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Models of representations for classical series of Lie algebras D. V. Artamonov Lomonosov Moscow State University
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     § 1. IntroductionA model of representations of a Lie algebra is a representation which is a direct sum of all its finite-dimensional irreducible representations taken with multiplicity $1$. Thus one can think about the classical Weyl construction as a model of representations of the algebra $\mathfrak{gl}_n$ (see [1]). Formally, the Weyl construction is an explicit embedding of a certain irreducible representation into a tensor power of a standard representation of $\mathfrak{gl}_n$. But then by taking the direct sum of these embeddings we obtain a model of representations in the sense of the present paper. There exist analogues of the Weyl construction for other classical Lie algebras [2], and even for some exceptional Lie algebras [3]. In physical literature, the models based on the language of creation and annihilation operators are used. Such an approach in the case of the series $A$ is used in [4] and in the subsequent studies of Baird and Biedenharn. But an attempt to generalize this language to the case of the series $C$ faces considerable difficulties. In doing so, one usually considers small dimensions only (see [5]–[8]). The Zhelobenko construction [9] is also a model. In the actual fact, all these three constructions are similar. Anticipating what will follow, it is worth pointing out that the construction of the present paper is also similar to these construction. Let discuss the known models. There exist numerous models of combinatorial nature, the complete list here is unfeasible. Let us return to the aforementioned papers by Biedenharn with coauthors [4]–[8]. In [10] (see also [11]), a special model of representations for the algebra $\mathfrak{sl}_3$ was constructed. These two papers concluded an extensive series of studies, where the authors tried to evaluate explicitly the Clebsh–Gordan and Racah coefficients that describe a splitting of tensor products of representations into irreducible summands. Flath was involved in this activity on its final stage, and, although he dealt with the classical objects, he called the explicit classical representation theory the “mathematical golden mine” [11]. Inspired by [10], Gelfand and Kapranov wrote the paper [12], where the notion of a model of representations was introduced. In [12], some models of geometric nature for all classical Lie algebras over $\mathbb{C}$ were constructed. These inspiring studies on the classical representation theory were a starting point for the present paper. We need also to mention that there exist other numerous models of geometric nature. Among such models we mention some subspaces in the space of functions on a homogeneous space or on an $HV$-variety [13]. The present paper can be considered as a continuation of [14] and as a generalization of its results to other classical Lie algebras. In [14], for the algebra $\mathfrak{gl}_n$, one considers functions of independent variables $A_X$, $X\subset\{1,\dots,n\}$, antisymmetric on $X$. It turns out that the polynomials in these variables that satisfy some system of partial differential equations (known as the antisymmetrized Gelfand–Kapranov–Zelevinsky system; A-GKZ for short; see [15]) form a model. This model is naturally called the A-GKZ model. Note that this system of equations is close to the hypergeometric system on the space $\Lambda^k\mathbb{C}^N$, which was constructed in [16]. So, this system can also be called a system of hypergeometric type. In this way, we obtain a model of representations whose space is the space of polynomial solutions of a hypergeometric type system. The existence of such a model sheds light on the fact that hypergeometric functions and constants appear very often in explicit calculations in the representation theory (see, for example, [17]). In addition, in the A-GKZ model, one can naturally construct a base in each representation. An advantage of the A-GKZ model is that it simultaneously contains both an explicit base and explicit formulas for the scalar product. This makes possible to do some non-trivial calculations. So, for example, in [18], using the A-GKZ model in the case $n=3$, explicit simple formulas for an arbitrary Clebsh–Gordan coefficient were obtained. In addition, in [19], using the same model, an explicit formula for an arbitrary $6j$-symbol for the algebra $\mathfrak{gl}_3$ was obtained. In addition, in [14] a base in the Zhelobenko model was obtained with the help of $A$-hypergeometric functions (in this construction, the A-GKZ system also plays a crucial role). In this base, it proves possible to explicitly write the action of generators of the algebra $\mathfrak{gl}_n$. This construction is called the GKZ base for the Zhelobenko model. In [14], this base plays an important role in establishing a relation between the constructed base of the A-GKZ model and the Gelfand–Tsetlin base. There are also analogues of the GKZ base for the Zhelobenko model for other Lie algebras of small dimensions (see [20] and [21]). In the present paper, in § 6, we construct an analogue of the GKZ and A-GKZ systems for the Lie algebras of the series $B$, $C$, $D$. This leads to an A-GKZ model for these algebras (see § 8.3). It is remarkable that the constructions for different series are essentially the same. In the A-GKZ model, a base can be naturally constructed. In addition, for the series $B$, $C$, $D$, we construct a GKZ base in the Zhelobenko model and establish formulas for the action of generators in this base (see § 8.5). Using this result, in § 8.5.6, we construct other bases in the Zhelobenko realization. We also investigate a relation between the constructed base in the A-GKZ model and the Gelfand–Tsetlin base. First, a new point of view to the notion of a Gelfand–Tsetlin diagram is suggested (see § 7, Definition 10). There exists a one-to-one correspondence between the objects introduced in Definition 10 and the traditional Gelfand–Tsetlin diagrams. (diagrams for all classical series in the usual sense can be found in [22]). In our new approach, the formulas for the action of generators and so on acquire a more natural form (see § 9). The Gelfand–Tsetlin diagrams in our sense index the base vectors in irreducible representations. Of course, there are numerous constructions of sets whose elements solve the same problem. Usually, these sets are constructed as sets of integer points in some polytopes (the Berenstein–Zelevinsky–Littelmann string polytopes [23], the Vinberg–Littelmann–Feigin–Fourier polytopes [24]). The objects introduced in Definition 10 can also be obtained as integer points in some polytope. But we use the term a Gelfand–Tsetlin diagram since one can easily reconstruct the highest weights from a chain of subalgebras that appear in the standard procedure of construction of the Gelfand–Tsetlin base. Second, we prove that the transition matrix between the base of the A-GKZ model and the Gelfand–Tsetlin base is triangular. We show that this transition is nothing else but the Gram–Schmidt orthogonalization in the A-GKZ base. The paper is organized as follows. Sections 2–4 are introductory. In § 2, the basic notions are introduced. In § 3, the Zhelobenko model is discussed; we also obtain an explicit description of the Zhelobenko model for the series $B$, $C$, and $D$, which supplements the results of the book [9]. The results in this section are formally new, but they can be obtained form the results of [9] via quite simple calculations. In § 4, we present results for the series $A$. These results are insignificant modifications of those from [14], there being no essentially new results in this section. The main results of the paper are given in §§ 6–8. In § 6, the A-GKZ system for the series $B$, $C$, $D$ is introduced; in § 7, a new definition of a Gelfand–Tsetlin diagram for this series is given; and in § 8, the A-GKZ model and the GKZ base for the Zhelobenko model are constructed. In § 9, a relation of the constructed base of the A-GKZ model and the Gelfand–Tsetlin base is discussed. To conclude this introduction, we also mention the paper [25]. The results of this paper are not directly related to those of the present paper, but the ideology of [25] is quite close. Here, an arbitrary GKZ function is interpreted as a matrix element in a representation of a special Lie algebra. § 2. The basic objectsIn this section, we give the definition of an important class of functions that plays a crucial role in the present paper. We present a system of equations satisfied by these functions. We also present the Lie algebras used in the present paper. 2.1. $\Gamma$-seriesA detailed information about a $\Gamma$-series can be found in [16]. Let $\mathcal{B}\subset \mathbb{Z}^N$ be a lattice, let $\gamma\in \mathbb{Z}^N$ be a fixed vector. The hypergeometric $\Gamma$-series in the variables $z_1,\dots,z_N$ is defined by
$$
\begin{equation}
\mathcal{F}_{\gamma}(z,\mathcal{B})= \sum_{b\in \mathcal{B}} \frac{z^{b+\gamma}}{\Gamma(b+\gamma+1)},
\end{equation}
\tag{2.1}
$$
where $z=(z_1,\dots,z_N)$, and the following multi-index notation is used
$$
\begin{equation*}
z^{b+\gamma}:=\prod_{i=1}^N z_i^{b_i+\gamma_i},\qquad \Gamma(b+\gamma+1):=\prod_{i=1}^N\Gamma(b_i+\gamma_i+1).
\end{equation*}
\notag
$$
Note that if at least one of the components of the vector $b+\gamma$ is negative integer, then the corresponding summand in (2.1) vanishes. Due to this fact, the $\Gamma$-series considered in the present paper contain finitely many non-zero terms. For simplicity, we will write factorials instead of $\Gamma$-functions. Any $A$-hypergeometric function satisfies a system of partial differential equations called the Gelfand–Karpanov–Zelevinsky (shortly, GKZ) system, which consists of equations of two types. 1. Let $a=(a_1,\dots,a_N)$ be a vector orthogonal to the lattice $\mathcal{B}$, then
$$
\begin{equation}
a_1z_1\, \frac{\partial}{\partial z_1}\mathcal{F}_{\gamma}+\dots+ a_Nz_N\, \frac{\partial}{\partial z_N}\mathcal{F}_{\gamma} = (a_1\gamma_1+\dots+a_N\gamma_N)\mathcal{F}_{\gamma};
\end{equation}
\tag{2.2}
$$
here, it suffices to consider only the base vectors of the lattice orthogonal to $\mathcal{B}$. 2. Let $b\in \mathcal{B}$ and $b=b_+-b_-$, where all coordinates of the vectors $b_+$, $b_-$ are non-negative. Choosing the non-zero elements $b_+ = (\dots, b_{i_1},\dots,b_{i_k},\dots)$ and $b_-=(\dots, b_{j_1},\dots,b_{j_l},\dots)$ in these vectors, we have
$$
\begin{equation}
\begin{gathered} \, \mathcal{O}_b \mathcal{F}_{\gamma}=0,\qquad \mathcal{O}_b= \biggl(\frac{\partial}{\partial z} \biggr)^{b_+}- \biggl(\frac{\partial}{\partial z}\biggr)^{b_-}, \\ \biggl(\frac{\partial}{\partial z}\biggr)^{b_+}:= \biggl(\frac{\partial}{\partial z_{i_1}}\biggr)^{b_{i_1}}\cdots \biggl(\frac{\partial}{\partial z_{i_k}}\biggr)^{b_{i_k}}, \\ \biggl(\frac{\partial}{\partial z}\biggr)^{b_-} := \biggl(\frac{\partial}{\partial z_{j_1}}\biggr)^{b_{j_1}}\cdots \biggl(\frac{\partial}{\partial z_{j_l}}\biggr)^{b_{j_l}}. \end{gathered}
\end{equation}
\tag{2.3}
$$
It suffices to consider only a finite collection of vectors1 $b \in \mathcal{B}$. The system of partial differential equations can be identified with an ideal in the ring of differential operators generated by the operators defining the equations of the system. Let us show a way for explicit construction of the ideal corresponding to system (2.3). To this end, we firstly describe some properties of the correspondence $b\in B\mapsto \mathcal{O}_b$. We have the following. 1. $kb\mapsto (\mathcal{O}_b)^k$, $k\in\mathbb{Z}_{\geqslant 0}$, $-b \mapsto -\mathcal{O}_b$. 2. Let $b=b_+-b_-$, $c=c_+-c_-$, where $c_{\pm}$ have only non-negative coordinates. Assume that the expansion of $b+c$ as a difference of vectors with non-negative coordinates is $(b_++c_+)-(b_-+c_-)$ (that is, in this equality, no reduction takes place in each coordinate). Then
$$
\begin{equation*}
b+c\mapsto \mathcal{O}_{b+c}=\biggl(\frac{\partial}{\partial z}\biggr)^{c_+} \mathcal{O}_b+\biggl(\frac{\partial}{\partial z}\biggr)^{b_+}\mathcal{O}_c.
\end{equation*}
\notag
$$
3. Let $b=(b_++u)-(b_-+v)$, $c=(c_++v)-(c_-+u)$, where $c_{pm}$, $u$, $v$ have only non-negative coordinates. Assume that the expansion of $b+c$ as a difference of vectors with non-negative coordinates is $(b_++c_+)-(b_-+c_-)$ (that is, in this equality in each coordinate no reduction takes place). Then
$$
\begin{equation*}
\biggl(\frac{\partial}{\partial z}\biggr)^{u+v} \mathcal{O}_{b+c}= \biggl(\frac{\partial}{\partial z}\biggr)^{c_+}\mathcal{O}_b + \biggl(\frac{\partial}{\partial z}\biggr)^{b_+}\mathcal{O}_c.
\end{equation*}
\notag
$$
We need the following definition. Definition 1. Assume that we have a collection of differential operators with constant coefficients. A system generated by this collection is a system defined by the following collection of differential operators. First, we take all operators form the initial collection. Second, we take all operators that belong to the ideal generated by these operators. Third, we take all operators obtained from the operators in this ideal by division (if possible) by a differential monomial. The above considerations show that the GKZ system is generated by operators corresponding to base vectors of the lattice. 2.2. Algebras $\mathfrak{o}_{2n+1}$, $\mathfrak{o}_{2n}$, $\mathfrak{sp}_{2n}$The Lie algebras $\mathfrak{o}_{2n}$, $\mathfrak{sp}_{2n}$ are considered as subalgebras in the Lie algebra of all $2n\times 2n$ matrices, whose rows and columns are indexed by $i,j=-n,\dots,-1,1,\dots,n$, and the algebra $\mathfrak{o}_{2n+1}$ is a subalgebra in the Lie algebra of all $(2n+1)\times (2n+1)$ matrices, whose rows and columns are indexed by $i,j=-n,\dots,-1,0,1,\dots,n$. The algebras $\mathfrak{o}_{2n+1}$ and $\mathfrak{o}_{2n}$ are generated by the matrices where $i,j=-n,\dots,-1,0,1,\dots,n$ in the case $\mathfrak{o}_{2n+1}$ and $i,j=-n,\dots,-1,1,\dots,n$ in the case $\mathfrak{o}_{2n}$.The algebra $\mathfrak{sp}_{2n} $ is generated by the matrices
$$
\begin{equation}
F_{i,j}=E_{i,j}-\operatorname{sign}(i)\operatorname{sign}(j)E_{-j,-i},
\end{equation}
\tag{2.5}
$$
where $i,j=-n,\dots,-1,1,\dots,n$. Let $g_n $ be the Lie algebras $\mathfrak{o}_{2n+1}$, $\mathfrak{o}_{2n}$ or $\mathfrak{sp}_{2n}$. The chosen realization is also a root realization of the corresponding Lie algebra. The elements $F_{i,j}$, $i<j$, correspond to positive roots, $F_{i,j}$, $i>j$, correspond to negative roots; and $F_{i,i}$ generate the Cartan subalgebra. An analogous choice of the Cartan subalgebra and the root elements will be used also for $\mathfrak{gl}_m$. To be able to talk about a Gelfand–Tsetlin base, we need to fix a chain of subalgebras. A subalgebra $g_{n-k}\subset g_n$ is defined as the span $\langle F_{i,j}\rangle_{i,j\neq \pm 1,\dots,\pm (k-1)}$. § 3. The Zhelobenko modelIn this section, we present a model of representations realized in the space of functions on the corresponding Lie group. Zhelobenko [9] proved Theorem 1, which describes the space of this model in the case of the series $A$. We formulate and prove Theorem 2, which is a straightforward generalization of the Zhelobenko result to other series. In these theorems, the representation spaces are described as solution spaces of some systems of partial differential equations. A more explicit description of the representation space is given in Theorem 3. 3.1. Functions on a groupConsider the space of functions on a group $G$. An element $X\in G$ acts on a function $f(g)$, $g\in G$ by right shifts Thus, the space $\mathrm{Fun}$ of all functions of the group $G$ is a representation of $G$, and so, of the Lie algebra $\operatorname{Lie} G$. Let2 $G=\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$, $\mathrm{SO}_{2n+1}$ and let $a_i^j$, $i,j=-n,\dots,n$, be a function of a matrix element on the group $G$. Here, $j$ is a row index, and $i$ is a column index. We also put
$$
\begin{equation}
a_{i_1,\dots,i_k}:= \det(a_i^j)_{i=i_1,\dots,i_k}^{j \in\text{the first $k$ rows}}.
