S Khoroshkin

O Ogievetsky

Diagonal reduction algebra and reflection equation

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

Introduction

Reduction algebras were introduced in [START_REF] Mickelsson | Step algebras of semi-simple subalgebras of Lie algebras[END_REF][START_REF] Asherova | Projection operators for simple Lie groups. II. General scheme for construction of lowering operators. The groups SU(n) (Russian)[END_REF] for a study of representations of a Lie algebra with the help of the restriction to the space of highest weight vectors with respect to a reductive subalgebra g. In an abstract setting, for an associative algebra A which contains U(g) as a subalgebra and satisfies certain finiteness conditions, the corresponding reduction algebra is the double coset (2) equipped with a nontrivial multiplication (3).

This associative multiplication, defined with the help of the extremal projector of Asherova-Smirnov-Tolstoy [AST], can be also described [K] by means of the so called universal dynamical twist J. This twist gives rise to a solution of the universal dynamical Yang-Baxter equation [ABRR].

The diagonal reduction algebra D(g) is a particular case, associated to the diagonal embedding of U(g) into U(g) ⊗ U(g), of reduction algebras. The algebra D(g) acts in the space of highest weight vectors of the tensor product of two representations of g, considered as the representation of g. In [START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF][START_REF] Khoroshkin | Structure constants of diagonal reduction algebras of gl type[END_REF] we presented a list of ordering defining relations for natural generators of the diagonal reduction algebra D(gl n ) of the Lie algebra gl n .

The main goal, proposition 4.1 of this paper, is to relate the algebra D(gl n ) to the R-matrix formalism. We exhibit a matrix L of certain generators of the algebra D(gl n ) such that the defining relations can be collected into the operator equation usually called the reflection equation.

To this end we study and use the reduction algebras Diff h (n, N) of algebras of differential operators in nN variables, which we call the algebras of h-deformed differential operators. These reductions were used in [KN] for the representation theory of Yangians. The algebras Diff h (n, N) are closely related, by means of the generalized Harish-Chandra isomorphism [KNV], to "relative Yangians" of A. Joseph [J] and "family algebras" of A. A. Kirillov [Kr].

In Section 3 we introduce a distinguished set of Heisenberg type generators of the algebras Diff h (n, N) and write down the defining relations for them. It turns out that the relations admit the R-matrix form, with the well-known solution R of the dynamical Yang-Baxter equation. Next we exhibit homomorphisms from the diagonal reduction algebra D(gl n ) to Diff h (n, N) (h-analogues of the "oscillator representations"), which lead to the presentation of D(gl n ) by means of the reflection equation.

The description of the algebra D(gl n ) as the reflection equation algebra has many advantages. As an application we find two natural families of central elements of D(gl n ) expressed as "quantum" traces of powers of L-operators, see Section 4.2. Also, the Rmatrix formalism reveals the braided bialgebra structure of D(g), see Section 4.3.

The modules over the algebra D(g) and the braided tensor structure on certain categories of D(g)-modules we study in a forthcoming publication.

Reduction algebras

In this section we recall the definition and some basic properties of reduction algebras. We restrict ourselves to reduction algebras related to general linear Lie algebra gl n which we denote further by g. Let e ij , i, j = 1, . . . , n, be the standard generators of the Lie algebra g, with the commutation relations [e ij , e kl ] = δ jk e il -δ il e kj .

(1)

We use the notation h and n ± for the Cartan and two opposite nilpotent subalgebras of g; h i denotes the element e ii ∈ h, and ε i the elements in h * , so that ε i (h j ) := δ i,j ;

h ij := h i -h j ∈ h and hi := h i -i, hij = hi -
hj are the elements of U(h). We define Ū(h) to be the ring of fractions of the commutative ring U(h) with respect to the multiplicative set of denominators, generated by the elements (h ij + k) -1 , k ∈ Z; and Ū(g) to be the ring of fractions of the universal enveloping algebra U(g) with respect to the same set of denominators.

1. Let A be an associative algebra which contains U(g) as a subalgebra, and the adjoint action of g on A is locally finite. In particular A is a U(g)-bimodule with respect to the multiplication by elements from U(g) on the left and on the right. Assume in addition that A is free as the left U(h)-module and the adjoint action of U(h) is semisimple. Let Ā be the localization Ā = A ⊗ U(h) Ū(h). The double coset space

à := n -Ā\ Ā/ Ān + (2)
equipped with a natural associative multiplication ⋄ , see e.g. [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF] for details, is usually called the reduction (double coset) algebra. The multiplication ⋄ is described by the prescription

x ⋄ y = x P y , ( 3
)
where P is the extremal projector [AST]. The projector P belongs to a certain extension of Ū(g), satisfies the properties

x P = P y = 0 for x ∈ n + , y ∈ n -, P = 1 mod n -Ū(g), P = 1 mod Ū(g)n + , P 2 = P .
and can be given by the explicit multiplicative formula [AST]. Alternatively, one can find representatives x ∈ Ā and ỹ ∈ Ā of coset classes x and y, such that x belongs to the normalizer of the left ideal Ān + , or ỹ belongs to the normalizer of the right ideal n -Ā. Then x ⋄ y is the image in the coset space à of the product x • ỹ.

In the sequel the algebra A is free as the left U(g)-module with respect to the multiplication by elements of U(g) on the left, and this U(g)-module is generated by a linear space V , invariant with respect to the adjoint action (which is supposed to be locally finite) of the Lie algebra g. In this case the reduction algebra is the free left Ū(h)-module, generated by V , see [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF][START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF]. Note also that the construction of the double coset space à does not use the multiplication in A but the U(g)-bimodule structure on A only. We denote sometimes by : x : the image in à of the element x ∈ Ā. This notation is useful to distinguish between the multiplication • in the algebra A and the multiplication ⋄ in the reduction algebra Ã.

2. The Weyl group of the root system of g is the symmetric group S n . Let σ 1 , . . . , σ n-1 be the generators of S n ; σ i corresponds to the permutation (i, i + 1). The group S n acts on vector spaces h * and h, so that σ i (h j ) = h σ i (j) and σ i (ε j ) = ε σ i (j) . These actions are related by λ(σ(h)) = σ -1 (λ)(h), σ ∈ S n , h ∈ h and λ ∈ h * . The action of S n on h extends to the action of a cover Sn of the group S n by automorphisms of the Lie algebra g. We denote by the same symbols σ i the following automorphisms of the algebra U(g):

σ i (x) := Ad exp(e i,i+1 ) Ad exp(-e i+1,i ) Ad exp(e i,i+1 ) (x) , so that σ i (e kl ) = (-1) δ ik +δ il e σ i (k)σ i (l) .
Let ρ = -n k=1 kε k . Then the shifted action • of the group S n on the vector space h * is defined by setting

σ • λ := σ(λ + ρ) -ρ . (4)
With the help of (4) we induce the action • of S n on the commutative algebra U(h) by regarding the elements of this algebra as polynomial functions on h * . We have

σ • hk = hσ(k) , σ • hij = hσ(i)σ(j) .
We also use the following notation for shift automorphisms of the rings U(h) and Ū(h). For any α ∈ h * and any element f ∈ Ū(h) denote by f [α] the image of f under the shift automorphism Ū(h) → Ū(h), defined by the rule h → h + (α, h). In particular,

h i [ε j ] = h i + δ ij .
3. Assume now that the action of Sn on U(g) extends to the action by automorphisms of the algebra A. For any i = 1, ..., n -1 define the map qi :

A → Ã by qi (x) = k≥0 (-1) k k! êk i,i+1 (σ i (x)) e k i+1,i k j=1 ( hi,i+1 -j) -1 (5)
Here êi,i+1 (x) = [e i,i+1 , x] is the adjoint action of e i,i+1 on x. The image of the sum n -A + An + of the ideals is a subspace of n -Ā + Ān + , see [Zh], so the formula (5) defines the map, denoted by the same symbol qi : n -A \ A/An + → Ã. The map qi satisfies the relations qi

(hx) = (σ i • h)q i (x), qi (xh) = qi (x)(σ i • h) (6)
for any h ∈ U(h) and x ∈ A. The use of (6) extends the map qi : n -A \ A/An + → Ã to the map qi : Ã → Ã. The maps qi satisfy the braid group relations [Zh] and are automorphisms of the reduction algebra Ã, see [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF].

3 Algebra of h-deformed differential operators

Reduction of the algebras of differential operators

There is a homomorphism ψ : U(g) → Diff(n) of the algebra U(g) to the algebra of polynomial differential operators in n variables x 1 , . . . , x n . The image of e ij ∈ U(g) is

ψ(e ij ) = x i ∂ j . (7)
Denote by Diff h (n) the reduction algebra of Diff(n) ⊗ U(g) with respect to the diagonal embedding of U(g). We call Diff h (n) algebra of h-deformed differential operators (the explicit definition, given below, makes it clear that this algebra admits a well-defined limit hij → ∞ for i < j and h1 > h2 > • • • > hn in which it becomes the usual algebra of differential operators with polynomial coefficients).

The algebra Diff h (n) is generated over Ū(h) by the classes of x i and ∂ j , which we denote by the same symbols. These elements are subject in Diff h (n) to quadratic-linear relations over Ū(h), which we now describe.

These relations can be computed directly. However, almost all the required information can be found in the description of the reduction algebra of U(gl n+1 ) with respect to U(gl n ) (this description is a basic step in the derivation of the Gelfand-Tsetlin basis in [Zh]). Indeed, the generators e i,n+1 and e n+1,i , i = 1, . . . , n, of the corresponding double coset algebra form the bases of the fundamental representation ω and its dual ω * with respect to the adjoint representation of g = gl n . The elements x i and ∂ i form the same bases of ω and ω * , the only difference is that the commutators [e i,n+1 , e n+1,j ] belong to the Cartan subalgebra of gl n+1 while [x i , ∂ j ] = δ ij . Thus, by [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF]4.5.3], we have

x i ⋄ x j = α ij x j ⋄ x i , ∂ j ⋄ ∂ i = α ij ∂ i ⋄ ∂ j , i < j, x i ⋄ ∂ j = ∂ j ⋄ x i i = j, x i ⋄ ∂ i = j β ij ∂ j ⋄ x j + µ i . (8)
Here

α ij = hij + 1 hij , β ij = 1 1 -hij ϕ j [ε j ] ϕ i with ϕ j = k:k>j hjk hjk -1 , (9)
and µ i are the elements of Ū(h) which are to be determined by another argument.

Lemma 3.1 We have

µ i = -ϕ -1 i .
Proof. Note first that µ n = -1. Indeed, since ∂ n is a highest weight vector with respect to the adjoint action of U(g) on Diff(n),

x n ⋄ ∂ n =: x n ∂ n :=: ∂ n x n : -1.
On the other hand, the products ∂ j ⋄ x j are equal to sums m a m : ∂ m x m : with some a m ∈ Ū(h) and do not contain a constant term. Thus

µ n = -1 = -ϕ -1 n .
For the derivation of other µ j we use the Zhelobenko automorphisms qi .

It is not difficult to check that qi

(x i ) = -x i+1 hi,i+1 hi,i+1 -1 , qi (x i+1 ) = x i , qi (x j ) = x j , j = i, i + 1, qi (∂ i+1 ) = ∂ i hi,i+1 hi,i+1 -1 , qi (∂ i ) = -∂ i+1 , qi (∂ j ) = ∂ j , j = i, i + 1. ( 10
)
Now we apply the automorphism qn-1 to the already known identity

x n ⋄ ∂ n = j β n,j ∂ j ⋄ x j -1.
We get the relation

hn-1,n hn-1,n -1 x n-1 ⋄ ∂ n-1 = j qn-1 β n,j ∂ j ⋄ x j -1.
Taking into account the last line in (8), we rewrite this relation in the form

x n-1 ⋄ ∂ n-1 = j β n-1,j ∂ j ⋄ x j - hn-1,n -1 hn-1,n , which implies that µ n-1 = - hn-1,n -1 hn-1,n = -ϕ -1 n-1 .
Next we apply qn-2 to the identity

x n-1 ⋄ ∂ n-1 = j β n-1,j ∂ j ⋄ x j -ϕ -1 n-1 and find that µ n-2 = -ϕ -1
n-1 and then further

µ i = -ϕ -1 i for all i.
The last line in (8) can be now rewritten as

x i ⋄ ∂ i = j 1 1 -hij ∂ j ϕ j ⋄ x j ϕ -1 i -ϕ -1 i ,
which suggests the change of variables

∂j := ∂ j ϕ j . (11)
In the new variables the relations ( 8) and ( 10) look as follows:

x i ⋄ x j = hij + 1 hij x j ⋄ x i , i < j, ∂i ⋄ ∂j = hij -1 hij ∂j ⋄ ∂i , i < j, x i ⋄ ∂j = ∂j ⋄ x i , i < j; x i ⋄ ∂j = hij ( hij -2) ( hij -1) 2 ∂j ⋄ x i , i > j, x i ⋄ ∂i = j 1 1 -hij ∂j ⋄ x j -1, (12) qi (x i ) = -x i+1 hi,i+1 hi,i+1 -1 , qi (x i+1 ) = x i , qi (x j ) = x j , j = i, i + 1, qi ( ∂i ) = - hi,i+1 -1 hi,i+1 ∂i+1 , qi ( ∂i+1 ) = ∂i , qi ( ∂j ) = ∂j , j = i, i + 1. (13)
We have other sets of Heisenberg type generators in the algebra Diff h (n). Set

ϕ ′ j = k:k<j hjk hjk -1 , (14)
and put ∂j :

= ∂ j ϕ ′ j -1 . ( 15
)
Arguments parallel to that of Lemma 3.1 show that

x i ⋄ x j = hij + 1 hij x j ⋄ x i , i < j, ∂i ⋄ ∂j = hij -1 hij ∂j ⋄ ∂i , i < j, ∂j ⋄ x i = x i ⋄ ∂j , i > j, ∂j ⋄ x i = hij ( hij -2) ( hij -1) 2 x i ⋄ ∂j , i < j, ∂i ⋄ x i = j 1 1 + hij x j ⋄ ∂j + 1. ( 16
)
and the same as in ( 13) coefficients in the action of Zhelobenko automorphisms. Alternatively, we can leave the variables ∂ j unchanged and rescale the variables x j . For the variables xj := ϕ j x j and ∂ j we get the relations of the form ( 12) with the reversed inequalities between i and j in first two lines of the relations (12); for the variables xj := ϕ ′ j -1 x j and ∂ j we get the relations of the form ( 16) with the reversed inequalities between i and j in first two lines of relations (16).

