP}
    \quad    \kappa (x, y\s ) =
     \sum_{m\geqslant 0} \,
     \dfrac{(-1)^m}{m!} \; \prod\limits_{i=1}^{m}
      ([x, y\s ]-(m-i\s )\ad x \ad  y /2\s ).\,
      \,\tag{ii}
\end{equation}
   In {\s \rm (ii)}, for $u, v \in {\mathscr U}(\fla \s )$ and
   $U, V \in End_{K} \bigr({\mathscr U}(\fla \s ) \bigl )$,
    the product in  \[{\mathscr U}(\fla \s )\oplus
   End_{K} \bigr({\mathscr U}(\fla \s ) \bigl )\]
   is defined  by
\[ (u \oplus U\s )(v \oplus V\s ) = (uv + U ( v \s ) \s ) \oplus
   (uV + U{\scriptstyle \circ} {V}\s ), \bb and  \]
\[     \text{} \qquad \prod\limits_{i=1}^{n} a_i =
     ( \cdots (\,(\s a_1 a_{2} \s )\s  a_{3} \s)\cdots a_n \s),
    \quad a_i \in
    {\mathscr U}(\fla )\oplus
   End_K \bigr({\mathscr U}(\fla ) \bigl ).\]
\end{Cor}


  The product above is interpreted  as $1$  for $ n <1 .$
   Now we fix  a general $3$-graded Lie algebra $\g$
 over  $K.$
   From Corollary \ref{C:AHCD}, one obtains
 commutation relations for  powers of   elements  of
 $\g_{-1}, \g_{1} \subset {\mathscr U} ( \g).$
       Let $ z \in \g_{1}$ and $w \in \g_{-1}$.
       Corollary \ref{C:AHCD} implies that (\cite{DO2})
\begin{equation}\label{E:commuta}
  \dfrac{ z^i }{i!} \dfrac{ w^j }{j!} =
   \sum\limits_{m = 0 }^{ \min (i, \s j)} \dec  \decc \decc \decc
    \sum\limits_{\substack
  { \bb \bl \qquad \quad  m_1 + m_2 + m_3  = m
   \\ \qquad \quad \bl  m_{1}, \s \s  m_{2}, \s \s m_{3} \s
  {\mathstrut  \geqslant 0}}}
   \decc \decc \dec \dec A^{j - m}_{m_1}\s
   B_{m_2} \s C^{i - m}_{m_3},
   \quad {\rm where}\\
\end{equation}
\begin{align*}
     A^{0}_{0} &= 1 \s , \quad   A^{0}_{m} = 0,  \quad  m > 0 \s . \\
     A^{k}_{m} &=
      \dfrac{ 1 }{k !}   \decc \decc \dec \dec \nbb
        \sum \limits_{\substack{
        \bb \bl \qquad \quad n_1 + \cdots +n_{k} = {m}
   \\ \qquad \quad n_1, \ldots ,n_{k} \geqslant \s 0 }}
   \decc \decc \dec \dec
   \dec \dec
   (\s (-\ad w \ad  z /2\s )^{n_1}w \s)
   \cdots (\s (-\ad w \ad  z /2\s)^{n_{k}}w \s),
   \bl  k > 0 \s , \s  m \geqslant 0.
     \quad \\ \\
   B_{m} &= \dfrac{ 1 }{m!} \; \prod\limits_{i=1}^{m}
      ([z, w\s ]-(m-i\s )\ad z \ad  w /2\s )\,, \quad m \geqslant 0.
       \quad \\ \\
   C^{0}_{0} &= 1 \s, \quad   C^{0}_{m} = 0,  \quad  m > 0 \s .\\
   C^{k}_{m} &=
      \dfrac{ 1 }{k !}   \decc \decc \dec \dec \nbb
        \sum \limits_{\substack{
        \bb \bl \qquad \quad n_1 + \cdots +n_{k} = {m}
   \\ \qquad \quad n_1, \ldots ,n_{k} \geqslant \s 0 }}
    \decc \decc \dec \dec
    \dec \dec
   (\s (-\ad z \ad  w /2\s )^{n_1}z \s)
   \cdots (\s(-\ad z \ad w /2\s)^{n_{k}}z \s),
   \bb k > 0 \s , \s  m \geqslant 0.
    \end{align*}
For instance, for $\g = \mathfrak{sl}_2(K )$ and
\[ H = \left( \begin{array}{rr}
                          1 &  0 \\
                          0 & -1
                        \end{array} \right) , \quad
          z = E =  \left( \begin{array}{rr}
                          0 & 1 \\
                          0 & 0
                        \end{array} \right),  \quad
                    w = F = \left( \begin{array}{rr}
                          0 & 0 \\
                          1 & 0
                        \end{array} \right), \]
one obtains the classical commutation relations:
\[ \dfrac {E^i}{i!} \dfrac {F^j}{j!}=
    \sum\limits_{m = 0 }^{ \bl  \min (i, \s j)}
  \dfrac { F^{j-m}}{(j-m)!} \s
   \s \binom { H -i - j + 2 m}{ m }   \s
   \dfrac {E^{i-m}}{(i-m)!} \bb \cdot  \]