\end{equation}
\tag{3.2}
$$
In other words, we consider the determinant of the submatrix in the matrix $(a_i^j)$, formed by the first $k$ rows and columns $i_1,\dots,i_k$. In the case of the series $D$, we also put
$$
\begin{equation}
\overline{a}_{i_1,\dots,i_n}:= \det(a_i^j)_{i=i_1,\dots,i_n}^{j=-n,\dots,-2,1}.
\end{equation}
\tag{3.3}
$$
It can be shown that
$$
\begin{equation}
(a_{-n,\dots,-2,-1})^{-1}= \overline{a}_{-n,\dots,-2,1}.
\end{equation}
\tag{3.4}
$$
Using (3.1), (3.2), we can obtain formulas for the action of generators of the algebra on these determinants. To formally write down these formulas,3 we introduce the action of an operator $E_{i,j}$ on a determinant,
$$
\begin{equation}
E_{i,j}a_{i_1,\dots,i_k}=a_{\{i_1,\dots,i_k\}|_{j\mapsto i}},
\end{equation}
\tag{3.5}
$$
where ${\cdot}\,|_{j\mapsto i}$ is a substitution of $j$ instead of $i$, and in the case $j\notin \{i_1,\dots,i_k\}$, we get zero. Now, for the Lie algebra of the series $B$, $C$, $D$, the action of $F_{i,j}$ is described via that of $E_{i,j}$. Now let us give an explicit formula for the highest vector of a given highest weight (see [4] and [9]). The vector
$$
\begin{equation}
v_0 = \begin{cases} {\displaystyle\prod_{k=-n}^{-2} (a_{-n,\dots,-k})^{m_{-k}-m_{-k+1}} a_{-n,\dots,-2,-1}^{m_{-1}}} &\text{for $B$, $C$, $D$, and } m_{-1}\geqslant 0, \\ {\displaystyle\prod_{k=-n}^{-2} (a_{-n,\dots,-k})^{m_{-k}-m_{-k+1}} \overline{a}_{-n,\dots,-2,1}^{-m_{-1}}} &\text{for the series $D$, and }m_{-1}< 0 \end{cases}
\end{equation}
\tag{3.6}
$$
is a highest vector for $g_n$ with highest weight $[m_{-n},\dots,m_{-1}]$ for all the series $B$, $C$, $D$. Note that, in the case of integer highest weight, this is a polynomial function. For $B$, $D$ and if the highest weight is half-integer, the last factor has a fractional exponent. In this case, we obtain a multi-valued function, which becomes single-valued when changing to the group $\mathrm{Spin}$. Definition 2. The direct sum of subrepresentations in the space $\mathrm{Fun}$ generated by highest vectors (3.6) is called the Zhelobenko model. The space of this model is denoted by $\mathrm{Zh}$. 3.2. Relations between determinant3.2.1. The Plücker relationsAll series between the minors $a_{i_1,\dots,i_k}$ of an $m\times m$ matrix satisfy the Plücker relations
$$
\begin{equation}
r_s=\sum_{t=1}^{k+1} (-1)^t a_{i_1,\dots,i_{k-1},j_t} a_{j_1,\dots,j_{t-1}, j_{t+1},\dots,j_{k+1}}=0,
\end{equation}
\tag{3.7}
$$
where $s$ is some abstract index numerating these relations. These relations are sufficient conditions that guarantee that the collection of numbers $a_{i_1,\dots,i_k}$ is a collection of minors of type (3.2) of some matrix. The following result holds. Lemma 1 (see [26]). In the case of the series $A$, the ideal $I_{\mathfrak{gl}_m}$ of relations between the determinants $a_X:=a_{i_1,\dots,i_k}$ is generated by the Plücker relations. 3.2.2. The Jacobi relationsIn the case of the series $B$, $C$, $D$, some additional relations exist. To derive them, we use the fact that the minors of a matrix $(a_i^j)$ and its inverse $((a^{-1})_i^j)$ are related by the Jacobi relations [27]
$$
\begin{equation}
a_{i_1,\dots,i_k}^{j_1,\dots,j_k}=\det(a)(-1)^{\sum_{p=1}^k i_p+ \sum_{q=1}^k j_q}(a^{-1})^{\widehat{i_1},\dots, \widehat{i_k}}_{\widehat{j_1},\dots,\widehat{j_k}}.
\end{equation}
\tag{3.8}
$$
Here, if a minor form composed, in the right-hand side of the equality we take all the columns with indices $\{-n,\dots,n\}$, except $i_1,\dots,i_k$, all the rows with indices $\{-n,\dots,n\}$, except $j_1,\dots,j_k$. Let us write the corollaries for determinants in the cases of series $B$, $C$, $D$. Let $X$ be an element of the corresponding Lie group. Then
$$
\begin{equation}
\begin{gathered} \, X^t\Omega X=\Omega \quad \Longleftrightarrow \quad X^{-1}=\Omega^{-1}X^t\Omega,\quad \Omega=(\omega_{i,j}), \\ \begin{aligned} \, \omega_{i,j} &=\begin{cases} +1, &i=-j,\ i<0, \\ -1, &i=-j,\ i>0, \\ 0 &\text{otherwise}, \end{cases} \quad \text{for the series }C, \\ \omega_{i,j} &=\begin{cases} +1, &i=-j, \\ 0 &\text{otherwise}, \end{cases} \quad\text{for the series }B, D. \end{aligned} \end{gathered}
\end{equation}
\tag{3.9}
$$
Note that $\Omega^{-1}=-\Omega$ for the series $C$, and $\Omega^{-1}=\Omega$ for the series $B$ and $D$. So, in this case, for the matrix elements, we have
$$
\begin{equation*}
a_i^j= \begin{cases} a_{-j}^{-i} &\text{for the series }B, D, \\ \operatorname{sing}(i)\operatorname{sign}(j)a_{-j}^{-i} &\text{for the series }C. \end{cases}
\end{equation*}
\notag
$$
Since $\det(a)=1$ for all groups, we obtain
$$
\begin{equation}
\begin{aligned} \, a_{i_1,\dots,i_k}^{-n,\dots,-n+k-1} &=(-1)^{i_1+\dots+i_k} (-1)^{-n+\dots+(-n+k-1)} (a^{-1})^{\widehat{i_1},\dots, \widehat{i_k}}_{\widehat{-n},\dots,\widehat{-n+k-1}} \nonumber \\ &=(-1)^{i_1+\dots+i_k}(-1)^{(-2n+k-1)k/2} \nonumber \\ &\qquad\times\begin{cases} (a_{-j}^{-i})^{\widehat{i_1},\dots, \widehat{i_k}}_{\widehat{-n},\dots,\widehat{-n+k-1}} &\text{for the series }B, D, \\ (\operatorname{sing}(i)\operatorname{sing}(j)a_{-j}^{-i})^{\widehat{i_1}, \dots, \widehat{i_k}}_{\widehat{-n},\dots,\widehat{-n+k-1}} &\text{for the series }C \end{cases} \nonumber \\ &=(-1)^{i_1+\dots+i_k}(-1)^{(-2n+k-1)k/2} \nonumber \\ &\qquad\times\begin{cases} (a_i^j)_{\widehat{-i_1},\dots, \widehat{-i_k}}^{\widehat{n},\dots,\widehat{n-k+1}} &\text{for the series }B, D, \\ (\operatorname{sing}(i)\operatorname{sing}(j)a_i^j)_{\widehat{-i_1},\dots, \widehat{-i_k}}^{\widehat{n},\dots,\widehat{n-k+1}} &\text{for the series }C. \end{cases} \end{aligned}
\end{equation}
\tag{3.10}
$$
Thus we have reached the following result. Lemma 2. For the series $B$, $C$, $D$, For the series $D$, we also have Here, where $s=1$ in the cases $B$, $D$, and $s$ is $-1$ to the power equal to the number of rows and columns with negative indices in the case of the series $C$.Assertion (3.12) for the series $D$ can be obtained via reordering or rows. Let us introduce a shorter notation for determinants. If $X\subset \{-n,\dots,n\}$ is an index set, we put $a_X:=a_{i_1,\dots,i_k}$. We also introduce the similar notation $\overline{a}_X$ in the case of the algebra $\mathfrak{o}_{2n}$ and $|X|=n$. In this notation, (3.11) assumes the form where, for $X={i_1,\dots,i_k}$, we put $-X:=\{-i_1,\dots,-i_k\}$, and, for $Y=\{j_1,\dots,j_k\}$, we put $\widehat{Y}:=\{-n,\dots,n\}\setminus Y$.The following result holds. Lemma 3. Let $G$ be one of the groups $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$, $\mathrm{SO}_{2n+1}$, that is, $G$ is a group of matrices that preserve the bilinear form with the matrix $\Omega=(\omega_{i,j})$, where $\omega_{i,j}$ are defined in (3.9). Then the ideal of relations between the determinants $a_X$ of type (3.2) is generated by the Plücker and the Jacobi relations. The same is true in the case of the series $D$, where the determinants $\overline{a}_X$ are taken instead of the determinants $a_X$ of order $n$. We first consider the determinants $a_X$. The following result holds. Proposition 1. Let $O_1$ and $O_2\in G$ be matrices. If all their minors constructed on columns that belong to an arbitrary subset $X$ and first consecutive rows (that is, minors of type (3.2)) are equal, then $O_1=TO_2$, where $T$ is a low-unitriangular matrix. Similarly, if all minors constructed on rows that belong to an arbitrary subset of $X$ and first consecutive columns are equal, then $O_1=O_2T$, where $T$ is an upper-unitriangular matrix. A close (but not exactly equivalent) result can be found in [26], Proposition 14.2. Proof of Proposition 1. Let us verify the first assertion, the second one is immediate from the first one.
We use the well-known fact that the $k$-dimensional subspace $L=\langle x^1,\dots,x^k\rangle\subset\mathbb{C}^N$, $x^i=(x^i_1,\dots,x^i_N)$, is uniquely defined by its Plücker coordinates $\{a_{j_1,\dots,j_k} =\det(x^i_j)^{i=1,\dots,k}_{j=j_1,\dots,j_k}\}$. Consider the matrices $O_1$, $O_2$. For an arbitrary $k$, let $x^1,\dots,x^k$ be the first $k$ rows of the matrix $O^1$, and let $y^1,\dots,y^k$ be the first $k$ rows of the matrix $O^2$. From the result formulated in the previous paragraph, we have $\langle x^1,\dots,x^k\rangle = \langle y^1,\dots,y^k\rangle$. Hence $O^1=TO^2$, where $T$ is low-triangular. Since the determinants of the submatrices of $O_1$ and $O_2$ constructed on the first consecutive rows and columns are equal, the matrix $T$ is low-unitriangular. This proves Proposition 1. Corollary 1. Let the minors of $O$ satisfy $a_X=\pm a_{\widehat{-X}}$, where the sign is defined in (3.13). Then $\pm\Omega^{-1} O^\top\Omega=O^{-1}T$, where “$\pm$” “$=$” “$-$” for the series $C$, and “$+$”, for the series $B$ and $D$, and $T$ is an upper-unitriangular matrix. Proof. The relation $a_X=\pm a_{\widehat{-X}}$ was obtained in (3.10) as an equality for the minors constructed on the first consecutive columns for the matrices $X^{-1}$ and $\pm\Omega^{-1}X^\top\Omega$. Now Corollary 1 is immediate from Proposition 1. Let us return to the proof of Lemma 3. Above (Lemmas 1, 2) it was shown that the Plücker and the Jacobi relations belong to the ideal $I_{g_n}$ of all relations between minors for the group $G$. Let us show that these relations generate the ideal $I_{g_n}$. Indeed, consider the mapping $\varphi\colon X\mapsto \{a_X\}$ that associates with a matrix the collection of its minors. The ideal $I_{gl_m}$ is an ideal in the ring of polynomials in the independent variables $A_X$ such that its null-space is the closure of the image $\varphi(\mathrm{GL}_m)$. When passing from the image to its closure, only the origin of the point with zero coordinates is added. Indeed, in each collection of coordinates $\{a_X\}_{|X|=k}$ we carry out a projectivization (separately in each collection). The image $\varphi(\mathrm{GL}_m)$ after these projectivizations coincides with the variety defined by the Plücker relations (see [26]). Now the required result follows due to homogeneity of the Plücker relations and homogeneity for each $k$ of the mapping $\operatorname{pr}_{\{a_X\}_{|X|=k}}\circ \varphi$ (where $\operatorname{pr}_{\{a_X\}_{|X|=k}}$ is a projection to the corresponding coordinates). Thus, if a collection of numbers $\{a_X\}$ is non-zero, then these numbers are minors of an invertible ($m\times m$)-matrix. Now let us take $m=2n$ or $2n+1$ and consider an embedding $G\subset \mathrm{GL}_m$. The ideal $I_{g_n}$ is an ideal whose null-space is the closure of the image $\varphi(G)$. Consider the ideal generated by the Plücker and Jacobi relations. Let $\{a_X\}$ be a non-zero element from it null-space. The Plücker relations guarantee that $\{a_X\}$ can be considered as a collection of minors of some matrix $O$. Corollary 1 shows that the Jacobi relations provide that $\pm \Omega O^\top\Omega=O^{-1}T$ for an upper-unitriangular matrix $T$, where “$\pm$” “$=$” “$-$” for the series $C$ and “$+$” for the series $B$, $D$. This equality is equivalent to the following one: $\pm O\Omega O^\top\Omega=T $, which implies that $T\Omega=O\Omega O^\top$ is a skew-symmetric matrix for the series $C$ and a symmetric matrix for the series $B$ and $D$. Hence, there exists a low-unitrangular matrix $X$ such that $XT \Omega X^\top=\Omega$. For the matrix $XO$, we have $\pm \Omega(XO)^\top\Omega=(XO)^{-1}$, that is, $XO$ belongs to the group $G$ under consideration. Since $X$ is a low-unitrangular matrix, the determinants of type (3.2) for the matrices $XO$ and $O$ coincide. Now from the relations from the ideal $I_{\mathfrak{gl}_m}$ and the Jacobi relations it follows that $a_X$ are minors of the matrix $XO\in G$. As noted above, this implies that the ideal generated by $I_{\mathfrak{gl}_m}$ and the Jacobi relations is the ideal of all relations between the determinants $a_X$ for the group $G$. The case of the series $D$ and determinants $\overline{a}_X$ is reduced to the above case by reordering of rows. This completes the proof of Lemma 3. 3.2.3. Some useful relationsLet us derive some relations that are corollaries of the Plücker and Jacobi relations that we will use below. Proof. Let us first consider the case $i=1$. Then the Plücker relation holds:
Substituting the Jacobi relations
$$
\begin{equation*}
\begin{gathered} \, a_{\pm n,\dots,\pm 2,1,0}= a_{\pm n,\dots,\pm 2,-1}, \\ a_{\pm n,\dots,\pm 2,-1,0}= a_{\pm n,\dots,\pm 2,-1},\qquad a_{\pm n,\dots,\pm 2,-1,1}=- a_{\pm n,\dots,\pm 2,0} \end{gathered}
\end{equation*}
\notag
$$
into (3.15), we readily obtain (3.14) with $i=1$. The case of an arbitrary $i$ is considered similarly. The additional signs due to permutations of indices cancel out. This proves Proposition 2. Proposition 3. In the case of the series $B$ and $D$, and in the case of the series $D$, an analogous relation holds for the determinants $\overline{a}_X$. Proof. We again consider first the case $i=1$, $j=2$. For notational simplicity, we omit the first indices. The following Plücker relation holds. Hence
In the case of the series $D$, we can naturally embed the Lie group into the Lie group of the series $B$ of plus one dimension. So, it what follows, we can work with the series $B$. Now, applying the Jacobi relation to the determinants $a_{\dots,-2,2,-1}$, $a_{\dots,-2,1,2}$ in the numerator, this gives
$$
\begin{equation*}
a_{\dots,-2,2}==\frac{ a_{\dots,-2,1}a_{\dots,0,-1}}{a_{\dots,-2,0}}+ \frac{ a_{\dots,-2,-1}a_{\dots,0,1} }{a_{\dots,-2,0}}.
