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This article is cited in 1 scientific paper (total in 1 paper) 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 gln (see [1]). Formally, the Weyl construction is an explicit embedding of a certain irreducible representation into a tensor power of a standard representation of gln. 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 sl3 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 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 gln, one considers functions of independent variables AX, X⊂{1,…,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 ΛkCN, 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 gl3 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 gln. 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. Γ-seriesA detailed information about a Γ-series can be found in [16]. Let B⊂ZN be a lattice, let γ∈ZN be a fixed vector. The hypergeometric Γ-series in the variables z1,…,zN is defined by where z=(z1,…,zN), and the following multi-index notation is usedNote that if at least one of the components of the vector b+γ is negative integer, then the corresponding summand in (2.1) vanishes. Due to this fact, the Γ-series considered in the present paper contain finitely many non-zero terms. For simplicity, we will write factorials instead of Γ-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=(a1,…,aN) be a vector orthogonal to the lattice B, then here, it suffices to consider only the base vectors of the lattice orthogonal to B.2. Let b∈B and b=b+−b−, where all coordinates of the vectors b+, b− are non-negative. Choosing the non-zero elements b+=(…,bi1,…,bik,…) and b−=(…,bj1,…,bjl,…) in these vectors, we have It suffices to consider only a finite collection of vectors1 b∈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∈B↦Ob. We have the following. 1. kb↦(Ob)k, k∈Z⩾0, −b↦−Ob. 2. Let b=b+−b−, c=c+−c−, where c± 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 3. Let b=(b++u)−(b−+v), c=(c++v)−(c−+u), where cpm, 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 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 o2n+1, o2n, sp2nThe Lie algebras o2n, sp2n are considered as subalgebras in the Lie algebra of all 2n×2n matrices, whose rows and columns are indexed by i,j=−n,…,−1,1,…,n, and the algebra o2n+1 is a subalgebra in the Lie algebra of all (2n+1)×(2n+1) matrices, whose rows and columns are indexed by i,j=−n,…,−1,0,1,…,n. The algebras o2n+1 and o2n are generated by the matrices where i,j=−n,…,−1,0,1,…,n in the case o2n+1 and i,j=−n,…,−1,1,…,n in the case o2n.The algebra sp2n is generated by the matrices where i,j=−n,…,−1,1,…,n.Let gn be the Lie algebras o2n+1, o2n or sp2n. The chosen realization is also a root realization of the corresponding Lie algebra. The elements Fi,j, i<j, correspond to positive roots, Fi,j, i>j, correspond to negative roots; and Fi,i generate the Cartan subalgebra. An analogous choice of the Cartan subalgebra and the root elements will be used also for glm. To be able to talk about a Gelfand–Tsetlin base, we need to fix a chain of subalgebras. A subalgebra gn−k⊂gn is defined as the span ⟨Fi,j⟩i,j≠±1,…,±(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∈G acts on a function f(g), g∈G by right shifts Thus, the space Fun of all functions of the group G is a representation of G, and so, of the Lie algebra LieG. Let2 G=Sp2n, SO2n+1, SO2n+1 and let aji, i,j=−n,…,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 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 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,
\begin{equation}
a_{i_1,\dots,i_k}=\pm a_{\widehat{-i_1},\dots,\widehat{-i_k}}.
\end{equation}
\tag{3.11}
For the series D, we also have
\begin{equation}
\overline{a}_{i_1,\dots,i_k}= \pm \overline{a}_{\widehat{-i_1},\dots,\widehat{-i_k}}.