Polarized form of relations

1. The first line of ( 12) can be written in the following polarized form:

x i ⋄ x j = 1 hij x i ⋄ x j + h2 ij -1 h2 ij x j ⋄ x i , i < j, x i ⋄ x j = 1 hij x i ⋄ x j + x j ⋄ x i , i > j , ( 17
) ∂i ⋄ ∂j = - 1 hij ∂i ⋄ ∂j + h2 ij -1 h2 ij ∂j ⋄ ∂i , i < j, ∂i ⋄ ∂j = - 1 hij ∂i ⋄ ∂j + ∂j ⋄ ∂i , i > j . (18)
Rewrite ( 17), ( 18) and the last two lines in (12) in an operator form2

x i ⋄ x j = Rij kl x k ⋄ x l , ∂i ⋄ ∂j = Rlk ji ∂k ⋄ ∂l , x i ⋄ ∂j = Tik jl ∂k ⋄ x l -δ i j , (19)
where Rij kl and Tik jl are matrix coefficients of operators R, T :

C n ⊗ C n → Ū(h) ⊗ C n ⊗ C n . Their nonzero values are Rij ij = 1 hij , i = j, Rij ji =      h2 ij -1 h2 ij , i < j, 1, i ≥ j (20) Tij ij = - 1 hij -1 , i = j, Tji ij =    hij ( hij + 2) ( hij + 1) 2 , i < j, 1, i ≥ j (21)
Note the following identity:

Tkl ij [-ε l ] = Rlk ji . (22)
Similarly, the relations ( 16) can be presented in an operator form as

x i ⋄ x j = Rij kl x k ⋄ x l , ∂i ⋄ ∂j = Rlk ji ∂k ⋄ ∂l , ∂j ⋄ x i = Ŝik jl x l ⋄ ∂k + δ i j , (23)
where the non-zero elements of Ŝ are

Ŝij ij = 1 hij + 1 , Ŝij ji =      1 , i > j hij ( hij -2) ( hij -1) 2 , i < j (24)
2. The relations in the last two lines of ( 16) can be obtained in another way, by inverting the last relation in ( 19). Let Ψ be the skew inverse to T (see e.g. [O], section 4.1.2 for details of the R-matrix technique needed here), that is,

Ψik jl Tlm kn = δ i n δ m j . (25)
Multiplying the relation

x l ⋄ ∂k = Tlm kn ∂m ⋄ x n -δ l k
by Ψik jl from the left and contracting repeated indices, we get the relation ∂j

⋄ x i = Ψik jl x l ⋄ ∂k + Ψik jk . ( 26
)
The relations ( 11), ( 15), and the last two lines of ( 16) imply that

∂i = ∂i (Q - i ) -1 = Q + i ∂i , (27)
where

Q ± i = k:k =i hik ± 1 hik , Q - j [ε j ] Q + j = 1 , (28)
and

Ψij ij = Q + i Q - j 1 hij + 1 , Ψij ji =      1 , i < j ( hij -1) 2 hij ( hij -2) , i > j (29)
Comparing ( 26) and the last line in ( 16) we conclude that

Ψik jk = Q + i δ i j . (30) Define operators Q ± : C n → C n by (Q ± ) i j = Q ± j δ i j .
Then the relation ( 30) can be rewritten as

Tr 2 Ψ12 = Q + 1 .
Here the lower index specifies the number of the copy of the space C n in which the corresponding operator acts nontrivially. For example, Q + 1 stands for the operator Q + acting in the first copy and Tr 2 means the contraction of indices in the second copy.

Reflection equation and copies

To lighten the notation, in the formulation of statements, the matrix multiplication of matrices with entries in a reduction algebra is written without the symbol ⋄ (which is always assumed).

Set Li

j := x i ⋄ ∂j ∈ Diff h (n).
Proposition 3.2 The matrix L satisfies the reflection equation

R12 L1 R12 L1 -L1 R12 L1 R12 = R12 L1 -L1 R12 . ( 31
)
Proof. Consider the monomial x i 1 ⋄ x i 2 ⋄ ∂j 1 ⋄ ∂j 2 . Reorder it in two ways. The first way:

x i 1 ⋄ x i 2 ⋄ ∂j 1 ⋄ ∂j 2 = Ri 1 i 2 kl x k ⋄ x l ⋄ ∂j 1 ⋄ ∂j 2 = Ri 1 i 2 kl x k ⋄ Tln j 1 m ∂n ⋄ x m ⋄ ∂j 2 - Ri 1 i 2 kl δ l j 1 x k ⋄ ∂j 2 = k,l,m,n Ri 1 i 2 kl x k • ∂n ⋄ Tln j 1 m [-ε n ]x m ⋄ ∂j 2 - Ri 1 i 2 kj 1 x k ⋄ ∂j 2 = Ri 1 i 2 kl Lk n ⋄ Rnl j 1 m Lm j 2 - Ri 1 i 2 kj 1 Lk j 2 .
The second way of reordering:

x i 1 ⋄ x i 2 ⋄ ∂j 1 ⋄ ∂j 2 = x i 1 ⋄ x i 2 ⋄ Rmn j 2 j 1 ∂n ⋄ ∂m = Rmn j 2 j 1 [-ε i 1 -ε i 2 ]x i 1 ⋄ Ti 2 t nk ∂t ⋄ x k ⋄ ∂m - Rmn j 2 j 1 [-ε i 1 -ε i 2 ]δ i 2 n x i 1 ⋄ ∂m = Rmn j 2 j 1 [-ε i 1 -ε i 2 ] Ti 2 t nk [-ε i 1 ]x i 1 ⋄ ∂t ⋄ x k ⋄ ∂m - Rmi 2 j 2 j 1 [-ε i 1 -ε i 2 ]x i 1 ⋄ ∂m = t,k,m,n x i 1 ⋄ ∂t ⋄ Ti 2 t nk [-ε t ]x k ⋄ ∂m Rmn j 2 j 1 [-ε t -ε i 2 + ε k -ε m ] - m x i 1 ⋄ ∂m Rmi 2 j 2 j 1 [-ε m -ε i 2 ] = t,k,m,n Li t Ti 2 t nk [-ε t ] ⋄ Lk m Rmn j 2 j 1 [-ε t -ε i 2 + ε k -ε m ] - m Li 1 m Rmi 2 j 2 j 1 [-ε m -ε i 2 ] .
Matrix elements Rij kl and Tij kl are nonzero only if i = k and j = l, or i = l and j = k. We can thus replace the shift by ε t + ε i 2 in the last displayed line with the shift by ε n + ε k .

Besides, the matrix coefficient Rmn j 2 j 1 depends only on the difference h n -h m and is invariant with respect to the shift by ε m + ε n . Using the relation ( 22) we rewrite the result as Let now Diff(n, N) be the algebra of differential operators in nN variables x iα , i = 1, . . . , n, α = 1, . . . , N. There is a homomorphism

x i 1 ⋄ x i 2 ⋄ ∂j 1 ⋄ ∂j 2 = Li 1 t Ri 2 t kn ⋄ Lk m Rmn j 2 j 1 - Li 1 m Rmi 2 j 2 j 1 (
ψ N : U(g) → Diff(n, N). The image of e ij ∈ U(g) is ψ(e ij ) = α x iα ∂ jα . ( 33
)
Denote by Diff h (n, N) the reduction algebra of Diff(n, N) ⊗ U(g) with respect to the diagonal embedding of U(g). The algebra Diff h (n, N) is generated over Ū(h) by the classes of all x iα and ∂ jβ , which we denote by the same symbols. Denote

∂jβ = ∂ jβ ϕ j . ( 34
)
The calculations from Sections 3.1 and 3.2 can be repeated for any N. The result is Proposition 3.3 The elements x iα and ∂jβ satisfy the following relations

x iα ⋄ x jβ = Rij kl x kβ ⋄ x lα , ∂iα ⋄ ∂jβ = Rlk ji ∂kβ ⋄ ∂lα , x iα ⋄ ∂jβ = Rki lj [ε l ] ∂kβ ⋄ x lα -δ α β δ i j . (35)
The same proof as that of proposition 3.2 shows that the combinations

Li j := α x iα ⋄ ∂jα (36)
satisfy the same reflection equation. We formulate this assertion in the separate proposition.

Proposition 3.4 The matrix L, defined in (36) satisfies the reflection equation

R12 L1 R12 L1 -L1 R12 L1 R12 = R12 L1 -L1 R12 . ( 37
)
Remark. Equally well one can define the grassmanian version of h-deformed differential operators, the reduction of the algebra U(g) ⊗ C[ξ iα , d jβ ] where ξ iα are anti-commuting variables and d jβ grassmanian derivatives in them. The defining relations differ by the following sign changes:

ξ iα ⋄ ξ jβ = - Rij kl ξ kβ ⋄ ξ lα , diα ⋄ djβ = - Rlk ji dkβ ⋄ dlα , ξ iα ⋄ djβ = - Rki lj [ε l ] dkβ ⋄ xi lα -δ α β δ i j . (38)
Here djβ := d jβ ϕ j . Then the operator L with entries Li j := α ξ iα ⋄ djα satisfies the same reflection equation (37).

R-matrix and its skew inverse

Using the first relation in ( 35) we can reorder a monomial x iα ⋄ x jβ ⋄ x kγ , α = β = γ = α, as a combination of monomials of the form x •γ ⋄ x •β ⋄ x •α in two ways, as (x iα ⋄ x jβ ) ⋄ x kγ or as x iα ⋄ (x jβ ⋄ x kγ ). The ordered products form a basis in the reduction algebra (see [START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF], section 2) so the two ways of reordering lead to the same result. This implies certain compatibility conditions for the operator R which are formulated in the following Proposition.

Proposition 3.5 The operator R is a solution of the dynamical Yang-Baxter equation

a,b,u Rij ab Rbk ur [-ε a ] Rau mn = a,b,u Rjk ab [-ε i ] Ria mu Rub nr [-ε m ] . ( 39
)
This solution has already appeared several times in different contexts (see e.g. [I, ES] and references therein).

In addition, the operator R satisfies the relations

R2 = Id C n ⊗C n , ( 40
) R21 = RT | h →-h, (41)
where ( R21 ) ik jl := Rki lj and ( RT ) ik jl := Rjl ik and

Q + i [-ǫ i ] Q + k [-ǫ i -ǫ k ] Rik jl = Rik jl Q + j [-ǫ j ] Q + l [-ǫ j -ǫ l ] ,
which is an immediate consequence of

Q - j Q - i [-ǫ j ] = Q - i Q - j [-ǫ i ] .
The operators Ψ, Ŝ and R are related by

Ψik jl = Q + i [ǫ k -ǫ l ] Ŝik jl (Q + l ) -1 [-ǫ l ] , (42)
Ŝij kl = Rij kl [ǫ k ] . (43)
By (41

), Ψ21 = ΨT | h →-h so Ψam an = ( Ψ21 ) ma na = ( ΨT | h →-h) ma na = ( Ψ| h →-h) na ma = Q + n | h →-hδ n m = Q - n δ m n , (44)
or Tr 1 Ψ12 = Q - 2 . It follows from (25) together with (30) and ( 44

) that a Q - a [-ǫ m ] Rma na = δ m n , (45)
a Q + a [ǫ m ] Ram an = δ m n .
Remark. The reordering of the monomial

( ∂iα ⋄ ∂jβ ) ⋄ ∂kγ = ∂iα ⋄ ( ∂jβ ⋄ ∂kγ )
leads to a compatibility condition for the operator R which is equivalent to the same dynamical Yang-Baxter equation for R.

The reordering of any of the monomials

(x iα ⋄ x jβ ) ⋄ ∂kγ = x iα ⋄ (x jβ ⋄ ∂kγ ), (x iα ⋄ ∂jβ ) ⋄ ∂kγ = x iα ⋄ ( ∂jβ ⋄ ∂kγ ), α = β = γ = α,
similarly leads to a compatibility condition for the operator R which (in each case) now is equivalent to the dynamical Yang-Baxter equation for R together with the equality (40). The verification of this statement uses the fact that the matrix element R ij kl can be nonzero only if i = k and j = l, or i = l and j = k.

4 Diagonal reduction algebra

R-matrix presentation

The diagonal reduction algebra D(g) is by definition the reduction algebra of A = U(g) ⊗ U(g) with respect to the diagonal embedding of U(g). Let e

(1)

ij and e

(2) ij be the standard generators e ij of the Lie algebra g in the first and the second tensor components. Denote by s i j the generators of the diagonal reduction algebra defined as the images in D(g) of e

(1) ij . In other words, s i j = P e

(1) ij P . We will also need another set of generators s ′ i j = P e

(2) ij P . The elements s i j and s ′ i j are related by

s i j + s ′ i j = h i δ i j . (46)
In addition to the elements ϕ j , defined in (9), we need, for any j = 1, . . . , n, and m > j, the following elements of Ū(h):

ϕ jm = k:j<k<m
hjk hjk -1 .

The description of the algebra D(g) in terms of generators s i j -s ′ i j was given in [START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF][START_REF] Khoroshkin | Structure constants of diagonal reduction algebras of gl type[END_REF]. Here we suggest another presentation. Let L i j , i, j = 1, . . . , n, be the following elements of D(g):

L i j :=    s i j ϕ j , i = j s i j - m:m>i s m m 1 him ϕ im ϕ j , i = j (47)
The elements L i j are linear combinations of s k l with the triangular transition matrix. Thus L i j generate D(g) as the algebra over Ū(h).

Proposition 4.1 Elements L i j ∈ D(g) satisfy quadratic-linear relations collected in reflection equation

R12 L 1 R12 L 1 -L 1 R12 L 1 R12 = R12 L 1 -L 1 R12 . ( 48
)
The relations (48) form a complete list of defining relations over the field of fractions of Ū(h).