\end{equation*}
\notag
$$
An application of (3.14) shows that
$$
\begin{equation*}
\begin{aligned} \, a_{\dots,-2,2} &= \frac{a_{\dots,-2,1}\sqrt{a_{\dots,2,-1}a_{\dots,-2,-1}}} {\sqrt{a_{\dots,-2,1}a_{\dots,-2,-1}}} +\frac{ a_{\dots,-2,-1} \sqrt{a_{\dots,-2,1}a_{\dots,2,1}}} {\sqrt{a_{\dots,-2,1}a_{\dots,-2,-1}}} \\ &=\sqrt{a_{\dots,-2,1}a_{\dots,2,-1}}+\sqrt{a_{\dots,-2,-1}a_{\dots,2,1}}. \end{aligned}
\end{equation*}
\notag
$$
Analogously, we have the Plücker relations
$$
\begin{equation*}
a_{\dots,-1,1}a_{\dots,-1,-2,2}+a_{\dots,-1,2}a_{\dots,-1,1,-2}+ a_{\dots,-1,-2}a_{\dots,-1,2,1}=0.
\end{equation*}
\notag
$$
Hence, proceeding as above,
$$
\begin{equation*}
\begin{aligned} \, a_{\dots,-1,1} &=-\frac{a_{\dots,-1,2}a_{\dots,-1,1,-2}+ a_{\dots,-1,-2}a_{\dots,-1,2,1}}{a_{\dots,-1,-2,2}} \\ &=\frac{a_{\dots,-1,2}a_{\dots,-2,0}}{a_{\dots,0,-1}} - \frac{a_{\dots,-1,-2}a_{\dots,0,2}}{a_{\dots,-1,0}}. \end{aligned}
\end{equation*}
\notag
$$
Now an application of (3.14) gives
$$
\begin{equation*}
\begin{aligned} \, a_{\dots,-1,1} &= \frac{a_{\dots,-1,2}\sqrt{a_{\dots,-2,-1}a_{\dots,-2,1}}} {\sqrt{a_{\dots,-2,-1} a_{\dots,2,-1}}} +\frac{a_{\dots,-1,-2} \sqrt{a_{\dots,-1,2} a_{\dots,1,2}}} {\sqrt{a_{\dots,-1,2} a_{\dots,-1,-2}}} \\ &=-\sqrt{a_{\dots,2,-1}a_{\dots-2,1}}+\sqrt{a_{\dots,-1,-2}a_{\dots,1,2}}. \end{aligned}
\end{equation*}
\notag
$$
So, we have
$$
\begin{equation*}
a_{\dots,-2,2}+a_{\dots,-1,1}=2\sqrt{a_{\dots,-1,-2}a_{\dots,1,2}}.
\end{equation*}
\notag
$$
The Jacobi relations imply that $a_{\dots,-1,1}=a_{\dots,-2,2}$, and so
$$
\begin{equation}
a_{\dots,-1,1}=\sqrt{a_{\dots,-1,-2}a_{\dots,1,2}}= \sqrt{a_{\dots,-2,-1}a_{\dots,2,1}}.
\end{equation}
\tag{3.17}
$$
This proves (3.16) with $i=1$, $j=2$. The case of arbitrary $i$, $j$ is considered similarly. The additional signs due to permutations of indices cancel out. Proposition 3 is proved. 3.3. Conditions to single out an irreducible representation in the Zhelobenko model3.3.1. The general theoremAn indicator system is a system of differential equations of the form
$$
\begin{equation}
\begin{alignedat}{2} &L_{-n,-n+1}^{r_{-n}+1}f=0,\ \dots, \ L_{-2,-1}^{r_{-2}+1}f=0,\ L_{-1,1}^{r_{-1}+1}f=0 &\ &\text{ for the series }A, C, \\ &L_{-n,-n+1}^{r_{-n}+1}f=0,\ \dots,\ L_{-2,-1}^{r_{-2}+1}f=0, \ L_{-1,0}^{r_{-1}+1}f=0 &\ &\text{ for the series }B, \\ &L_{-n,-n+1}^{r_{-n}+1}f=0,\ \dots,\ L_{-2,-1}^{r_{-2}+1}f=0, \ L_{-2,1}^{r_{-1}+1}f=0 &\ &\text{ for the series }D. \end{alignedat}
\end{equation}
\tag{3.18}
$$
Here, $L_{-i,-j}$ is an operator acting on a function $f(a)$ as the left infinitesimal shift by $F_{-i,-j}$ for the series $B$, $C$, $D$ and by $E_{-i,-j}$ for the series $A$. The exponent $r_{-i}$ is defined as follows:
$$
\begin{equation}
\begin{aligned} \, &r_{-n} = m_{-n}\,{-}\,m_{-n+1},\, \dots,\, r_{-2} = m_{-2}\,{-}\,m_{-1},\, r_{-1} = m_{-1} \text{ for the series }A, C, \\ &r_{-n} = m_{-n}\,{-}\,m_{-n+1},\, \dots,\, r_{-2} = m_{-2}\,{-} \,m_{-1},\, r_{-1} = 2m_{-1} \text{ for the series }B, \\ &r_{-n} = m_{-n}\,{-}\,m_{-n+1},\, \dots,\, r_{-2} = m_{-2}\,{-}\,|m_{-1}|, \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad r_{-1} = m_{-2}\,{+}\,|m_{-1}| \text{ for the series }D. \end{aligned}
\end{equation}
\tag{3.19}
$$
The following result can be found in [9]. Theorem 1. For the series $A$, let the group $\mathrm{GL}_{n+1}$ act in the space with coordinates indexed by4 $-n,\dots,-1,1$. Then, in the space of functions on the group $\mathrm{GL}_{n+1}$, an irreducible representation with highest weight $[m_{-n},\dots,m_{-1},0]$ and highest vector (3.6) is singled out by the following conditions. 1. $L_-f=0$, where $L_-$ is the left infinitesimal shift by an arbitrary element of $\mathrm{GL}_{n+1}$ corresponding to a negative root. 2. $L_{-i,-i}f=m_{-i}f$, where $L_{-i,-i}$, $i=-1,1,\dots,n$, is the left infinitesimal shift by an element of the group $\mathrm{GL}_{n+1}$ corresponding to a Cartan element $E_{-i,-i}$. 3. $f$ satisfies an indicator system. This theorem describes explicitly the space of the model $\mathrm{Zh}$ in the case of the series $A$. Let us prove an analogous result for the series $B$, $C$, and $D$. Theorem 2. In the Zhelobenko realization for the series $B$, $C$, $D$, an irreducible representation with highest vector (3.6) is singled out by conditions 1, 3 and condition 2, where $E_{-i,-i}$ is changed to $F_{-i,-i}$ in the definition of $L_{-i,-i}$. Our next aim is to prove Theorem 2. Proof of Theorem 2. The scheme of the proof is as follows. First of all, we give formulas for the action of an operator of left infinitesimal shift on determinants. These formulas are used to verify that the highest vector satisfies conditions 1–3. The main difficulty here is to prove that the highest vector satisfies an indicator system. Once this is done, it is easily to show that an arbitrary vector of the representation satisfies conditions 1–3. Finally, we show that the set functions that satisfy conditions 1–3 consists precisely of the functions which are vectors of the representation with highest vector (3.6).
Let us proceed with this plan. Consider the determinant $a_{i_1,\dots,i_k}$ and let us find the action of the left infinitesimal shift on this determinant. For convenience of the description, we introduce a temporary notation for the determinants $a^{-n,\dots,-n+k-1}_{i_1,\dots,i_k}$, where the superscripts indices the numbers of rows from which the elements of the determinant are taken. The operator $L_{-i,-j}$ of left infinitesimal shift acts on the row indices $-n,\dots,-n+k-1$ by the following rule. For the series $A$, the operator of left infinitesimal shift $E_{-i,-j}$ acts by the rule
$$
\begin{equation}
L_{-i,-j}a^{-n,\dots,-n+k-1}_{i_1,\dots,i_k}=a^{\{-n,\dots,-n+ k-1\}|_{-i\mapsto -j}}_{i_1,\dots,i_k}.
\end{equation}
\tag{3.20}
$$
The action of the operator of left infinitesimal shift by $F_{-i,-j}$ for the series $B$, $C$, $D$ can be expressed in terms of these operators. A similar argument shows that
$$
\begin{equation}
L_{-i,-i}a^{-n,\dots,-n+k-1}_{i_1,\dots,i_k}=\begin{cases} a^{-n,\dots,-n+k-1}_{i_1,\dots,i_k} &\text{if } -i\in\{-n,\dots,-n+k-1\}, \\ 0 &\text{otherwise}. \end{cases}
\end{equation}
\tag{3.21}
$$
On the product of determinants the operators $L_{-i,-i}$, $L_{-i,-j}$ act by the Leibniz rule. Let us verify that conditions 1–3 for a representation with highest weight (3.6) are met. We first show that conditions 1–3 are satisfied for the highest vector (3.6). Conditions 1 and 2 are proved directly. Let us verify condition 3. From (3.20) it follows that the operators $L_{-i,-i+1}$ for $i=-n,\dots,-3$, when act on (3.6), do actually act only on $a_{-n,\dots,-i}^{m_{-i}-m_{-i+1}}$. The operator $L_{-2,-1}$ for the series $A$, $B$, $C$ and $D$ in the case $m_{-1}\geqslant 0$, and the operator $L_{-2,1}$ for the series $D$ in the case $m_{-1}<0$, when acting on (8.7), do actually act only on $a_{-n,\dots,-2}^{m_{-2}-m_{-1}}$. The operator $L_{-1,1}$ for the series $A$, $C$, $L_{-1,0}$ for the series $B$ act on determinants of order $n$. In the case of the series $D$, there are also special operators: the operator $L_{-2,1}$ in case $m_{-1} \geqslant 0$, and the operator $L_{-2,-1}$ in the case $m_{-1}< 0$. Both these operators act on determinants of orders $n-1$ and $n$. The powers $r_{-i}=m_{-i}-m_{-i+1}$ of the determinants $a_{-n,\dots,-n+i-1}$ are integer, and hence by (3.20) the equation $L_{-i,-i+1}^{r_{-i}+1}v_0 = 0$, $i=-n,\dots,-2$ is satisfied. The same argument shows that $L_{-1,1}^{r_{-1}+1}v_0=0$ for the series $A$ and $C$. Let us now check that $L_{-1,0}^{m_{-1}+1}v_0=0$ holds for the series $B$. To this end, new formulas are required. So, for the series $B$, we have
$$
\begin{equation*}
\begin{gathered} \, L_{-1,0}a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}= a_{-n,\dots,-2,-1}^{-n,\dots,-2,0},\qquad L_{-1,0}^2a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}= -a_{-n,\dots,-2,-1}^{-n,\dots,-2,1}, \\ L_{-1,0}^3a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}=0. \end{gathered}
\end{equation*}
\notag
$$
Hence the equation $L_{-1,0}^{2m_{-1}+1}v_0=0$ holds for the series $B$ and an integer highest weight. We also have
$$
\begin{equation*}
\begin{aligned} \, L_{-1,0}\bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}\bigr)^{1/2} &= \frac{1}{2}\, a_{-n,\dots,-2,-1}^{-n,\dots,-2,0} \bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}\bigr)^{-1/2}, \\ L_{-1,0}^2a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1} &= -\frac{1}{2}\, a_{-n,\dots,-2,-1}^{-n,\dots,-2,1} \bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}\bigr)^{-1/2} \\ &\qquad-\frac{1}{4}\, \bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,0}\bigr)^2 \bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}\bigr)^{-3/2}=0. \end{aligned}
\end{equation*}
\notag
$$
In the derivation of the last equality, we used the relation
$$
\begin{equation*}
a_{-n,\dots,-2,-1}^{-n,\dots,-2,1}a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1} = -\frac{1}{2}\, \bigl(a_{-n,\dots,-2,-1}^{-n,\dots,-2,0}\bigr)^2.
\end{equation*}
\notag
$$
From these equalities it follows that $L_{-1,0}^{2m_{-1}+1}v_0=0$ for the series $B$ with half-integer highest weight.
Now consider the series $D$ with $m_{-1}\geqslant 0$. Let us verify that $L_{-2,1}^{m_{-2}+m_{-1}+1}v_0=0$. We have
$$
\begin{equation*}
\begin{gathered} \, L_{-2,1}a_{-n,\dots,-2}^{-n,\dots,-2}=a_{-n,\dots,-2}^{-n,\dots,1},\qquad L_{-2,1}a_{-n,\dots,-2}^{-n,\dots,1}=0, \\ L_{-2,1}a_{-n,\dots,-2,-1}^{-n,\dots,-2,-1}= 2a_{-n,\dots,-2,-1}^{-n,\dots,1,-1},\qquad L_{-2,1}a_{-n,\dots,-2,-1}^{-n,\dots,1,-1}= a_{-n,\dots,-2,-1}^{-n,\dots,1,2}, \\ L_{-2,1}a_{-n,\dots,-2,-1}^{-n,\dots,1,2}=0. \end{gathered}
\end{equation*}
\notag
$$
So, $v_0$ is mapped to zero under the operator $L_{-2,1}$ raised to the power equal to the sum of $1$ and the size of the determinant $a_{-n,\dots,-2}$ and twice of the size of the determinant $a_{-n,\dots,-2,-1}$, that is, under the action of $L_{-2,1}$ raised to the power $1+(m_{-2}-m_{-1})+2m_{-1}=1+m_{-2}+m_{-1}$. For the series $D$ and $m_{-1}<0$, let us verify that $L_{-2,-1}^{m_{-2}+m_{-1}+1}v_0=0$. We have
$$
\begin{equation*}
\begin{gathered} \, L_{-2,-1}a_{-n,\dots,-2}^{-n,\dots,-2}=a_{-n,\dots,-2}^{-n,\dots,-1},\qquad L_{-2,-1}a_{-n,\dots,-2}^{-n,\dots,-1}=0, \\ L_{-2,-1}a_{-n,\dots,-2,1}^{-n,\dots,-2,1}= 2a_{-n,\dots,-2,-1}^{-n,\dots,-1,1},\qquad L_{-2,-1}a_{-n,\dots,-2,-1}^{-n,\dots,-1,1}= -a_{-n,\dots,-2,-1}^{-n,\dots,-1,2}, \\ L_{-2,1}a_{-n,\dots,-2,-1}^{-n,\dots,-1,2}=0. \end{gathered}
\end{equation*}
\notag
$$
Thus $v_0$ is mapped to zero under the action of the operator $L_{-2,-1}$ raised to the power $1+(m_{-2}-m_{-1})+2m_{-1}=1+m_{-2}+m_{-1}$. So, the highest vector satisfies an indicator system. That conditions 1–3 hold for an arbitrary vector follows from the fact that the left and right shifts commute and an arbitrary vector of the representation can be written as a linear combination of right shifts of the highest vector. It remains to verify that the functions satisfying conditions 1–3 are precisely the functions that form an irreducible representation with the highest vector (3.6). Let a function on $G$ satisfy conditions 1–3. Then its restriction to the subgroup $Z\subset G$ of upper-unitriangular matrices satisfies an indicator system. According to [9], this restriction belongs to a realization of the representation in the space of functions on the group $Z$ (in [9], this realization in the space of functions on $Z $ is given for all series). From [9] it also follows that the initial function on $G$ belongs to the representation with highest weight (3.6). This proves Theorem 2. 3.3.2. Solution of the indicator system and of the equations $L_{-i,-i}f=m_{-i}f$In the proof of Theorem 2, we obtained formulas for the action of $L_{-i,-i}$. The following lemma is immediate from these formulas. Lemma 4. The solutions of the system $L_{-i,-i}f = m_{-i}f$ that are functions of determinants are described as follows. If this function is represented as a sum of monomials in determinants, then, in each monomial, the sum of exponents of determinant of size $n-i+1$ is $r_{-i}$ for $i=n,\dots,2$. The sum of exponents of determinants of size $n$ is also $|m_{-1}|$. It remains to single out from these functions the solutions of the indicator system. Lemma 5. If the highest weight is integer non-negative, then the solutions of the indicator system are polynomials in determinants that satisfy the condition of Lemma 4. Proof. The highest vector (3.6) is a polynomial in determinants, and, the space of polynomials of determinants is invariant under the action of elements of the algebra, and hence an arbitrary vector of the representation can also be presented as a polynomial of determinants. Now the conclusion of the lemma is immediate from (3.20). The same conclusion also holds for the series $D$ with $m_{-1}<0$ if as determinants of size $n$ one takes $\overline{a}_{i_1,\dots,i_n}$. Let us now consider the case of half-integer highest weight. Lemma 6. If the highest vector in half-integer, then, in the class of the functions that satisfy the conditions of Lemma 4, the solutions of the indicator system are precisely the functions All $f_{\alpha}$ are polynomials of determinants and the sum of exponents of determinants of size $n-i+1$ is $r_{-i}$ for $i=n,\dots,2$. In the case of the series $D$ and $m_{-1}<0$, the determinants of type $\overline{a}_X$ are taken as determinants of order $n$. In all cases, the sum of exponents of determinants of order $n$ is $|m_{-1}|-1/2$. Proof. Consider the case $B$. At the first step, we will show that an arbitrary vector of the representation with highest weight (3.6) has the form of function (3.22).