\end{equation}
\tag{3.12}
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. The choice of the sign on both sides is the same. 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,
\begin{equation}
a_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,-i,i}= \sqrt{a_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots, \pm 1,-i,-j}a_{\pm n,\dots,\widehat{\pm i},\dots,\widehat{\pm j},\dots, \pm 1,i,j}}\,,
\end{equation}
\tag{3.16}
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}fIn 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
\begin{equation}
f= \begin{cases} {\displaystyle\sum_{\alpha}\sqrt{a_{\pm n,\dots,\pm 2,\pm 1}}\cdot f_{\alpha}} &\textit{for the series $B$ and of $D$ with }m_{-1}\geqslant 0, \\ {\displaystyle\sum_{\alpha}\sqrt{\overline{a}_{\pm n,\dots,\pm 2,\pm 1}}\cdot f_{\alpha}} &\textit{for the series $D$ with }m_{-1}<0, \end{cases}
\end{equation}
\tag{3.22}
here, \alpha denotes a choice of + or - for all the indices \pm n,\dots,\pm 1. In the case of the series B, no conditions on the choice of the signs are imposed. For the series D with m_{-1}\geqslant 0, the sign + is chosen at positions where the parity coincides with that of n, and, for m_{-1}< 0 the sign “+” is chosen at positions whose parity coincides with that n-1. 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 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
\begin{equation*}
\begin{aligned} \, &\langle\sqrt{a_{\pm n,\dots,\pm 2,\pm1}}\,\rangle, \textit{ the parity of the number of $-$ is equal to that of }n, \\ &\langle\sqrt{\overline{a}_{\pm n,\dots,\pm 2,\pm1}}\,\rangle, \textit{ the parity of the number of $-$ is equal to that of }n-1. \end{aligned}
\end{equation*}
\notag
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:
\begin{equation}
\begin{gathered} \, \delta\in \mathcal{B}_{\mathrm{GC}}^{\mathfrak{gl}_m} \quad \Longleftrightarrow \quad \forall\, p,q \in \{-n,\dots,n\},\quad p\preceq q \\ \sum_{X\text{ contains }\geqslant (p+n+1-s(p,q))\text{ elements}\,\preceq \,q} \delta_X=0. \end{gathered}
\end{equation}
\tag{4.3}
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 ALet 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 AThe 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}_mThe 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}_mConsider 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, DTo 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_XWe 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, DUsing 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
\begin{equation}
\begin{gathered} \, \mathcal{O}_{\alpha}\mathcal{F}:=\biggl(\frac{\partial^{\tau_1+\tau_2}} {\partial^{\tau_1} B_{i,Z}\,\partial^{\tau_2} B_{j,y,Z}} - \frac{\partial^{\tau_3+\tau_4}}{\partial^{\tau_3}B_{j,Z}\,\partial^{\tau_4} B_{i,y,Z}}\biggr)\mathcal{F}=0, \\ \tau_i=\begin{cases} 2 &\text{if the corresponding variable } \\ &\text{is of type }B_{\pm n,\dots,\pm 1}\text{ or } B_{\pm n,\dots,\pm 1,0}, \\ 1 &\text{otherwise}, \end{cases} \\ \biggl(\frac{\partial}{\partial B_X}- \pm\frac{\partial}{\partial B_{\widehat{-X}}}\biggr) \mathcal{F}=0, \text{ the sign is defined in (3.13)}, \\ \begin{aligned} \, &\biggl(\frac{\partial^2 }{\partial^2 B_{\pm n,\dots,\widehat{\pm i},\dots, \widehat{\pm j},\dots,\pm 1,-i,i}} \\ &\qquad-\frac{\partial^2 }{\partial B_{\pm n,\dots,\widehat{\pm i},\dots, \widehat{\pm j},\dots,\pm 1,-i,-j} \, \partial B_{\pm n,\dots, \widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,i,j}}\biggr)F=0, \end{aligned} \\ \begin{gathered} \, \biggl(\frac{\partial^2} {\partial B_{\pm n,\dots,\widehat{i},\dots,\pm 1,-i}\, \partial B_{\pm n,\dots,\widehat{i},\dots,\pm 1,i}}- \frac{1}{\sqrt{-2}}\, \frac{\partial^2}{\partial^2 B_{\pm n,\dots,\widehat{i},\dots,\pm 1,0}}\biggr)\mathcal{F}=0 \\ \text{ only for the series }B. \end{gathered} \end{gathered}
