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\begin{document}

\title{Family algebras}
\author{A. A. Kirillov}
\address{Department of Mathematics, The University of Pennsylvania, 
Philadelphia, PA 19104}
\email{kirillov@math.upenn.edu }
\subjclass[2000]{Primary 15A30, 22E60}
\keywords{Enveloping algebras, invariants, representations of 
semisimple Lie algebras}
%\issueinfo{6}{1}{}{2000}
\copyrightinfo{2000}{American Mathematical Society}
\commby{Svetlana Katok}
\date{December 31, 1999}
\begin{abstract}A new class of associative algebras  is introduced and 
studied. These 
algebras  are related to simple complex Lie algebras (or root systems). 
Roughly speaking, they are finite dimensional approximations to the 
enveloping algebra $U(\mathfrak{g})$ viewed as a module over its center. 

It seems that several important questions on semisimple algebras and 
their representations can be formulated, studied and sometimes solved 
in terms of our algebras. 

Here we only start this program and hope that it will be continued and 
developed.
\end{abstract}
\maketitle

\section*{0. Introduction}

The aim of this paper is to introduce and study a new class of 
associative algebras: the so-called family algebras. We assume that the 
reader is acquainted with the general background of the theory of 
semisimple Lie algebras (see e.g. \cite{H}).

Let $\g $ be a simple complex Lie algebra with the canonical 
decomposition
\begin{equation*}\g = \n _{-} \oplus \h \oplus \n _{+}.
\end{equation*}
We denote by $P$  (resp. $Q$)  the weight (resp. root) 
lattice in $\h ^{*}$ and by $P_{+}$ 
(resp.  $Q_{+}$) the semigroup generated by fundamental weights 
$\omega _{1},\, \omega _{2},\dots ,\,  \omega _{l}$ (resp.  by simple 
roots 
$\alpha _{1},\, \alpha _{2}, \dots ,\, \alpha _{l})$.

For every $\lambda \in P_{+}$ let ($\pi _{\lambda },\, V_{\lambda }$) 
be an 
irreducible representation of 
$\g $ with highest weight $\lambda $. We denote by  $d(\lambda )$ the 
dimension of  $V_{\lambda }$.

Let $\lambda ^{*}$ denote the highest weight of the dual (or 
contragredient) representation which 
acts in $V_{\lambda }^{*}$ by $\pi _{\lambda ^{*}}(X)=-(\pi _{\lambda 
}(X))^{*}$. It 
is clear that $d(\lambda )= d(\lambda ^{*})$.

The space $\End \,V_{\lambda }$ is isomorphic to the matrix space 
$\Mat _{d(\lambda )}(\C )$ and has a   $\g $-module structure defined by
\begin{equation*}X\cdot A= [\pi _{\lambda }(X),\,A].
\end{equation*}
Recall that the symmetric algebra $S(\g )$ and the enveloping algebra 
$U(\g )$ also have 
(isomorphic) $\g $-module structures.

Let $G$ be a connected and simply connected Lie group with 
$\Lie (G)=\g $. The action of $\g $ on $\End \, V_{\lambda },\ S(\g )$ 
and 
$U(\g )$ gives rise to the corresponding action of $G$. We define  two 
kinds of  {\bf family algebras}: the {\bf classical} algebra 
$\mathcal{C}_{\lambda }(\g )$ and the {\bf quantum} algebra 
$\mathcal{Q}_{\lambda }(\g )$ by 
\begin{equation*}\mathcal{C}_{\lambda }(\g ):= (\End \,V_{\lambda 
}\otimes S(\g ))^{G},\qquad \mathcal{Q}_{\lambda }(\g ):= (\End 
\,V_{\lambda }\otimes U(\g ))^{G}. \tag{1}
\end{equation*}

We hope to apply the theory of family algebras to several important 
questions on semisimple algebras and their representations. 

Here we only start this program and formulate some preliminary results.