Consider a representation with highest vector $\sqrt{a_{-n,\dots,-2,-1}}$, that is, the spinor representation. Proposition 4. For the algebra $\mathfrak{o}_{2n+1}$ a representation with the highest vector $\sqrt{a_{-n,\dots,-2,-1}}$ is the span of functions of type Proof. Let us prove that an application of $F_{p,q}$ to $\sqrt{a_{\pm n,\dots,\pm 2,\pm 1}}$ gives a linear combination of function of the same type. Hence the above span is a representation that contains the spinor representation, since $\sqrt{a_{-n,\dots,-2,-1}}$ is the highest vector of the spinor representation. The coincidence of the above span and the spinor representation follows from the fact that both these linear spaces are of dimension $2^n$.
The proofs of the result in the above paragraph are the same for all choices of signs. For simplicity of notation, we will assume choose the sign $-$. In this case, the result under the action on $\sqrt{a_{-n,\dots,-2,-1}}$ is non-zero only for the generators $F_{0,-i}$, $F_{j,-i}$, $i,j>0$. In these cases, we have
$$
\begin{equation}
\begin{aligned} \, F_{0,-i}&\colon \sqrt{a_{-n,\dots,-2,-1}} \mapsto\frac{(-1)^i a_{-n,\dots,\widehat{-i},\dots,-1,0}}{2\sqrt{a_{-n,\dots,-1}}}, \\ F_{j,-i}&\colon \sqrt{a_{-n,\dots,-2,-1}} \mapsto\frac{(-1)^i a_{-n,\dots,\widehat{-i},\dots,-1,j}-(-1)^j a_{-n,\dots,\widehat{-j},\dots,-1,i} }{2\sqrt{a_{-n,\dots,-1}}}. \end{aligned}
\end{equation}
\tag{3.24}
$$
Consider the first equality in (3.24). Using (3.14) and taking the square root of this relation, we find that
$$
\begin{equation}
F_{0,-i}\sqrt{a_{-n,\dots,-2,-1}}=\frac{\sqrt{-2}}{2} \cdot \sqrt{a_{-n,\dots,\widehat{-i},\dots,-1,i}}.
\end{equation}
\tag{3.25}
$$
Now let us consider the second equality in (3.24). We claim that
$$
\begin{equation*}
\frac{(-1)^i a_{-n,\dots,\widehat{-i},\dots,-1,j}-(-1)^j a_{-n,\dots,\widehat{-j},\dots,-1,i}}{2\sqrt{a_{-n,\dots,-1}}}
\end{equation*}
\notag
$$
is a linear combination of functions (3.23). Without loss of generality we can assume that $i=1$, $j=2$, that is, we consider the fraction
$$
\begin{equation}
\frac{a_{-n,\dots,-3,-2,2}- a_{-n,\dots,-3,1,-1}}{2\sqrt{a_{-n,\dots,-1}}}.
\end{equation}
\tag{3.26}
$$
Using (3.16) (for $i=1$, $j=2$, and expressing $a_{-n,\dots,-3,-1,1}$, and also, for $i=2$, $j=1$, and expressing $a_{-n,\dots,-3,-2,2}$), we find that (3.26) is equal to $\sqrt{a_{\dots,2,1}}$. In other words, (3.24) for $i=1$, $j=2$, is equal to
$$
\begin{equation}
F_{2,-1}\sqrt{a_{-n,\dots,-1}}= \sqrt{a_{-n,\dots,2,1}}.
\end{equation}
\tag{3.27}
$$
This proves Proposition 4. Now consider the case of an arbitrary highest weight. The highest vector can be represented as where $v'_0$ is a polynomial of determinants. An arbitrary vector $f$ can be written as a linear combination of vectors obtained by the action on the highest vector of the operators $\prod_{i<j} F_{i,j}^{p_{i,j}}$. As a result, we obtain a vector of type (3.22).So, the Zhelobenko model is contained in the space of functions of type (3.22). Now we need to show that all function of this type belong to the Zhelobenko model. For this it is enough to check that each function of the form (3.22) that satisfies the conditions of Lemma 4 is a solution of the indicator system. The operator $L_{-i,-i+1}$, $i=n,\dots,2$, acts only on the determinants of size $n- i+ 1$. So, these operators act only on the determinants of size $1,\dots,n-1$. Such determinants occur only in $f_{\alpha}$, and hence their exponents are non-negative integers. The sum of the exponents of determinants of size $i$ is $r_{-n+i-1}$, because thus the conditions of Lemma 4 are satisfied. Hence the conditions $L_{-i,-i+1}^{r_{-i}+1}f=0 $ for $i=n,\dots,2$ hold. Now let us consider the equation $L_{-1,0}^{2m_{-1}+1}f=L_{-1,0}^{2[m_{-1}]+1+1}f=0$, where $[m_{-1}]$ is the integer part. The operator $L_{-1,0}^{2[m_{-1}]+1+1}$ may act on each summand in (3.22) by the Leibniz rule as follows. – Either $L_{-1,0}^{2[m_{-1}]+1+1}$ acts entirely on the second factor $f_{\alpha}$. In this case, we get $0$, since in the polynomials $f_{\alpha}$ the sum of exponents of determinants of size $n$ is $[m_{-1}]$, but such polynomials are annihilated already in $L_{-1,0}^{2[m_{-1}]+1}$. – Or $L_{-1,0}^{2[m_{-1}]+1}$ acts on the second factor $f_{\alpha}$, and $L_{-1,0}$ acts on the first factor. In this case, we get $0$ by the same reason. – Or $L_{-1,0}^{2[m_{-1}]+2-k}$ acts on the second factor and $L_{-1,0}^{k}$ acts on the first factor; here, $k\geqslant 2$. The first factor is a vector of the representation with highest weight $[1/2,\dots,1/2]$, and so it vanishes under the action of $L_{-1,0}^{2}$. Thus the vectors of type (3.22) vanishes under the action of $L_{-1,0}^{2m_{-1}+1}$. This proves the lemma in the case of series $B$. Now consider the series $D$. We argue as for the series $B$. First, we consider the spinor representations with highest weights $[1/2,\dots,1/2,1/2]$, $[1/2,\dots,1/2,-1/2]$. Proposition 5. For the algebra $\mathfrak{o}_{2n}$, the representations with the highest vectors $\sqrt{a_{-n,\dots,-2,-1}}$, $\sqrt{\overline{a}_{-n,\dots,-2,1}}$ coincide with the spans of the functions Proof. We argues as in Proposition 4. We have an embedding of the Lie algebras $\mathfrak{o}_{2n}\subset\mathfrak{o}_{2n+1}$ induced by an embedding of the root systems; a similar embedding for Lie groups also holds. Thanks to this, the determinants $a_{\pm n,\dots,\pm 1}$ and also $\overline{a}_{\pm n,\dots,\pm 1}$ can be considered as functions on the group $\mathrm{SO}_{2n+1}$, and so we can use (3.27).
From this formula we see that under the action of the algebra $\mathfrak{o}_{2n}$ on the function $\sqrt{a_{\pm n,\dots,\pm 2,\pm1}}$, $\sqrt{\overline{a}_{\pm n,\dots,\pm 2,\pm1}}$, we obtain a linear combination of functions of the same type. At the same time, the parity of the number of minuses in the indexes is preserved. Thus, both linear spans under consideration are representations. The dimension of each of these linear spans is $2^{n-1}$. The irreducible representations with highest vectors $\sqrt{a_{-n,\dots,-2,-1}}$, $\sqrt{\overline{a}_{-n,\dots,-2,1}}$ are also of dimension $2^{n-1}$. Now Proposition 5 follows. Further considerations in the case of series $D$ proceed as in the case of series $B$. The only change is the replacement of $L_{-1,0}$ by $L_{-2,1}$. Lemma 6 is proved. To summarize, we have proved the following theorem. Theorem 3. The space of an irreducible representation with highest vector (3.6) in the Zhelobenko model is described as follows. In the cases of the series $A$, $B$, $C$ and the series $D$ with $m_{-1}\geqslant 0$, we consider the space of functions of determinants $a_X$, $|X|\leqslant n$. In the case of the series $D$ with $|m_{-1}|<0$, we consider the space of functions of determinants $a_X$, $|X|< n$ and $\overline{a}_X$, $|X|=n$. In addition, the functions must satisfy the following condition. In the case of integer highest weight, they are polynomials of the determinants satisfying the conditions of Lemma 4. In the case of an integer highest weight, they are functions of type (3.22) satisfying the conditions of Lemma 6. § 4. The GKZ system for the series $A$. The A-GKZ systemIn this section, we introduce two important systems of partial differential equations. Solutions of one of these systems (the GKZ system) are used to form a base in the Zhelobenko model, and solutions of the second system (the A-GKZ system) are used to build a new representation model. The results of this section concern only the series $A$; to a large extent, all these facts were obtained in [14], but we insignificantly modify them to adapt for consideration of other series. We also show how the proofs of the modified results can be obtained from those of similar results in [14]. 4.1. The Gelfand–Tsetlin lattice. The vectors $v_{\alpha}$. The number $\mathcal{K}$Consider the Lie algebra $\mathfrak{gl}_m$, which we identify with the algebra of matrices whose rows and columns are indexed by the numbers $-n,\dots,\widehat{0},\dots,n$ (in the case $m=2n$) or $-n,\dots,0,\dots,n$ (in the case $m=2n+1$). With this algebra we associate a shifted lattice in the space $\mathbb{Z}^{N}$, whose coordinates are numbered by proper subsets $X\subset \{-n,\dots,n\}$. Here, $N$ is the number of possible proper subsets of $X\subset\{-n,\dots,n\}$. Such a strange (at a first glance) indexing is taken in order to obtain a lattice that will be used in further analysis for the series $B$, $C$, and $D$. The definition of the Gelfand–Tsetlin lattice for the series $A$, which is given below, differs slightly from that of [14]. The essence of the difference is as follows. The idea of a homogeneous system defining the Gelfand–Tsetlin lattice is that an inhomogeneous versions of this system represent naive conditions on the vector of exponents of a monomial of determinants, which can appear in the decomposition of a function corresponding to the vector of the Gelfand–Tsetlin base. Accordingly, this system should depend on the chain of subalgebras underlying the construction of the Gelfand–Tsetlin base. In [14], a chain is used in which the subalgebra $\mathfrak{gl}_{m-k}$ is formed by matrices having non-zero elements of the first $m-k$ rows and columns relative to the standard order on the set $\{-n,\dots,n\}$. In the present paper, we use a chain in which the subalgebra $\mathfrak{gl}_{m-k}$ is formed by matrices with non-zero elements in $m-k$ rows and columns with the smallest indices relative to the ordering This is done in order to use the results for the series $A$ in further consideration of other series. Consider the quantity Definition 3. The Gelfand–Tselin lattice $\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}\subset \mathbb{Z}^N$ for the series $A$ is defined by the following homogeneous system of equations: The resulting lattice is called the Gelfand–Tsetlin lattice for the series $A$. Let us construct the generators of this lattice. Consider the subsets $Y_i$, $i=-n,\dots,n$, of the following type
$$
\begin{equation}
\begin{gathered} \, Y_{-n}=\{-n\},\quad Y_n=\{-n,n\}, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots \\ \qquad\qquad Y_{-i}=\{-n,\dots,-i\},\quad Y_i=\{-n,\dots,-i,i\},\qquad i>0. \end{gathered}
\end{equation}
\tag{4.4}
$$
Note that $Y_{-i}=Y_{-i-1}\cup\{-i\}$, $Y_i=Y_{-i}\cup\{i\}$. We also note that
$$
\begin{equation*}
\{j\colon j\succ Y_{-i}\}=\{j:j\succeq i\},\qquad \{j\colon j\succ Y_i\}= \{j\colon j\succeq -i+1\},
\end{equation*}
\notag
$$
where the index is greater than a subset if it is greater than any element of the subset. Consider the vectors (for $i\geqslant 0$)
$$
\begin{equation}
v_{\alpha}= \begin{cases} e_{Y_{-i-1},-i,X}-e_{Y_{-i-1},j,X}-e_{Y_{-i-1},-i,y,X} + e_{Y_{-i-1},j,y,X}, &-i\prec j\prec y \prec X, \\ e_{Y_{-i},i,X}-e_{Y_{-i},j,X}-e_{Y_{-i},i,y,X} +e_{Y_{-i},j,y,X}, &i\prec j\prec y \prec X. \end{cases}
\end{equation}
\tag{4.5}
$$
Here, $e_{Z}$ is the unit base vector corresponding to a coordinate $Z$. One checks directly that these vectors satisfy system (4.3). To select base vectors, we fix $\pm i$, $j$, $y$, and consider vectors (4.5) for all possible $X$. Let us construct a graph in which these subsets $X$ will be vertices, and the edges will be pairs of subsets of the form $X_1=X$, $X_2=\{x\}\cup X$. Let us choose a collection of subsets such that the corresponding subgraph is a tree, which is maximal with respect to the extension with preservation of this property. For fixed $\pm i$, $j$, $y$ and the chosen $X$, we construct vectors (4.6). Lemma 7. The chosen vectors $v_{\alpha}$ form a base in the lattice $\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$. The proof of this lemma is similar to that of an analogous result in [14] and hence omitted. For brevity, we set (for $i\in \{-n,\dots,n\}$) where $Z=Y_{i-1}\cup X$ for $i\leqslant 0$ and $Z=Y_{-i}\cup X$ for $i >0$.Let $\mathcal{K}$ be the number of vectors in the base of $\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$. 4.2. The GKZ system for the series $A$Let us write down the GKZ system associated with the lattice $\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$. In § 2.1, a GKZ system associated with an arbitrary lattice was defined. In this section, we will write only equations of the second type, which are determined from the lattice $\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$. This will be the GKZ system for the series $A$. Namely, let $A_X$ be the variables which are skew-symmetric over the set $X$, and which not obey any other relation. In this case, by the GKZ system for the series $A$ we mean the system generated (see Definition 1) by the differential equations constructed from vectors (4.6):
$$
\begin{equation}
\biggl(\frac{\partial^2}{\partial A_{i,Z}\, \partial A_{j,y,Z}}- \frac{\partial^2}{\partial A_{j,Z}\, \partial A_{i,y,Z}}\biggr)\mathcal{F}.