\end{equation}
\tag{6.3}
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, DDefinition 5. The A-GKZ system for the series B, D is a system generated by the equations (see Definition 1)
\begin{equation}
\begin{gathered} \, \begin{aligned} \, \overline{\mathcal{O}}_{\alpha}\mathcal{F} &:= \biggl(\frac{\partial^{\tau_1+\tau_2}}{\partial^{\tau_1} B_{i,Z}\, \partial^{\tau_2} B_{j,y,Z}}- \frac{\partial^{\tau_3+\tau_4}}{\partial^{\tau_3} B_{j,Z}\, \partial^{\tau_4} B_{i,y,Z}} \\ &\qquad+ \frac{\partial^{\tau_5+\tau_6}}{\partial^{\tau_5} B_{y,Z}\, \partial^{\tau_6} B_{i,j,Z}}\biggr)F=0, \end{aligned} \\ \tau_i= \begin{cases} 2 &\text{if the corresponding variable is of type} \\ &A_{\pm n,\dots,\pm 1}\text{ or }A_{\pm n,\dots,\pm 1,0}, \\ 1 &\text{otherwise}, \end{cases} \\ \biggl(\frac{\partial}{\partial B_X}- \pm\frac{\partial}{\partial B_{\widehat{-X}}}\biggr)F=0, \text{ the sign is defined in (3.13)}, \\ \begin{aligned} \, &\biggl(\frac{\partial^2}{\partial^2 B_{\pm n,\dots,\widehat{\pm i}, \dots,\widehat{\pm j},\dots,\pm 1,-i,i}} \\ &\qquad-\frac{\partial^2}{\partial B_{\pm n,\dots,\widehat{\pm i},\dots, \widehat{\pm j},\dots,\pm 1,-i,-j} \, \partial B_{\pm n,\dots, \widehat{\pm i},\dots,\widehat{\pm j},\dots,\pm 1,i,j}}\biggr)F=0, \end{aligned} \\ \begin{gathered} \, \biggl(\frac{\partial^2}{\partial B_{\pm n,\dots,\widehat{i},\dots,\pm 1,-i}\, \partial B_{\pm n,\dots,\widehat{i},\dots,\pm 1,i}}-\frac{1}{\sqrt{-2}}\, \frac{\partial^2}{\partial^2 B_{\pm n,\dots,\widehat{i},\dots, \pm 1,0}}\biggr)F=0 \\ \text{only for the series }B. \end{gathered} \end{gathered}
\end{equation}
\tag{6.4}
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 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, DIt 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)
\begin{equation}
\begin{gathered} \, \mathcal{O}_{\alpha}\mathcal{F}:=\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}=0, \\ \biggl(\frac{\partial}{\partial A_X}-\pm\frac{\partial} {\partial A_{\widehat{-X}}}\biggr)\mathcal{F}=0, \text{ the sign is defined in (3.13)}. \end{gathered}
\end{equation}
\tag{6.11}
Definition 8. The A-GKZ system for tensor representations is the system generated by the equations (see Definition 1)
\begin{equation}
\begin{gathered} \, \overline{\mathcal{O}}_{\alpha}F=\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=0, \\ \biggl(\frac{\partial}{\partial A_X}- \pm\frac{\partial}{\partial A_{\widehat{-X}}}\biggr)F=0. \end{gathered}
\end{equation}
\tag{6.12}
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_nIn 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, D8.1. 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 nth 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 lemmaLemma 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
\begin{equation*}
v_{\alpha}=\tau_1\cdot e_{X_1}+\tau_2\cdot e_{X_2}- \tau_3\cdot e_{X_3}-\tau_4\cdot e_{X_4},
\end{equation*}
\notag
where e_X is the unit vector corresponding to the coordinate a_X, \tau_i\in\mathbb{Z}_{\geqslant 0}. In addition, for each such a generative, there exists a relation between the variables of the form
\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}
\tag{8.8}
for some X_5, X_6, \tau_5, \tau_6. With each generator v_{\alpha} we associate the vector
\begin{equation*}
r_{\alpha}=\tau_1\cdot e_{X_1}+\tau_2\cdot e_{X_2}- \tau_5\cdot e_{X_5}-\tau_6\cdot e_{X_6}.
\end{equation*}
\notag
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]).
\begin{equation*}
\biggl(\lambda_1 g_1\biggl(\frac{\partial}{\partial C}\biggr)+\dots+ \lambda_l g_l\biggl(\frac{\partial}{\partial C}\biggr)\biggr)f_{\delta}(C)=0
\end{equation*}
\notag
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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\Bibitem{Art24} |
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