It is plausible that the relations (48) form a complete list of defining relations over the ring Ū(h) itself. Proof of proposition 4.1 is based on the properties of homomorphisms of the diagonal reduction algebra D(g) to Diff h (n, N) and the proposition 3.4.

Using the map ψ, see ( 33), we define a homomorphism

ψ 1 := ψ ⊗ 1 : U(g) ⊗ U(g) → Diff(n, N) ⊗ U(g). ( 49
)
The map ψ 1 sends the diagonal g to the diagonal g and thus defines the homomorphism ψ1 : D(g) → Diff h (n, N).

Lemma 4.2 The map ψ1 sends the generator

L i j ∈ D(g) to the element Li j ∈ Diff h (n, N), (see(36)) ψ1 (L i j ) = Li j .
Proof of proposition 4.1. The statement of proposition 4.1 follows from proposition 3.4, lemma 4.2 and the injectivity of the map ψ for N ≥ n (consider the map ψ as the tangent map for the group action GL n × Mat n×N → Mat n×N ; for N = n we obtain the classical isomorphism of U(g) with the ring of right invariant differential operators on GL n ⊂ Mat n×n ).

In particular, for N ≥ n the map ψ1 is injective on the subspace of D(g), generated by polynomials in L i j of degree not bigger than two. This proves that the relations (48) are satisfied. In [START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF], we presented a complete list of relations for the generators s i j -s ′ i j of the algebra D(g). These are ordering relations of degree at most two in these generators. This means, in particular, that the number of linearly independent over Ū(h) quadratic monomials in these generators equals the number of ordered quadratic monomials in n 2 generators. Since the transition to the generators L i j has a triangular form, the relations are of degree at most two on generators L i j and the number of linearly independent quadratic monomials in L i j is as before. Therefore it is left to prove the completeness of relations (48) in degree at most two. To show this we write the relations (48) in the form Z A c µ A ( h) = linear in L i j terms, where {Z A } is the set of quadratic monomials in L i j . In the asymptotic regime h1 > h2 > • • • > hn and hij → ∞ for i < j these relations are well defined and become

Z A c µ A (∞) = linear in L i j terms.
Indeed, in this limit the matrix R12 turns into the permutation matrix P 12 so the relations (48) turn into the standard defining relations for the Lie algebra g,

Λ 2 Λ 1 -Λ 1 Λ 2 = P 12 (Λ 1 -Λ 2 ) .
By general deformation arguments, rk c µ A ( h) ≥ rk c µ A (∞). Hence the number of linearly independent quadratic monomials for generic h is not bigger than that in the limit. But in both cases, for generic h and asymptotically, this number equals the number of ordered (for any linear order) quadratic monomials in L i j . Proof of lemma 4.2. Denote by : x iα ∂ jα : the image of the element x iα ∂ jα ⊗1 ∈ Diff(n, N)⊗ U(g) in the reduction algebra Diff h (n, N). Due to the definitions of elements L i j , Li j and of the map ψ 1 it is sufficient to establish the equalities

x iα ⋄ ∂jβ =    : x iα ∂ jβ : ϕ j , i = j : x iα ∂ jβ : - m:m>i : x mα ∂ mβ : 1 him ϕ im ϕ j , i = j (50)
for fixed α and β and then sum them up over α = β = 1, . . . , N. First we note that

x nα ⋄ ∂ iβ =: x nα ∂ iβ : and x iα ⋄ ∂ nβ =: x iα ∂ nβ : (51)
for any i = 1, . . . , n. This is because x nα and ∂ nβ are respectively lowest and highest weight vectors with respect to the adjoint action of the diagonal U(g):

[x nα , e ji ] = 0 , [e ij , ∂ nβ ] = 0 for any i < j, so that x nα P ∂ iβ =: x nα ∂ iβ : and x iα P ∂ nβ =: x iα ∂ nβ : in the double coset space Diff h (n, N).

Next we apply Zhelobenko operator qn-1 to both sides of equality

x nα ⋄ ∂ jβ =: x nα ∂ jβ : , j = n -1, n.
We have, using (10) and homomorphism property of Zhelobenko operators,

qn-1 (x nα ⋄ ∂ jβ ) = qn-1 (x nα) ) ⋄ qn-1 (∂ jβ ) = x n-1,α ⋄ ∂ jβ , j = n -1, n.
On the other hand, we can apply qn-2 to : x n-1,α ∂ nβ : as to elements of the adjoint representation of g, see [KO2, eq. (4.5)]

qn-1 (:

x nα ∂ jβ ) =: x n-1,α ∂ j,β :
. This implies, due to (51), the equality

x n-1,α ⋄ ∂ jβ =: x n-1,α ∂ jβ : for j = n -1.
Proceeding further with application of other Zhelobenko automorphisms qi we obtain similarly

x iα ⋄ ∂ jβ =: x iα ∂ jβ : for i = j . (52)
Due to (34) this is equivalent to the first line of (50). For the derivation of the rest of the relations (50) we employ the action of the Zhelobenko operators on the elements : x iα ∂ iβ : (no sum in i). Note that the sum x iα ∂ iβ + x i+1,α ∂ i+1,β is invariant with respect to the adjoint action of the sl 2 -subalgebra generated by e ii+1 and e i+1i , while the difference x i,α ∂ iβ -x i+1,α ∂ i+1,β spans the zero weight subspace of the three-dimensional representation of this sl 2 -subalgebra, with the highest weight vector x iα ∂ i+1,β . This implies the relations qi :

x iα ∂ iβ : = -: x iα ∂ iβ : 1 hi,i+1 -1 + : x i+1,α ∂ i+1,β : hi,i+1 hi,i+1 -1 , qi : x i+1,α ∂ i+1,β : =: x iα ∂ iβ : hi,i+1 hi,i+1 -1 -: x i+1,α ∂ i+1,β : 1 hi,i+1 -1 . (53)
Besides, qi :

x jα ∂ jβ : =: x jα ∂ jβ : for j = i, i + 1. ( 54
)
On the other hand, by (10) we have qi

x iα ⋄ ∂ iβ = qi x iα ⋄ qi (∂ iβ ) = x i+1,α hi,i+1 hi,i+1 -1 ⋄ ∂ i+1,β = x i+1,α ⋄ ∂ i+1,β hi,i+1 + 1 hi,i+1 , qi x i+1,α ⋄ ∂ i+1,β = qi x i+1,α ⋄ qi (∂ i+1,β ) = x i ⋄ ∂ i hi,i+1 hi,i+1 -1 . ( 55
)
We apply qn-1 to both sides of the equality x n ⋄ ∂ n =: x n ∂ n :. Using ( 53) and ( 55) we get

x n-1,α ⋄ ∂ n-1,β hn-1,n hn-1,n -1 =: x n-1,α ∂ n-1,β : hn-1,n hn-1,n -1 -: x nα ∂ nβ : 1 hn-1,n -1 ,
which implies the equality

x n-1,α ⋄ ∂ n-1,β =: x n-1,α ∂ n-1,β : -: x nα ∂ nβ : 1 hn-1,n . ( 56
)
Applying qn-2 to (56) we get

x n-2,α ⋄ ∂ n-2,β hn-2,n-1 hn-2,n-1 -1 =: x n-2,α ∂ n-2,β : hn-2,n-1 hn-2,n-1 -1 -: x n-1,α ∂ n-1,β : 1 hn-2,n-1 -1 -: x nα ∂ nβ : 1 hn-2,n , which gives x n-2,α ⋄ ∂ n-2,β =: x n-2,α ∂ n-2,β -x n-1,α ∂ n-1,β 1 hn-2,n-1 -x nα ∂ nβ hn-2,n-1 -1 hn-2,n-1 hn-2,n
: .

Proceeding further we obtain for any i ≤ n the relation

x iα ⋄ ∂ iβ =: x iα ∂ iβ : - m>i 1 him ϕ im : x mα ∂ mβ : (57)
This is precisely the second line of (50).

Note. Due to the realization L i j → x iα ⋄ ∂jα , the action of the automorphisms qi on the generators L i j can be directly read off the formulas (13). In particular, the action on the diagonal generators is standard, qi (L j j ) = L σ i (j)

σ i (j) .

Central elements

1. Let Λ be a n × n matrix with noncommutative entries belonging to some Ū(h)bimodule, such that the weight of Λ i j equals ε i -ε j , that is, hk Λ i j = Λ i j ( hk + δ i k -δ j k ). Assume that Λ verifies the reflection equation

R12 Λ 1 R12 Λ 1 -Λ 1 R12 Λ 1 R12 = R12 Λ 1 -Λ 1 R12 .
Proposition 4.3 For any nonnegative integer N the elements tr (Λ N Q -) commute with Λ i j for all i and j.

This immediately implies

Corollary 4.4 For any nonnegative integer N the elements tr (L N Q -) are central in the algebra D(g).

Proof of proposition 4.3. The defining relation

( R12 Λ 1 R12 -R12 )Λ 1 = Λ 1 ( R12 Λ 1 R12 -R12 ) implies that ( R12 Λ 1 R12 -R12 )Λ N 1 = Λ N 1 ( R12 Λ 1 R12 -R12
) for any non-negative integer N, or, using (40),

Λ 1 R12 Λ N 1 R12 -Λ N 1 R12 = R12 Λ N 1 R12 Λ 1 -R12 Λ N 1 . (58)
Let Tr (h)2 be the linear map from the space of tensors Ξ i 1 i 2 j 1 j 2 to the space of tensors Υ i 1 j 1 such that Tr (h)2 (Ξ)

i 1 j 1 = (Q -) u v [-ǫ i 1 ]Ξ i 1 v j 1 u
. This is the h-analogue of the R-matrix trace in the second space. We calculate Tr (h)2 of each term of (58).

• The image of the expression Λ 1 R12 Λ N 1 R12 under the map Tr (h)2 is u,a,b,c,d

Q - u [-ǫ i 1 ] Λ i 1 c (Λ N ) a b Rcu ad [ǫ a -ǫ b ] Rbd j 1 u = u,a,b,c,d Λ i 1 c (Λ N ) a b Rcu ad [ǫ a -ǫ b ] Rbd j 1 u Q - u [ǫ a -ǫ b -ǫ c ]
which we rewrite, using ( 25) and ( 42), as = a,b,c

Λ i 1 c (Λ N ) a b Q - a [ǫ a -ǫ b ] δ b a δ c j 1 = Λ i 1 j 1 Tr (Λ N Q -) . • The image of R12 Λ N 1 R12 Λ 1 is u,a,b,d,f (Λ N ) a b Q - u [-ǫ i 1 + ǫ a -ǫ b ] Ri 1 u ad [ǫ a -ǫ b ] Rbd f u Λ f j 1 ,
which, using again ( 25) and ( 42), equals = a,b,f

(Λ N ) a b Q - a δ b a δ i 1 f Λ f j 1 = Tr (Λ N Q -) Λ i 1 j 1 . • The image of R12 Λ N 1 is u,a Q - u [-ǫ i 1 ] Ri 1 u au (Λ N ) a j 1 .
Thus, by (45),

Tr (h)2 ( R12 Λ N 1 ) = Λ N 1 . • The image of Λ N 1 R12 is u,a Q - u [-ǫ i 1 ] (Λ N ) i 1 a Rau j 1 u = (Λ N ) i 1 a Q - u [-ǫ a ] Rau j 1 u
and we again obtain Λ N 1 . Combining these calculations we find that the application of the map Tr (h)2 to the relation (58

) gives ΛTr (Λ N Q -) = Tr (Λ N Q -)Λ as stated. Notes. 1. Tr (L N Q -) = Tr (Q -L N ) since the diagonal elements of L N have weight 0.
2. The reflection equation ( 48) admits shifts L i j → L i j +const • δ i j . 3. Tr Q + = Tr Q -= n. Indeed, with the explicit form (29) of the tensor Ψ, the relation ( 30) is

j 1 1 + hij Q - j = 1 for any i .
Write this relation for gl n+1 , with indices in the range {0, 1, . . . , n}, for i = 0:

n l=1 h0l -1 h0l + n j=1 1 h0l Q - j = 1
(here Q - j corresponds to gl n ). Decomposing into a power series in h-1 0 and comparing coefficients at 1 h0 we find Tr Q

-= n. Since Q - a | h →-h = Q + a , we have Tr Q + = n as well.
2. The images, in the reduction algebra, of e

(2) ij satisfy the same relations as the images of e

(1) ij . We have therefore another set of generators of D(g)

L ′ i j =    s ′ i j ϕ j , i = j s ′ i j - m:m>i s ′ m m 1 him ϕ im ϕ j , i = j (59)
which satisfy the same algebra

R12 L ′ 1 R12 L ′ 1 -L ′ 1 R12 L ′ 1 R12 = R12 L ′ 1 -L ′ 1 R12 .
Using (46), one can check that the matrices L and L ′ are related by

L ′ + L = H .
Here H is the operator C n → Ū(h) ⊗ C n with matrix coefficients

H i j := ( hj + n)δ i j .
By proposition 4.3, for any nonnegative integer N the elements Tr (L ′N Q -) are central in the algebra D(g).

Substituting into the reflection equation the expression for L ′ in terms of L we find

R12 H 1 R12 H 1 -H 1 R12 H 1 R12 = R12 H 1 -H 1 R12 , (60) R12 L 1 R12 H 1 + R12 H 1 R12 L 1 -L 1 R12 H 1 R12 -H 1 R12 L 1 R12 = 2 R12 L 1 -2 L 1 R12 .
Note that the latter equality (one can check it directly) holds for any matrix L whose matrix element L i j has the weight ε i -ε j . Presumably the center is generated by the elements Tr (L N Q -) and Tr (L ′N Q -).

Notes. 1. The center of the algebra of h-deformed differential operators is non-trivial. It is described in [HO].

2. The relation (60) shows that the assignment L → H is a realization of the reflection equation algebra (48).

Braided bialgebra structure

Consider the Ū(h)-bimodule D(g) ⊗Ū (h) D(g). The elements M i j generate the first copy of D(g) and Mi j generate the second copy. These elements satisfy the relations R12

M 1 R12 M 1 -M 1 R12 M 1 R12 = R12 M 1 -M 1 R 12 , R12 M1 R12 M1 -M1 R12 M1 R12 = R12 M1 -M1 R 12 . (61)
We impose the commutation relation

R12 M 1 R12 M1 = M1 R12 M 1 R12 . ( 62
)
By virtue of the dynamical Yang-Baxter equation, with this setting, the Ū(h)-bimodule D(g) ⊗Ū (h) D(g) becomes the associative algebra, which we denote by D(g) ⊚ D(g).