\section*{1. Generalities about family algebras}
\subsection*{1.1. Main definitions}

First of all, we make  the basic definition (1) more visual and practical. 
For a given pair $(\g ,\,\lambda )$ let us consider the set of all 
matrices $A$ of order $d(\lambda )$ with elements from $S(\g )$ or from 
$U(\g )$. 

We can  define two different actions of an element $g\in G$ on $A$: 

--- the right action via conjugation by the matrix $\pi (g)$: 
\begin{equation*}A\mapsto A\cdot g:=\pi (g)^{-1} A \pi (g);
\end{equation*}

--- the left action by application of Ad$(g)$ to  all  matrix elements 
of $A$:
\begin{equation*}
A\mapsto g\cdot A,\qquad \text{where}\qquad (g\cdot A)_{ij}=\Ad 
(g)A_{ij}.
\end{equation*}

The statement that for any $g\in G$ the results of these two actions on 
$A$ coincide (i.e. $A\cdot g =g\cdot A$)  is equivalent to the claim 
that $A$ belongs to the family algebra. 

The infinitesimal version of this condition is
\begin{equation*}[A,\,\pi (X)]_{ij}= \ad \,X (A_{ij}).  \tag{2}
\end{equation*}
For the classical family algebras the condition (2) can be rewritten in 
terms of the canonical Poisson structure on $\G $. For this end  we 
identify $S(\g )$ with Pol$(\G )$ and consider the basic elements 
$X_{1},\,\dots ,\,X_{n}$ of $\g $ as coordinates on $\G $. By $\partial 
^{i}$ we 
denote the partial derivative with respect to $X_{i}$. Then the Poisson 
bracket has the form
\begin{equation*}\{f_{1},\,f_{2}\}=c_{ij}^{k}\,X_{k}\,\partial 
^{i}\!f_{1}\,\partial ^{j}\!f_{2}.
\end{equation*}
The condition (2) is equivalent to 
\begin{equation*}[A,\,\pi (X)]_{ij}= \{X,\, A_{ij}\}\qquad \text{or 
simply}\qquad [A,\,\pi (X)]= \{X,\, A\}.  \tag{2$'$}
\end{equation*}

We shall use yet another interpretation of the definition of classical 
family algebras. Consider elements of  $\mathcal{C}_{\lambda }(\g )$ as 
polynomial matrix-valued functions on $\G $. Then the condition that a 
function $A$ corresponds to an element of the  family algebra means
\begin{equation*}A(K(g)F)= \pi _{\lambda }(g) A(F) \pi _{\lambda 
}(g)^{-1}. \tag{2$''$}
\end{equation*}

 The useful corollary of this definition is 

\begin{theorem1} Assume that $\pi _{\lambda }$ has a simple spectrum 
(i.e. all weights have multiplicity 1). Then  $\mathcal{C}_{\lambda 
}(\g )$ is 
commutative.
\end{theorem1}
\begin{proof} It is enough to check the commutativity of $A(F)$ and 
$B(F)$ for generic $F$. Since a generic element of $\G \cong \g $ is 
conjugate to an element of the Cartan subalgebra, we can assume that 
$F\in \h $. But then (2$''$) implies that the values of $A(F)$ and $B(F)$ 
are diagonal matrices, hence they commute.
\end{proof}


Now we recall some properties of  Lie algebras $\g $ which possess an 
$\Ad(G)$-invariant symmetric bilinear form $(\cdot \ ,\,\cdot )$. 
For a simple Lie algebra $\g $ such a form always exists and is unique 
up to a scalar factor. We will usually use the form 
\begin{equation*}(X,\,Y)=\tr \,(\pi (X)\pi (Y)), \tag{3}
\end{equation*}
where $\pi $ is a non-trivial irreducible representation of $\g $ of 
minimal dimension.

The $\Ad(G)$-invariant form allows one to identify adjoint and coadjoint 
modules and thus identify 
$S(\g )$ not only with the polynomial algebra $P(\G )$ but also with the 
algebra $S(\G )$, which we regard as the algebra of differential 
operators on $\G $ with constant coefficients. We denote by $\Delta (P)$ 
the differential operator on $\G $ associated with $P\in S(g)$.

