Weyl-Schr\"odinger representations of Heisenberg groups in infinite dimensions

A complexified Heisenberg matrix group $\mathrm{H}_\mathbb{C}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schr\"odinger type irreducible representations of $\mathrm{H}_\mathbb{C}$ on the space $L^2_\chi$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $\chi$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty)$ irreducible acted on $H$. The space $L^2_\chi$ is generated by Schur polynomials in variables on Paley--Wiener maps over $U(\infty)$. Therewith, the Fourier-image of $L^2_\chi$ coincides with a space of Hilbert--Schmidt entire analytic functions on $H$ generated by suitable Fock space. Applications to linear and nonlinear heat equations over the group $\mathrm{H}_\mathbb{C}$ are considered.


Introduction
An aim of this work is to investigate irreducible Weyl-Schrödinger representations of the complexified Heisenberg group H (see [16, n.9]), consisting of matrix elements X(a, b, t) with any a, b ∈ H and t ∈ C such that where H is an infinite-dimensional complex Hilbert space and ½ is its identity map.
The group H has the unit X(0, 0, 0) and inverse elements of the form X(a, b, t) −1 = X (−a, −b, −t + a | b ).
In what follows, we consider the infinite-dimensional unitary group U(∞) = Ť U(i), containing all subgroups U(i) of unitary i × i-matrices, which acts irreducibly on a complex Hilbert space {H, · | · } with an orthonornal basis (e i ) i∈N .
To find the desired representation, we use the space L 2 χ of C-valued functions that are quadratically integrable with respect to the probability measure χ. Wherein, according to our Institute of Mathematics, University of Rzeszów, e-mail: ovlopusz@ur.edu.pl Mathematics Subject Classification (2010): 81R10;43A65;46E50;35R03 assumption χ has a structure of the projective limit χ = lim ← − χ i of probability Haar's measures χ i on U(i), satisfying the Kolmogorov consistency conditions in an abstract Bochner's formulation (see [22,26]).
In [20,23] it was shown that the projective limit χ = lim ← − χ i is well defined over the projective limit U = lim ← − U(i) with respect to the Livšic transforms π i+1 i : U(i + 1) → U(i) such that χ i = π i+1 i (χ i+1 ). In this paper, we prove that for such χ each function from L 2 χ admit a superposition (linearization in the sense of [4]) on PaleyWiener maps associated with U(∞). As a result, it is shown that Schur polynomials form an orthonormal basis in L 2 χ and the Fourier-image of L 2 χ consists of Hilbert-Schmidt analytic functions on H. Note also that projective limits of probability measures over various infinite-dimensional manifolds with similar properties were investigated in [29,30,31].
If instead of the unitary group U(∞) we take the infinite-dimensional linear space with a Gaussian measure γ, a similar construction of the appropriate space L 2 γ can be found in the well-known works [1]. In this case, the Fourier-image of L 2 γ coincides with the Segal-Bargmann space of entire analytic functions over which the Schrödinger type irreducible representations of Heisenberg groups are well defined. In the present paper, we change γ by the unitarily-invariant projective limit χ = lim ← − χ i and, as a result, we obtain another irreducible representation, called to be the Weyl-Schrödinger type.
Infinite-dimensional Heisenberg groups over R was considered in [18] by using the reproducing kernel Hilbert spaces. The Schrödinger representation of such groups using Gaussian measures over a real Hilbert space was described in [2]. Since the group H in the case of matrix entries a, b, t ∈ R coincides with the classical Heisenberg group over R (see, e.g. [11]), the results of the present paper can be considered as a complexification of previous studies. The Weyl-Schrödinger representation obtained here is not equivalent to that was described earlier.
Further, let us briefly describe the main results. Consider the following mapping φ : H ∋ h −→ φ h ∈ L 2 χ defined by Paley-Wiener maps where e * i (·) := · | e i and the projections π i : U ∋ u → u i ∈ U(i) are uniquely defined by π i+1 i . Every function φ h of variable u ∈ U satisfies the equality (Cor.6.6) ż exp Re φ h dχ = exp 1 4 h 2 , h ∈ H.
Our further goal is to analyze the inverse isomorphism Ψ −1 which can be described by the Fourier transform under the measure χ in following waŷ The Fourier transform F acts isometrically on the Hardy space of analytic functions H 2 β (Thm 6.8). So, F acts as an analytic extension of the mapping φ.