In other words, D(g)⊚D(g) is the associative algebra over Ū(h), generated by elements M i j and Mi j of weight ε i -ε j subject to the defining relations ( 61)-( 62). It is isomorphic to D(g) ⊗Ū (h) D(g) as a Ū(h)-bimodule. In a similar fashion we define the product ⊚ of three and more copies of D(g). The coproduct (63) is clearly coassociative. We have therefore defined the braided bialgebra structure on D(g).

The coproduct (63) has a natural interpretation in terms of differential operators. Divide the interval {1, 2, . . . , N} into two subintervals, {1, 2, . . . , ν} and {ν + 1, 2, . . . , N} for an arbitrary ν, 1 ≤ ν < N. Then M i j → ν α=1 x iα ⋄ ∂jα and Mi j → N α=ν+1 x iα ⋄ ∂jα is a realization of the algebra D(g) ⊚ D(g) while L i j → N α=1 x iα ⋄ ∂jα is a realization of the algebra D(g).

Appendix. Basic relations for n = 2 Denote h = h12 = h 1 -h 2 +1. The defining relations between different copies of generators in the algebra Diff h (2, 2), see ( 35), look as follows:

x 1 ⋄ x ′ 2 = 1 h x ′ 1 ⋄ x 2 + h2 -1 h2 x ′ 2 ⋄ x 1 , ∂1 ⋄ ∂′ 2 = - 1 h ∂′ 1 ⋄ ∂2 + h2 -1 h2 ∂′ 2 ⋄ ∂1 , x 2 ⋄ x ′ 1 = x ′ 1 ⋄ x 2 - 1 h x ′ 2 ⋄ x 1 , ∂2 ⋄ ∂′ 1 = ∂′ 1 ⋄ ∂2 + 1 h ∂′ 2 ⋄ ∂1 , x i ⋄ x ′ i = x ′ i ⋄ x i , ∂i ⋄ ∂′ i = ∂′ i ⋄ ∂i , i = 1, 2, x 1 ⋄ ∂′ 2 = ∂′ 2 ⋄ x 1 , x 2 ⋄ ∂′ 1 = h( h + 2) ( h + 1) 2 ∂′ 1 ⋄ x 2 , x 1 ⋄ ∂′ 1 = ∂′ 1 ⋄ x 1 + 1 1 - h ∂′ 2 ⋄ x 2 -1, x 2 ⋄ ∂′ 2 = 1 1 + h ∂′ 1 ⋄ x 1 + ∂′ 2 ⋄ x 2 -1.
Here the elements {x 1 , x 2 , ∂1 , ∂2 } belong to the first copy and the elements {x ′ 1 , x ′ 2 , ∂′ 1 , ∂′ 2 } belong to the second copy.

The ordering form of the relations (48) is

L 1 1 L 1 2 = h -3 h -2 L 1 2 L 1 1 + 1 h -2 L 1 2 L 2 2 + L 1 2 , L 2 2 L 1 2 = h -3 ( h -2)( h + 1) L 1 2 L 1 1 + ( h -1) 2 ( h -2)( h + 1) L 1 2 L 2 2 - h -1 h + 1 L 1 2 , L 1 1 L 2 1 = ( h + 1) 2 ( h -1)( h + 2) L 2 1 L 1 1 - h + 3 ( h -1)( h + 2) L 2 1 L 2 2 - h + 1 h -1 L 2 1 , L 2 2 L 2 1 = - 1 h + 2 L 2 1 L 1 1 + h + 3 h + 2 L 2 1 L 2 2 + L 2 1 , L 1 1 L 2 2 = L 2 2 L 1 1 , L 1 2 L 2 1 = L 2 1 L 1 2 - 1 h(L 1 1 -L 2 2 ) 2 + L 1 1 -L 2 2 .
The central elements of Corollary 4.4 have the form

h -1 h (L N ) 1 1 + h + 1 h (L N ) 2 2 .

Lemma 4. 5

 5 The matrix M + M satisfies the reflection equation (48D(g) → D(g) ⊚ D(g) of algebras.

  For instance, D(g) ⊚ D(g) ⊚ D(g) is generated by M i j , Mi j and Mi j , with the defining relations (61)-(62) and, in addition,R12 M1 R12 M1 -M1 R12 M1 R12 = R12 M1 -M1 R 12 , R12 M 1 R12 M1 = M1 R12 M 1 R12 , R12 M1 R12 M1 = M1 R12 M1 R12 .

Unless the opposite is stated, we adopt the Einstein convention: if a tensor index appears in an expression twice, once as an upper index and once as a lower index, the summation over this index is assumed.

Acknowledgments. The work of S. K. was supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program, by the RFBR grant 14-01-00547 and joint RFBR-Ukrainian Academy of Sciences grant 14-01-90405-Ukr.


S Khoroshkin

O Ogievetsky

Contravariant form for reduction algebras and Pieri rule

We study properties and constructions of contravariant forms on reduction algebras. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gl n with a symmetric or wedge tensor power of its fundamental representation. Their zeroes describe Pieri rules.

Introduction

The contravariant (or Shapovalov) form on highest weight modules is a powerful tool in the representation theory of reductive Lie algebra. It is used for the construction of irreducible representations, description of singular vectors of Verma modules etc [D]. In this paper we define an analogue of the Shapovalov form for certain reduction algebras, compute it and apply to the the well-known problems of classical representation theory, calculating the norms of n + -invariant vectors in tensor products of irreducible finitedimensional representation of the Lie algebra gl n and symmetric or exterior powers of its fundamental representation. Here n + is a Lie algebra of upper triangular matrices. Zeros of these norms describe Pieri rules. The norms themselves can be regarded as a generalization of particular Clebsch-Gordan coefficients.

To perform these calculations we pass to three particular reduction algebras: diagonal reduction algebra D(gl n ), and Diff h (n) together with its odd analogue GDiff h (n), see Section 2.3 for definitions. The first algebra may be regarded as a deformation of U(gl n ) with coefficients in the localized universal enveloping algebra Ū(h), and the latter two as analogous deformations of the algebras of polynomial differential operators in even or odd variables. These algebras possess their own Ū(h)-valued contravariant forms whose specializations to dominant weights is then used for the calculation of norms of n +invariant vectors in tensor products of finite-dimensional representations of gl n .

The main ingredient of our applications of the theory of reduction algebras is the connection of the contravariant form on the reduction algebra to the Zhelobenko automorphism ξw 0 , where w 0 is the longest element of the Weyl group of gl n . The origin of this connection goes back to Zhelobenko, see [Zh]. It was reformulated, proved and used in [KN] for the description of irreducible representations of Yangians. We reproduce here the arguments of [KN] and then compute the contravariant form on polynomial representations of the algebras Diff h (n) and GDiff h (n) in two ways: using ξw 0 and by direct computations in the latter reduction algebras.

The paper is organized as follows. In Sections 2.1-2.2 we recall the definition of Mickelsson algebras and their localizations called reduction algebras, introduce Zhelobenko automorphisms and describe in Section 2.3 our basic examples -reduction algebras D(g), Diff h (n) and GDiff h (n). In Section 3.3 we introduce a natural class of ( Z, h)-modules over reduction algebras and a notion of Ū(h)-valued contravariant forms on them. We establish a connection of these forms with the contravariant forms on n + -invariants and n --coinvariants of certain g-modules. Here g is a reductive Lie algebra, n ± are their opposite nilpotent subalgebras. In Section 3.4 we describe analogues of the Harish-Chandra map for our basic examples of reduction algebras and define with their help contravariant forms on these algebras. Section 3.5 is devoted to the calculation of these forms on basic polynomial representations of the algebras Diff h (n) and GDiff h (n). Sections 4.1-4.2 are devoted to the justification of the evaluations of the computed contravariant forms and their use for the norms of n + -invariant vectors in tensor products of irreducible finitedimensional representations of the Lie algebra gl n and symmetric or exterior powers of its fundamental representation. In Section 4.3 we deduce the Pieri rules. Appendices contain an alternative derivation of norms of n + -invariant vectors.

Reduction algebras 2.1 Three types of reduction algebras

Let g be a finite-dimensional reductive Lie algebra with a fixed triangular decomposition g = n + + h + n -, where h is Cartan subalgebra, n + and n -are two opposite nilpotent subalgebras. We denote by ∆ the root system of g and by ∆ + the set of positive roots. Let A be an associative algebra which contains the universal enveloping algebra U(g). In particular, A is a U(g)-bimodule with respect to the left and right multiplications by elements of U(g). We assume that A is free as the left U(g)-module and, moreover, that A contains a subspace V , invariant with respect to the adjoint action of U(g) such that A is isomorphic to U(g) ⊗ V as the left U(g) module. The action on U(g) ⊗ V is diagonal. The adjoint action of g on V is assumed to be reductive.

In this setting we have three natural reduction algebras. The Mickelsson [M] algebra Z + = Z(A, n + ) is defined as the quotient of the normalizer of the left ideal J + = An + modulo J + . The Mickelsson algebra Z -= Z(A, n -) is defined as the quotient of the normalizer of the right ideal J -= n -A modulo J -.

In the following we assume that A is equipped with an anti-involution ε whose restriction to U(g) coincides with the Cartan anti-involution:

ε(e αc ) = e -αc , ε(h) = h for any h ∈ h , (1)
where α c , c = 1, ..., r, are simple roots in ∆ + and e ±αc and h αc = αc are Chevalley generators of g, normalized by the conditions

[h αc , e ±αc ] = ±2e ±αc , [e αc , e -αc ] = h αc .
Due to (1), ε(J + ) = J -and ε(Z + ) = Z -so that ε establishes an anti-isomorphism of the associative algebras Z + and Z - Denote by K the multiplicative set, which consists of finite products of elements

h α + k , k ∈ Z . (2)
Here h α ∈ h is the coroot corresponding to a root α of the root system ∆ of the Lie algebra g. For the construction of the third reduction algebra we localize with respect to K the enveloping algebras U(h), U(g) and the algebra A, denoting by Ū(h), Ū(g) and Ā the corresponding rings of fractions. Define Z and Z = Z(A, n ± ), Z ⊂ Z, as the double coset spaces Z = A/(J -+ J + ) , Z = Ā/( J-+ J+ ) ,

where J+ = Ān + and J-= n -Ā. The localized double coset space Z is an associative algebra with respect to the multiplication ⋄ , see e.g. [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF] for details. The multiplication ⋄ is described by the rule

x ⋄ y = x P y mod J+ + J-, (3)
where P is the extremal projector [AST] for g, P 2 = P. The projector P belongs to a certain extension of Ū(g) (see [Zh] for details), satisfies the properties

x P = P y = 0 for x ∈ n + , y ∈ n -, (4)
P = 1 mod n -Ū(g), P = 1 mod Ū(g)n + , (5)
ε(P) = P , (6)
and can be given [AST] by the explicit multiplicative formula (45). Alternatively, one can take representatives x ∈ Ā and ỹ ∈ Ā of coset classes x and y such that either x belongs to the normalizer of the left ideal Ān + or ỹ belongs to the normalizer of the right ideal n -Ā. Such representatives exist, see Lemma 2.1 (ii) below. Then x ⋄ y is the image in the coset space à of the product x • ỹ. The latter description shows that the maps ι ± : Z ± → Z, defined as compositions of natural inclusions and projections

ι + : Z + = Norm(J + )/ J + → A/ J + → Z ⊂ Z , ι -: Z -= Norm(J -)/ J -→ J -/A → Z ⊂ Z , (7)
are homomorphisms of algebras.

For each root α of the root system ∆ of the Lie algebra g denote by hα

∈ Ū(h) the element hα = h α + (ρ, h α ),
where ρ ∈ h * is the half sum of positive roots. Denote by K + ⊂ K the multiplicative set, which consists of finite products of elements ( hα + k) where k is a positive integer.

Lemma 2.1 (i) The maps ι ± are injective.

(ii) For each z ∈ Z there exist polynomials d + , d -∈ K + such that d + • z belongs to the image of i + and z • d -belongs to the image of i -.

(iii) The anti-involution ε induces an anti-automorphism of the double coset algebra Z, leaves invariant the subspace Z and maps the images of Z ± to the images of Z ∓ .

Proof. (i) If x ∈ Norm(J + ) then due to (5), P x ≡ x mod J+ (in the above mentioned extension of Ā). If ι + (x) = 0 then x ∈ J + + J -, but P J -= 0 by the properties of the projector thus x ∈ J + .

(ii) For any x ∈ A the element P x (which is in the above extension of Ā) belongs to the normalizer of J + by the properties of the projectors. Present P as a series P = i d i f i e i , where d i are elements of Ū(h),

f i ∈ U(n -), e i ∈ U(n + ). Then P x ≡ i d i f i êi (x) mod J+ ,
where êi (x) is the adjoint action of e i on x. Since the adjoint action of n + in Ā is locally finite, the latter sum is finite and belongs to the normalizer of J + in Ā. Multiplying this sum by the common multiple of d i we get the element of Norm(J + ) in A.

(iii) Straightforward.

Zhelobenko operators

It follows that the adjoint action of g on A, x(a) := xa -ax, x ∈ g, a ∈ A, is locally finite and semisimple. That is, A can be decomposed into a direct sum of finite-dimensional g-modules with respect to the adjoint action of g. We assume also that simple reflections σ c , c = 1, ..., r, of h, generating the Weyl group of g are extended to automorphisms of the algebra A, preserving U(g). We denote them by the same symbols and assume that they still satisfy the corresponding braid group relations

σ a σ b σ a • • • m ab = σ b σ a σ a • • • m ab , a, b = 1, ..., r, a = b , (8)
where m ab = 2 if c ab = 0, m ab = 3 if c ab c ba = 2 and m ab = 6 if c ab c ba = 3 with c ab the Cartan matrix of g. Since the adjoint action of g in A is reductive, there is a common choice of such an extension2 , see e.g. [K],

σ c (x) = e ede αc • e -ade -αc • e ade αc (x) . (9)
Denote by qc the linear map qc : A → Ā/ J+ given by the relation

qc (x) := k≥0 (-1) k k! êk αc (σ c (x))e k -αc k j=1 (h αc -j + 1) -1 mod J+ . ( 10
)
Properties of the operator qc are listed in the following proposition [Zh], see also [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF].