In particular, if $\{X_{i}\}$ and $\{X^{i}\}$ are any dual bases in $\g $ 
with respect to the chosen 
Ad-invariant bilinear form on $\g $, then we have 
\begin{equation*}\Delta (X^{i})=\partial ^{i}\qquad \text{and}\qquad 
X_{i}\Delta (X^{i})= E\ 
\text{(the Euler operator)}. \tag{4}
\end{equation*}
The form $(\cdot \ ,\,\cdot )$ extends to a non-degenerate 
$\Ad(G)$-invariant form on  $S(\g )\cong P(\G )$ (also denoted by 
parentheses)  via
\begin{equation*}(P,\,Q)= \Delta (P)Q \bigm |_{X=0}.  \tag{5} 
\end{equation*}
 One can check that if $\{X_{i}\}_{i=1,2,\dots , n=\dim \g }$ is an 
orthonormal basis in $\g $, then the monomials 
\begin{equation*}\frac{X^{k}}{\sqrt {k!}}:=\frac{X_{1}^{k_{1}}\cdots 
X_{n}^{k_{n}}}{\sqrt {(k_{1})!\cdots (k_{n})!}} 
\end{equation*} 
form an orthonormal basis in $S(\g )$. 

The bilinear form (5) extends further to the space 
$\Mat _{d(\lambda )}(\C )\otimes S(\g )$ (which can be viewed as the 
space of matrix-valued polynomials on $\G $). For decomposable elements 
this extension is given by
\begin{equation*}(A_{0}\otimes P,\,B_{0}\otimes Q) :=\tr 
(A_{0}B_{0})(P,\,Q)\quad \text{for }\ A_{0},B_{0}\in \Mat 
_{d(\lambda }(\C ), \ P,Q\in S(\g ). \tag{5$'$}
\end{equation*}
Note also that the extended form  has the following useful 
property:
\begin{equation*} (AB,\,C)=(A,\,\Delta (B)C). \tag{6}
\end{equation*}
In other words, the matrix differential operator $\Delta (B)$ acting 
from the left is adjoint to the operator of right multiplication by 
$B$. 

\begin{theorem2} Let $P\in S(\g )$ be an invariant polynomial (i.e. we 
assume that $P$ belongs to $I(\g )=S(\g )^{G}$). Then the matrix 
\begin{equation*}M_{P}:= \pi _{\lambda }(X_{i})\otimes \partial ^{i}P 
\tag{7}
\end{equation*}
belongs to $\mathcal{C}_{\lambda }(\g )$. 
\end{theorem2}
\begin{proof} Direct verification of (2$'$).
\end{proof}


We call $M_{P}$ an  {\bf $M$-type element} of $\mathcal{C}_{\lambda 
}(\g )$. 

Note that for $P=C:=\frac{1}{2} X_{i}X^{i}$ we get a special element
\begin{equation*}M:=M_{C}= \pi _{\lambda }(X_{i})\otimes X^{i}, \tag{8}
\end{equation*}
which belongs to $\mathcal{C}_{\lambda }(\g )$ (resp. to 
$\mathcal{Q}_{\lambda }(\g )$) if we interpret $X^{i}$ as an element of 
$S(\g )$ (resp. of 
$U(\g )$).

The remarkable fact is 

\begin{theorem3} The $M$-type elements belong to the center of 
$\mathcal{C}_{\lambda }(\g )$.
\end{theorem3}


\begin{proof} For $A\in \mathcal{C}_{\lambda }(\g ),\ P\in I(\g )$ we 
have 
\begin{equation*}[M_{P},\,A]=[\pi _{\lambda }(X_{i}),\,A]\, \partial 
^{i}P \overset {{(2'), (4)}}{=} 
\{A,\,X_{i}\}\, \partial ^{i}P= \{A,\,P \}=0.
\end{equation*}
\end{proof}


Below we use this theorem  to show that many important classical family 
algebras are commutative.

