Restriction of representations of $GL(n+1,C)$ to $GL(n,C)$ and action of the Lie overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of $\GL(n+1,C)$ to the subgroup $GL(n,C)$ (it is determined by the Gelfand-Tsetlin branching rule). We write explicitly formulas for generators of the Lie algebra $gl(n+1)$ in the direct sum of representations of $\GL(n,C)$. Nontrivial generators act as differential-difference operators, the differential part has order $(n-1)$, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of $GL(n,C)$


The Gelfand-Tsetlin formulas.
It is well known that restrictions of finite dimensional holomorphic representations of the general linear group GL(n, C) to the subgroup GL(n − 1, C) is multiplicity free. Considering a chain of restrictions GL(n, C) ⊃ GL(n − 1, C) ⊃ GL(n − 2, C) ⊃ · · · ⊃ GL(1, C) we get a canonical decomposition of our representation into a direct sum of one-dimensional subspaces. Taking a vector in each line we get a basis of the representation. In [7] Gelfand and Tsetlin announced formulas for action of generators of the Lie algebra gl(n) in this basis. It turns out that the Lie algebra gl(n) acts by difference operators in the space of functions on a certain convex polygon in the lattice Z n(n−1)/2 . In particular, this gives an explicit realization of representations of the Lie algebra gl(n).