\end{equation}
\tag{4.7}
$$
Below, when referring to system (4.7), we will mean the system generated by these equations. This agreement will be used for all GKZ and A-GKZ systems introduced in the present paper. Lemma 8 (see [16]). The space of polynomial solutions of system (4.7) has a base $\mathcal{F}_{\gamma}(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m})$ consisting of $\Gamma$-series. Here, we take the shift vectors $\gamma$ that represent different elements of the quotient space $\mathbb{C}^N /\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$, and such that one of the representatives of this class has only non-negative coordinates (see § 5.1). In Lemma 8, the term “polynomial” can be replaced by “expanding in power series”, but it should be understood that the system also has other solutions that expand in power-logarithmic series. 4.3. The A-GKZ system for the series $A$The A-GKZ system for the series $A$ is the system generated (see Definition 1) by the equations
$$
\begin{equation}
\biggl(\frac{\partial^2}{\partial A_{i,Z}\, \partial A_{j,y,Z} }- \frac{\partial^2}{\partial A_{j,Z}\, \partial A_{i,y,Z} }+ \frac{\partial^2}{\partial A_{y,Z}\, \partial A_{i,j,Z}}\biggr)F.
\end{equation}
\tag{4.8}
$$
Remark 1. An important observation is that with an equation of the A-GKZ system (that is, with vector (4.6)), a Plücker relation for determinants is associated With the vector $v_{\alpha}$ of type (4.6) we associate the vector Now let us give formulas for some solutions of the A-GKZ system. Let $t,s\in\mathbb{Z}^\mathcal{K}_{\geqslant 0}$, where $\mathcal{K}$ be the number of independent vectors $v_{\alpha}$. Consider the functions
$$
\begin{equation}
\mathcal{F}_{\gamma}^s(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}) = \sum_{t\in\mathbb{Z}^{\mathcal{K}}} \frac{(t+1)\cdots(t+s)A^{\gamma+tv}}{s!\, (\gamma+tv)!},
\end{equation}
\tag{4.11}
$$
in the formula we use the multi-index notation $tv:=t_1v_1+\dots+t_{\mathcal{K}}v_{\mathcal{K}}$, $sr:=s_1r_1+\dots+s_{\mathcal{K}}r_{\mathcal{K}}$ and also the multi-index notation for factorials. We set
$$
\begin{equation}
F_{\gamma}(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m})= \sum_{s\in \mathbb{Z}^{\mathcal{K}}} (-1)^s\mathcal{F}_{\gamma-sr}^s (A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}).
\end{equation}
\tag{4.12}
$$
Note that this series depends on the choice of the vector $\gamma$, not only on the class $\gamma$ $\operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$. The proof of the following result proceeds precisely as in [14] (see also [15]). Proposition 6. The functions $F_{\gamma}(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m})$ for vectors $\gamma$ chosen in Lemma 8, form a base in the space of polynomial solutions of the A-GKZ system. Remark 2. In [14] the following result is proved. Its proof can again be carried out verbatim in the situation considered here. Let $a_{i_1,\dots,i_k}$ be a function of the form (3.2) for the group $\mathrm{GL}_m$. Let $I_{\mathfrak{gl}_m}$ be the ideal of relations between these minors, which is considered as an ideal in the polynomial ring $\mathbb{C}[A]$. To each relation there corresponds the differential operator via This gives us an ideal $\overline{I}_{\mathfrak{gl}_m}$ in the ring of differential operators with constant coefficients. Note that the equations of the A-GKZ system are obtained by rule (4.13) from some Plücker type relations.In [14], it was shown that, nevertheless, the solution space of the ideal $\overline{I}_{\mathfrak{gl}_m}$ coincides with that of the A-GKZ system. § 5. Models of representations for $\mathfrak{gl}_m$The results of this section, which were obtained in [14], also concern only the series $A$. As before, consider the Lie algebra $\mathfrak{gl}_m$, where $m=2n$ or $2n+1$, acting in a space with coordinates $-n,\dots,n$, where $0$ is thrown out in the first case, and in the second cases $0$ remains. Next, let $\{-n,\dots,n\}$ be such an index set. 5.1. Gelfand–Tsetlin diagrams and shift vectorsWith a Gelfand–Tsetlin diagram $(m_{p,q})$ for the algebra $\mathfrak{gl}_m$ we can associates the shifted lattice $\Pi=\gamma+\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$, defined by the inhomogeneous system of equations
$$
\begin{equation}
\begin{gathered} \, \delta\in \Pi \quad \Longleftrightarrow\quad \forall\, p,q\in\{-n,\dots,n\},\quad p\preceq q \text{ one has} \\ \qquad\qquad\sum_{X\text{ contains }\geqslant (p+n+1-s(p,q)) \text{ elements} \, \preceq \,q} \delta_X=m_{p,q}, \end{gathered}
\end{equation}
\tag{5.1}
$$
where $s(p,q)$ was defined in (4.2). Lemma 9. There exists a one-to-one correspondence between the Gelfand–Tsetlin diagrams $(m_{p,q})$ and the shifted lattices $\Pi=\gamma+\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$ that contain at least one vector with non-negative coordinates. The elements of the Gelfand–Tsetlin diagram understood in the usual sense are restored using equalities (5.1). In this approach, the diagrams related to the representation of the highest weight $[m_{-n},\dots,m_n]$ are defined as the class $\operatorname{mod}\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$ of the vectors $\gamma$ such that
$$
\begin{equation*}
\sum_{X\colon |X|=p}\gamma_X=m_{p+n-1-s(p,1)},\qquad p\in\{-n,\dots,n\}.
\end{equation*}
\notag
$$
Thus, it is possible to define the Gelfand–Tsetlin diagram as the shifted lattice $\Pi=\gamma+\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$ of integer vectors such that in this shifted lattice there is a representative with only non-negative coordinates. Now a Gelfand–Tsetlin diagram can be called $\mathfrak{gl}_{m-q}$-maximal if in its representative $\gamma$ (and, therefore, in all its vectors) the coordinates corresponding to subsets $X$ are zero if the monomial $A_X$ is not a $\mathfrak{gl}_{m-q}$ highest vector under the action of $\mathfrak{gl}_{m}$ on these variables defined by (5.2). 5.2. The GKZ base in the Zhelobenko realizationIn [14], the following result was proved. Theorem 4. Consider a $\Gamma$-series $\mathcal{F}_{\gamma}(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m})$ such that $\Pi=\gamma+ \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$ are Gelfand–Tsetlin diagrams in the sense of the previous section. Then $\mathcal{F}_{\gamma}(A;\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m})$ is a base of the Zhelobenko model $\mathrm{Zh}$, and this base is consistent in the decomposition of $\mathrm{Zh}$ into the sum of irreducible representations. 5.3. An A-GKZ model of representations $\mathfrak{gl}_m$Consider the action of the Lie algebra $\mathfrak{gl}_m$ on the independent variables $A_X$ defined by
$$
\begin{equation}
E_{i,j}=\sum_XA_{X,i}\, \frac{\partial}{\partial A_{X,j}}.
\end{equation}
\tag{5.2}
$$
The following result holds (see [14]). Theorem 5. The space of polynomial solutions of the A-GKZ system is invariant under the action of $\mathfrak{gl}_m$. This space is a direct sum of finite-dimensional irreducible representations taken with multiplicity $1$. Thus, the A-GKZ solution space in the case of $\mathfrak{gl}_m$ is a model of finite-dimensional irreducible representations. It is not difficult to observe that this model is naturally identified with a subspace in the tensor product of standard representations. 5.4. A relation with the Gelfand–Tselin baseThe following result can be found in [14]. On the set of vectors considered by $\operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$, consider the ordering
$$
\begin{equation}
\gamma \preceq \delta \quad \Longleftrightarrow\quad \gamma=\delta-s r\ \operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m},\qquad s\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0},
\end{equation}
\tag{5.3}
$$
where ${\mathcal{K}}$ is the number of independent vectors $v_{\alpha}$ of type (4.6), and the vectors $r_{\alpha}$ are defined in (4.10). We use the notation $sr:=s_1r_1+\dots+s_{\mathcal{K}}r_{\mathcal{K}}$. According to § 5.1, the Gelfand–Tsetlin diagrams are in a one-to-one correspondence with the shift vectors considered by $\operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m}$ so that there is a vector with non-negative coordinates in this class. In this case, we can introduce the notation $G_{\delta}$ for the Gelfand–Tsetlin base vector corresponding to the diagram $\delta$. Theorem 6. 1. The base $F_{\delta}$ and $G_{\delta}$ are related by a lower-triangular transformation relative to the ordering $\prec$. 2. $G_{\delta}$ is a orthogonalization of $F_{\delta}$. In [14], a transformation that relates $F_{\delta}$ and $G_{\delta}$ was found explicitly. § 6. The GKZ and A-GKZ systems for the series $B$, $C$, $D$To obtain systems for which the analogues of Theorems 4, 5 hold, we need to add new equations to the GKZ and A-GKZ systems for the series $A$. These equations are related to new (non-Plücker) relations arising for the series $B$, $C$, $D$. 6.1. An auxiliary lattice $\mathcal{B}^{g_n}$. The variables $B_X$We first consider the lattice
$$
\begin{equation}
\overline{\mathcal{B}}^{\,g_n}=\mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_n} \subset \overline{L}\simeq \mathbb{Z}^N.
\end{equation}
\tag{6.1}
$$
The lattice $\overline{L}\simeq \mathbb{Z}^N$ is interpreted as the set of exponents of the above Laurent monomials in the variables $A_X$. The lattice $\overline{\mathcal{B}}^{\,g_n}$ thus constructed is the auxiliary lattice $\mathcal{B}^{g_n}$ for the series $C$. However, some additional constructions are needed for the series $B$ and $D$. Namely, consider the variables $B_X$ associated with the above introduced variables $A_X$ according to the following rule. If $X$ or $\widehat{-X}$ coincides with $\{\pm n,\dots,\pm 1\}$, then $B_X=\sqrt{A_X}$; otherwise, $B_X=A_X$. Note that a construction of this type (a transition to variables, some of which coincide with the determinants, and some of which are the roots of the determinants in order for the spinor representation to be realized in the space of polynomials in these variables) for the case of $\mathfrak{o}_5$ is implemented in all details in [21]. Consider the lattice $L=\mathbb{Z}^N$ of exponents of monomials in the variables $B_X$. Using the substitution from the previous paragraph, we can naturally embed the lattice $\bar{L}$ in $L$. This gives us the embedding $\overline{\mathcal{B}}^{g_n}\subset \overline{L}\subset L=\mathbb{Z}^N$. We can now define $\mathcal{B}^{g_n}$ as a sublattice in $L$ which is the image of $\overline{\mathcal{B}}^{g_n}$ under this embedding. Let us introduce the degrees of the variables $B_X$:
$$
\begin{equation}
\deg B_X= \begin{cases} 1, &X\neq \bigl\{\{\pm n,\dots,\pm 1\},\, \{\pm n,\dots,\pm 1,0\}\bigr\}, \\ \dfrac{1}{2} &\text{otherwise}. \end{cases}
\end{equation}
\tag{6.2}
$$
6.2. The GKZ system for $g_n$ in the case of the series $B$, $D$Using the lattice $\mathcal{B}^{g_n}$, it is possible to build a GKZ system,, and then, an A-GKZ system. But then one needs to add some new equations, so that, for the series $B$ and $D$, the analogue of the result formulated in Remark 2 would hold. These new equations would correspond to the Jacobi relations (3.11) and to the root of (3.14). Despite the addition of these equations, we call the resulting systems the GKZ and A-GKZ systems for the series $B$, $D$. Consider functions of independent variables $B_X$ indexed by proper subsets in $\{-n,\dots,n\}$. In the case of the $B$ series, the index $0$ is included. Definition 4. The GKZ system for the series $B$, $D$ is a system generated by the equations The equation of the second type is called the Jacobi equation. For the idea of construction of this system, see Remark 3. 6.3. The A-GKZ system for the series $B$, $D$ Definition 5. The A-GKZ system for the series $B$, $D$ is a system generated by the equations (see Definition 1) Remark 3. The idea of construction of these A-GKZ systems is as follows. This system is constructed from some relations between $a_X$ and $\sqrt{a_{\pm n,\dots, \pm 1}}$. In these relations, the determinants are first replaced by the variables $A_X$, and then, by the variables $B_X$, after which the substitution $B_X\mapsto \partial/\partial B_X$ is applied. The first equations are constructed using some Plücker relations, the chosen relations corresponding to vectors (4.6). The equations of the second type are obtained via the Jacobi relations. The equations of the third and fourth types are written using (3.14) and (3.16). The GKZ system is a simplification of the A-GKZ system. Let us construct bases in the spaces of solutions of the GKZ and A-GKZ systems. For an arbitrary vector $\delta\in\mathbb{Z}^N$, we introduce the functions analogous to (4.11),
$$
\begin{equation}
\mathfrak{f}_{\delta}^s(B)=\sum_{t\in\mathbb{Z}^{\mathcal{K}}} \frac{(t+1)\cdots(t+s) B^{\delta+tv}}{s!\, (\delta+tv)!},
\end{equation}
\tag{6.5}
$$
where we use the chosen base $v_{\alpha}$, $\alpha=1,\dots,\mathcal{K}$, of the auxiliary lattice $\mathcal{B}^{g_n}$, and the multiindex notation from § 4.3. We also set
$$
\begin{equation*}
\mathfrak{f}_{\delta}(B):= \mathfrak{f}_{\delta}^0(B).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
f_{\delta}(B):=\sum_{s\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0}}(-1)^s \mathfrak{f}_{\delta}^s(B).