\subsection*{1.2. The structure of the family algebras}

Let us look more attentively at the $\g $-module structure of 
End\,$V_{\lambda }\cong \Mat _{d 
(\lambda )}(\C )$. It splits into irreducible components:
\begin{equation*}\Mat _{d(\lambda )}(\C )=\bigoplus _{i} W_{i},  \tag{9}
\end{equation*}  
where $W_{i}$ is a simple  $\g $-module with highest weight $\mu 
_{i}$. 

We shall call representations $(\pi _{\mu _{i}},\,W_{i})$ the {\bf 
children} 
of $\pi _{\lambda }$ (resp. the dominant weights $\mu _{i}$ will be 
called the 
children of $\lambda $). 

In the case where all $\mu _{i}$ are different, we shall say that there are no 
{\bf twins}. 

Suppose that $V_{\lambda }$ admits a $G$-invariant bilinear form. In 
other 
words, assume that 
$\pi _{\lambda }$ belongs to orthogonal  or symplectic type. 

In the orthogonal case for an appropriate basis in $V_{\lambda }$ all 
matrices $\pi _{\lambda }(X),\,X\in \g ,$ are skew-symmetric.   Then 
$\Mat _{d(\lambda )}(\C )$ as a $\g $-module splits into symmetric and 
antisymmetric parts. We  call {\bf boys} those children that belong to 
antisymmetric part, and {\bf girls} those that are in the symmetric 
part. 

In the symplectic case, $d(\lambda )=2n$ is even and there is a 
$\pi _{\lambda }(G)$-invariant 
skew-symmetric form in $V_{\lambda }$. We can choose a  basis such that 
this form is given by the matrix 
\begin{equation*}J=\begin{pmatrix}0 & 1_{n} \\
-1_{n}& 0 \\
\end{pmatrix}
.
\end{equation*}
In this basis all matrices $\pi _{\lambda }(X),\ X\in \g ,$ are 
$J$-symmetric, i.e. have the form $J S$, where $S$ is symmetric. 

The $\g $-module $\Mat _{d(\lambda )}(\C )$ again splits into two parts: 
$J$-symmetric and 
$J$-antisymmetric. 
Here we also call girls those children that are in the $J$-symmetric 
part, and boys those in the
$J$-antisymmetric part. 

Note, that in both cases there is a distinguished girl---the trivial 
representation $(\pi _{0},\, 
\C \cdot 1)$ or $(\pi _{0},\,\C \cdot J)$.

We also observe that the adjoint representation Ad is a common child to 
all non-trivial 
representations $\pi _{\lambda }$; the corresponding subspace is 
spanned by the
matrix elements of $M$ (see (8) above). For representations of 
orthogonal  type it is a boy, while for those  of symplectic type it is 
a girl. We shall see below that in some cases there are twins of type 
Ad.

The algebra $\Mat _{d(\lambda )}\otimes S(\G )$ of matrix differential 
operators on $\G $ acts on the algebra $\Mat _{d(\lambda )}\otimes S(\g 
)$ 
of matrix polynomials on  $\G $  according to the rule 
\begin{equation*}(A_{0}\otimes D)\cdot (B_{0}\otimes P)= 
A_{0}B_{0}\otimes D(P).
\end{equation*}
Since $\Delta $ is a $G$-equivariant \pagebreak\ map (it is an isomorphism of 
$\g $-modules), we conclude that the subalgebra of $G$-invariant matrix 
operators
\begin{equation*}\mathcal{D}_{\lambda }(\g ):=\left (\Mat _{d(\lambda 
)}\otimes S(\G )\right )^{G} 
\end{equation*}
coincides with $(1\otimes \Delta )\, \mathcal{C}_{\lambda }(\g )$.

