Polynomial Quantization and Overalgebra

We construct polynomial quantization (a variant of quantization in the spirit of Berezin) on para-Hermitian symmetric spaces. For that we use two approaches: (a) using a reproducing function, (b) using an “overgroup”. Also we show that the multiplication of symbols is an action of an overalgebra.


Para-Hermitian Symmetric Spaces
Let G/H be a semisimple symmetric space. Here G is a connected semisimple Lie group with an involutive automorphism σ = 1, and H is an open subgroup of G σ , the subgroup of fixed points of σ . We consider that groups act on their homogeneous spaces from the right, so that G/H consists of right cosets Hg.
Let g and h be the Lie algebras of G and of H respectively. Let B g be the Killing form of G. There is a decomposition of g into direct sums of +1, −1-eigenspaces of the involution σ : The subspace q is invariant with respect to H in the adjoint representation Ad. It can be identified with the tangent space to G/H at the point x 0 = H e.
The dimension of Cartan subspaces of q (maximal Abelian subalgebras in q consisting of semisimple elements) is called the rank of G/H . Now let G/H be a symplectic manifold. Then h has a non-trivial center Z(h). For simplicity we assume that G/H is an orbit Ad G · Z 0 of an element Z 0 ∈ g. In particular, then Z 0 ∈ Z(h).
Further, we can also assume that G is simple. Such spaces G/H are divided into 4 classes (see [3,4]): We focus on spaces of class (c). Here the center Z(h) is one-dimensional, so that Z(h) = RZ 0 , and Z 0 can be normalized so that the operator I = (adZ 0 ) q on q has eigenvalues ±1. A symplectic structure on G/H is defined by the bilinear form ω(X, Y ) = B g (X, I Y ) on q.
The ±1-eigenspaces q ± ⊂ q of I are Lagrangian, H -invariant, and irreducible. They are Abelian subalgebras of g. So g becomes a graded Lie algebra: , Z], see [5]. Let κ be the genus of this Jordan pair.
Set Q ± = exp q ± . The subgroups P ± = H Q ± = Q ± H are maximal parabolic subgroups of G. One has the following decompositions: 2) mean that almost any element g ∈ G can be decomposed as ( the Gauss decomposition): or (the anti-Gauss decomposition): (1.4) where h ∈ H , ξ ∈ q − , η ∈ q + , all three factors in (1.3) and (1.4) are defined uniquely. We also use the Gauss decomposition (1.3) in a little different form:  4) generate actions of G on q − and q + respectively, namely, ξ → ξ = ξ • g and η → η = η • g: where X ∈ q − , Y ∈ q + . These actions are defined on open and dense sets depending on g. Therefore, G acts on q − ×q + : (ξ, η) → ( ξ, η). The stabilizer of the point (0, 0) ∈ q − × q + is P + ∩ P − = H , so that we get an embedding It is defined on an open and dense set, its image is also an open and dense set. Therefore, we can consider (ξ, η) ∈ q − × q + as coordinates on G/H , let us call them horospherical coordinates.
Let us write explicit formula for embedding (1.8). We use a redecomposition "anti-Gauss" to "Gauss". We take ξ ∈ q − , η ∈ q + and decompose the anti-Gauss product exp ξ · exp (−η) according to formula (1.5) (the "Gauss"): where X ∈ q − , Y ∈ q + . The obtained element h ∈ H depends on ξ and η only, denote it by h(ξ, η). Using (1.9), let us form the following element g ∈ G: Then the pair ξ, η goes just to the point x = x 0 g where g is defined by (1.10). Under the action of the group G the element h(ξ, η) is transformed as follows: is a polynomial in ξ, η. Moreover, see [5], it is the power N(ξ, η) κ of an irreducible polynomial N(ξ, η) of degree r in ξ and η separately. In horospherical coordinates the G-invariant measure on G/H is: where dξ and dη are Euclidean measures on q − and q + respectively.