Actions of overalgebras in the spectral decompositions.
We can write images of many operators under the classical Fourier transform. It is commonly accepted that Plancherel decompositions of representations are higher analogs of the Fourier transform.
Consider a group G and its subgroup H. Restrict an irreducible unitary representation of G to H. Generally, an explicit spectral decomposition of the restriction seems hopeless problem. However, there is a collection of explicitly 1 Supported by the grant FWF, P25142 solvable problems of this type 2 . In [20] it was conjectured that in such cases the action of the Lie algebra g can be written explicitly as differential-difference operators. In fact, in [20] there was considered the tensor product V s ⊗ V * s of a highest weight and a lowest weight unitary representations of SL(2, R) (i.e., G ≃ SL(2, R) × SL(2, R) and H ≃ SL(2, R) is the diagonal). This representation is a multiplicity free integral over the principal series of SL (2, R). It appears that the action of all generators of the Lie algebra sl(2, R) ⊕ sl(2, R) in the spectral decomposition can be written explicitly as differential-difference operators. The differential part of these operators has order two and the difference operators are difference operators in the imaginary direction 3 .
Molchanov [12]- [15] solved several problems of this type related to rank-one symmetric spaces 4 . In all his cases the Lie overalgebra acts by differentialdifference operators; difference operators act in the imaginary dimension. However, the order of the differential part in some cases is 4.
In the present paper, we consider restrictions of holomorphic finite-dimensional representations GL(n + 1, C) to GL(n, C) and write explicit formulas for the action of the overalgebra in the spectral decomposition. Also, we formulate a conjecture concerning restrictions of unitary principal series.
1.3. Notation. The group GL(n, C) and its subgroups. Denote by GL(n, C) the group of invertible complex matrices of size n. By gl(n) we denote its Lie algebra, i.e. the Lie algebra of all matrices of size n. Let U(n) ⊂ GL(n, C) be the subgroup of unitary matrices.
By E ij we denote the matrix whose ij's entry is 1 and other entries are 0. The unit matrix is denoted by 1 or 1 n , i.e. 1 = j E jj . Matrices E ij can be regarded as generators of the Lie algebra gl n .
Denote by N + = N + n ⊂ GL(n, C) the subgroup of all strictly upper triangular matrices of size n, i.e., matrices of the form By N − n we denote the group of strictly lower triangular matrices. 2 Also, some explicitly solvable spectral problems in representation theory can be regarded as special cases of the restriction problem. In particular, a decomposition of L 2 on a classical pseudo-Riemannian symmetric space G/H can be regarded as a special case of the restriction of a Stein type principal series of a certain overgroup G to the symmetric subgroup G, [21]. So, for L 2 on symmetric spaces the problem of action of the overalgebra discussed below makes sense.
3 On Sturm-Liouville in imaginary direction, see [23] and further references in that paper, see also [10]. The most of known appearances of such operators are related to representation theory and spectral decompositions of unitary representations. 4 In particular, he examined the restrictions of maximally degenerate principal series for the cases GL(n + 1, By B − n ⊂ GL(n, C) we denote the subgroup of lower triangular matrices, Let ∆ be the subgroup of diagonal matrices, we denote its elements as δ = diag(δ 1 , . . . , δ n ).
Next, we need a notation for sub-matrices. For a matrix X denote by the left upper corner of X of size α × β. Let I : 0 < i 1 < i 2 < . . . i α α, J : 1 j 1 . . . j β be sets of integers. Denote by the matrix composed of entries x iµ,jν . By we denote matrix composed of i i ,. . . , i α -th rows of X (the order is not necessary increasing, also we allow coinciding rows). 1.4. Holomorphic representations of GL(n, C). Recall that irreducible finite dimensional holomorphic representations ρ of the group GL(n, C) are enumerated by collections of integers (signatures) This means that there is a cyclic vector v (a highest weight vector) such that Denote such representation by ρ p = ρ n p . Denote the set of all signatures by Λ n . The dual (contragradient) representation to ρ p has the signature p * := (−p n , . . . , −p 1 ). (1.3) 1.5. Realizations of representations by differential operators. Recall a model for irreducible finite dimensional holomorphic representations for details, see Section 2). We consider the space Pol(N + n ) of polynomials in the variables z ij . The generators of the Lie algebra gl(n) act in this space via differential operators All other generators can be expressed in the terms of E kk , E k(k+1) , E (k+1)k (also it is easy to write explicit formulas as it is explained in Section 2). In this way, we get a representation of the Lie algebra gl(n).
There exists a unique finite dimensional subspace V p invariant with respect to such operators. The representation ρ p is realized in this space. The highest weight vector is f (Z) = 1. In the next subsection we describe this space more explicitly.
Remark. This approach arises to [6]. Of course, this model is a coordinatization of the constructions of representations of GL(n, C) in sections of line bundles over the flag space B − n \ GL(n, C) (as in the Borel-Weil-Bott theorem). The space N + n ≃ C n(n−1)/2 is an open dense chart on this space, the terms with ∂ in (1.4)-(1.5) correspond to vector fields on the flag space, the zero order terms are corrections corresponding to the action in the bundle. Elements of the space V n p are precisely polynomials, which are holomorphic as section of bundle on the whole flag space. However, we need explicit formulas and prefer a purely coordinate language. Denote by dµ(Z) the measure on N + n given by the formula Proof is given in Section 2. Description-2. We define the Zhelobenko operators by  On Hilbert spaces determined by reproducing kernels, see, e.g., [22], Sect. 7.1. The proposition is more-or-equivalent to the Borel-Weil theorem.
1.7. The restriction of representations GL(n+1, C) to GL(n, C). Consider the representation ρ n+1 r of GL(n+1, C) with a signature r = (r 1 , . . . , r n+1 ). It is well-known (see Gelfand, Tsetlin [7]) that the restriction of ρ n+1 r to GL(n, C) is multiplicity free and is a direct sum of all representations ρ q of GL(n, C) with signatures satisfying the interlacing conditions r 1 q 1 r 2 q 2 r 3 . . . q n r n+1 . (1.9) Our purpose is to write explicitly the action of Lie algebra gl(n + 1) in the space ⊕V n q . 1.8. Normalization. First, we intend to write a GL(n, C)-invariant pairing as an integral where Z ranges in N + n+1 , U ranges in N + n , and the kernel is a polynomial in the variables z ij , u kl . Denote Z cut := [Z] (n+1)n , this matrix is obtained from Z by cutting of the last column. Denote by U ext the n × (n + 1)-matrix obtained from U by adding the zero last column, Consider the (n + 1) × (n + 1)-matrix composed from the first (n + 1 − α) rows of the matrix Z and the first α rows of the matrix U ext . Denote by Φ α the determinant of this matrix: Consider the n × n-matrix composed of the first n − α rows of the matrix Z cut and the first α rows of the matrix U , denote by Ψ α its determinant: (1.13) Proposition 1.4 Consider signatures p = (p 1 , . . . , p n+1 ), q = (q 1 , . . . , q n ) such that p and q * are interlacing. Then the expression (1.10) with the kernel (1.14) determines a GL(n, C)-invariant nonzero pairing between V n+1 p and V n q . For instance, for n = 3 we get Next we pass to the dual signature r = p * (below p and q are rigidly linked by this restraint), and represent L p,q in the form (1.15) 1.9. Action of the overalgebra. Fix a signature r = p * ∈ Λ n+1 as in the previous subsection. Consider the space We can regard elements of this space as 'expressions' of continuous variables u kl , where 1 k < l n, and of integer variables q 1 , . . . , q n . Integer variables range in the domain r j q j r j+1 . More precisely, for a fixed q the expression F (U, q) is a polynomial in the variables u kl , moreover, this polynomial is contained in V q .
Our family of forms L p,q determines a duality between V r * and the space V r , hence we get a canonical identification of V r and V r . Therefore, we have a canonically defined action of the Lie algebra gl(n + 1) in V r . We preserve the notation E kl for operators in V p and denote operators in V r by F kl . For 1 k, l n the operators F n kl act in the space V r by the first order differential operators in U with coefficients depending on q according the standard formulas (see (1.4)-(1.6)). For instance, where ∂ kl := ∂ ∂u kl . The purpose of this work is to present formulas for the generators F 1(n+1) , F (n+1)n . Together with (1.17)-(1.20) they generate the Lie algebra gl(n), formulas for the remaining generators F j(n+1) , F (n+1)j consist of similar aggregates as below, but are longer.
Denote by T ± j the following difference operators (the remaining variables do not change). We will write expressions, which are polynomial in u kl , ∂ kl , linear in T ± j and rational in q j . These expressions satisfy the commutation relations in gl(n) and preserve the space V r . For 1 k < l n denote by [k, l] the set [k, l] := {k, k + 1, . . . , l}.