The action of this subalgebra preserves the subalgebra 
$\mathcal{C}_{\lambda }(\g )\subset \Mat _{d(\lambda )}\otimes S(\g ).$ 
For our goals the most important 
examples of  elements of  
$\mathcal{D}_{\lambda }(\g )$ are 
\begin{equation*}D_{P}:=\Delta (M_{P})=\sum _{i}{\pi _{\lambda 
}(X_{i})\otimes \Delta (\partial ^{i}P)},
\end{equation*}
and in particular
\begin{equation*}D:=\Delta (M)=\sum _{i}{\pi _{\lambda }(X_{i})\otimes 
\partial ^{i}}.
\end{equation*}
\subsection*{1.3. Generalized exponents}

Here we recall some remarkable results mostly due to B. Kostant \cite{K}. 

The first result describes the structure of $\g $-module $S(\g )$. Let 
$I(g)=S(\g )^{G}$. We identify $I(g)$ with the algebra $P(\G )^{G}$ of 
$G$-invariant polynomials on $\G $. 

Let $I_{+}(g)$ be the augmentation ideal in $I(\g )$ (i.e. the ideal of 
polynomials vanishing at the origin). 
Denote by $J(\g )$ the  ideal in $S(\g )$ generated by $I_{+}(g)$. The 
space $H(\g )$ of {\bf harmonic polynomials} on $\G $ is defined as the 
orthogonal complement to $J(\g )$ in $S(\g )$. 

According to (6), $H(\g )$ can also be defined as the space of solutions 
$h$ to the system of 
equations
\begin{equation*}\Delta (P) h=0,\qquad P\in I_{+}(g). \tag{10}
\end{equation*}
(Of course, it is enough to consider the generators of $I_{+}(\g )$ in 
the 
role of $P$.)

\begin{theorem4}[Kostant]  a) There is an isomorphism of graded 
$\g $-modules:
\begin{equation*}S(\g )\cong I(\g )\otimes H(\g ). \tag{11}
\end{equation*} 
b) Each irreducible representation $\pi _{\lambda }$ has finite 
multiplicity in $H(\g )$. 

More precisely, if $s=m_{\lambda }(0)$ is the multiplicity of the zero 
weight in $V_{\lambda }$, then there exist numbers $e_{1}(\lambda 
),\,\dots \,,e_{s}(\lambda )$ (not necessarily distinct) such that $\pi 
_{\lambda }$ 
occurs in the homogeneous components $H^{e_{1}(\lambda )}(\g ),\, \dots 
,H^{e_{s}(\lambda )}(\g )$.
\end{theorem4}


The numbers $e_{1}(\lambda ),\,\dots \,,e_{s}(\lambda )$  are called 
the {\bf generalized exponents} related to the representation $\pi 
_{\lambda }$. 

Since $H(\g )$ is a self-dual $\g $-module, the generalized exponents 
are 
the same for $\lambda $ and $\lambda ^{*}$.

The {\bf ordinary exponents} correspond to the adjoint representation
for which $\lambda =\psi $, the highest root of $\g $, and $s=l$, the 
rank 
of $\g $.

The multiplicity $m_{\lambda }(\mu )$ of the weight $\mu $ in 
$V_{\lambda }$  
is often denoted by $K_{\lambda \mu }$ and called {\bf Kostka 
number}. It has the remarkable 
$q$-analog:  the so-called {\bf Kostka polynomial} $K_{\lambda \mu } (q)$,
which
 can be expressed in terms of the $q$-analog of the Kostant formula for 
ordinary multiplicities:
\begin{equation*}K_{\lambda \mu } (q)= \sum _{w\in W}{(-1)^{l(w)} 
\mathcal{Q}(w(\lambda +
\rho )-(\mu +\rho )\bigm | q)},
\tag{12}
\end{equation*}
where $\mathcal{Q}$ is the $q$-analog of the Kostant partition function 
and 
is defined by 
\begin{equation*}\sum _{\nu \in Q_{+}}{\mathcal{Q}(\nu \bigm |q)e^{\nu 
}} 
=\prod _{\alpha >0}{(1-qe^{\alpha })^{-1}}. \tag{13}
\end{equation*} 

\begin{theorem5}[Hesselink]  The polynomials $K_{\lambda \mu } 
(q)$  for $\mu =0$ coincide with  generating functions for generalized 
exponents:
\begin{equation*}K_{\lambda 0} (q)=\sum _{i=1}^{K_{\lambda 0}} 
{q^{e_{i}(\lambda )}}.  \tag{14}
\end{equation*}
\end{theorem5}


We observe that though the Hesselink result looks very natural and 
elegant,  its practical use is rather restricted. The reason is that 
the explicit formula for generalized exponents which follows from 
Hesselink result is too involved. For Lie algebras of rank $\geq 3$ it 
is practically uncomputable.