Maximal Degenerate Series Representations
In this Section we introduce two series of representations induced by characters of maximal parabolic subgroups P ± of G (maximal degenerate series representations).
Let λ ∈ C. We take the following character ω λ of H : and then we extend this character to the subgroups P ± , setting it equal to 1 on Q ± . We consider induced representations π ± λ of G: π ± λ = Ind G, P ∓ , ω ∓λ . They act on the space D ± λ (G) of functions f ∈ C ∞ (G) having the uniformity property: by translations from the right: . Realize them in the non-compact picture: we restrict functions from D ± λ (G) to the subgroups Q ± and identify them (as manifolds) with q ± , we obtain Let us write intertwining operators. Introduce operators A ∓ λ by: The operator A ± λ intertwines π ± λ with π ∓ −λ−κ . Their composition is a scalar operator: where lower or upper indexes are taken, c(λ) is a meromorphic function of λ, invariant with respect to the change λ → −λ − κ.
The representation π ± λ of the universal enveloping algebra Env (g) of the Lie algebra g (we preserve the same symbols) is given by some differential operators. In particular, for L ∈ g these operators have the first order. On the product ϕψ of functions they act as follows: Let us introduce the following bilinear form on functions on q ± : It is invariant with respect to the pair (π ± −λ−κ , π ± λ ): The principal anti-automorphism X → X ∨ in Env (g) corresponds to the map g → g −1 in the group G: if X = L 1 L 2 . . . L k , where L i ∈ g, then It is an anti-involution: (XY ) ∨ = Y ∨ X ∨ . Formula (2.4) gives: We rename the kernel of intertwining operators: Formula (1.11) gives λ ( ξ, η)) = λ (ξ, η) · ω λ ( h) −1 · ω λ ( h) , which can be interpreted as an invariance property of the function λ (ξ, η) : . This formula can be rewritten as For elements L of the Lie algebra g, formula (2.7) gives:

Symbols and Transforms
In this Section we apply to a para-Hermitian symmetric space G/H the scheme of quantization in the spirit of Berezin offered in [6]. We consider the most algebraic version of the quantization, we call it the polynomial quantization. For an initial algebra of operators we take here the algebra of operators π − λ (Env (g)), λ ∈ C. The role of the Fock space is played by a space of functions ϕ(ξ ), ξ ∈ q − , so that our operators act in functions ϕ(ξ ). We introduce covariant and contravariant symbols of operators, the Berezin transform etc.
As a (an analog of) supercomplete system we take the function (2.6). Introduce covariant and contravariant symbols of operators D = π − λ (X), where X ∈ Env (g). We start from formula (2.1), it is reproducing formula:

Theorem 3.1 The function Df (ξ )
is expressed in terms of f (ξ) by one of two following formulas: Proof Let us apply the operator D to both hand sizes of (3.1) (as functions of ξ ) we at once get (3.2). Now let us write formula (3.1) replacing f by π − λ (g)f . We obtain By (2.4) the inner integral here becomes as follows: then we use (2.7) and turn this integral into: hence formula (3.4) becomes as follows: Let us pass here from the group G to the algebra Env (g): we change g by X and obtain just (3.3).
Comparing (3.2), (3.3) with (3.1), we introduce the following two functions: Then we write Therefore, remembering invariant measure dx on G/H , see (1.12), we can write (3.2) and (3.3) as follows: Thus, to the operator D two functions F and F correspond: D −→ F and D −→ F . Let us call them covariant and contravariant symbols of the operator D, respectively. We shall denote them also co λ D and contra λ D. The following theorem converts these correspondences. (3.7) and (3.8). Therefore, the maps co λ and contra λ are one-to-one.

Theorem 3.2 The operator D is recovered by its covariant and contravariant symbols by means of equalities
For generic λ, the space of symbols of both types is the space Establish a connection between co-and contravariant symbols. Comparing (3.6) with (3.5) and using (2.7), we obtain Formula (2.5) says that the operator conjugate to an operator Since ξ , η are horospherical coordinates on G/H , symbols (co-and contra-) become functions on G/H and, moreover, polynomials on G/H ⊂ g. It is why we call this variant of quantization the polynomial quantization. For generic λ, the space of symbols is the space of all polynomials on G/H . In particular, the symbol of the identity operator is the function on G/H equal to 1 identically. If X belongs to the Lie algebra g itself, then the symbol of the operator π − λ (X) is a linear function B g (X, x) of x ∈ G/H ⊂ g up to a factor depending on λ.
The multiplication of operators gives rise to the multiplication of covariant symbols, denote it by * . Namely, let F 1 and F 2 be covariant symbols of operators D 1 and D 2 respectively. Then the covariant symbol F 1 * F 2 of the product D 1 D 2 is .
Let us call this function B λ the Berezin kernel. It can be regarded as a function B λ (x, y) on G/H × G/H . It is invariant with respect to G: Thus, the space of covariant symbols is an associative algebra with 1.
In particular, let us write formulas for multiplication by symbols V corresponding to elements L of the Lie algebra g.