For a set
More generally, if I ⊂ J ⊂ [1, n], we write I ⊳ J if the minimal (resp. maximal) element of I coincides with the minimal (resp. maximal) element of J.
For any I ⊳ [1, n] we define the operator

Theorem 1.5 a)
The generator E 1(n+1) acts by the formula where coefficients a m are given by The generator E (n+1)n is given by the formula 1.10. Further structure of the paper. Section 2 contains preliminaries on holomorphic representations of GL(n, C). In Section 3, we verify the formula for the kernel L(Z, U ). Our main statement is equivalent to a verification of differential-difference equations for the kernel L(Z, U ), this is done in Section 4. In Section 5, we formulate a conjecture about analog of our statement for unitary representations.
Acknowledgements. Fifteen years ago the topic of the paper was one of aims of a joint project with M. I. Graev, which was not realized in that time (I would like to emphasis his nice paper [9]). I am grateful to him and also to V. F. Molchanov for discussions of the problem.
2.1. Realization in the space of functions on GL(n, C). For details and proof, see [28]. We say that a function f (g) on GL(n, C) is a polynomial function if it can be expressed as a polynomial expression in matrix elements g ij and det(g) −1 . Denote the space of polynomial functions by C[GL(n, C)]. The group GL(n, C) × GL(n, C) acts in C[GL(n, C)] by the left and right shifts': Fix a signature p and consider the space of N − n -invariant functions H p such that for any diagonal matrix δ, we have The group GL(n, C) acts in H p by the right shifts, This representation is irreducible and equivalent to ρ p . The vector ∆ p is its highest weight vector.

Realization in the space of functions on N −
n . An element g ∈ GL(n, C) satisfying the condition det[g] jj = 0 for all j admits a unique Gauss decomposition Restrict a function f ∈ H p to the subgroup N + n . We get a polynomial in the variables z ij , where i < j. By the Gauss decomposition, f is uniquely determined by this restriction. Therefore the space V p can be regarded as a subspace of the space of polynomial in z ij . The description of this space is given by the Zhelobenko Theorem 1.2.
Denote the factors in the Gauss decomposition of Zg in the following way Then where b jj (Z, g) are the diagonal matrix elements of the lower triangular matrix b(Z, g). We present formulas for transformations Z → Z [g(t)] , where g(t) are standard one-parametric subgroups in GL(n, C).
Proof. We say that a function The Jacobian satisfies this identity. Let us verify that the right hand side of (2.8) also satisfies it. It suffices to consider one factor det[Zg] −2 jj . Let us evaluate det[Z [g] h] jj . Represent Z, g, h etc. as block j + (n − j)matrices, Then Since det U 11 = 1, we have Next, this proves the desired statement.
Since the both sides of (2.8) satisfy the chain identity (2.9), it suffices to verify the identities for a system of generators of GL(n, C). Thus we can verify (2.8) for elements one-parametric subgroups exp(tE k(k+1) ), exp(tE kk ), exp(tE (k+1)k ). Formulas for the corresponding transformations are present in the previous subsection. Only the case exp(tE (k+1)k ) requires a calculation. In this case, the Jacobi matrix is triangular. Its diagonal values are On the other hand, in the product (2.8) the j-th factor is (1 + tz k(k+1) ) −2 , other factors are 1.