\end{equation}
\tag{6.6}
$$
Both functions $\mathfrak{f}_{\delta}(B)$ and $f_{\delta}(B)$, $\delta\in\mathbb{Z}^N$, are polynomials. The following differentiation rule holds:
$$
\begin{equation}
\frac{\partial}{\partial B_X} \mathfrak{f}^s_{\delta}= \mathfrak{f}^s_{\delta-e_X},\qquad \frac{\partial}{\partial B_X} f_{\delta}=f_{\delta-e_X},
\end{equation}
\tag{6.7}
$$
where $e_X$ is the unit base vector corresponding to the coordinate $B_X$. Also, as in the case of the series $A$, it is proved that $\mathfrak{f}_{\delta}(B)$ and $f_{\delta}(B)$ are solutions of the systems consisting of equations of the first type for systems (6.3) and (6.4), respectively. Consider the vectors
$$
\begin{equation}
\begin{aligned} \, h_{\kappa} &=e_X-e_{\widehat{-X}},\qquad \kappa=1,\dots,T, \\ x_{\chi} &=2e_{\pm n,\dots,\widehat{\pm i},\dots, \widehat{\pm j},\dots,\pm 1,-i,i}- e_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,-i,-j} \\ &\qquad-e_{\pm n,\dots,\widehat{\pm i},\dots, \widehat{\pm j},\dots,\pm 1,i,j}, \qquad \chi=1,\dots,Z, \\ w_{\epsilon} &=e_{\pm n,\dots,\widehat{i},\dots,\pm 1,-i}+ e_{\pm n,\dots,\widehat{i},\dots,\pm 1,i}-2e_{\pm n,\dots,\widehat{i}, \dots,\pm 1,0}, \qquad \epsilon=1,\dots,E. \end{aligned}
\end{equation}
\tag{6.8}
$$
Note that by fixing $\kappa$ we fix a subset $X=\{i_1,\dots,i_t\}$. For a given $\kappa$, let the sign $(\pm_{\kappa} 1)$ be defined according to (3.13). Consider the functions
$$
\begin{equation}
\mathcal{F}^s_{\delta}(B):= \begin{cases} {\displaystyle\sum_{t_1\in \mathbb{Z}^T,\, t_2\in\mathbb{Z}^{E},\, t_{3}\in \mathbb{Z}^{Z}} \biggl( \prod_{\kappa}(\pm_{\kappa} 1)^{t_1^{\kappa}} \cdot (\sqrt{-2})^{\sum_{\epsilon} t_2^{\epsilon}} \biggr)} \\ \qquad\qquad\qquad\qquad\qquad \times\, \mathfrak{f}^s_{\delta+t_1 h+t_2 w+t_3x}(B) &\text{for the series } B, \\ {\displaystyle\sum_{t_1\in \mathbb{Z}^K,\, t_{3}\in \mathbb{Z}^{Z}} \biggl( \prod_{\kappa}(\pm_{\kappa} 1)^{t_1^{\kappa}} \biggr) \cdot \mathfrak{f}^s_{\delta+t_1 h+t_3x}(B)} &\text{for the series }D, \end{cases}
\end{equation}
\tag{6.9}
$$
$$
\begin{equation}
F_{\delta}(B):=\sum_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}}(-1)^s \mathcal{F}^s_{\delta}(B).
\end{equation}
\tag{6.10}
$$
Here, we set $t_1 h:=t_1^1 h_1+\dots+t_1^{T} h_T$, $t_2 w:=t_2^1 w_1+\dots+t_2^{E} w_E$, and $t_3 \chi:=t_3^1 x_1+\dots+t_3^{Z} x_{Z}$. We need the following important definition. In both cases, $\mathcal{B}^{g_n}_{\mathrm{GC}}$ is considered as a sublattice of the lattice $L$ of exponents of the monomials of $B_X$. Now the following result is secured by (6.7). Theorem 7. The space of polynomial solutions of the GKZ system for the series $B$, $D$ has a base consisting of the function $\mathcal{F}_{\delta}(B)=\mathcal{F}^0_{\delta}(B)$ for various $\operatorname{mod} \mathcal{B}^{g_n}_{\mathrm{GC}}$ vectors $\delta\in \mathbb{Z}^N$ for which that this function is non-zero. The space of polynomial solutions of the A-GKZ system for $g_n$ has the base (with the same choice of $\delta$) of the function $F_{\delta}(B)$. Corollary 2. There is a one-to-one correspondence between the spaces of polynomial solutions of the GKZ system (6.3) and the A-GKZ system (6.4). Remark 4. This theorem is a manifestation of the following general idea in the analytic theory of differential equations: from a given system of equations, a simplified system is constructed, which is solved explicitly (GKZ is a “simplification” of A-GKZ). Then, from each solution of the simplified system, the solution of the initial system is constructed. There are several formalizations of this idea (see [28] and [29]). Remark 5. A relation between the GKZ and A-GKZ systems is an example of the toric degeneration of differential operators. The theory the toric degenerations considers degenerations of Plücker relations in the variables $A_X$. In this case, the basic relations are selected and some terms are removed [30]. This theory, which involves the Gelfand–Tsetlin diagrams, is related to the representation theory (see the original paper [31] and the paper [32], which is closest to the present study). Nevertheless, we do not formalize this relation between the theory of toric degenerations in the present study. In the present paper, we consider degenerations of differential operators, their solution spaces are investigated, and Gelfand–Tsetlin diagrams are used for indexing base solutions. All this has no analogues in the studies on toric degenerations. 6.4. The systems for the tensor representations in the case of the series $B$, $C$, $D$It is known that in the case of the series $C$, any finite-dimensional representation5 is realized as a subrepresentation in the tensor power of the standard representation. In the case of series $B$, $D$, it is possible to pose the problem of construction of models only for such representations. To build a model of tensor representations, we will use more simple GKZ and A-GKZ systems than in the previous section. Consider the independent variables $A_X$ which are antisymmetric in $X$. When passing to the Zhelobenko model, $a_X$ is substituted in place of $A_X$ for all $X$. Let us take the auxiliary lattice $\overline{\mathcal{B}}^{\,g_n}$ constructed above. We denote it now also by $\mathcal{B}^{g_n}$. From this lattice, we construct a GKZ system and augment it with the Jacobi equation. Definition 7. By the GKZ system for tensor representations we mean the system generated by the equations (see Definition 1) Definition 8. The A-GKZ system for tensor representations is the system generated by the equations (see Definition 1) This lattice is naturally considered as a sublattice in the lattice of exponents of monomials $A_X$. The solutions of the A-GKZ system are constructed as follows:
$$
\begin{equation}
\mathcal{F}^s_{\delta}(A):=\sum_{t_1\in \mathbb{Z}^\mathcal{K}} \biggl( \prod_{\kappa}(\pm_{\kappa} 1)^{t_1^{\kappa}} \biggr) \cdot \mathfrak{f}^s_{\delta+t_1 h}(A),
\end{equation}
\tag{6.13}
$$
where $\mathfrak{f}^s_{\delta}(A)$ is defined by the same formula (6.5), where, however, the variables $A_X$ and another auxiliary lattice $\mathcal{B}^{g_n}$ are used,
$$
\begin{equation}
F_{\delta}(A):=\sum_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}}(-1)^s \mathcal{F}^s_{\delta}(A).
\end{equation}
\tag{6.14}
$$
An analogue of Theorem 7 and Corollary 2 holds. § 7. The Gelfand–Tsetlin diagrams for the algebra $g_n$In analogy with § 5.1, we define a Gelfand–Tsetlin diagram for the algebra $g_n$. We first define the Gelfand–Tsetlin diagrams as follows. Definition 10. A Gelfand–Tsetlin diagram for the algebra $g_n$ is a shifted lattice $\Pi=\gamma+ \mathcal{B}_{\mathrm{GC}}^{g_n}$ consisting of integer vectors $\gamma\in\mathbb{Z}^N$ such that $\Pi$ contains a vector with only non-negative coordinates. In this case, to a diagram $\delta$ there corresponds the set of highest weights for the algebras $g_{n-k+1}$, $k=1,\dots,n$, constructed by
$$
\begin{equation}
\begin{aligned} \, m_{p,q} &=\sum_{X\colon X\text{ contains } \geqslant (n+p+1) \text{ indices of absolute value } \geqslant q}\gamma_X \\ &\qquad -\sum_{X\colon X\text{ contains } \geqslant (-p-1) \text{ indices of absolute value } \geqslant q}\gamma_X, \\ &\qquad p=-n,\dots,-1,\qquad q=1,\dots,n,\qquad p \leqslant -q+1. \end{aligned}
\end{equation}
\tag{7.2}
$$
This definition is correct, since it does not depend on the choice of a representative of $\gamma$ in the equivalence class $\operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{\mathfrak{g}_n}$. Another definition of Gelfand–Tsetlin diagrams is more familiar, in which some tables are constructed that encode the base vectors of a finite-dimensional irreducible representation of $g_n$ (see [22]). At the same time, these tables contain the rows $m_{p,q}$ defined in (7.2), as well as some other rows. Below it will be shown that the diagrams in the sense of Definition 10 are also in a one-to-one correspondence with the base vectors of finite-dimensional irreducible representations of $g_n$. As a result, the objects defined in Definition 10 are manifestations of the same essence that is encoded by the Gelfand–Tsetlin diagram in the classical (combinatorial) sense. The advantage of Definition 10 becomes clear for evaluation of formulas for the action of generators. In terms of the vectors $\gamma$, these formulas look very simple and natural (see (8.15)). § 8. Models of representations for the series $B$, $C$, $D$8.1. The actionThere is an action of the algebra $g_n$ on the variables $A_X$ defined by (8.15), where $F_{i,j}$ is substituted in terms of $E_{i,j}$. The action of $g_n$ on the variables $B_X$ in the case of the series $B$, $D$ for $X$ and $\widehat{-X}\neq \{\pm n, \dots,\pm 1\}$ is defined in the same way as on $A_X$. In the case of $X$ or $\widehat{-X}=\{\pm n,\dots,\pm 1\}$, the action is defined so that $B_X$ for these $X$’s would form a spinor representation. To obtain such formulas, we should take the formulas of the action $g_n$ on the roots of the determinants $\sqrt{a_X}$ (or $\sqrt{\overline{a}_X}$ if we would like to obtain a representation with $m_{-1}<0$ for the series $D$), and then replace $\sqrt{a_X} $ (or $\sqrt{\overline{a}_X}$ ) by $B_X$. Thus, in the case of the series $D$, there are two ways to construct an action. An explicit form of the resulting formulas was discussed when proving Proposition 4 (see (3.25) and (3.27)). Thus, the polynomial spaces $\mathbb{C}[A]$ and $\mathbb{C}[B]$ become representation spaces of the algebra $g_n$. 8.2. The planOur next aim is to obtain the following results. 1. Prove that Gelfand–Tsetlin diagrams in the sense of Definition 10 encode the base vectors of irreducible finite-dimensional representations. 2. Verify that the solution space of the A-GKZ system is a representation model. 3. Construct bases of the Zhelobenko model consisting of $\Gamma$-series or monomials. To achieve these goals, we first prove that the solution space of the A-GKZ system coincides with that of the ideal of relations between determinants (see the construction of this solution space in the proof of Proposition 7). As a result, the solution space A-GKZ contains a representation model. We next prove that the space of polynomial solutions of the A-GKZ model coincides with this model (it is reasonable to call this model an A-GKZ model). In addition, the base solutions $F_{\gamma}$ form a base of an A-GKZ model. As a corollary, we find that the number of Gelfand–Tsetlin diagrams in the sense of Definition 10 for a fixed highest weight is equal to the dimension of a representation with this highest weight. Once this is done, we pass to the Zhelobenko model. We prove that the span of $\Gamma$-series of the determinants constructed from Gelfand–Tsetlin diagrams in the sense of Definition 10 coincides with the Zhelobenko model. Hence, by dimensional arguments, it follows that the $\Gamma$-series of the determinants constructed from the Gelfand–Tsetlin diagrams (in the sense of Definition 10) form a base for the A GKZ model. The first and second goals posed here will be achieved in § 8.4, and the third goal will be dealt with in § 8.5.6. 8.3. The A-GKZ model8.3.1. Invariance of the solution space of the A-GKZ systemThis section discusses the A-GKZ systems defined in §§ 6.3 and 6.4. The variables $A_X$ being independent, we can introduce an invariant scalar product between polynomials $f(A)$, $g(A)$ by the formula
$$
\begin{equation}
\langle f(A),g(A)\rangle := f\biggl(\frac{\partial}{\partial A}\biggr) g(A)|_{A=0},
\end{equation}
\tag{8.1}
$$
where $f(\partial/\partial A) $ is the result of substitution into $f$ of the differential operators $\partial/\partial A_X$ in place of the variables $A_X$. An analogous scalar product is introduced also for the variables $B$. Now let us prove the following result. Proof. Let us consider separately the A-GKZ systems from §§ 6.3 and 6.4. In both cases, let $\overline{I}$ be the ideal contained in $\mathbb{C}[\partial/\partial B]$ or $\mathbb{C}[\partial/\partial A]$, respectively, and which corresponds to the A-GKZ system under consideration.
The action of $g_n$ on the variables $A_X$ induces an action on $\mathbb{C}[\partial/\partial A_X]$. The action of $g_n$ on $\mathbb{C}[\partial/\partial B_X]$ is defined similarly. Let us show that, in both cases, the ideal $\overline{I}$ is invariant. From this fact it immediately follows that the space $\mathrm{Sol}_{\overline{I}}$ of polynomial solutions of the ideal under consideration is invariant. An important observation is worth making before proceeding with consideration of the A-GKZ systems from §§ 6.3 and 6.4. Suppose that a system of partial differential equations corresponds to the ideal $\overline{J}\subset \mathbb{C}[\partial/\partial A]$, which, in turn, is obtained by the substitution $A_X\mapsto \partial/\partial A_X$ from the ideal $J\subset \mathbb{C}[A]$. In terms of the scalar product (8.1), the fact that $f(A)$ is a solution of a system of partial differential equations is equivalent to saying that $f(A)$ is orthogonal to the ideal $\overline{J}\subset \mathbb{C}[A] $ corresponding to the system. Thus, for the ideal $J\subset \mathbb{C}[A]$, we have $\mathrm{Sol}_{\overline{J}}=(\overline{J})^{\perp}$. Note that $(J^{\perp})^{\perp}$ is the closure of $J$ in the topology induced by the scalar product (8.1). The monomials being orthogonal, we have $(J^{\perp})^{\perp}=J$. So, the coincidence of the spaces of polynomial solutions for the system A-GKZ for the series $A$ and the ideal $\overline{I}_{\mathfrak{gl}_m}$ implies the coincidence of the ideals
$$
\begin{equation}
\overline{I}_{\mathfrak{gl}_m}=\overline{I}^A \subset \mathbb{C}[A],
\end{equation}
\tag{8.2}
$$
where $\overline{I}^A$ is the ideal that defines the A-GKZ system for the series $A$. This implies, in particular, that the ideal $\overline{I}^A$ is invariant.
Let us now prove Proposition 7 for the A-GKZ system from § 6.4. Let $I_{g_n}\subset \mathbb{C}[A]$ be an ideal of relations between determinants $a_X$. Lemma 3 asserts that the ideal $I_{g_n}$ is generated by the ideal $I_{\mathfrak{gl}_m}$ ($m=2n$, for the series $C$, $D$, and $m=2n+1$, for the series $B$) and the Jacobi relations. Hence $\overline{I}_{g_n}$ is generated by the ideal $\overline{I}_{\mathfrak{gl}_m}$ and the differential operators that correspond to the Jacobi relations. By (8.2), $\overline{I}_{g_n}$ is generated by $\overline{I}^A$ and the operators that appear from the Jacobi relations. But the ideal generated by $\overline{I}^A$ and the Jacobi relations is just the ideal $\overline{I}$ for the system from § 6.4. So, for the system from § 6.4, we have As a consequence of (8.3), the ideal $\overline{I}$ is invariant.Now consider the A-GKZ system for $g_n$ from § 6.3. Consider the substitution of the variables $B_X$ for the variables $A_X$ by the rule:
$$
\begin{equation}
A_X\mapsto \begin{cases} B_X, &X\neq \bigl\{ \{\pm n,\dots,\pm 1\}, \{\pm n,\dots,\pm 1,0\} \bigr\}, \\ B^2_X &\text{ othewise}. \end{cases}
\end{equation}
\tag{8.4}
$$
It can be verified directly (using transformations leading (3.24) to (3.25) and (3.27)) that this substitution is consistent with the action modulo the ideal $I'_{g_n}\subset \mathbb{C}[B]$, which is generated by the Plücker relations (4.9), the Jacobi relations for the variables $B$, and the relations
$$
\begin{equation}
\begin{gathered} \, B_{\pm n,\dots,\widehat{i},\dots,\pm 1,i} B_{\pm n,\dots,\widehat{i},\dots,\pm 1,-i}=\frac{\sqrt{-1}}{\sqrt{2}} B^2_{\pm n,\dots,\widehat{i},\dots,\pm 1,i,0} \text{ only for the series }B, \\ B^2_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,-i,i}= B_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,-i,-j} B_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,i,j}, \end{gathered}
\end{equation}
\tag{8.5}
$$
whose square is a consequence of the Plücker and Jacobi relations (see (3.14) and (3.16)).