Here the shifted action of the Weyl group on h is used:

w • hα = hw(α) .
Proposition 2.2 (i) qc (J + ) = 0;

(ii) qc (J -) ⊂ J-+ J+ / J+ ;

(iii) qc (hx) = (σ c • h)q c (x) for any x ∈ A and h ∈ h;

(iv) qc (xh) = qc (x)(σ c • h) for any x ∈ A and h ∈ h.

The last two properties allow to extend the map qc to the map of the localized algebras qc : Ā → Ā/ J+ . The properties (i) and (ii) show that the map qc defines a linear map of the double coset algebra Z to itself. The Zhelobenko maps satisfy the braid group relations [Zh]:

qa qb qa • • • m ab = qb qa qa • • • m ab , a, b = 1, ..., r, a = b (11)
and the inversion relation [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF]:

q2 c (x) = (h αc + 1) -1 σ 2 c (x) (h αc + 1) mod J+ . (12)
In [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF] we established the following homomorphism properties of the Zhelobenko maps qc .

Proposition 2.3 Zhelobenko map qc defines a homomorphism of the Mickelsson algebra Z + to the double coset algebra Z and an automorphism of the double coset algebra Z.

One can equally start from the right ideal J -and define Zhelobenko operators ξc = εq c ε :

A → J-\ Ā: ξc (x) := k≥0 1 k! k j=1 (h αc -j + 1) -1 e k αc êk -αc (σ c (x)) mod J-. (13)
As well as qc the maps ξ c determine the automorphisms ξ c : Z → Z of the double coset algebra, satisfying the braid group relations (8).

Proposition 2.4 The following relation between automorphisms qc and ξc of the double coset algebra Z takes place ξc

(x) = q-1 c (σ c ε) 2 (x) , ( 14
)
where x is a representative in Ā of the double coset.

Proof. It is sufficient to check (14) for the sl 2 subalgebra g c of g related to the simple root α c . The operators q and ξ are automorphisms of the algebra P h (2) (see precise definitions below) so it is sufficient to check (14) for the 2-dimensional representation since all other representations arise as the homogeneous components of P h (2). With the explicit formulas for q, see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF], the calculation for the 2-dimensional representation is immediate. See also [KNS].

Note that the automorphism (σ c ε) 2 is the involution which is -1 on even-dimensional irreducible representations of g c , and +1 on odd-dimensional irreducible representations of g c .

For g = gl n , the symmetric group S n acts on the universal enveloping algebra U(g) by permutation of indices. In the sequel we shall use this action to extend the automorphisms σ c of the Weyl group action on h. In this situation the automorphism (σ c ε) 2 is identical.

Reduction algebras Diff h (n) and D(g)

In the sequel we use the following notation for the Lie algebra gl n . The standard generators are denoted by e ij , the Cartan elements e ii by h i . We set h ij = h i -h j , hi = h i -i and hij = hihj . The space h * is spanned by the elements ǫ i , ǫ i (h j ) = δ j i . Let Diff(n) be an associative ring of polynomial differential operators in n variables x i , where i = 1, ..., n. It is generated by the elements x i and ∂ i , i = 1, ..., n, subject to the defining relations

[x i , x j ] = [∂ i , ∂ j ] = 0 , [∂ i , x j ] = δ j i . (15)
Let ψ : U(gl n ) → Diff(n) be the homomorphism of associative algebras, such that

ψ(e ij ) = x i ∂ j . ( 16
) Set A = Diff(n) ⊗ U(gl n ) . ( 17
)
This algebra contains U(gl n ) as a subalgebra generated by the elements

ψ(e ij ) ⊗ 1 + 1 ⊗ e ij , i, j = 1, ..., n .
The corresponding double coset reduction algebra Z is denoted further by Diff h (n) and is called the algebra of h-differential operators. The algebra Diff h (n) is generated over Ū(h) by the images of the elements 1 ⊗ x i and 1 ⊗ ∂ i , which we denote for simplicity by the same letters x i and ∂ i . They satisfy quadratic relations which can be written in the R-matrix form, see [START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF]Proposition 3.3].

As an Ū(h)-module, Diff h (n) is freely generated by images in Z of elements 1 ⊗ d, where d ∈ Diff(n). To distinguish elements in Diff(n) and in Diff h (n), we use sometimes the notation : d : for the image in Diff h (n) of a polynomial differential operator d. The anti-involution ε : Diff(n) → Diff(n) is given by the rule

ε(x i ) = ∂ i , ε(∂ i ) = x i .
For the definition of the Zhelobenko operators we use the action of the symmetric group S n , which permutes indices of the generators x i and ∂ i .

The same construction applied to the ring GDiff(n) of Grassmann differential operators, generated by the odd generators ζ i and δ i , i = 1, ..., n, with the defining relations

ζ i ζ j + ζ j ζ i = δ i δ j + δ j δ i = 0 , ζ i δ j + δ j ζ i = δ i j ( 18
)
and the homomorphism ϕ : U

(gl n ) → GDiff(n), such that ϕ(e ij ) = ζ i δ j (19)
gives rise to the reduction algebra GDiff h (n).

For any reductive Lie algebra g one can define the diagonal reduction algebra as follows. Set A = U(g) ⊗ U(g) and use the diagonally embedded U(g) as U(g)-subalgebra of A. This subalgebra is generated by the elements x (1) + x (2) , where, for x ∈ g, x (1) := x ⊗ 1, x (2) := 1 ⊗ x. The Chevalley anti-involution ε and the braid group action on U(g) is naturally extended to its tensor square. The corresponding reduction algebra is denoted by D(g) and is called the diagonal reduction algebra.

There are two families of natural generators of D(g). The first family is given by the images of the elements x (1) , x ∈ g. In particular, we denote the images of Cartan-Weyl generators e

(1)

α by s (1)
α , and the images of the elements h

(1) α , h α ∈ h by t (1) α .
The second family is given by the projections of the elements x (2) , x ∈ g, where we use analogous notations with the change of the upper index. Clearly,

s (1) α + s (2) α = 0 , and
t (1) α + t (2) α = h α , α ∈ ∆ .
We will be mainly interested in the diagonal reduction algebra D(gl n ). The algebraic structure of the D(gl n ) was studied in [START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF][START_REF] Khoroshkin | Diagonal reduction algebra and reflection equation[END_REF]. Note that the homomorphisms ( 16) and ( 19) define the homomorphisms of the reduction algebras

ψ : D(gl n ) → Diff h (n) and ϕ : D(gl n ) → GDiff h (n) . ( 20
)
3 Contravariant forms

Extremal projector and n ∓ -(co)invariants

Let M be an A-module. Then the space M • = M n + of n + -invariants (or singular vectors or highest weight vectors) is a Z + -module, and the space

M • = M n -= M/n -M of n -- coinvariants is a Z --module.
Assume further that M is locally n + -finite, and the action of h is semisimple with non-singular (sometimes called dominant) weights, that is

M = ⊕ λ∈h * M λ , hv = (h, λ)v v ∈ M λ , h ∈ h , and (h α , λ + ρ) = -1, -2, ... , α ∈ ∆ + . (21)
Equivalently, the eigenvalues of all elements hα , α ∈ ∆ + , are not negative integers. In this case, the action of the extremal projector P on M is well defined, and the properties (4) of P imply that its image in End M establishes an isomorphism of n + -invariants

M • and n --coinvariants M • : P : M • M • , x → P x . ( 22
)
If in addition the eigenvalues of all elements hα , α ∈ ∆ + , are generic, that is,

(h α , λ + ρ) ∈ Z , α ∈ ∆ + , (23)
then each of these isomorphic spaces comes equipped with a Z-module structure. The multiplication by elements z ∈ Z, which we sometimes denote by the symbol ⋄ of the multiplication in the double coset algebra, can be described in several ways. First, using Lemma 2.1, we can multiply z by a polynomial d -from the right and get an element of Z -, which we use for the action on coinvariants M • ; the multiplication by d -on each weight space is then given as the multiplication by a nonzero number thus is an invertible operator on M • , so this allows to define the action of z itself. Second, we can multiply any representative of M • by P z P (or z P ) and get another element of M • . We can use analogous arguments for M • with the passage from z to an element of Z + . Finally, we can multiply an element of M • directly by P z P (or P z ) and get another element of M • .

There is another special case of a natural identification of n + -invariants with n -coinvariants. Assume that the restriction of an A-module M to g is decomposed into a direct sum of finite-dimensional g-modules. In this case not all weights of M are nonsingular, but the weights of M • and of M • are dominant, that is, (h α , λ) = 0, 1, 2, 3, . . . , α ∈ ∆ + , due to the structure of irreducible finite-dimensional g-modules. Thus we have a well defined action of P on M • and M • , establishing an isomorphism of them. The action of Z + on M • can be extended to the action of elements from Z, and the action of Z -on M • can be also extended to the action of elements from Z due to Lemma 2.1.

The functor, attaching to a A-module M, whose restriction to g decomposes into a direct sum of finite-dimensional g-modules, the Z + -module M • is faithful and sends irreducible representations to irreducible representations. To show the latter property, we choose two highest weight vectors v, u ∈ M • . If M is irreducible, then there exists a ∈ A, such that av = u. Then P av = u as well. Repeating the arguments used in the proof of Lemma 2.1 we can replace P a by an element d -1 a ′ , where a ′ ∈ Norm(J + ) and the denominator d ∈ K + is such that a ′ v = du. Since any highest weight of the finite dimensional module is non-singular, d acts on u by multiplication by a nonzero scalar c, so the element a ′′ = c -1 a ′ ∈ Norm(J + ) maps v to u, a ′′ v = u. An analogous picture holds for the space M • of coinvariants and the algebra Z -.

Contravariant forms

Let M be an A-module. A symmetric bilinear form (, ) :

M ⊗ M → C is called contravari- ant 3 if (ax, y) = (x, ε(a)y) (24)
for any x, y ∈ M and a ∈ A. Let M be an A-module equipped with a contravariant form (, ). Then this form induces a pairing ( , ) :

M • ⊗ M • → C
which is contravariant for a pair of reduction algebras Z -and Z + , that is,

(gx, y) = (x, ε(g)y) , x ∈ M • , y ∈ M • , g ∈ Z -, ε(g) ∈ Z + . ( 25
)
If M is locally n + -finite, and the action of h is semisimple with non-singular weights, see (21), then due to the isomorphism of the spaces M • and M • the contravariant form on M induces the contravariant form on the space M • of n --coinvariants, so that its value (u, u ′ ) • on two elements u and u ′ of M • is equal to

(u, u ′ ) • = (u, P u ′ ) .
3 The content of this section can be equally repeated for a sesquilinear contravariant form.

This form satisfies the following contravariant property:

(gu, u ′ ) = (u, ε(g) P u ′ ) (26)
for any u, u ′ ∈ M • and g ∈ Z -.

On the other hand, a contravariant form on M defines a symmetric bilinear form (, ) • on M • by restriction. Under the above assumptions it satisfies the contravariant property If the eigenvalues of all elements hα , α ∈ ∆ + , in M are generic, then both forms on isomorphic spaces

(gv, v ′ ) • = (v, P ε(g)v ′ ) • (
M • and M • are Z -contravariant, (gu, v) = (u, ε(g)v), for any g ∈ Z and u, v ∈ M • (or u, v ∈ M • ).

( Z, h)-modules

Let now M be a left module over the reduction algebra Z. We call it a ( Z, h)-module, or h-module over the reduction algebra Z if, in addition, M has a structure of a free right Ū(h)-module such that:

-(z ⋄ m) • h = z ⋄ (m • h) for any z ∈ Z, m ∈ M and h ∈ Ū(h);
-the adjoint action of h on M is semisimple. These conditions imply that M is also free as a left Ū(h)-module. For example, the reduction algebra Z itself is the h-module over itself with respect to the left multiplication by elements of Z and the right multiplication by elements of Ū(h).

Assume that the weights of the adjoint action of h are generic, see (23). Then for any µ ∈ h * we can define the "evaluation" Z-module M(µ),

M(µ) = M/MI µ ( 28
)
where I µ is the (maximal) ideal in Ū(h) generated by elements h -(µ, h) for all h ∈ h.

We define a contravariant form on an h-module M as a contravariant map ( , ) : M ⊗ M → Ū(h), which is linear with respect to the right action of Ū(h),

(g ⋄ u, v) = (u, ε(g) ⋄ v) , g ∈ Z , (uh, v) = (u, vh) = (u, v)h , h ∈ Ū(h) , (29)
for any u, v ∈ M. For a generic µ ∈ h * , the evaluation of a contravariant form on an h-module M determines a C-valued contravariant form on M(µ).

Here is the main example, which we use in this paper, of h-modules over reduction algebras. Assume that we are given a pair (B, γ), which consists of an associative algebra B and an algebra homomorphism γ : U(g) → B. Let A = B ⊗ U(g). Then A contains the diagonally embedded subalgebra U(g), generated by the elements γ(x) ⊗ 1 + 1 ⊗ x, x ∈ g.