Proof of Proposition 1.1.
The U(n)-invariance of the L 2 (N + n , µ p )-inner product. We must check the identity (2.11) Fix g ∈ U(n). Substituting Z = U [g] to the right-hand side, we get In notation (2.10) we have Therefore [U [g] ] jn has the form where C 11 is a lower-triangular matrix, and Next, The matrix g is unitary, gg * = 1. Therefore the last expression equals to Keeping in mind Let us represent U g in the form DW , where D ∈ B − n , g ∈ N + n . Then here b ii (U, g) are the diagonal elements of U g. Therefore, Since g ∈ U(n), we have | det(g)| = 1. Keeping in mind (2.13), we come to Therefore the right-hand side of (2.11) equals i.e., coincides with the left-hand side. Thus the group U(n) acts in L 2 (N + n , dµ p ) by the unitary operators (2.6). Intersection of the space of L 2 and the space of polynomials. Denote this intersection by W .
First, W is U(n)-invariant. Indeed, for ψ ∈ W we have ρ p (g)ψ ∈ L 2 . By definition, ρ p (g)ψ is a rational holomorphic function. Represent it as an irreducible fraction α(Z)/β(Z). Let Z 0 be a non-singular point of the manifold β(Z) = 0, let O be neighborhood of Z 0 . It is easy to see that O |ρ p (g)ψ| 2 dŻ = ∞. Therefore ρ p (g)ψ is a polynomial.
Second, the space W is non-zero. For instance it contains a constant function. Indeed, N + n dµ p were evaluated in [24] and it is finite. The third, W is finite-dimensional. Indeed, our measure has the form |r(Z)| −2 dŻ, where r(Z, Z) is a polynomial in Z, Z. A polynomial ϕ ∈ W satisfies the condition Clearly, degree of ϕ is uniformly bounded.
Next, the operators (2.6) determine a unitary representation in L 2 (N + n ) and this representation is an element of the principal non-unitary series (see, e.g., [6] or [28], Addendum). But a representation of the principal series can not have more than one finite-dimensional subrepresentation.

The formula for the kernel
In this section we prove Proposition 1.4.
We have two collections of variables z ij , u i ′ j ′ . Zhelobenko operators R kl in z and u we denote by R z kl , R u kl .
if k ∈ I, l ∈ I; (−1) θ(I,k,l) Φ α [I • ; J], if k ∈ I, l / ∈ I. where I • is obtained from I by replacing of k by l, and the corresponding change of order. If k = i s and i t < l < i t+1 , when and θ(I, k, l) = t − s.
The same properties (with obvious modifications) hold for If k / ∈ I, then Φ α [I, J] does not depend on variables z km . Therefore we get 0. If k ∈ I, these variables are present only in one row. Application of R z kl is equivalent to a change of the row 0 . . . If l ∈ I, then this row is present in the initial matrix, and again det = 0. Otherwise, we come to Φ α [I • ; J] up to the order of rows.
Corollary 3.2 For any k, l, α, The same property holds for operators R u and for Ψ α [I; J].

Verification of the Zhelobenko conditions.
Proof. By the interlacing conditions, L p,q is a polynomial. We must verify the identities To be definite verify the first equality, The sum of exponents (the both exponents are positive) is and we obtain the desired 0. If k = n the factor Ψ n−k is absent, we have By the interlacing condition −p n+1 q 1 and we again obtain 0.

Invariance of the kernel.
Consider matrices Φ α , Ψ α defined by (1.13). Lemma 3.5 Let g ∈ GL(n, C). Denote g = g 0 0 1 ∈ GL(n + 1, C). Then Proof. We write the Gauss decompositions of Z g, U g, Then the Gauss decomposition of U ext (see (1.11)) is denote factors in the right-hand side by C, Q. Then We pass to determinants in the left-hand side and the right-hand side. Keeping in mind we come to (3.6). Proof of (3.7) is similar.
Thus L p,q is a non-zero GL(n, C)-invariant vector in V p ⊗ V q . We also know that such a vector is unique up to a scalar factor.