\section{$3$-graded Lie algebras in a pair of
generators}\label{S:3gr}

  We assume  throughout this section that
  the field $K$ has  characteristic $0.$
   Given $p(t) \in K[t]$, we write $\idep \s \equiv \s  p(t)\cdot
   K[t].$

\begin{Th}[\cite{DOL}] \label{T:Emb} Any $3$-graded Lie algebra
  generated by an element of
  degree $1$ and another of degree $-1$
   over a field $K$ of characteristic zero can be realized as a
   $3$-graded Lie subalgebra of $\slpoq$ for some \hbox{$p(t)\in K[t]$}.
   Moreover,
   it is symmetric if and only if
   it  is isomorphic, as a graded Lie algebra, to  the
   $3$-graded
   Lie subalgebra
   generated by
\[ (t \s + \dec \idep) E
    \text{ and } (t \s + \dec \idep) F \]
inside $\slpoq$ for some
   \hbox{$p(t)\in K[t]$}.
  \end{Th}

The characterization  obtained above can be used to find a
classification of  $3$-graded Lie algebras in a pair of generators
over $\com \;$(\cite{DO3}). Furthermore, since the same $3$-graded
Lie algebra can be
   embedded in  \hbox{$\slpoq$} in different ways,
   it can be  of interest to understand how
    it depends on the  chosen polynomial
    \hbox{$p(t)\in K[t]$} and
     representatives for the
   generators in  \hbox{$\slpoq$.} The following example
illustrates that issue.

  Let $p(t) = t^3 - c\s t, \; c \in \com \, ,$
\[ x = (t + \idep) \s E \, ;
     \qquad y =  (t + \idep) \s F,  \]
and let
  $\g$ be the $3$-graded Lie algebra
 spanned by $x$ and $y$
  in \hbox{$\slpoq.$} Then
\[ [x, y] \equiv h,\;
  [h,x]= 2 \s c\s x; \; [h,y]= -2 \s c\s y, \]
which shows that $\g$ is a linear combination of $x, y, h$.

 If $c \neq 0$, then it is easy  to see that $x, y, h$ can be
 normalized  in order to produce
\[ [x, y] = h,\; [h,x]= 2 \s  x; \; [h,y]= -2 \s  y .\]
In other words, $\g$ is isomorphic to $\mathfrak{sl}_2(\com \,).$
 On the other hand, if $c=0$, we have
\[ [x, y] = h,\; [h,x]=   [h,y]= 0, \]
and  $\g$ is the   $3$-dimensional Heisenberg algebra.

\begin{figure}[h]
 \centerline{ \qquad $\boxed{p(t)= t^3 -c\s t}$ }
\hspace*{-3in}\begin{picture}(100,100)(-50,-50)