Let M be a B-module, given as a quotient of B over its left ideal I, which contains all the elements γ(x), x ∈ n + . Assume that the action of elements of γ(h), h ∈ h, is semisimple and all the weights ν of this action are integers, ν(h α ) ∈ Z, for any α ∈ ∆. Consider the left A -module N = M ⊗ (U(g)/ U(g)n + ). Let N be the localization of N, which consists of the left fractions d -1 n, where n ∈ N and d is an element of the multiplicative set K, generated by the elements (b α + k), k ∈ Z, where b α = γ(h α ) ⊗ 1 + 1 ⊗ h α are elements of the diagonally embedded Cartan subalgebra. Define M (h) to be the space of n --coinvariants of N with respect to the diagonally embedded n -,

M (h) = N/n -N , where N = M ⊗ (U(g)/ U(g)n + ) .
By construction, M (h) is a quotient of Ā by the sum of the right ideal Jand the left ideal containing J+ . Thus M (h) is a quotient of the double coset space Z = Ā/( J+ + J-) by the image in Z of some left ideal in Ā. Due to the structure of the multiplication in Z, a ⋄ b = a P b, this image is also a left ideal in Z, so

M (h) is a left Z-module. For any m ∈ M and h ∈ h we set m • h := m(1 ⊗ h) . (30)
Since elements 1 ⊗ h normalize all the ideals defining M (h) , this is a well defined free right action of U(h), commuting with the Z-action on M (h) . Moreover, due to the integer conditions on the weights of the initial module M, this action has a natural extension to the action of Ū(h),

m • (h α + k) -1 := (h α + k -ν(m)(h α )) -1 m ,
where ν(m) ∈ h * is the weight of m. For a generic µ ∈ h * (that is, µ(h α ) ∈ Z for any α ∈ ∆) the specialization M (h) (µ) is isomorphic to the space of n --coinvariants of the tensor product M ⊗ M µ , where M µ is the Verma module of g with the highest weight µ,

M (h) (µ) ≃ (M ⊗ M µ ) • , which is isomorphic, in its turn, to the space (M ⊗ M µ ) • of n + - invariants, see (22). Denote by π µ : M (h) → (M ⊗ M µ )
• the composition of the evaluation map with the above isomorphisms. Then

π µ (xh) = π µ (x)µ(h) for any x ∈ M (h) , h ∈ h, and π µ ((m ⊗ 1) • f ) = P •(m ⊗ 1 µ ) • f (µ) (31)
for any m ∈ M and f ∈ Ū(h). Here f (γ) is the evaluation of f , regarded as a rational function on h * at the point γ ∈ h * . The space (M ⊗ M µ ) • has a natural structure of the module over the corresponding reduction algebra Z + which extends, due to conditions on µ, to the structure of Z -module. Note also that the homomorphism γ : U(g) → B induces the homomorphism of the reduction algebras γ : D(g) → Z (32) so that M (h) carries as well the structure of a ( D(g), h)-module. Assume that the module M (h) is equipped with a contravariant form (, ). For a generic µ this form induces a contravariant form on (M ⊗ M µ ) • by the rule

(π µ (x), π µ (y)) = (x, y)(µ) , x, y ∈ M (h) .
(33)

The ( Z, h)-modules which we use in this paper arise from the rings P(n) = C[x 1 , ..., x n ] of polynomial functions in commuting variables and from the ring G(n) = C[ζ 1 , ..., ζ n ] of polynomial functions in anti-commuting variables. The ring P(n) is a module over the ring Diff(n) and over the Lie algebra gl n . Analogously, the ring G(n) is a module over the ring GDiff(n) and over the Lie algebra gl n .

We define Diff h (n)-module P h (n) as a quotient of the reduction algebra Diff h (n) over the left ideal I ∂ , generated by all ∂ i , i = 1, ..., n. Since the Cartan subalgebra normalizes the ideal I ∂ , I ∂ h ⊂ I ∂ for any h ∈ h, and the weight of any monomial is integer, we have the right action of Ū(h) on Diff h (n) which supplies P h (n) with a structure of h-module over the reduction algebra Diff h (n). We define analogously the GDiff h (n)-module G h (n).

In terms of the constructions above we set

B = Diff(n), γ = ψ, see (16), M = Diff(n)/ Diff(n){∂ 1 , ..., ∂ n } in the even case and B = GDiff(n), γ = ϕ, see (19), M = GDiff(n)/GDiff(n){δ 1 , ..., δ n } in the odd case.
Example. Let V be the two-dimensional tautological representation of gl 2 with the basis {v 1 , v 2 }. The ( D (gl 2 ), h)-module V (h) is free as a one sided Ū(h)-module of rank 2. Its left D (gl 2 )-module structure is described by the following formulas:

s (1) 12 v 1 = 0 , s (1) 12 v 2 = v 1 h 12 h 12 + 1 , s (1)
21 v 1 = v 2 , s (1)
21 v 2 = 0 , s (1)
11 v 1 = v 1 , s (1)
11 v 2 = v 2 1 h 12 + 1 , s (1)
22 v 1 = 0 , s

(1)

22 v 2 = v 2 h 12 h 12 + 1 , hi v j = v j ( hi + δ j i ) .

Harish-Chandra maps

Constructions of contravariant forms for reduction algebras refer to analogues of Harish-Chandra map for the universal enveloping algebras of reductive Lie algebras. We describe this map in our two basic examples.

Lemma 3.1 (i) The left ideal I ∂ = Diff h (n) ⋄ {∂ 1 , . . . , ∂ n } of Diff h (n) is generated over Ū(h) by the classes of elements Y ∂ i where Y ∈ Diff(n), i = 1, ..., n.
(ii) The right ideal

I x = {x 1 , . . . , x n } ⋄ Diff h (n) of Diff h (n) is generated over Ū(h) by the classes of elements x i X where X ∈ Diff(n), i = 1, ..., n. (iii) The natural inclusion Ū(h) → Diff h (n) establishes the isomorphism of Ū(h)- modules Ū(h) and Diff h (n)/(I ∂ + I x ).
Proof. (i) This is a corollary of the property (4) of the extremal projector, together with the ad n + -invariance of the linear span of ∂ i .

(ii) Parallel to (i).

(iii) Follows from the Poincaré-Birkhoff-Witt property of the ring Diff h (n): elements

: (x 1 ) a 1 . . . (x n ) an ∂ b 1 1 . . . ∂ bn n : form a basis of Diff h (n) over Ū(h), see [KO4].
The map Diff h (n) → Ū(h), which attaches to any element x ∈ Diff h (n) the unique element x (0) ∈ Ū(h) such that x -x (0) ∈ I ∂ + I x is an analogue of the Harish-Chandra map U(g) → U(h). With its use we define in a standard way the Ū(h)-valued bilinear form on Diff h (n) and on its left module P h

(n) = Diff h (n)/ I ∂ : (x, y) = (ε(x) ⋄ y) (0) . ( 34
) Recall that ε(x i ) = ∂ i , ε(∂ i ) = x i ; in particular, ε(I ∂ ) = I x .
It is not difficult to show that this form is contravariant, see ( 29), and symmetric (x, y) = (y, x)

for any x, y ∈ Diff h (n) or x, y ∈ Pn . The same statements take place for (GDiff

h (n), h)- module G h (n).
The diagonal reduction algebra D(g) contains a family of commuting, see [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF][START_REF] Khoroshkin | Diagonal reduction algebras of gl type; Funktsional[END_REF], elements t

(1) α (in the notation of Section 2.3). Let C[t] be the ring of polynomials in t

(1)

α , α ∈ ∆ + . Lemma 3.2 (i) The left ideal I + = D(g) ⋄ {s (1) α , α ∈ ∆ + } of D(g) is generated over Ū(h) by the classes of elements Y e (1) α , where Y ∈ U(g) (1) , α ∈ ∆ + . (ii) The right ideal I -= {s (1) -α , α ∈ ∆ + } ⋄ D(g) of D(g) is generated over Ū(h) by the classes of elements e (1) -α X, where X ∈ U(g) (1) , α ∈ ∆ + . (iii) The natural inclusion Ū(h) ⊗ C[t] → D(g) establishes the isomorphism of Ū(h)- modules Ū(h) ⊗ C[t] and D(g)/(I + + I -).
Proof. (i) This is a corollary of the property (4) of the extremal projector, together with the ad n + -invariance of the linear space n + ⊗ 1.

(ii) Parallel to (i).

(iii) As in Lemma 3.1, this follows from Poincaré-Birkhoff-Witt property of D(g), see [START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF].

The map D(g) → Ū(h) ⊗ C[t], which attaches to any element u ∈ D(g) the unique element u (0) ∈ Ū(h) ⊗ C[t] such that u -u (0) ∈ I + + I -is an analogue of the Harish-Chandra map U(g) → U(h). With its use we define in a standard way the Ū(h) ⊗ C[t]valued bilinear form on D(g):

(u, v) = (ε(u) ⋄ v) (0) . ( 35
)
The contravariant forms ( 34) and ( 35) are compatible. Namely, the bilinear form (34) on P h (n) is also contravariant with respect to D(gl n ) due to the homomorphism ( 16). As a D(gl n )-module, P h (n) decomposes into a direct sum of homogeneous components. The component P h (n; k) of degree k is generated by the element : (x 1 ) k : annihilated by the ideal I + . We have (:

(x 1 ) k : , : (x 1 ) k :) = k! .
The restriction of the form (34) to P h (n; k) can be obtained by the evaluation, see (17),

t 11 := e (1)
11 → k!, t jj := e

(1) jj → 0, j > 1, of the form (35). For the anti-commuting variables, the homogeneous component of degree k is generated by the element : ζ 1 ζ 2 . . . ζ k : annihilated by the ideal I + , for which

(: ζ 1 ζ 2 . . . ζ k : , : ζ 1 ζ 2 . . . ζ k :) = 1 .
Now the evaluation is t ii → 1, i = 1, . . . , k, and t jj := e

(1) jj → 0, j > k. The bilinear form ( 34) is covariant with respect to the action of Zhelobenko operators in the following sense.

Lemma 3.3 For any elements x, y ∈ P h (n) or x, y ∈ G h (n) we have qc (x, y) = ( ξc (x), qc (y)) .

(36)

Proof consists in the following calculation: qc (x, y) = qc ((ε(x) ⋄ y) 0 ) = (q c (ε(x) ⋄ qc (y)) 0 = (ε( ξc (x) ⋄ qc (y)) 0 = ( ξc (x), qc (y)) .

Calculations of contravariant form on P h (n) and G h (n)

We use the notation x ↑a = x(x + 1) . . . (x + a -1) and x ↓a = x(x -1) . . . (x -a + 1) for the Pochhammer symbols.

Proposition 3.4 Images of monomials :

x ν :=: (x 1 ) ν 1 • • • (x n ) νn : in P h (n)
have the following scalar products:

(: x ν :, :

x ν ′ :) = δ ν,ν ′ n k=1 ν k ! • i,j:i<j ( hij -ν j ) ↑ν i +1 h↑ν i +1 ij = δ ν,ν ′ n k=1 ν k ! • i,j:i<j Γ( hij -ν j + ν i + 1)Γ( hij ) Γ( hij -ν j )Γ( hij + ν i + 1) . ( 37
)
Proposition 3.5 Images of monomials :

ζ ν :=: (ζ 1 ) ν 1 • • • (ζ n ) νn : in G h (n)
have the following scalar products:

(: ζ ν :, : ζ ν ′ :) = δ ν,ν ′ i,j:i<j ( hij -ν j ) 1-ν i ( hij ) 1-ν i = δ ν,ν ′ i,j:i<j ( hij -1 + ν i ) ν j ( hij ) ν j (38)
We present two different proofs of Propositions 3.4 and 3.5. The first proof is based on the description of the contravariant form for certain reduction algebras given in [KN]. We reproduce it in the particular case of the reduction algebra Diff h (n). Let w 0 be the longest element of the symmetric group S n , regarded as the Weyl group of Lie algebra gl n , w 0 = (n, n -1, ..., 2, 1). Let

w 0 = s c 1 s c 2 • • • s c N , N = n(n-1)

2

, be a reduced decomposition of w 0 . Set

qw 0 = qc 1 qc 2 • • • qc N , ξw 0 = ξc 1 ξc 2 • • • ξc N .
Due to the braid relation ( 11), definitions of qw 0 and ξw 0 do not depend on a reduced decomposition of w 0 and qw 0 ξw 0 = ξw 0 qw 0 = 1 .

For any two monomials

x ν = (x 1 ) ν 1 (x 2 ) ν 2 ...(x n ) νn and x ν ′ = (x 1 ) ν ′ 1 (x 2 ) ν ′ 2 ...(x n ) ν ′ n in com- muting variables x 1 , ..., x n , and elements ϕ 1 , ϕ 2 ∈ Ū(h) set <: x ν : ϕ 1 , : x ν ′ : ϕ 2 >= ϕ 1 ϕ 2 δ ν,ν ′ n k=1 ν k ! . (39)
This defines a Ū(h)-valued pairing on P h (n), linear with respect to the right multiplication by elements of Ū(h). Analogously, for any two monomials

ζ ν = (ζ 1 ) ν 1 (ζ 2 ) ν 2 ...(ζ n ) νn and ζ ν ′ = (ζ 1 ) ν ′ 1 (ζ 2 ) ν ′ 2 ...(ζ n ) ν ′ n in anti-commuting variables ζ 1 , ..., ζ n , and elements ϕ 1 , ϕ 2 ∈ Ū(h) set <: ζν : ϕ 1 , : ζν ′ : ϕ 2 >= ϕ 1 ϕ 2 n k=1 δ ν k ,ν ′ k . (40)
This defines a Ū(h)-valued pairing on G h (n), linear with respect to the right multiplication by elements of Ū(h). We have, see also eq. (3.18) in [KN, Section 3.3],

Proposition 3.6 (i) For any two monomials x ν , and x ν ′ the contravariant pairing of their images in P h (n) is equal to

(: x ν :, : x ν ′ :) = qw 0 <: w 0 (x ν ) :, ξw 0 (: x ν ′ :) > . (41)
(ii) For any two monomials ζ ν , and ζ ν ′ the contravariant pairing of their images in Ḡn is equal to (: ζ ν :, : ζ ν ′ :) = qw 0 <: w 0 (ξ ν ) :, ξw 0 (:

ζ ν ′ :) > . (42)
Here in the right hand side of ( 41) and ( 42) we use the action of the symmetric group on monomials in P n and G n by permutations of indices. The outer action of qw 0 is simply the shifted Weyl group action on Ū(h). Proposition 3.6 reduces the calculation of contravariant forms in P h (n) and G h (n) to the calculation of Zhelobenko operator ξw 0 , which is a simple technical exercise; the result, e.g. for P h (n), is ξw 0 (:

x ν :) =: x w 0 ν : qw 0 i,j:i<j ( hij -ν j ) ↑ν i +1 h↑ν i +1 ij , cf (37).
Remark. The following general statement holds. Let V be an irreducible finite-dimensional gl n -module and <, > a contravariant form on V . Instead of (39) take its Ū(h)-linear extension to the ( D(gl n ), h)-module V (h) . Then the formula (41) defines a contravariant form on V (h) . This can be also deduced from [KN].