The calculation
Below q ∈ Λ n , r = p * ∈ Λ n+1 are signatures. The signatures r and q are interlacing. The kernel L is the same as above (1.14), We must verify the differential-difference equations Remark. For α = 1 we immediately get ∂Φ1 ∂z 1(n+1) = −1. On the other hand, taking formally α = 1 in the sum in (4.1) we get R 11 Ψ 0 /Φ 1 . At first glance, we must assume that Ψ 0 = 1 and that R 11 is the identical operator. Under this assumption, R 11 Ψ 0 /Φ 1 = 1/Φ 1 . However, this gives an incorrect sign. ⊠ In the following two proofs we need manipulations with determinants. To avoid huge matrices or compact notation, which are difficult for reading, we expose calculations for matrices having a minimal size that allows to visualize picture.
Proof. We have (see (2.7)) Take n = 3 and α = 2, Then Proof. Split E (n+1)n as a sum Obviously, Let us evaluate all DΦ α , DΨ α . 1) DΦ α = 0 for all α = 1. Take n = 3, α = 2, Variables z jn (in our case z j4 ) are present only in the next-to-last column. We apply the operator D to this column and come to a matrix with coinciding columns.
Thus we know all DΦ α , DΨ α . This gives us the desired statement.

Quadratic relations. Now we wish to write a family of quadratic relations between different functions of type
We do not introduce operators R kk and R kk Φ k−1 , R kk Ψ k−1 are only symbols used in formulas. In particular, formula (4.1) now can be written the form without a term with abnormal sign. Below it allows to avoid numerous anomalies in the formulas, duplications of formulas and branchings of calculations. and this proves (4.7). We wish to apply one of the Plucker identities to the product of these minors (see, e.g., [4], Section 9.1). We take m from the collection (1, . . . , n + 1 − α; 1, . . . , α), exchange it with an element of the collection (1, . . . , n−β; 1, . . . , m, . . . , β+ 1; ξ) and consider the product of the corresponding minors. Next, we take the sum of all such products of ∆. In this way, we obtain Look to the summands of our expression up to signs. 1) After exchanging of m with 1, . . . , n − β we get a matrix ∆[. . . ] with coinciding rows and therefore we obtain 0, see (4.9).
3) The sum (4.13)-(4.16) corresponds to the sum in the left hand side of the the desired identity (4.4) with j < β + 1.

5) The summand (4.19)-(4.20) corresponds to the expression
Next, let us watch signs. Denote by (i 1 i 2 . . . i k ) a cycle in a substitution. Denote by σ(·) the parity of a substitution.
Remark. By definition, the right-hand side is a rational expression of the It is easy to observe (see below) that all the exponents s α , t β are equal 1. In a calculation below we decompose this function as a sum of 'prime fractions' with denominators Φ α , Ψ β and finally get an unexpectedly simple expression. ⊠ Let 1 k < l n, µ < n. Denote (q n − q j + j − n); (4.23) Notice that the additional term in the last row is 0 if l = n.
Proof. Obviously, By Lemma 3.1, only terms with k µ < l give a nonzero contribution. This gives the first row. Next, The second term in the right-hand side is 0 for l < n. For l = n this term equals R kn Ψ n−1 /Φ n .
Proof. First, we evaluate Each R I is a product R i1i2 . . . R is−1is . Each R itit+1 is a first order differential operator without term of order zero, therefore we can expand R I L according the Leibniz rule. Many summands of this expansion are zero by a priory reasons. Indeed, Therefor the expression (R I T − n L)/L is a sum of products of the following type where i c−1 µ < i c . These expressions are equal correspondingly Next, we represent Θ 1n as Each summand in the big brackets splits into a product of the form where H[m γ , m γ+1 ] is an expression, which depend on m γ , m γ+1 and does not depend on other m i . Since the coefficients also are multiplicative in the same sense, the whole sum in the big brackets also splits in a product Thus Z kl equals to ζ kl . Let ζ kl be as above (4.24). Denote By the previous lemma, this notation is compatible with the earlier notation Θ 1n . Also, Θ nn = Ψn−1 Φn .