Theorem 3.3 We have (the point means pointwise multiplication)
Proof To prove (3.10), we take in formula (3) D 1 = π − λ (L) and F 2 = F , then we differentiate by (2.2), as a result we get (3.10). Now by (3.5) we have then by (2.8) we can change here the latter operator by the operator {−(1 ⊗ π + λ (L))} and then transpose it with D ⊗ 1 since they act on different variables. We obtain The multiplication of contravariant symbols is obtained from the multiplication of these functions as covariant symbols by the permutation of multipliers and by change λ → −λ − κ. Corresponding kernel is get from the Berezin kernel B λ by the same change. Thus, we have two maps: co λ and contra λ which connect polynomials on G/H and operators acting on functions f (ξ).
The composition B λ = co λ • contra λ maps the contravariant symbol of an operator D to its covariant symbol. Let us call B λ the Berezin transform. The kernel of this transform is just the Berezin kernel.
If a polynomial F on G/H is the covariant symbol of an operator D = π − λ (X), X ∈ Env (g), and is the contravariant symbol of an operator A simultaneously, then . Such a map was absent in Berezin's theory for Hermitian symmetric spaces.

Polynomial Quantization and the Overgroup
As an overgroup for G we take the direct product G = G × G. It contains G as the diagonal {(g, g), g ∈ G}. First we describe a series of representations R λ of G.
Let P be a parabolic subgroup P consisting of pairs (zh, hn), z ∈ Q − , h ∈ H , n ∈ Q + . Let ω λ be a character of P equal to ω λ (h) at these pairs. The representation of G induced by the character ω λ of the subgroup P is denoted R λ .
Let us give some realizations of representations R λ . Denote by C (a "cone") the manifold of "double" cosets y = s −1 1 Q − Q + s 2 , s 1 , s 2 ∈ G. The group G acts on C as follows: Denote by D λ (C) the space of functions f on C of class C ∞ satisfying the following homogeneity condition The representation R λ acts on D λ (C) by Let us take in C two sections: "hyperbolic" section X and "parabolic" section . The manifold X ⊂ C consists of cosets The group G acts on X by x → g −1 xg. The stabilizer of the initial point x 0 = Q − Q + is H , so that X can be identified with G/H .
The manifold ⊂ C consists of cosets This manifold can be identified with q − × q + . We can embed → X repeating the embedding q − × q + → G/H , see (1.8)-(1.10). Namely, let a point x = s −1 Q − Q + s, s ∈ G, has horospherical coordinates ξ, η. By (1.9) we find the element h(ξ, η) and then by (1.10) we obtain an element g. We rename it by s, so that where X ∈ q − , Y ∈ q + . Therefore Thus, the embedding above assigns to a point γ ∈ , given by (4.3), the point x ∈ X , given by (4.5).
The representation R λ can be realized in functions on these manifolds X and .
First consider X . A point x = s −1 Q − Q + s in X under action (4.1) goes to the point g −1 Take the element sg 2 (sg 1 ) −1 , i. e. the element sg 2 g −1 1 s −1 , and decompose it "by Gauss": Here the element h * ∈ H depends on the point x only and does not depend on its representative s. Let us form an element s * ∈ G: It gives the point x * = (s * ) −1 Q − Q + s * in X . By (4.7) we have , so that R λ acts in functions on X = G/H as follows: (4.8)

Theorem 4.1 In horospherical coordinates ξ, η on X the representation R λ is
where h 2 and h 1 are taken from decompositions (1.6) and (1.7) with g = g 2 and g = g 1 respectively.

Theorem 4.2 In horospherical coordinates ξ, η on the representation R λ is
It shows that R λ is equivalent to a tensor product: The group G contains three subgroups isomorphic to G. The first one is the diagonal consisting of pairs (g, g), g ∈ G. The restriction of the representation R λ to this subgroup is the representation U by translations on G/H : Indeed, (4.6) and (4.7) with g 1 = g 2 = g give h * = e and s * = sg.
Two other subgroups G 1 and G 2 consist of pairs (g, e) and (e, g), where g ∈ G.
By virtue of Theorem 4.1, the restriction of the representation R λ to the subgroup G 2 is given by Similarly, the restriction of the representation R λ to the subgroup G 1 is given by Let us go from the group G to the universal enveloping algebra Env(g). Then from (4.14) and (4.15) for X ∈ Env(g) we obtain Let us take as f the function f 0 equal to the 1 identically. Then we have Right hand sides of formulae (4.16) and (4.17) are just covariant and contravariant symbols of operator D = π − λ (X), see Section 3. Let us change the position of arguments in R λ , then we have a new representation R λ of G, namely, R λ (g 1 , g 2 ) = R λ (g 2 , g 1 ). Using the realization of R λ on the section , we see that the tensor product A λ ⊗ B λ intertwines the representation R λ with the representation R −λ−κ . Passing from to X and replacing λ by −λ − κ, we obtain that the operator c(λ)A −λ−κ ⊗ B −λ−κ intertwines the representation R −λ−κ with the representation R λ and transfers contravariant symbols to covariant ones. It has the kernel B λ (ξ, η; u, v), i. e. it is precisely the Berezin transform.
This theorem adjoins to themes of [7,8,10] and gives a new point of view on the multiplication of symbols.