Applying the superposition with Ψ , we describe two different representations of the additive group (H, +) over L 2 χ defined by shift and multiplicative groups (Lem. 7.1). Using this we show (in Thm 8.1) that an irreducible representation of the Heisenberg group H C can be realized on L 2 χ in the Weyl-Schrödinger form for all a, b ∈ H and z ∈ C, where T † b and M † a * are defined by shift and multiplicative groups, respectively. It is also proved that the Weyl system W † (a, b) has the densely-defined generator p † a,b := ∂ † b +φ a which satisfies the commutation relation where the groups M † a * and T † b are generated byφ a and ∂ † b , respectively. Applying the Weyl-Schrödinger representation to the associated with H C heat equation, we prove (Thm 9.1) that the following Cauchy problem with has the unique solution w(r) = G † r f for any function f from a finite sum À L 2,n χ , where the 1-parameter Gaussian semigroup G † r has the form Here τ = (τ i ) belongs to the abstract Wiener space {w 0 , · w 0 } defined by the injections l 2 w 0 c 0 of real Banach spaces and endowed with the Wiener measure w in according to the known Gross' theorem [10], whereas the sequence of projectors (p ∼ n ) onto R n is convergent to the identity map on w 0 .
Finally, note that this work is a continuation of previous publications [15,16]. The novelty results from the observation that the system of Schur polynomials with variables on Paley-Wiener maps form an orthonormal basis in L 2 χ . This allowed us to investigate irreducible Weyl-Schrödinger representations and Weyl systems of the Heisenberg group H C on the whole space L 2 χ .
2 Invariant probability measure Their elements u ∈ U are called the virtual unitary matrices. The right action There exists the dense embedding U(∞) U (see [23, n.4]) which assigns the stabilized sequence u = (u k ) to each u m ∈ U(m) such that We always assume that the group U(m) is endowed with the probability Haar measure χ m . Using the Kolmogorov consistency theorem (see, e.g. [23,Lem.4 ), we determine the probability measure on U to be the projective limit where π m+1 m (χ m+1 ) means an image-measure and χ 0 = 1. As is known [28,Thm 2.5], the measure χ is Radon. We now describe the necessary properties of χ.
Consider the Hilbert space L 2 χ of functions f : U → C with the following norm and inner product Lemma 2.1. For any f ∈ L ∞ χ there exists the limit Moreover, the measure χ is invariant under the right action, which means that Restrict χ m toǓ(m). By [28,Thm 6], a probability measureχ satisfying conditions π m (χ) = χ m |Ǔ (m) is well defined iff for every ε > 0 there exists a compact set K ⊂ lim Then by the Prokhorov theoremχ is uniquely determined aš Let ε > 0 and K 1 ⊂Ǔ (1) be a compact set such that Assume that K 1 , . . . , K m are constructed. Since χ m = π m+1 m (χ m+1 ), we get By regularity of χ m+1 |Ǔ (m) , there exists a compact set The induction is complete. Then K = lim ← − K m with K 0 = ½ is compact. By virtue of (2.7), we The measureχ can be extended to lim ← − U(m) \ lim ← −Ǔ (m) as zero, since each χ m is zero on U(m) \Ǔ (m). The uniqueness of the projective limits yieldsχ = χ. So, χ = lim ← − χ m is also well defined and by (2.6) and (2.7) we get By the known Portmanteau theorem [13,Thm 13.16] it follows that the limit (2.3) exists. Whereas, the property (2.4) is a consequence of the equalities by the Fubini theorem. It yields (2.5) since the internal integral on the right-hand side is independent of g by (2.4) and We now note the concentration property of Haar measures sequence (χ m ) satisfying the Kolmogorov conditions χ m = π m+1 m (χ m+1 ) if each group U(m) is endowed with the normalized Hilbert-Schmidt metric As is well known (see [9,33]), (U(m), d HB , χ m ) is a Lévy family. Namely, the following sequence of isoperimetric constants dependent on ε > 0 Taking into account the Lemma 2.1, we can formulate the following conclusion.