Lemma 4.6
The Θ mn satisfies the following recurrence relation, This statement is obvious.
In particular, this gives an explicit expression for Θ 1n .
Proof. We prove our statement by induction. Assume that for Θ nn , Θ (n−1)n , . . . , Θ (m+1)n the formula is correct. We must derive the equality We have We stress that the expression ξ µ mν does not depend on γ. Our sum transforms to µ,ν:µ<ν Denote by B µ mν the expression in the square brackets and write it explicitly: Next, we apply the quadratic relations (4.4) and (4.5) and transform the last expression to Thus, We collect similar terms and get ξ µν σ ν n (q µ+1 − q µ ), (4.32) First, we transform S 2 , The sum in the second line is evaluated in the following lemma.
In particular, (q n −r n+2−j −k+1)· (q n −q n−k −k)+(q n−k −r n−k+1 ) and the big bracket joins to the product k j=2 . Let us return to our calculation. We must evaluate S 1 + S 2 , An evaluation of this sum is similar to the proof of Lemma 4.8. We verify the following identity by induction, Substituting k = ν − 1 we get the desired coefficient in (4.31).
It remains to notice that formulas for the coefficient in the front of 1 Φn are slightly different. In this case, the sum S 2 is absent, but ζ mn has an extra term 1 Φn . Starting this place, the calculation is the same, the extra term replaces the additional term in (4.37).
This completes a proof of Lemma 4.7.
4.5. The extension of calculations. Next, we must evaluate other summands in F 1(n+1) L. Denote Proof. We decompose T − τ L as Factors R kl of operators R I act as zero on the factors of the second bracket. Therefore this bracket can be regarded as a constant. After this, we get the same calculations as for Θ 1n but n is replaced by τ .
We must verify the following identities n τ =α A summand in the left-hand side equals (q m − r n+2−k + n − α − k + 1).
The factor α−1 i=1 (q m − q i + i − m) cancels, the factors n−τ +1 j=1 and n−α k=n−m+2 join together and we come to .
Next, we write a rational function Multiplying this identity by (q α − r α+1 ) we get (4.39).

4.7.
Invariance of the space V r .

Lemma 4.11
The space V r (see (1.16)) is invariant with respect to the operator F 1(n+1) .
First, functions ℓ Z (U ) generate the space V r (because the pairing V p × V r → C determined by L is nondegenerate). Next, Differentiating a family of elements of V r with respect to a parameter we get elements of the same space V r .
4.8. Formula for F (n+1)n . Here a calculation is more-or-less the same, we omit details (in a critical moment we use the quadratic identities (4.4), (4.6)).
5 Infinite dimensional case. A conjecture 5.1. Principal series. Now consider two collections of complex numbers (p 1 , . . . , p n ) and (p • 1 , . . . , p • n ) such that Consider a representation ρ n p|p • of the group GL(n, C) in the space L 2 (N + n ) determined by the formula (5.1) We get the unitary (nondegenerate) principal series of representations of GL(n, C) (see [6]). Denote by Λ unitary n the space of all parameters p, p • .
Remark. In formula (5.1), we have complex numbers in complex powers. We understand them in the following way: In the right hand side, the first factor has a positive base of the power, the second factor has an integer exponent. Hence the product is well defined. ⊠ Remark. Formula (5.1) makes sense if p j − p • j ∈ Z, and gives a nonunitary principal series of representations. The construction of holomorphic representation discussed above corresponds to p ∈ Λ n , p • = 0. If both p, p • ∈ Λ n (5.2) then our representation contains a finite-dimensional (nonholomorphic) representation ρ p ⊗ ρ p • , where ρ denotes the complex conjugate representation. ⊠

5.2.
Restriction to the smaller subgroup. Consider a representation ρ n+1 p|p • of the group GL(n + 1, C). According [1], the restriction of ρ p|p • to the subgroup GL(n, C) is a multiplicity free integral of all representations of ρ n p|p • of unitary nondegenerate principal series of GL(n, C). Moreover, the restriction has Lebesgue spectrum. Thus the restriction U p|p • can be realized in the space L 2 (N + n × Λ unitary n ) by the formula 5.3. Intertwining operator. Next, we write an integral operator (n+1)n the operators obtained from F p 1(n+1) , F p (n+1)n by replacing Define the operators The statement seems doubtless, since we have an analytic continuation of our finite-dimensional formulas from the set (5.2). However, this is not an automatic corollary of our result. In particular, it is necessary to find the Plancherel measure on Λ unitary n (i.e. a measure making the operator J unitary).