Proof of Proposition 3.6 (i). To find (: x ν :, :

x ν ′ :) we should calculate, see Lemma 3.1, (ε(x ν ) ⊗ 1)∆(P)(x ν ′ ⊗ 1) 0 (43) in A = Diff(n) ⊗ Ū(gl n ).
Here () 0 means the projection of A to Ū(h) parallel to the sum of the left ideal generated by all ∂ i and diagonally embedded e ij , i < j (equivalently, by all ∂ i and 1 ⊗ e ij , i < j) and of the right ideal, generated by all x i and diagonally embedded e ij , i > j (equivalently, by all x i and 1 ⊗ e ij , i > j). The symbol ∆ stands for the diagonal embedding of U(gl n ). Present P in an ordered form

P = i d i (h)x i y i , where d i ∈ Ū(h), x i ∈ U(n -), y i ∈ U(n + ) .
Then ∆(P ) = i d i h (1) + h (2) ∆(x i )∆(y i ). Moving ∆(x i ) to the left and ∆(y i ) to the right, we conclude that their components in the second tensor factor do not affect the result so we can rewrite (43) as

(ε(x ν ⊗ 1))(P[h (2) ] ⊗ 1)(x ν ′ ⊗ 1) 0 ( 44
)
where P[h (2) ] means the shift of Ū(h)-valued coefficients in P. The factorized expression for P, see [AST] for details, reads

P = → γ∈∆ + P γ where P γ = n≥0 (-1) n n!(h γ + ρ(h γ ) + 1) ↑n e n -γ e n γ . (45)
Then

P[h (2) ] = → γ∈∆ + P γ [h (2) ] where P γ [h (2) ] = n≥0 (-1) n n!(h γ + h (2) γ + ρ(h γ ) + 1) ↑n e n -γ e n γ .
In (44) the elements h γ , e -γ and e γ should be understood as differential operators, see ( 16). Thus, the formula (44) defines a pairing in the Ū(gl n )-module P(n) with coefficients in 1 ⊗ Ū(h), so that (: x ν :, :

x ν ′ :) =< x ν , ψ P[h (2) ] x ν ′ > (46)
with the subsequent identification of elements h (2) = 1 ⊗ h with elements h ∈ h.

Next we compute ξw 0 (: x ν ′ :). Since the space P(n) ⊗ 1 is an ad gl n -invariant subspace of A = Diff(n) ⊗ Ū(gl n ), the consecutive application of statements (iii) and (iv) of Proposition 2.2 leads to the following expression for ξw 0 (x ν ′ ), see [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF]Section 8.1]: ξw 0 (:

x ν ′ :) = → γ∈∆ + n≥0 (-1) n n!(h -γ + ρ(h -γ ) + 1) ↑n ên γ ên -γ (w 0 (x ν ′ ⊗ 1)) . (47)
Here ĝ means as before the operator of adjoint action of g ∈ gl n . The adjoint action of g ∈ gl n on P(n) coincides with its action by the left multiplication by ψ(g) on P(n), realized as the quotient of Diff(n) over the left ideal generated by ∂ i . Therefore the equality (47) can be understood as the relation in the gl n -module P(n) with coefficients in 1 ⊗ Ū(h): ξw 0 (:

x ν ′ :) = → γ∈∆ + n≥0 (-1) n n!(-h (1) γ -h (2) γ -ρ(h γ ) + 1) ↑n e n γ e n -γ (w 0 (x ν ′ )) ⊗ 1 = = (w 0 ⊗ 1) → γ∈∆ + n≥0 (-1) n n!(h (1)
γ + h

(2)

w 0 (γ) -ρ(h γ ) + 1) ↑n e n -γ e n γ (x ν ′ ) ⊗ 1 .
of the spaces of n + -invariant vectors and n + -invariant vectors of the weight λ.

Corollary 4.2 (i) The map τλµ is an epimorphism.

(ii) The square of the norm of each n + -invariant vector of the weight λ = µ + ν of the U(gl n )-module P n ⊗ L µ is given by the relation (50).

Proof. The statement (i) follows from Lemma 4.1(ii).

For the proof of the statement (ii) we note that the projection map 1 ⊗ τ µ transforms the contravariant form on P n ⊗ M µ to the contravariant form on P n ⊗ L µ . In particular, 1 ⊗ τ µ transforms the restriction of the contravariant form to the space of n + -invariant vectors of weight λ in P n ⊗ M µ to the restriction of the contravariant form to the space of n + -invariant vectors of weight λ in

P n ⊗ L µ , (u, v) = (τ λµ (u), τλµ (v)) , u, v ∈ (P n ⊗ L µ ) •
λ . Thus for any non-singular λ the square of the norm of the vector P(x ν ⊗ 1µ ) ∈ (P n ⊗ L µ ) • λ is given by the relation (50).

Assume now that both µ and λ are non-singular. For any ν ∈ h * denote by Z ν := Diff h (n) ν the subspace of the reduction algebra Z = Diff h (n) generated by images in the double coset space Diff h (n) of elements in A = Diff(n) ⊗ U(gl n ) of the weight ν,

Z ν := {x mod (J + + J -) | x ∈ A , [h, x] = ν(h)x for any h ∈ h} . Since the Diff(n) ⊗ U(gl n )-module P n ⊗ L µ is irreducible, for any vector v ∈ (P n ⊗ L µ ) • λ there exists y ∈ (Diff(n) ⊗ U(gl n )) µ-λ such that y • v = 1 ⊗ 1µ . Then z ⋄ v := P y • v = 1 ⊗ 1µ ( 52
)
where z ∈ Z µ-λ is the image of y in Z . Due to Corollary 4.2, the map τλµ :

(P n ⊗ M µ ) • λ → (P n ⊗ L µ ) •
λ is an epimorphism. We now describe its kernel for dominant λ and µ in two equivalent ways. Consider any element u ∈ (P n ⊗ M µ ) • λ . Lemma 4.3 (i) u ∈ Ker τλµ iff z ⋄ u = 0 for any z ∈ Z µ-λ .

(ii) u ∈ Ker τλµ iff it is in the kernel of the contravariant form ( , ).

Proof. Let u ∈ Ker τλµ . Then for each z ∈ Z µ-λ we have z ⋄ u = 0. Indeed, the space (P n ⊗ M µ ) • µ is one-dimensional and is generated by the vector 1 ⊗ 1 µ . The map 1 ⊗ τ µ is Diff(n) ⊗ U(gl n )-equivariant thus the map τµ commutes with action of elements of Diff h (n). Moreover τµ (1 ⊗ 1 µ ) = 1 ⊗ 1µ . Then the vanishing of the left hand side of the equality τµ (z

⋄ u) = z ⋄ τµ (u) (53)
implies the relation z ⋄ u = 0 since z ⋄ u is proportional to 1 ⊗ 1 µ by the weight reasons.

On the other hand, if z ⋄ u = 0 for any z ∈ Z µ-λ then (53) implies that z ⋄ τµ u = 0 for any z ∈ Z µ-λ . Then, by (52), we have τµ u = 0. This proves (i).

Next, Lemma 4.1 says that each vector v ∈ (P n ⊗ M µ ) • λ can be presented as

x ⋄ (1 ⊗ 1 µ ) for some x ∈ Z λ-µ . If v = P(x ν ⊗ 1 µ ) then x = x ν ⊗ 1. Here ν = λ -µ. Then for each u ∈ Ker τλµ , (u, v) = (u, x ⋄ (1 ⊗ 1 µ )) = (ε(x) ⋄ u, 1 ⊗ 1 µ ) = 0 since ε(x) ∈ Z µ-λ . On the other hand, if (u, ε(x)(1 ⊗ 1 µ )) = 0 for any x ∈ Z µ-λ then (x ⋄ u, 1 ⊗ 1 µ ) = 0 and thus x ⋄ u = 0.
Let λ = µ + ν be the weight of the n + -invariant vector P(x ν ⊗ 1 µ ). The denominator of

B (ν) i,j (µ) is positive. The numerator of B (ν) i,j (µ) is (μ i -λj )(μ i -λj + 1) . . . ( λi -λj ) .
The last factor is a positive integer. So the product vanishes iff μi -λj ≤ 0 for some i, j , i < j .

It is sufficient to analyze only the neighboring indices, that is, to replace (55) by the condition μi -λi+1 ≤ 0 for some i .

Indeed, if μi -λi+1 > 0 and j > i then μi -λj > 0 since λi+1 -λj ≥ 0. We conclude that the following lemma holds. In other words, the tensor product

L (m) ⊗ L µ contains the representation L λ , µ ⊂ λ and |λ| = |µ| + m, if µ i ≥ λ i+1 for all i . (57)
The condition (57) says exactly that the difference λ\µ is a horizontal strip so we obtain the Pieri rule.

The situation with the odd variables is different. The right hand side of ( 54) is i,j:i<j C (ν) i,j (µ) where

C (ν) i,j (µ) := hij (µ) -ν j hij (µ) 1-ν i .
The denominator of C (ν) i,j (µ) is positive and the numerator is zero iff ν i = 0 and ν j = 1 for some i, j, i < j, and hij (µ) = 1, or µ i -µ j = i -j + 1 which may occur only if j = i + 1 and µ i+1 = µ i . But then λ i = µ i and λ i+1 = µ i+1 + 1 = λ i + 1 which cannot happen for a diagram λ. Thus all irreducible representations L λ such that ν = λ -µ is a weight of Λ m (h) do appear in the tensor product L (1 m ) ⊗ L µ which is exactly the statement of the dual Pieri rule about the vertical strip.

A Rings

Diff h (n) and GDiff h (n) A.1 Diff h (n)
The ring is generated by the elements x i and ∂ i . We shall use, instead of the set of generators generators {x i , ∂ i } the set {x i , ∂i } where, see [START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF],

∂j = ∂ j ϕ ′ j -1 with ϕ ′ j = k:k<j hjk hjk -1 . ( 58
)
The defining relations for the variables x i are

x i ⋄ x j = hij + 1 hij x j ⋄ x i , i < j . (59)
The remaining defining relations read

∂i ⋄ ∂j = hij -1 hij ∂j ⋄ ∂i , i < j , ∂j ⋄ x i = x i ⋄ ∂j , i > j , ∂j ⋄ x i = hij ( hij -2) ( hij -1) 2 x i ⋄ ∂j , i < j , ∂i ⋄ x i = j 1 1 + hij x j ⋄ ∂j + 1 . (60)
We have, for i < j:

(x i ) ⋄ a ⋄ (x j ) ⋄ b = (x j ) ⋄ b ⋄ (x i ) ⋄ a ( hij + 1) ↑a ( hij -b + 1) ↑a . ( 61
)
The proof is by induction, say, first on a and then on b.

We have

x j 1 ⋄ x j 2 ⋄ . . . ⋄ x j k = : x j 1 x j 2 . . . x j k : if j 1 ≥ j 2 ≥ • • • ≥ j . ( 62
)
The proof is by induction on k. Write the extremal projector in the form P = A 2 A 3 ...A n where A m = P 1,m P 2,m ... P m-1,m , m = 2, ..., n , with the notation P i,j := P ǫ i -ǫ j , see (45). By the induction hypothesis, x j 2 ⋄ . . . ⋄ x j k =:

x j 2 . . . x j k :, so

x j 1 ⋄ x j 2 ⋄ . . . ⋄ x j k = x j 1 ⋄ : x j 2 . . . x j k : ≡ x j 1 P : x j 2 . . . x j k : .

The assertion (62) follows because A l : x j 2 . . . x j k : ≡ : x j 2 . . . x j k : for l = j 2 + 1, . . . , n, and

x j 1 A 2 ...A j 1 ≡ x j 1 . A.2 GDiff h (n)
Now, the defining relations for the variables ζ i are

ζ i ⋄ ζ j = - hij -1 hij ζ j ⋄ ζ i , i < j . ( 63
) Let δj = δ j ϕ ′ j -1 . The remaining defining relations read δi ⋄ δj = - hij + 1 hij δj ⋄ δi , i < j , δj ⋄ ζ i = -ζ i ⋄ δj , i > j , δj ⋄ ζ i = - hij ( hij -2) ( hij -1) 2 ζ i ⋄ δj , i < j , δi ⋄ ζ i = - j 1 1 + hij ζ j ⋄ δj + 1 . (64)
Similarly to (62), for the h-Grassmann variables,

ζ j 1 ⋄ ζ j 2 ⋄ . . . ⋄ ζ j k =: ζ j 1 ζ j 2 . . . ζ j k : if j 1 > j 2 > • • • > j k . (65)

A.3 Zhelobenko automorphisms

Recall that we use the action of the symmetric group as the extension by the automorphisms of the Weyl group action. The action of Zhelobenko automorphisms on generators is qi (

x i ) = x i+1 hi,i+1 hi,i+1 -1 , qi (x i+1 ) = x i , qi (x j ) = x j , j = i, i + 1 , qi ( ∂i ) = hi,i+1 -1 hi,i+1 ∂i+1 , qi ( ∂i+1 ) = ∂i , qi ( ∂j ) = ∂j , j = i, i + 1 , qi ( hj ) = hs i (j) . (66)
The Zhelobenko automorphisms act on ζ i with the same coefficients as on x i .

Let q be the Zhelobenko automorphism for gl 2 . Then

q (x 2 ) ⋄ a ⋄ (x 1 ) ⋄ b = (x 2 ) ⋄ b ⋄ (x 1 ) ⋄ a ( h) ↑a+1 ( h -b) ↑a+1 (67) and q
-1 (x 2 ) ⋄ a ⋄ (x 1 ) ⋄ b = (x 2 ) ⋄ b ⋄ (x 1 ) ⋄ a ( h) ↑a ( h -b) ↑a . ( 68
)

B Calculation of norms

Even variables. Let ν = (ν 1 , . . . , ν n ) be a milti-index, ν! = ν 1 ! . . . ν n ! and x ν = (x n ) ⋄ νn ⋄ . . . ⋄ (x 1 ) ⋄ ν 1 . The monomials x ν for ν ∈ Z n ≥0 form a basis of P h (n). Define a bilinear form on P h (n) by (x ν , x ν ′ ) = δ ν,ν ′ ν!γ ν where γ ν := i,j:i<j

B (ν) i,j and B (ν) i,j := ( hij -ν j ) ↑ν i +1 h↑ν i +1 ij . ( 69
)
We denote by s i ν the multi-index (ν 1 , . . . , ν i-1 , ν i+1 , ν i , ν i+2 , . . . , ν n ).