As before, {H, · | · } is a separable complex Hilbert space with an orthonormal basis {e i : i ∈ N} and · = · | · 1/2 . For its adjoint space H * the conjugate-linear isometry * : with ı λ ⊢ n and η = η(λ), additionally indexed by all σ ∈ S n . The symmetric tensor power H ⊙n ⊂ H ⊗n is defined to be a range of the orthogonal projector S n : H ⊗n ∋ ψ n −→ h 1 ⊙ . . . ⊙ h n := (n!) −1 ř σ∈Sn σ(ψ n ). We assume that H ⊗n is completed and that H ⊗0 = C. Let ψ n := h ⊗n for h = h i . The embedding {h ⊗n : h ∈ H} ⊂ H ⊙n is total by the polarization formula [6, n.1.5] Let H η ⊂ H be spanned by e ı 1 , . . . , e ıη . We can uniquely assign to any semistandard Taking all ı ∈ I , we conclude that the system indexed by semistandard λ-tabloids : ı λ ⊢ n, λ ∈ Y n , ı ∈ I , additionally indexed by all σ ∈ S n , forms an orthonormal basis in the whole tensor power H ⊗n .
As usually, the symmetric Fock space is defined to be the Hilbertian orthogonal sum Γ(H) = À n≥0 H ⊙n with the orthogonal basis e Y := Ť e Yn : n ∈ N 0 of elements ψ = À ψ n with ψ n ∈ H ⊙n endowed with the inner product and norm Note that by tensor multinomial theorem the Fourier expansion under e Yn (3.4) Definition 3.1. For any h ∈ H and u ∈ U the Paley-Wiener maps are defined to be These maps satisfy the orthogonal conditions φ e i = φ i and have the natural extension φ h * =φ h onto the adjoint space H * .
The family of finite alphabets ı ∈ I is directed and for any ı, ı ′ there exists ı ′′ such that ı ∪ ı ′ ⊂ ı ′′ . This means that the whole system s Y n is orthonormal in L 2 χ . The property s µ  ⊥ s λ ı with |µ| = / |λ| for any ı,  ∈ I follows from (3.9), since for all Borel Ω inǓ or otherwiseχ|Ǔ = χ|Ǔ. Consequently, In particular,χ = lim ← −χ m is regular onŨ by the Riesz-Markov theorem [19, 1.1]. As a consequence, the space L 2 χ coincides with the completion of C(Ũ) and for any f ∈ L 2 χ there exists a sequence (f n ) ⊂ span(s Y ) such that ş |f − f n | 2 dχ → 0. Hence, the system s Y forms an orthogonal basis in L 2 χ . Finally, s Y n ∩ L 2 χ is total in L 2,n χ and s Y n ⊥ s Y m if n = / m. This yields (4.1).

Unitarily-weighted symmetric Fock space
Define on the tensor power H ⊗n the unitarily-weighted norm · H ⊗n Definition 5.1. The unitarily-weighted symmetric Fock space is defined to be the Hilbertian orthogonal sum Γ β (H) = À n≥0 H ⊙n β of elements ψ = À ψ n , ψ n ∈ H ⊙n β with the orthogonal basis e Y = Ť e Yn : n ∈ N 0 and the following inner product and norm We immediately notice that Proof. Taking into account that β λ ≤ 1, we get the following inequalities with summations over semistandard tableaux [ı λ ], [ µ ] and ı,  ∈ I . Let (λ, µ) ∈ N η(λ,µ) be the smallest partition of number n with the length η(λ, µ) containing the partitions λ for m and µ for n − m. Then η(λ, µ) ≥ max{η(λ), η(µ)} and so Thus, the following inequality holds. Using this inequality and that S n/m ≤ 1, we find Summing with coefficients 1/m!, we get T a exp(h) 2 β ≤ exp a 2 exp(h) 2 β . This inequality and totality of {exp(x) : h ∈ H} in Γ β (H) yield the required inequality (5.3). It also follows that Γ β (H) is invariant under T a and that the group property (5.3) holds, since ∂ a+b = ∂ a + ∂ b for all a, b ∈ H by linearity.
has the unique isometric conjugate-linear extension χ may be extended to Φ in following way χ . Now, we consider the ordinary irreducible representation of permutation group S n on the Specht λ-module S λ ı that is corresponded to the standard Young tableau [ı λ ]. The following known hook formula (see [7, I.4.3]) holds, The analyticity of H ∋ h → ψ * (h) is a result of the composition exp(·) and ψ * (·).