Now let us check that the ideal $I'_{g_n}$ is invariant. Indeed, it can be shown by a direct calculation that (8.5) is invariant. Now the invariance of $I'_{g_n} $ is a consequence of this fact, of the invariance of the ideal generated by the Plücker relations in the variables $A$, and of the remark about the above change $A\mapsto B$. Now let us consider the ideal $\overline{I}^{\,\prime}_{g_n}$. By (8.2), for the A-GKZ sysytem for $g_n$ from § 6.3, we have This proves Proposition 7Now the invariance of $\overline{I}$ follows from (8.6). Proof. By Proposition 7, the solution space of the A-GKZ system for $g_n$ is a representation of $g_n$. In the space $\mathrm{Sol}_{\mathrm{AGKZ}}$ of solutions of A-GKZ for $g_n$, consider the finite-dimensional subspace of functions $\mathrm{Sol}_{\mathrm{AGKZ}}^{l_{-n},\dots,l_{-1}}$ of fixed homogeneous degree $l_{p}$ with respect to $A_X$ (or $B_X$) with $|X|=(p+n+1)$ and $(-p-1)$, $p=-n,\dots,-1$. We have
Under the action of $g_n$, the homogeneous powers of determinants of the same size are preserved. Note also that base (6.10) is consistent with this decomposition of the solution space A-GKZ for $g_n$. We set $m_{-p}=(l_{-p-1}+l_{p+n+1})+(l_{-p-2}+l_{p+n+2})+\cdots$. The subspace $\mathrm{Sol}_{\mathrm{AGKZ}}^{l_{-n},\dots,l_{-1}}$ contains the following $g_n$-highest vector of weight $[m_{-n},m_{-n+1},\dots,m_{-1}]$:
$$
\begin{equation}
(A_{-n}+\pm A_{\widehat{n}})^{m_{-n}-m{-n+1}}( A_{-n,-n+1}+ \pm A_{\widehat{n-1,n}})^{m_{-n+1}-m_{-n+2}}\cdots,
\end{equation}
\tag{8.7}
$$
For the case of series $B$, $D$, we get a similar expression with the variables $B$. In addition, in the case of the series $D$, the last $n$th factor is taken to the power $|m_{-1}|$. The signs in brackets are defined in (3.13).
Thus, $\mathrm{Sol}_{\mathrm{AGKZ}}^{l_{-n},\dots,l_{-1}}$ contains an irreducible representation of the highest weight $[m_{-n},m_{-n+1},\dots,m_{-1}]$. This proves Lemma 10. 8.3.2. The minimality of the solution space of the A-GKZ systemWe need the followings result. Lemma 11. The solution space of the A-GKZ system and the model of representations constructed in the proof of Lemma 10 coincide. Let $\mathrm{Mod}$ be the direct sum of subrepresentation in the space of polynomials with respect to the variables $A_X$ (or $B_X$) generated by the highest vectors (8.7). Proof of Lemma 11. Assume on the contrary that $\mathrm{Mod}\neq \mathrm{Sol}_{\mathrm{AGKZ}}$. If the variables $A_X$ of the determinants $a_X$ (or $B_X$, for definiteness, we consider the case of the function $A_X$; the case where $f$ depends on $B_X$ is considered similarly) is substituted for the independent variables, the representation $\mathrm{Sol}_{\mathrm{AGKZ}}$ is mapped to the Zhelobenko model, and the subrepresentation $\mathrm{Mod}$, is isomorphically isomorphically to the Zhelobenko model. As a result (considering, for example, the finite-dimensional subspaces with fixed homogeneous degree with respect to $A_X$, $|X|=l_{p+n+1}$ or $l_{-p-1}$), we find that the kernel of this mapping is non-trivial. Hence there exists a non-zero function $f\in \mathrm{Sol}_{\mathrm{AGKZ}}$ such that that is, $f(A_X)\in I_{g_n}$. However (see (8.3)), $f(\partial/\partial A)g(A)=0$ for all $g\in \mathrm{Sol}_{\mathrm{AGKZ}}$. In particular, $f(\partial/\partial A)f(A)=0$, but this is possible only if $f=0$. This contradiction proves Lemma 11. Since $\mathrm{Sol}_{\mathrm{AGKZ}}^{l_{-n},\dots,l_{-1}}$ contains a base (6.10) indexed by Gelfand–Tsetlin diagrams for $g_n$ in the sense of Definition 10, we arrive at the following result. Corollary 3. Let a highest weight $[m_{-n},\dots,m_{-1}]$ be fixed. Then the number of Gelfand–Tsetlin diagrams in the sense of Definition 10 with this highest weight is equal to the dimension of this irreducible representation. 8.4. The A-GKZ modelSo, the solution space of the A-GKZ system is a model of representations of $g_n$, and the highest vectors are given by (8.7). Let us formulate this result as the following theorem. Theorem 8. There exists an A-GKZ model formed by the solution space of an A-GKZ system. In the case of series $B$, $D$, this system has a base consisting of the functions $F_{\gamma}(B)$, and, in the case of series $C$, a base of the functions $F_{\gamma}(A)$. Here, $\gamma$ runs over all possible Gelfand–Tsetlin diagrams in the sense of Definition 10. The functions for which the diagram has the highest weight $[m_{-n},\dots,m_{-1}]$ form a basis base in the representation space of highest weight $[m_{-n},\dots,m_{-1}]$. Thus, the first and the second goals from § 8.2 are achieved. 8.5. $\Gamma$-eries in the Zhelobenko realizationFor brevity, we will use the notation
$$
\begin{equation*}
\mathcal{F}_{\gamma}:=\mathcal{F}_{\gamma}(A) \text{ or }\mathcal{F}_{\gamma}(B),\qquad F_{\gamma}:=F_{\gamma}(A) \text{ or } F_{\gamma}(B),
\end{equation*}
\notag
$$
for the solution of GKZ and A-GKZ systems. 8.5.1. The result: $\Gamma$-series generate the Zhelobenko realizationConsider $\Gamma$-series that are solutions of the GKZ system for the corresponding algebra $g_n$. Let us substitute in these series the determinants $a_X$ in place of $B_X$ and $A_X$ according to the above substitution rule Let $W\subset \mathrm{Fun}$ be their linear span. This theorem immediately implies the coincidence of $W$ and the Zhelobenko model. Proof of Theorem 9. Consider the functions $\mathcal{F}_{\gamma}$ of fixed homogeneous degree $l_i$ with respect to the determinants of size $p+n+1$ and $-p-1$, $p=-n,\dots,-1$. Let us check that the action of the generator $F_{i,j}$ on a function $\mathcal{F}_{\gamma}$ produces a linear combination of the functions $\mathcal{F}_{\delta}$ of the same homogeneous degree. To this end, we need the following principal lemma.
8.5.2. The principal lemma Lemma 12. Let $\mathfrak{f}_{\gamma}(c)$ be a $\Gamma$ series in the variables $c_X=a_X$ or $\sqrt{a_X}$. The lattice from which this series is constructed has a base $v_{\alpha}$, $\alpha=1,\dots,K$, such that, for $\alpha=1,\dots,\mathcal{K}$, $\mathcal{K}<K$, these generators have the form Then, modulo (8.8), 8.5.3. The proof of the principle LemmaConsider the independent variables $C_X$ which are antisymmetric with respect to $X$. For a polynomial $g(C)$, let $g(\partial/\partial C)$ denote the differential operator obtained by replacing each variable $C_{X}$ by the differentiation $\partial/\partial C_X$. We let $\mathrm{Plk}$ denote the ideal generated by the relations
$$
\begin{equation*}
C^{\tau_1}_{X_1}C^{\tau_2}_{X_2}-C^{\tau_3}_{X_3}C^{\tau_4}_{X_4} + C^{\tau_5}_{X_5}C^{\tau_6}_{X_6}=0.
\end{equation*}
\notag
$$
With this relation we associate the A-GKZ system generated by the equations (see Definition 1)
$$
\begin{equation*}
\begin{aligned} \, &\biggl(\biggl(\frac{\partial}{\partial C_{X_1}}\biggr)^{\tau_1} \biggl(\frac{\partial}{\partial C_{X_2}}\biggr)^{\tau_2}- \biggl(\frac{\partial}{\partial C_{X_3}}\biggr)^{\tau_3} \biggl(\frac{\partial}{\partial C_{X_4}}\biggr)^{\tau_4} \\ &\qquad+\biggl(\frac{\partial}{\partial C_{X_5}}\biggr)^{\tau_5} \biggl(\frac{\partial}{\partial C_{X_6}}\biggr)^{\tau_6}\biggr)f=0, \end{aligned}
\end{equation*}
\notag
$$
and also the equation of the GKZ system corresponding to the remaining generators $v_{\alpha}$, $\alpha=k+1,\dots,K$. There is a base in the space of solution of this system
$$
\begin{equation*}
f_{\delta}(C)=\sum_{s\in \mathbb{Z}_{\geqslant 0}^{\mathcal{K}}} (-1)^s \mathfrak{f}_{\delta-sr}^s(C),
\end{equation*}
\notag
$$
where $\mathfrak{f}_{\delta-sr}^s(C)$ is defined by analogy with (6.5). We need the following result (similar to [14]). Lemma 13. If for all $C$, then $\lambda_1 g_1(c)+\dots+\lambda_l g_l(c)=0\ \operatorname{mod} \mathrm{Plk}$. We set $\delta+v:=\delta+v_1+\dots+v_{\mathcal{K}}$. The proof of the following result repeats verbatim that of [14]. Lemma 14. If $\mathfrak{f}_{\gamma+v+sr}^s(1)$ is the result of substitution of units instead of all $C$, then Let us show that a similar equality also holds for the functions $\mathcal{F}_{\gamma}(c)$ defined in (6.9). Indeed, this function is a sum of $\Gamma$-series of the form $\mathfrak{f}_{\delta}$, for each of which an equality of the form (8.10) holds. Summing these equalities, we get
$$
\begin{equation}
\mathcal{F}_{\gamma}\biggl(\frac{\partial }{\partial C}\biggr)f_{\delta}(C) = \sum_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}} \mathcal{F}_{\gamma+v+sr}^s(1) f_{\delta-\gamma-sr}(C).
\end{equation}
\tag{8.11}
$$
Proceeding as in [14], we obtain
$$
\begin{equation}
c_X\mathcal{F}_{\gamma}(c)=\sum_{s\in\mathbb{Z}^k_{\geqslant 0}} \mathrm{const}^{\gamma}_s\cdot \mathcal{F}_{\gamma+e_X+sr}(c).
\end{equation}
\tag{8.12}
$$
This proves the principal Lemma 10. 8.5.4. Coefficient in the principal lemmaLet us give a formula for the coefficients in (8.9). As in [14], we have
$$
\begin{equation}
\begin{aligned} \, \mathrm{const}^{\gamma}_s &=\frac{\mathcal{F}_{\gamma+v}^{s}(1)} {\mathcal{F}^s_{\gamma+v+e_X+sr}(1)} \nonumber \\ &\qquad-\sum_{p\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0}\colon s- p\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0},\, p\neq s} \frac{\mathcal{F}_{\gamma+v}^{p}(1)\mathcal{F}_{\gamma+v+pr+e_X+ (s-p)r}^{s-p}(1)} {\mathcal{F}_{\gamma+v+pr+e_X+(s-p)r}(1) \mathcal{F}_{\gamma+v+pr+e_X}(1)} \nonumber \\ &= \frac{\mathcal{F}_{\gamma+v}^{s}(1)}{\mathcal{F}^s_{\gamma+v+e_X+sr}(1) }- \sum_{p\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0}\colon s-p\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0}, \, p\neq s} \frac{\mathcal{F}_{\gamma+v}^{p}(1) \mathcal{F}_{\gamma+v+e_X+sr}^{s-p}(1)} {\mathcal{F}_{\gamma+v+e_X+sr} (1)\mathcal{F}_{\gamma+v+pr+e_X}(1)}. \end{aligned}
\end{equation}
\tag{8.13}
$$
8.5.5. The action of generators. The completion of the proof of Theorem 9Recall that $c_X=a_X$ or $\sqrt{a_X}$. The action of $F_{i,j}$ on $a_X$ is given by (3.5), the action on $\sqrt{a_X}$ is described in the proof of Proposition 4 (see (3.25)). Above, we have constructed the functions $\mathcal{F}$ depending on $A_X$ or on $B_X$. In §§ 6.3 and 6.4, in the definition of $A_X$, $B_X$, we have also pointed out which substitutions of the determinants $a_X$ in these variables should be taken when changing to the Zhelobenko realization. Let us perform this substitution. In both cases, the resulting function will be denoted by $\mathcal{F}_{\gamma}(a)$. In both cases, we have
$$
\begin{equation}
F_{i,j}=\sum_{Y_1,Y_2} c_{Y_1} \, \frac{\partial}{\partial c_{Y_2}}.
\end{equation}
\tag{8.14}
$$
The following principle Lemma 12 is obtained by the rule of differentiation of $\Gamma$-series. Let us return to the proof of Theorem 9. Theorem 3 gives necessary and sufficient conditions for a function to be included in the Zhelobenko model. An application of this lemma shows that $W\subset \mathrm{Zh}$ is a subrepresentation containing all finite-dimensional irreducible representations. As a result, $W=\mathrm{Zh}$ (cf. Theorem 3). Theorem 9 is proved. 8.5.6. Bases in the Zhelobenko realizationTheorem 10. The functions $\mathcal{F}_{\gamma}(a)$ constructed for various Gelfand–Tsetlin diagrams in the sense of the Definition 10 form a base of the Zhelobenko realization. Proof. Since all these functions satisfy the conditions of Theorem 9, they all belong to the Zhelobenko model and it remains to prove the linear independence of these functions.
The linear span of the function $\mathcal{F}_{\gamma}(a)$ for Gelfand–Tsetlin diagrams in the sense of Definition 10 of highest weight $[m_{-n},\dots,m_{-1}]$, form a subrepresentation in the Zhelobenko model of highest weight $[m_{-n},\dots,m_{-1}]$. Their number is equal to the dimension of this representation. So, they are independent, and therefore, form a base. This proves Theorem 10. Note that we have not only proved that the linear span of $\mathcal{F}_{\gamma}(a)$ is a model of representations of $g_n$, but we also explicitly wrote down formulas for the action of algebra generators. Let us prove another result. In the definition of the Gelfand–Tsetlin lattice $\mathcal{B}_{\mathrm{GC}}^{g_n}$, its embedding into the lattice $\mathbb{Z}^N$ of exponents (integers or half-integers) of monomials in determinants was given (see § 6). The image of a vector $\gamma\in \mathcal{B}_{\mathrm{GC}}^{g_n} $ with this embedding is denoted by $\overline{\gamma}$. A similar construction can be implemented with vectors of shifted lattices. So, for a Gelfand–Tsetlin diagram $\Pi$, we choose a representative of $\gamma$, and then construct a monomial in determinants $a^{\overline{\gamma}}$. Theorem 11. For every Gelfand–Tsetlin diagram in the sense of Definition 10, we construct a monomial $a^{\overline{\gamma}}$ according to the above procedure. These monomials form a base in the Zhelobenko realization. Proof. Indeed, by Lemma 14,
$$
\begin{equation*}
\mathcal{F}_{\gamma}(a)=a^{\overline{\gamma}}+\mathrm{h.o.t.},
\end{equation*}
\notag
$$
where $\mathrm{h.o.t.}$ is the sum of monomials $a^{\overline{\delta}}$, and $\gamma\prec \delta$, where the ordering $\prec$ defined in the same way as in (5.3), with replacement of the lattice by $\mathcal{B}_{\mathrm{GC}}^{g_n}$.
