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To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package %% Translation via Omnimark script a2l, February 21, 2000 (all in one %day!) \controldates{28-FEB-2000,28-FEB-2000,28-FEB-2000,28-FEB-2000} \documentclass{era-l} \issueinfo{6}{02}{}{2000} \dateposted{March 1, 2000} \pagespan{7}{20} \PII{S 1079-6762(00)00075-5} \def\copyrightyear{2000} \usepackage{amssymb} \usepackage{amsxtra} \usepackage{amsmath} \newcommand{\texorpdfstring}[2]{#1} \theoremstyle{plain} \newtheorem*{theorem1}{Theorem S} \newtheorem*{theorem2}{Lemma 1} \newtheorem*{theorem3}{Theorem M} \newtheorem*{theorem4}{Theorem 1} \newtheorem*{theorem5}{Theorem 2} \newtheorem*{theorem6}{Theorem 1} \newtheorem*{theorem7}{Theorem 2} \newtheorem*{theorem8}{Theorem 3} \newtheorem*{theorem9}{Theorem A} \newtheorem*{theorem10}{Corollary} \newtheorem*{theorem11}{Theorem B} \newtheorem*{theorem12}{Corollary} \newtheorem*{theorem13}{Theorem C} \newtheorem*{theorem14}{Corollary} \newtheorem*{theorem15}{Theorem D} \newtheorem*{theorem16}{Corollary} \newtheorem*{theorem17}{Lemma 2} \newtheorem*{theorem18}{Theorem G} \newtheorem*{theorem19}{Corollary} \newtheorem*{theorem20}{Theorem $\mathcal{Q}$} \theoremstyle{remark} \newtheorem*{remark1}{Remark 1} \newtheorem*{remark2}{Remark 2} \newtheorem*{remark3}{Remark 3} \newcommand\N{\mathbb{N}} \newcommand\C{\mathbb{C}} \newcommand\tr{\mathrm{tr}} \newcommand\Z{\mathbb{Z}} \newcommand\G{\mathfrak{g}^*} \newcommand\g{\mathfrak{g}} \newcommand\gl{\mathfrak{gl}} \newcommand\ssl{\mathfrak{sl}} \newcommand\ssp{\mathfrak{sp}} \newcommand\so{\mathfrak{so}} \newcommand\h{\mathfrak{h}} \newcommand\n{\mathfrak{n}} \newcommand\End{\mathrm{End}} \newcommand\Lie{\mathrm{Lie}} \newcommand\Mat{\mathrm{Mat}} \newcommand\ad{\mathrm{ad}} \newcommand\Ad{\mathrm{Ad}} \newcommand\ra{\rightarrow} \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