Example: A Hyperboloid of One Sheet
Here we determine explicitly an action of the overalgebra on the space of covariant symbols.
The group G is SL(2, R), the subgroup H consists of diagonal matrices, the space G/H is a hyperboloid of one sheet in R 3 . The overgroup G = G × G contains three subgroups G d , G 1 G 2 isomorphic to G, see Section 4. Let g be the Lie algebra of G. Then the Lie algebras of G and G d , G 1 , G 2 are g = g + g and g d , g 1 , g 2 , respectively.
In order to write an action of the overalgebra g, it is sufficient to take some subspace complementary to g d . Now we take the subalgebra g 2 . It consists of pairs (0, X), where X ∈ g.
In this example we take a few greater store of representations: besides of parameter λ there is a discrete parameter ν = 0, 1. We shall use the notation t λ,ν = |t| λ sgn ν t .
The group G = SL(2, R) consists of real matrices of the second order with unit determinant: For λ ∈ C, ν = 0, 1, denote by D λ,ν (R) the space of functions f in C ∞ (R) such that the function f (t) = t λ,ν f (1/t) belongs to C ∞ (R) too. The representation π λ,ν of the group G acts on D λ,ν (R) by (we consider that G acts from the right): Any irreducible finite-dimensional representation ρ k of the group G is labeled by the number k (the highest weight) such that 2k ∈ N = {0, 1, 2, . . .}. It acts on the space V k of polynomials ϕ(t) in t of degree 2k (so that dim V k = 2k + 1) by The Lie algebra g of the group G consists of real matrices of the second order with zero trace. A basis in g consists of matrices: The commutation relations are: Operators corresponding to elements of g and Env (g) in representations π λ,ν etc. do not depend on ν, so we do not write ν in indexes. For basis elements (5.1) we have Replacing here λ by 2k, we obtain formulas for ρ k . Let us realize the space R 4 of vectors x = (x 0 , x 1 , x 2 , x 3 ) as the space of real 2 × 2 matrices: The overgroup G acts as follows: x → g −1 1 xg 2 , (g 1 , g 2 ) ∈ G. Let C be the cone det x = 0, x = 0. For λ ∈ C, ν = 0, 1, let D λ,ν (C) denote the space of C ∞ functions f on the cone C homogeneous of degree λ and parity ν: Let R λ,ν be the representation of G by translations on the space D λ,ν (C) (in fact, it is a representation of the group SO 0 (2, 2) associated with a cone, G covers SO 0 (2, 2) with multiplicity 2): The section X of C by plane (tr x) = 1 can be identified with a hyperboloid of one sheet −x 2 1 + x 2 2 + x 2 3 = 1 in R 3 . Restrictions of functions in D λ,ν (C) to X form a space D λ,ν (X ) of functions on X . It is contained in C ∞ (X ) and contains D(X ). In the realization on X the representation R λ,ν is: The section X is invariant with respect to the action x → g −1 xg of G d = G, it is just the space G/H . The restriction of R λ,ν to G d = G is the quasiregular representation U of G on X . It preserves the space S(X ) of polynomials on X and decomposes in the direct sum: U = ρ 0 +ρ 1 +ρ 2 +. . . with the corresponding decomposition S(X ) = H 0 +H 1 +H 2 +. . .
Covariant symbols of operators π λ (X), X ∈ Env (g), we define by (3.5): In particular, covariant symbols for basis elements (5.1) are multiplied by (−λ)/2 polynomials For k ∈ N, we define the Poisson kernel P k (x; t) as follows. Denote This kernel is a fixed vector in the tensor product U ⊗ ρ k : (U(g) ⊗ ρ k (g)P k ) (x; t) = P k (x; t), g ∈ G.
Therefore, P k (x; t) is a generating function for polynomials in H k . Let us introduce the following differential operators S k (X), k ∈ N, and E(X) in variable t, linearly depending on X ∈ g, for basic elements (5.1) they are Expressing from (5.8) and (5.9) the second summands in right hand sides, substituting in (5.7) and remembering (5.3), we obtain (5.6) for X = L − . Now for X = L 1 and X = L + , we use equality (5.6) with X = L − already proved and commutation relations -successively the first and the second ones in (5.2), and corresponding relations for operators S k (X) and E(X) in (5.43) and (5.44).
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Theorem 5.1 Let X ∈ g. The operator R λ (0, X) acts on the Poisson kernel P k (x; t) as follows: R λ (0, X) P k = α k · S k (X) P k+1 + β k · ρ k (X) P k + γ k · E(X) P k−1 , (5.6) in the left hand side the operator acts on a function of ξ, η, and in the left hand side the operators act on functions of t.