It is very interesting to compare this approach with the so-called 
Gelfand-Tsetlin patterns, which label a basis in $V_{\lambda }$. These 
patterns have the form
\begin{align*}
m_{1,1} \hskip 20pt m_{1,2} \hskip 10pt \hskip 10pt \dots \hskip 10pt 
&\dots \hskip 10pt \dots \hskip 20pt    
m_{1,n-1} \hskip 20pt  m_{1,n}\\
 \hskip 10pt  m_{2,1} \hskip 20pt m_{2,2} \hskip 10pt  &\dots \hskip 
10pt 
m_{2,n-2}  \hskip 20pt  
m_{2,n-1}\\
\dots \phantom{ m_{n-1,1}\hskip 10pt} &\dots \phantom{ \hskip 10pt  
m_{n-1,2}}\dots \\
 m_{n-1,1}\hskip 5pt &\hskip 15pt  m_{n-1,2}\\
&m_{n,1}
\end{align*}
with integers $m_{ij}$ satisfying
\begin{equation*}m_{i,j}\geq m_{i+1,j}\geq m_{i,j+1}.
\end{equation*}
The weight of a vector $v_{M}$ corresponding to a pattern $M$ is equal 
to 
\begin{equation*}m_{n,1},\ m_{n-1,1}+m_{n-1,2}-m_{n,1},\ \dots ,\ \sum 
_{j}{m_{1j}}-\sum _{j} 
{ m_{2,j}}.
\end{equation*}
One can show that each vector $v_{M}$ of zero weight contributes to 
$K_{\lambda \mu } (q)$  a 
summand of the form $q^{c(M)}$, where $c(M)$ is a certain combinatorial 
function (the so-called {\bf charge}) of the pattern $M$.

 For the case $\g =sl(4)$ a pattern  of zero weight looks like
\begin{align*}
\phantom{\quad m_{3}+m_{4}=0)}m_{1} \hskip 20pt m_{2} \hskip 10pt 
&\hskip 10pt m_{3} 
\hskip 20pt m_{4} \quad (m_{1}+m_{2}+m_{3}+m_{4}=0) \\
  \phantom{\qquad (l_{1}+l_{2}+l_{3}=0)}l_{1} \hskip 25pt &l_{2} \hskip 
22pt l_{3}  
\hskip 20pt  \qquad (l_{1}+l_{2}+l_{3}=0)\\
p\hskip 10pt &\hskip 10pt  -p\\
&0
\end{align*}

I succeeded to compute the final expression only in several particular 
cases. 

For example, for the pattern of the form
\begin{align*}
\phantom{\quad m_{3}+m_{4}=0)}m \hskip 20pt 0 \hskip 10pt &\hskip 10pt 0 
\hskip 20pt -m  \phantom{\quad m_{3}+m_{4}=0)}\\
l \hskip 25pt &0 \hskip 20pt -l  \\
p\hskip 10pt &\hskip 8pt  -p\\
&0
\end{align*}
we have  $c(M)= m+l+p$, $K_{\lambda 0} =\frac{(m+1)(m+2)}{2}={\binom{m+
2}{2}}$, and  $K_{\lambda 0} (q)$ is equal to the sum of all monomials 
from 
the following triangle table:
\begin{align*}
q^{m} \hskip 20pt q^{m+1} \hskip 10pt &\dots \hskip 10pt q^{2m-1} 
\hskip 20pt 
q^{2m}\\ 
\hskip 20pt q^{m+2} \hskip 20pt \hskip 10pt &\dots \hskip 10pt \hskip 
17pt  
q^{2m+1}  \\
\dots \hskip 10pt&\dots \hskip 10pt \dots \\
q^{3m-2}&\hskip 18pt  q^{3m-1}\\
&q^{3m}
\end{align*}
which leads  to the expression
\begin{equation*}\begin{split}
K_{\lambda 0} (q)&=\  q^{m} \begin{bmatrix}m+2\\
2 \end{bmatrix}
_{q}=q^{m}\cdot \frac{q^{m+2}-1)(q^{m+1}-1)}{(q-1)(q^{2}-1)}\\
&=q^{m}+ q^{m+1}+2q^{m+2}+2q^{m+3}+3q^{m+4}+3q^{m+5}+\cdots \\
& \quad +\left [\frac{m}{2} \right ] q^{2m}+\dots+
2q^{3m-3}+2q^{3m-2} 
+q^{3m-1}+q^{3m}.
\end{split}\end{equation*}