 \put(-50,0){\vector(1,0){100}}
  \put(0,-50){\vector(0,1){100}}

   \put(-40, 40){$\com$}
   \put(10,50){${\mathfrak{sl}_2}(\com \,)$}



  \put(0,0){\circle{4}}
  \put(2,2){$t_1 =0$}

  \put(30,30){\circle{4}}
  \put(34,32){$t_2 = \sqrt{c}$}

  \put(-30,-30){\circle{4}}
  \put(-60,-46){$t_3 = -\sqrt{c}$}



  \put(210,50){Heisenberg}


 \put(150,0){\vector(1,0){100}}
  \put(200,-50){\vector(0,1){100}}
  \put(160, 40){$\com$}

  \put(200,0){\circle{4}}
  \put(201.4,1.4){\circle{4}}
  \put(198.6,-1.4){\circle{4}}

   \put(204,10){$t_1= t_2 =t_3 =0$}
 \label{Fig1}
  \end{picture}
\caption{Dependence of $\mathbf{\g}$ on the roots of
$p(t)$.}
\end{figure}


 One can realize from the example above that the  polynomial dependence
 of the Lie algebra  occurs at the level of root multiplicities
 rather than with respect to the polynomial itself. Indeed, such
 behavior occurs  in the general case. Moreover, the polynomial
 used in the symmetric case can be conveniently chosen, as
 described next:


  \begin{Th}[\cite{DO3}]\label{E:zeros} $\;$
 {\rm (a)} $\;$
  Let $\g$ be a symmetric
    $3$-graded Lie algebra in a pair\linebreak[4] of generators
  over  $ \com \,.$ Then $\g$ is isomorphic to the $3$-graded
   Lie subalgebra of \linebreak[4] $\slpoqc$
   generated by
  \[ (t \s + \dec \idep) E
    \text{ and } (t \s + \dec \idep) F \]
   for some
   $p(t) =  t^m q(u)$, $u = t^2,$ 
   $q(u) \in K[u],$ $ q(0) \neq 0. $


 If $\g = {\mathfrak l} \oplus  {\mathfrak n}$ is a  Levi decomposition
  of $\g$ as the direct sum of a semisimple subalgebra
  ${\mathfrak l}$  and the radical  ${\mathfrak n}$ of $\g$,
  then ${\mathfrak n}$ is nilpotent and
   \[ {\mathfrak l} \cong \bigoplus\limits_{i=1}^{k}
   \mathfrak{sl}_2(\com ),\]
  where $k$ is the number of distinct roots of $q(u).$\\

{\rm (b)} Let $\g_i, \, i=1,2,$ be
  generated by
\[ (t \s + \langle p_i(t) \rangle) E
    \text{ and } (t \s + \langle p_i(t) \rangle) F \] in
    $\slpoqci$
   for polynomials
   $p_i(t)$ such that
\begin{align*}
   & p_1(t) = t^l (u-a_1)^{m_1}\cdots(u-a_k)^{m_{k}}, \; a_i \neq a_j
   \neq 0 \text{ for } i \neq j, \, u = t^2,\\
   & p_2(t) = t^{\tilde l} (u-b_1)^{n_1}
   \cdots(u-b_{\tilde k})^{n_{\tilde k}},
    \; b_i \neq b_j
   \neq 0 \text{ for } i \neq j, \, u = t^2.
\end{align*}
  Then $\g_i$ are isomorphic as Lie algebras iff
   $k = \tilde k$, $l = \tilde l$  and
   the sequences $(m_j)$ and $(n_j)$ coincide after some
   reordering. In that case, the isomorphism can be chosen
   to preserve the gradation.
\end{Th}

  Such results raise some questions: suppose $X$ is  an algebraic
  variety over $\com \,,$   $A$ is a subalgebra of the
  regular functions on $X$ and $I$ is an ideal of $A.$
  Consider a finitely generated  $3$-graded Lie subalgebra $\g$ of
  $ \mathfrak{sl}_2(A\, /I \, )$ over $\com \,.$
  How does $\g$ depend on the zero set of $I$
   and  the geometry of $X$
  (possibly under extra
  hypotheses)?

  There seem to be interesting mathematical connections among
  these concepts far beyond the results of Theorem \ref{E:zeros}.






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\end{document}