Proposition B.1 The form (69) coincides with the contravariant form on P h (n). ( hi,i+1 -ν i ) ↑ν i+1 +1 and q-1 i (x ν ) = x s i ν ( hi,i+1 + ν i+1 ) ↓ν i +1 h↓ν i +1 i,i+1

.

(71)

Therefore, the transformation laws (x ν , x ν ) → q-1 i (x ν ), qi (x ν ) and (x ν , x ν ) → qi (x ν , x ν ) are the same so it is sufficient to prove (69) for an arbitrary permutation of (ν 1 , . . . , ν n ).

2. We prove the assertion by induction on degree |ν| = ν 1 +• • •+ν n , the induction base is (1, 1) = 1. Assume that |ν| > 0. By part 1, it is sufficient to verify the statement for ν such that ν 1 > 0. By induction hypothesis, (x υ , x υ ) = υ!γ υ where υ = (ν n , . . . , ν 2 , ν 1 -1). We have

x ν = x υ ⋄ x 1 and γ ν = γ υ j>1 h1j -ν j + ν 1 h1j + ν 1 .
It follows from (61) that

x υ ⋄ x 1 = x 1 ⋄ x υ j>1 h1j -ν j + ν 1 h1j + ν 1 .
Therefore,

(x υ ⋄ x 1 , x ν ) = (x 1 ⋄ x υ , x ν ) j>1 h1j -ν j + ν 1 h1j + ν 1 = (x υ , ∂ 1 ⋄ x ν ) j>1 h1j -ν j + ν 1 h1j + ν 1 .
We used the contravariance in the last equality. By (58), ∂ 1 = ∂1 . Now, ∂1 ⋄ x ν = (x n ) ⋄ νn ⋄ . . . ⋄ (x 2 ) ⋄ ν 2 ⋄ ∂1 ⋄ (x 1 ) ⋄ ν 1 by (60). We have ∂1 ⋄ x 1 = 1 + x 1 ⋄ ∂1 + linear combination of x j ⋄ ∂j with j > 1 .

The h-derivative ∂j , j > 1, then moves to the right through remaining x 1 without a constant term, so ∂1 ⋄ (x 1 ) ⋄ ν 1 = ν 1 + (a linear combination of terms F i ⋄ ∂i ) which does not contribute to the scalar product. .

We denote by s i ν the string (ν 1 , . . . , ν i-1 , ν i+1 , ν i , ν i+2 , . . . , ν n ).

Proposition B.2

The form (72) coincides with the contravariant form on G h (n).

By (65), this proposition is equivalent to Proposition 3.5. The proof is along the same lines as for Proposition B.1.

Proof. 1. Analogues of formulas ( 70) and ( 71) are qi (κ ν ) = κ s i ν hi,i+1 + ν i+1 hi,i+1

1-ν i hi,i+1 hi,i+1 -ν i 1-ν i+1 , qi (ζ ν ) = (-1) ν i ν i+1 ζ s i ν hi,i+1 hi,i+1 -ν i 1-ν i+1
, q-1 i (ζ ν ) = (-1) ν i ν i+1 ζ s i ν hi,i+1 + ν i+1 hi,i+1

1-ν i .

Again, the transformation laws (ζ ν , ζ ν ) → q-1 i (ζ ν ), qi (ζ ν ) and (ζ ν , ζ ν ) → qi (ζ ν , ζ ν ) are the same so it is sufficient to prove (72) for an arbitrary permutation of (ν 1 , . . . , ν n ).

2. Induction now is on degree |ν| = ν 1 + • • • + ν n . Assume that |ν| > 0. By part 1, it is sufficient to verify the statement for ν such that ν 1 = 1. By induction hypothesis, (ζ υ , ζ υ ) = κ υ where υ = (ν n , . . . , ν 2 , 0). We have

ζ ν = ζ υ ⋄ ζ 1 and κ ν = κ υ j>1 h1j h1j -ν j .
On the other hand,

ζ υ ⋄ ζ 1 = (-1) |υ| ζ 1 ⋄ ζ υ j>1 h1j h1j -1 ν j = (-1) |υ| ζ 1 ⋄ ζ υ j>1 h1j h1j -ν j .
The rest of the proof follows, as for the even variables, from the covariance and the fact that δ1 = δ 1 .

  27) for any u, v ∈ M • and g ∈ Z + . The forms on M • and M • are related as follows. For any u, v ∈ M • vectors P u and P u ′ belong to M • and (u, u ′ ) = (P u, P u ′ ).

Lemma 4. 6

 6 The tensor product L (m) ⊗L µ does not contain the representation L λ , µ ⊂ λ and |λ| = |µ| + m, iff the condition (55) holds.

  By (62), this proposition is equivalent to Proposition 3.4. Proof of Proposition B.1. Since the subspaces of different h-weight are orthogonal with respect to a contravariant form, it is sufficient to analyze the products (x ν , x ν ). 1. We first check the covariance (36) of the form (69). Collecting pairs B (ν) m,i and B (ν) m,i+1 for m < i and pairs B (ν) i,m and B (ν) i+1,m for m > i + 1 in the product for γ ν , we find qi(γ ν ) = γ s i ν qi (B (ν) i,i+1 ) B (s i ν) i,i+1 = γ s i ν ( hi,i+1 + ν i+1 ) ↓ν i +1 h↓ν i +1 i,i+1 h↑ν i+1 +1 i,i+1( hi,i+1 -ν i ) ↑ν i+1 +1 .

  Odd variables. Let now ν = (ν 1 , . . . , ν n ) where ν j ∈ {0, 1}, j = 1, . . . , n, andζ ν = (ζ n ) νn ⋄ . . . ⋄ (ζ 1 ) ν 1 . The monomials ζ ν form a basis of G h (n). Define a bilinear form on G h (n) by (ζ ν , ζ ν ′ ) = δ ν,ν ′ κ ν where κ ν :=

Other extensions by automorphisms of A of the Weyl group action on h can be used here. First, one can use the inverse to (9) or switch the positive and negative roots in (9).

Acknowledgments. We thank CIRM, Marseille, where a part of this work was done, for the hospitality. The work of both authors was supported by the grant RFBR 17-01-00585. The work of O. O. was also supported by the Program of Competitive Growth of Kazan Federal University and the work of S. K. by the Russian Academic Excellence Project "5-100".

Here in the second line we changed the summation index from γ to w 0 (-γ) and used the property ρ(h γ ) = -ρ(h w 0 (γ) ). We can rewrite the result using the shifted Weyl group action on the second tensor component: ξw 0 (:

The comparison of ( 44) and ( 48) gives the desired statement. The proof of the part (ii) is similar.

Appendix B contains the second proof of Propositions 3.4 and 3.5 based on explicit calculations in the rings Diff h (n) and GDiff h (n).

Specializations

Specialization to non-singular weights

The tensor product P n ⊗ M µ , where M µ is the gl n -Verma module with the highest weight µ, is a A = Diff(n) ⊗ U(gl n )-module, generated by the vector v µ = 1 ⊗ 1 µ . Since this vector satisfies the conditions

and (1 ⊗ h)v µ = µ(h)v µ for any h ∈ h, there is a unique A-contravariant C-valued form on P n ⊗ M µ , normalized by the condition (v µ , v µ ) = 1. This contravariant form can be constructed by means of the Harish-Chandra map (0) : A → U(h), given by the prescription x-x (0) ∈ I +ε(I), where I is the left ideal of A, generated by ∂ i ⊗1, i = 1, ..., n, and 1 ⊗ x, x ∈ n + . Then

Here (ε(x)y) (0) (µ) in the right hand side of (49) means the evaluation of a polynomial on elements of the Cartan subalgebra at the point µ ∈ h * . The restriction of this form to the space (P n ⊗ M µ ) • of n + -invariants defines on this space a bilinear form, satisfying the contravariance property (27). For a generic µ, the action of the Mickelsson algebra Diff(n) + = Norm(J + )/ J + on (P n ⊗ M µ ) • extends to the action of its localization, the reduction algebra Diff h (n). The actions of Diff(n) + and Diff h (n) satisfy the contravariance property (24). Due to Lemma 3.1, the Harish-Chandra maps, defining the contravariant forms for Diff h (n) and for A, are compatible, that is, they commute with the natural map from Diff(n) ⊗ U(gl n ) to its double coset Diff h (n); thus the contravariant form on (P n ⊗ M µ ) • coincides with the evaluation at µ of the contravariant form on P h (n) under the isomorphism (31), see (33).

We conclude that for generic µ the square of the norm of the n + -invariant vector P

On the other hand, the space P n decomposes into a direct sum of the spaces S m of polynomials of degree m, each being an irreducible gl n -module,

The gl n -module S m gives rise to ( D(gl n ), h)-module S m (h) , see Section 3.3. It possesses a D(gl n )-contravariant form, which is the restriction of Diff h (n)-contravariant form on P h (n). The evaluation of this form at generic µ is a C-valued D(gl n )-contravariant form on S m (h) . Up to a normalization, the map π µ transforms it to the restriction to

Thus the formula (50) describes norms of highest weight vectors in the tensor product of the m-th symmetric power of the fundamental representation normalized so that the square of the norm of the vector (x 1 ) m ⊗ 1 µ is equal to m!.

Denote λ = µ + ν. Since the denominators of the extremal projector P belong to the set K + , defined in Section 2.1, the n + -invariant vector P(x ν ⊗ 1 µ ) is well defined for any non-singular λ. The square of the norm of this vector is a rational function in λ. This function is equal to the right hand side of (50) for generic µ (that is, for generic λ). So for any non-singular λ the right hand side of ( 50) is finite and gives the square of the norm of P(x ν ⊗ 1 µ ).

The similar considerations hold for the decomposition of the space G(n) into a direct sum of its homogeneous components,

Specialization to irreducible representations

Throughout this section the weight λ ∈ h * is assumed to be non-singular. Let M and N be two g-modules from the category O, that is, they are n + -locally finite, h-semisimple and finitely generated g-modules. Assume that M is generated by a highest weight vector 1 µ of the weight µ (that is, M is a quotient of the Verma module M µ ). Consider the g-module N ⊗ M. Lemma 4.1 (i) The space (N ⊗ M) • is spanned by the images of vectors v ⊗ 1 µ , v ∈ N.

(ii) For any non-singular λ ∈ h * the space (N ⊗ M) • λ of n + -invariant vectors of the weight λ is spanned by the vectors P(v ⊗ 1 µ ), where v has the weight ν = λ -µ.

Proof. Each element in M can be presented as g • 1 µ with g ∈ U(n -). Using the relation

we prove by induction on degree of g ∈ U(n -) that for any v ∈ N we have an equality

The statement (ii) follows from (i) since for a non-singular λ the extremal projector P establishes an isomorphism of n --coinvariants and n + -invariants of the weight λ, see Section 3.1.

Consider the tensor product P n ⊗ L µ of the Diff(n)-module P n and an irreducible U(gl n )-module L µ of the highest weight µ with the highest weight vector 1µ . The natural projection 1 ⊗ τ µ :

Corollary 4.4 For non-singular λ and µ the vector P(x ν ⊗ 1µ ) is a nonzero element of (P n ⊗ L µ ) • λ iff its norm is nonzero.

Let now µ be the highest weight of a finite-dimensional irreducible gl n -module L µ . In particular, µ is dominant. Then the weights λ of all n + -invariant vectors of P n ⊗ L µ are highest weights of finite-dimensional irreducible gl n -modules and are dominant; so they are non-singular. Corollaries 4.2 and 4.4 describe all nonzero n + -invariant vectors of P n ⊗ L µ together with their norms.

The considerations are valid for Grassmann variables, where now the relation

describes the norms and nonvanishingness of n + -invariant vectors in the tensor product

We summarize the results in the following proposition.

Proposition 4.5 Assume that the weight µ is dominant.

(i) The square of the norm of an n + -invariant vector in P n ⊗L µ is given by the relation (50). The square of the norm of an n + -invariant vector in G n ⊗ L µ is given by the relation ( 54).

(ii) Any n + -invariant vector in P n ⊗L µ has a form P(x ν ⊗1 µ ) (with dominant λ = µ+ν) and is nonzero iff its norm is nonzero.

Pieri rule

We recall some terminology concerning finite-dimensional representations of gl n . A finitedimensional irreducible representation L µ of highest weight µ = (µ 1 , . . . , µ n ) is visualized by the Young diagram with µ j boxes in the j-th row. Let |µ| := µ 1 +. . . µ n and μi := µ i -i.

The m-th symmetric power L (m) of the tautological representation of gl n corresponds to the one-row diagram with m boxes. The m-th wedge power L (1 m ) of the tautological representation of gl n corresponds to the one-column diagram with m boxes.

For two diagrams µ and λ, µ ⊂ λ, the set-theoretical difference λ \ µ is called a horizontal strip if it contains no more than one box in any column. The difference λ\µ is called a vertical strip if it contains no more than one box in any row. The Pieri rule says that for any µ the product L (m) ⊗ L µ is multiplicity free and is a direct sum of L λ such that λ\µ is a horizontal strip of cardinality m. The dual Pieri rule says that for any µ the product L (1 m ) ⊗ L µ is multiplicity free and is a direct sum of V λ such that λ\µ is a vertical strip of cardinality m. The multiplicity freeness follows since the h-weights of the ( D(gl n ), h)-modules S m (h) and Λ m (h) are multiplicity free. We keep the notation of Proposition 4.5. Rewrite the last product in the right hand side of the formula (50) in the form i,j:i<j B (ν) i,j (µ) where B (ν) i,j (µ) := ( hij (µ) -ν j ) ↑ν i +1 hij (µ) ↑ν i +1 .