\begin{remark1} The general formula for $c(M)$ in the case of $sl(4)$ has been 
recently found by my student R. Masenten. It looks like 
\begin{equation*}c(M)= p+|l_{2}|+\max \,( m_{1}+l_{1}+l_{2},\  
-l_{2}-l_{3}-m_{4},\ m_{1}+m_{2}+p). \tag{15}
\end{equation*}
It is a challenging problem to simplify, explain and generalize this 
cumbersome formula.
\end{remark1}


\begin{remark2} The ordinary multiplicity of the zero weight in a 
unirrep $\pi _{\lambda }$ of the compact form $K\cong SU(n)$ of $G$ can 
be 
written as follows. Let  $M(K)$ be the algebra of (signed) measures on 
$K$ with convolution product. Pick up a maximal torus $T\subset K$ and 
denote by $\delta _{T}$  the element of $M(K)$ given by the normalized 
Haar measure on $T$.

Then $\pi _{\lambda }(\delta _{T})$ is the projector to $V_{\lambda 
}^{0}$ and 
\begin{equation*}K_{\lambda 0} (q)= \tr \, \pi _{\lambda }(\delta _{T}).
\end{equation*} 
Of course, we can replace the chosen torus $T$ in this formula by any 
other (they are all conjugate in $K$). We can also  take the average 
over all possible tori. Then we get a measure which is absolutely 
continuous with respect to Haar measure and has density  
\begin{equation*}f(u)=\frac{n!}{\prod _{i\neq j}{|\lambda _{i}-\lambda 
_{j}|}},
\end{equation*}
where $\lambda _{i}$ are eigenvalues of $u$.

It is natural to ask if there exists a simple graded version 
$f(u\,|\,q)$ of $f(g)$ such that 
\begin{equation*}K_{\lambda 0} (q)= \tr \, \pi (f(u\,|\,q)).
\end{equation*} 
For  the case $K=SU(2)$ we can put 
\begin{equation*}f(u\,|\,q)=\frac{1+q}{1-q\cdot \tr (u^{2}) +
q^{2}}.\tag{16}
\end{equation*}
\end{remark2}


\subsection*{1.4. Relation  between family algebras and generalized 
exponents}
Now let $\lambda \in P_{+}$ be a dominant weight such that $\pi 
_{\lambda }$ 
has $k$ children with no twins.  We denote by $W_{1}, \, \dots ,W_{k}$ 
the 
corresponding irreducible subspaces in 
$\Mat _{d(\lambda )}(\C )$ and by $\mu _{1}, \, \dots ,\mu _{k}$ the 
highest 
weights of corresponding representations.

We denote by  $p_{i}$ the projection to $W_{i}$ in $\Mat _{d(\lambda )} 
(\C )$ 
and use the same notation for its extension to $\mathcal{C}_{\lambda 
}(\G )$ 
(which is the restriction of $p_{i}\otimes 1$).