Definition 6.1. Let H 2 β be defined as a Hardy space of unitarily-weighted Hilbert-Schmidt analytic functions ψ * (h) of variable h ∈ H endowed with the inner product The conjugate-linear surjective isometry from H 2 β onto Γ β (H) is realized by the conjugatelinear mapping On the other hand, the correspondence Φ : e ⊙λ ı ⇄ φ λ ı with λ ∈ Y and ı ∈ I η(λ) allows us to determine the conjugate-linear isometry from Γ β (H) onto L 2 χ . As a result, the mapping where e * ∅ ı = 1, form orthogonal bases in P n β (H) and H 2 β , respectively, such that Every function ψ * ∈ H 2 β with ψ ∈ Γ β (H) has the expansion with respect to e * Y with summation in the inner sum over all semistandard tabloids [ı λ ] such that ı λ ⊢ n. Each function ψ * ∈ H 2 β is entire Hilbert-Schmidt analytic and can be also written as The following linear isometries, defined by linearization via coherent states, hold Proof. Taking into account (3.3) and (5.2), we conclude that every ψ * ∈ H 2 β such that ψ = À ψ n ∈ Γ β (H) with ψ n ∈ H ⊙n β has the following expansion On the other hand, in relative to the inner product · | · Γ , we have .
with u ∈ U take values in L 2 χ and can be represented as follows where the last exponential function has the power series expansion with coefficients in the form of complex Hermite polynomials h n,m (z,z), z ∈ C.
. Thus, It particularly follows that for all h = αx with x ∈ H, Using the n-homogeneity of derivatives, we find żφ n x f n dχ.
Finally, we notice that the isometry L 2 χ F ≃ H 2 β holds, since the isometry Φ * is surjective by Lemma 6.2. Similarly, we get L 2,n χ F ≃ P n β (H).

Intertwining properties of Fourier transform
The shift group on H 2 β is defined as Hence, by Lemma 6.4, As a conclusion, ∂ † a = − ∂ † a . Moreover, the following commutation relations hold, Proof. Using that T a and M a * are adjoint, we find that This together with the group property by applying F and F −1 yields (7.3). Now, we prove the commutation relations. For any f ∈ L 2 χ and h ∈ H, we have For eachf ∈ D(b * 2 ) ∩ D(∂ 2 a ) and t ∈ C by differentiation, we obtain Hence, for eachf from the dense subspace D(b * 2 ) ∩ D(∂ 2 a ) ⊂ H 2 β , which includes all polynomials generated by finite sums Ψ * (f ) = À ψ n ∈ Γ β (H) with ψ n ∈ H ⊙n β , Corollary 6.7 yields F = * • Φ * and F −1 = Φ • * −1 . The equality (7.5) for m = 0 can be rewritten as which includes all functions generated by finite sums Φ ( À ψ n ) with ψ n ∈ H ⊙n β .
On the other hand, let us define the auxiliary Weyl system Using group properties and the commutation relation (7.7), we obtain Hence, the mapping C × (H ⊕ H ) ∋ (t, h) −→ exp(t)W (h) acts as a group isomorphism into the operator algebra over H 2 β . So, the representation S : H ∋ X(a, b, t) −→ exp(t)W (h) = exp t + 1 2 a | b M b * T a is also well defined over H 2 β , as a composition of group isomorphisms. Let us check the irreducibility. Suppose the contrary. Assume there exist an element h 0 = / 0 in H and an integer n > 0 such that But, this is only possible for h 0 = 0. It gives a contradiction. Finally, using that we obtain that S † = F −1 S F is irreducible. Applying F , F −1 to (8.5) we get (8.1). Consider the Weyl system W † on the space L 2 χ . By (8.1) we obtain the equality Using this equality, we get (8.2) for any fixed h = a + b ∈ H ⊕ H . The 1-parameter group W † (τ a, τ b) = W † (τ h) with real τ has the generator p † h = p † a,b , since Taking into account the inequalities (7.1) and that F is isometric, we get Hence, the group W † (τ a, τ b) in variable τ ∈ R is strongly continuous on L 2 χ and therefore has the dense domain D(p † h ) = f ∈ L 2 χ : p † h f ∈ L 2 χ . Moreover, its generator p † h is closed (see, e.g., [34]). Note also that p † τ h = τ p † h for τ ∈ R. Finally, applying the commutation relation (7.4) and commutability of group generators in different directions over the dense set D(φ 2 a ) ∩ D(∂ †2 b ) ⊂ L 2 χ , we have The relation T †