In particular, if $\gamma=\gamma'\ \operatorname{mod} \mathcal{B}_{\mathrm{GC}}^{g_n}$, then
$$
\begin{equation*}
a^{\overline{\gamma}}=a^{\overline{\gamma'}}+\mathrm{h.o.t.}
\end{equation*}
\notag
$$
Consider a set of vectors $\gamma\in\mathbb{Z}_{\geqslant 0}^N$ which are representatives of Gelfand–Tsetlin diagrams with highest weight $[m_{-n},\dots,m_{-1}]$. The functions $ \mathcal{F}_{\gamma}(a)$ form a base of this representation in the Zhelobenko model, and the set of functions $a^{\overline{\gamma}}$ is related to $ \mathcal{F}_{\gamma}(a)$ by an upper-triangular transformation. Hence, the set $a^{\overline{\gamma}}$ also forms a base. This proves Theorem 11. Thus, the third goal from § 8.2 isa chieved. § 9. A relation to the Gelfand–Tsetlin baseLet us establish a relation between the bases $F_{\gamma}(A)$ (or $F_{\gamma}(B)$) in the A-GKZ realization and the base $\mathcal{F}_{\gamma}(a)$ in the Zhelobenko realization with the Gelfand–Tsetlin base $G_{\gamma}$. A Gelfand–Tsetlin type base can be defined as an eigenbasiss for the following maximal commutative subalgebra in $GT\subset U(g_n)$ (called the Gelfand–Tsetlin algebra). For the subalgebra chain $g_1\subset g_2\subset \dots\subset g_n$, we take the centres of universal wrappers $Z(U(g_1)),\dots,Z(U(g_n))$ and generate a subalgebra with them in $GT\subset U(g_n)$. This is the Gelfand–Tsetlin algebra. The generators can be explicitly write down:
$$
\begin{equation}
C_{p}^q=\sum_{i_1,\dots,i_p\text{ modulo }\geqslant q} F_{i_1,i_2}F_{i_2,i_3}\cdots F_{i_{2p},i_1}.
\end{equation}
\tag{9.1}
$$
It is directly verified that these generators are self-adjoint with respect to the scalar product (8.1). It is known that the eigenbase for the algebra $GT$ (that is, the Gelfand–Tsetlin base) is not unique in the case of the series $B$, $C$, $D$. Let us give a construction of some Gelfand–Tsetlin base. To this end, we first prove the following result. Lemma 16. The scalar product $\langle \mathcal{F}_{\gamma_1}(a),\mathcal{F}_{\gamma_2}(a) \rangle$ can be non-zero only if there exists $\omega$ such that $\gamma_1\preceq \omega$, $\gamma_2\preceq \omega$. The order $\prec$ is defined as in (5.3) with the lattice replaced by $\mathcal{B}_{\mathrm{GC}}^{g_n}$. Proof. For definiteness, we conduct arguments using the variables $A_X$. These arguments hold verbatim for the variables $B_X$,.
To calculate the scalar product, let us go to the A-GKZ realization. Then the vector $\mathcal{F}_{\gamma_1}(a)$ in the Zhelobenko realization is written in A-GKZ realization as a function of the form $\mathcal{FF}_{\gamma_1}(A) :=\mathcal{F}_{\gamma_1}(A)+h(A)$, where $h(A)\in I_{g_n}$. Using (8.1) and taking into account that $F_{\delta}(A)$ is annihilated by the ideal $\overline{I}_{g_n}$, we obtain
$$
\begin{equation*}
\langle \mathcal{F}_{\gamma_1}(A)+h(A),F_{\delta}(A) \rangle = \langle \mathcal{F}_{\gamma_1}(A),F_{\delta}(A)\rangle.
\end{equation*}
\notag
$$
By the definition, $\langle\mathcal{F}_{\gamma_1}(A),F_{\delta}(A)\rangle$ can be non-zero only if $\gamma_1\preceq \delta$. Hence $\mathcal{FF}_{\gamma_1}=\sum_{s\in \mathbb{Z}^{\mathcal{K}}_{\geqslant 0}} c_{\gamma_1}^s F_{\gamma_1+sr}(A)$. Therefore,
$$
\begin{equation*}
\langle \mathcal{F}_{\gamma_1}(a),\mathcal{F}_{\gamma_2}(a) \rangle = \sum_{s_1,s_1\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}} c_{\gamma_1}^{s_1} c_{\gamma_2}^{s_2} \langle F_{\gamma_1+s_1r}(A), F_{\gamma_2+s_2r}(A)\rangle.
\end{equation*}
\notag
$$
Considering the supports we conclude that the scalar product of $F_{\gamma_1+s_1r}(A)$ and $F_{\gamma_2+s_2r}(A) $ can be non-zero only if there exists $\omega$ such that $\gamma_1+s_1r$ and $ \gamma_2+s_2r\preceq \omega$. This condition is equivalent to the condition from the lemma. This proves Lemma 16. Corollary 5. There exists an orthogonal base $\mathcal{G}_{\gamma}$ in the Zhelobenko model expressed in terms of the base $\mathcal{F}_{\gamma}$ in an upper-triangular way with respect to the ordering $\prec$, that is, Proof. The Gelfand–Tsetlin base is an eigenbase for the Gelfand–Tsetlin algebra. Its generators (9.1) are self-adjoint with respect to the scalar product (8.1), so that the space of a finite-dimensional irreducible representation $V\subset \mathrm{Zh}$ is represented as an orthogonal direct sum of eigenspaces for $GT$. The set of eigenvalues defining one of the direct summands is given by the set of $g_{n-k}$-higher weights arising from the decomposition of $V$ into the sum of irreducible representation under the standard procedure of restriction of algebras $g_n\downarrow g_{n-k}$. We let $\mu=\{\mu_n,\dots,\mu_1\}$ denote the set of $(g_n,\dots,g_1)$-higher weights. The corresponding term in $V$ is denoted by $V_{\mu}$.
The Gelfand–Tsetlin diagram $\gamma$ is denoted by (see (7.1))
$$
\begin{equation*}
V^{\gamma}:=V_{\mu},\quad\text{where }\ \mu=\{wt_n(\mu),\dots,wt_1(\mu)\},
\end{equation*}
\notag
$$
where $wt_n(\delta)$ are defined in (7.1).
Let us prove the following result. Proposition 8. $\mathcal{F}_{\gamma}(a)\in \bigoplus_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}} V^{\gamma+sr}$. Proof. Consider first the monomial $a^{\delta}$. Applying the Jacobi relation if necessary, we can assume that $a^{\delta}$ depends only on the determinants $a_X$ with $|X|\leqslant n$.
Let us introduce the raising operation. Let us first consider a realization in the space of functions on the subgroup $Z$ of upper-unitriangular matrices on the corresponding group $G$ (see [9]). We let $z_{i,j}$ denote the functions of matrix elements on $Z$. In this case, a finite-dimensional irreducible representation of $V$ is realized in the space of polynomials in variables $z_{i,j}$ (including in the case of a half-integer of the highest weight) satisfying the indicator system (3.18), the exponents in which are determined by the highest weight according to the rule (3.19). The restriction to $Z$ of the monomial $a^{\delta}$ is written as a function $f(z_{i,j})$. In this case, one can assume that $i<-j$ in the case of series $B$, $D$ and $i\leqslant -j$ in the case of series $C$. The remaining variables $z_{i,j}$ are expressed in terms of these variables as polynomials. A function that is, a $g_{n-k}$-the highest vector depends only on the variables $z_{i,j}$, $j\in \{-k,\dots,\widehat{0},\dots,k\}$. Let us define the raising procedure as follows. The operators $F_{i,j}$, $i<j$, are applied until a $g_{n-k}$-highest vector is obtained. Let us agree that this is done in the following order. First, we apply $F_{-n,-(n-1)}$ to the maximum possible power (until the result is a non-zero), then apply $F_{-n,-(n-2)}$ to the maximum possible power, then apply $F_{-(n-1),-(n-2)}$, etc. That is, the operators are applied according to the order
$$
\begin{equation}
F_{-n,-(n-1)},\quad F_{-n,-(n-2)},\quad F_{-(n-1),-(n-2)}, \quad F_{-(n,-(n-3))},\quad \dots\,.
\end{equation}
\tag{9.2}
$$
In general, the positive root operators $F_{i,j}$, $i<j$, in the realization under consideration are written as
$$
\begin{equation*}
F_{i,j}=\biggl(\frac{\partial}{\partial z_{i,j}} +\sum_{t<i}z_{t,i}\, \frac{\partial}{\partial z_{t,j}}\biggr)- \pm\biggl(\frac{\partial}{\partial z_{-j,-i}}+ \sum_{t<-j}z_{t,-i}\, \frac{\partial}{\partial z_{t,j}}\biggr),
\end{equation*}
\notag
$$
where “$\pm$” “$=$” “$+$” for series $B$, $D$ and $\operatorname{sign}(i)\operatorname{sign}(j)$ for series $C$. But since we apply the operators $F_{i,j}$ in a certain order, they are written simply as
$$
\begin{equation*}
F_{i,j}=\frac{\partial}{\partial z_{i,j}}- \pm\frac{\partial}{\partial z_{-j,-i}}.
\end{equation*}
\notag
$$
Considering the arguments $z_{i,j}$ of our function, we get $F_{i,j}=\partial/\partial z_{i,j}$. The action of the raising procedure on a monomial $a^{\delta}$ can be described without going to the realization in the space of functions on $Z$. This operation acts as a substitution for $a_X\mapsto a_{X'}$, where $X'$ is obtained by the maximum possible left shift of all indexes of absolute value $\geqslant k$. Each function $f(z_{i,j})$ can be written as
$$
\begin{equation}
f=\sum_{\beta}c_{\beta}\cdot \prod_{|s|>k} \frac{z_{r,s}^{\beta_{r,s}}}{\beta_{r,s}!}\cdot f_{\beta}(z_{i,j}),
\end{equation}
\tag{9.3}
$$
where $\beta$ is some index listing the terms in $f$ of the specified type. In this case, the function $f_{\beta}(z_{i,j})$ depends only on the variables $z_{i,j}$, $j\in \{-k,\dots,\widehat{0},\dots,k\}$. When applying the raising procedure in (9.3), we get $\sum_{\beta}c_{\beta}f_{\beta}(z_{i,j})$, where the sum is taken over all $\beta$ such that the vector
$$
\begin{equation*}
\! [\beta]:=(\beta_{-n,-(n-1)}+ \beta_{(n-1),n},\beta_{-n,-(n-2)}+\beta_{(n-2),n}, \beta_{-(n-1),-(n-2)}+\beta_{(n-2),(n-1)},\dots)
\end{equation*}
\notag
$$
is maximal relative to the lexicographic order. Such terms we call the maximal. According to our assumption (made at the beginning of the proof) on $z_{i,j}$ that participate in $f$, we have $\beta_{(n-1),n}=\beta_{(n-2),n}=\dots=0$. So, the maximum term is unique, and corresponds to the maximum in the lexicographic sense of the vector of exponents $\beta$.
The fact that $f\in V_{\mu_0}\oplus V_{\mu}\oplus\dotsb$ means the following. When using the positive root operators $F_{p,q}$ in some other order, at one of the steps we get a function which is the sum of a highest vector from $V_{\mu}$ and some other summand. This situation occurs exactly when there are terms in (9.3) that are not maximal. The weight $\mathrm{weight}(f_{\beta})$ of the resulting $g_{n-k}$-highest vector is calculated as follows. Let $\mathrm{h.weight}$ denote the $g_n$-highest weight of the representation in question. Then
$$
\begin{equation}
\mathrm{weight}(f_{\beta})=\mathrm{h.weight}- \sum_{r<s< -k} \beta_{r,s}(e_{r}-e_{s}),
\end{equation}
\tag{9.4}
$$
where $e_{r}$, $e_{s}$ are unit vectors for the corresponding weight components. At the same time, the first $(n-k)$ coordinates are taken in the resulting vector $\mathrm{weight}(f_{\beta})$.
When changing from a maximum term to a non-maximal one, $\mathrm{weight}(f_{\beta})$ is augmented with a vector of the form $[0,\dots,1,\dots,-1,\dots,0]$. The same change in $\mathrm{weight}(f_{\beta})$ occurs when calculating the weight of $\mathrm{weight}(f_{\beta})$, which corresponds to the maximum term, but for the vector of exponents $\delta+r_{\beta}$, where $r_{\alpha}=e_{-n,\dots,-r-1,-k}-e_{-n,\dots,-r-1,-r} -e_{-n,\dots,-r-1,-s,-k} +e_{-n,\dots,-r-1,-r,-s}$. It follows that $a^{\delta}\in \bigoplus_{s\in\mathbb{Z}^{{\mathcal{K}}}_{\geqslant 0}}V^{\delta+sr}$. Let us return to the proof of Theorem 12. Using Proposition 8, we find that $\mathcal{G}_{\gamma}(a)$ also belongs to $\bigoplus_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}}V^{\gamma+sr}$. Since $\mathcal{G}_{\gamma}(a)$ are obtained by the orthogonalization procedure, $\mathcal{G}_{\gamma}(a)$ is orthogonal to all vectors of the form $\mathcal{F}_{\gamma+sr}(a)$, $s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0}$, $s\neq 0$. By Lemma 16, $\mathcal{G}_{\gamma}(a)$ is also orthogonal to all other $\mathcal{F}_{\delta}(a)$, $\delta\neq \gamma+sr$ $\operatorname{mod}\mathcal{B}_{\mathrm{GC}}^{g_n}$. So, $\mathcal{G}_{\gamma}(a)$ is orthogonal to $\bigoplus_{s\in\mathbb{Z}^{\mathcal{K}}_{\geqslant 0},\, s\neq 0}V^{\gamma+sr}$. Hence $\mathcal{G}_{\gamma}(a)\in V^{\gamma}$. As a result, the base $\mathcal{G}_{\gamma}(a)$ is consistent with the orthogonal decomposition of $V$ into the direct sum of the eigenspaces of $\mathrm{GT}$. So, $\mathcal{G}_{\gamma}(a)$ is the Gelfand–Tsetlin base. This proves Proposition 8. The next result is proved similarly to that in [14] for the series $A$. Theorem 13. The lower-triangular orthogonalization of the base $F_{\gamma}$ with respect to the order (5.3) is a Gelfand–Tsetlin type base $\mathcal{G}_{\gamma}$. To prove this result, we consider the function $G_{\gamma}$ in the variables $A_X$ or $B_X$, which represents the vector $\mathcal{G}_{\gamma}$ in the A-GKZ realization. Next, using the same arguments as for the series $A$, it can be shown that there exists an expression of the form
$$
\begin{equation*}
G_{\gamma}=\sum_{s\in\mathbb{Z}_{\geqslant 0}^{\mathcal{K}} } d_{\gamma}^s\cdot F_{\gamma-sr}.
\end{equation*}
\notag
$$
This expression is the Gram–Schmidt orthogonalization procedure. The Gram matrix for the $F_{\gamma}$ base can be explicitly written down using the same arguments as in [14] for the series $A$. From this matrix, we can explicitly write down the transition matrix from $F_{\gamma}$ to $G_{\gamma}$. 
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