Then the subspace $\mathcal{C}_{\lambda }(\G )_{i}:= 
p_{i}(\mathcal{C}_{\lambda }(\G ))=(W_{i}\otimes S(\g 
))^{G}=(W_{i}\otimes I(\g )\otimes H(\g ))^{G}$ is a free $I(\g 
)$-module of 
rank $K_{\mu _{i} 0}$. In fact, it has the grading inherited from $S(\g 
)$ 
and its Poincar\'{e} series is given by
\begin{equation*}ch_{\mathcal{C}_{\lambda }(\G )_{i}}(q)=ch_{I(\g 
)}(q)\cdot ch_{H(\g )_{\mu _{i}^{*}}}(q)=\frac{K_{\mu _{i} 0} 
(q)}{\prod _{k=1}^{l}(1-q^{d_{k}})},
\end{equation*}
where $d_{k},\,1\leq k \leq l,$ are the degrees of the generators of $I(\g 
)$. 
(It is well known that  
$d_{k}=e_{k}+1$, where $e_{k}=e_{k}(\psi )$ are ordinary exponents.)

So, we get the following result:

\begin{theorem6} The algebra $\mathcal{C}_{\lambda }(\g )$ is a free 
$I(\g )$-module of rank 
$m=\sum _{i=1}^{k} 
{m_{\mu _{i}} (0)}$.
\end{theorem6}


\begin{remark3} Assume that all weights of the representation 
$\pi _{\lambda }$ are simple (i.e. 
have multiplicity 1). Then  the zero weight subspace in 
$\Mat _{d(\lambda )} (\C )\cong \End \, V_{\lambda }$ is the subspace of 
diagonal matrices so that $m=d(\lambda )$. I do not know the meaning of 
the corresponding $q$-analog.
\end{remark3}


\begin{theorem7} The algebra $\mathcal{C}_{\lambda }(\g )$ is 
commutative  
if  and only if all weights of the representation $\pi _{\lambda }$ have 
multiplicity 1.
\end{theorem7}


It follows from the following general result:

\begin{theorem8} Let $K(\g )$ be the field generated by the zero 
weight subalgebra $S(\g )^{0}$ of  $S(\g )$, let $\mu _{i}$ be children of 
$\lambda $, and $\delta (\mu _{i})$ their multiplicities. Then the 
algebra 
$\mathcal{K}_{\lambda }(\g ):=\mathcal{C}_{\lambda }(\g )\otimes _{S(\g 
)^{0}} K(\g )$ is 
isomorphic to the direct sum of matrix algebras:
\begin{equation*}\mathcal{K}_{\lambda }(\g )\cong \bigoplus _{i} \Mat 
_{\delta (\mu _{i})}(K(\g )). \tag{17}
\end{equation*}
\end{theorem8}
\begin{proof}[Sketch of the proof] We shall regard elements of 
$\mathcal{K}_{\lambda }(\g )$ as rational matrix-valued functions on $\G $ 
with values 
in  $\Mat _{\lambda }(\C )$. The condition (2$''$) implies that 
this 
function is uniquely determined by its restriction to $\h ^{*}\subset 
\G $. 

Conversely, this restriction can be any rational function on $\h ^{*}$ 
with values in the zero weight subspace $\Mat _{\lambda }(\C )^{0}$. (It 
follows from the birational isomorphism $G \cong G/B \times H$.)  But 
the latter is exactly the direct sum $\bigoplus _{i} 
\Mat _{\delta (\mu _{i})}(\C ).$
\end{proof}


\section*{2. Family algebras for standard representations of classical 
simple Lie algebras}

Here we illustrate the general theory described above by examples 
related to the simplest 
representations of classical simple Lie algebras. It can be viewed
as a beautiful exercise in linear algebra which ties together many 
classical results.
 
One of interesting consequences of the results listed below is the fact 
that all  family algebras in question are commutative. It is not true 
for the quantum algebras and for some classical algebras related to 
other representations.
 
\subsection*{\texorpdfstring{2.1. The 
case $\mathfrak{g}=\mathbf{A_{n}}
\protect\cong\mathfrak{sl} (n+1,\,\mathbb{C})$}{2.1. The 
case