Weyl–Schrödinger Representations of Heisenberg Groups in Infinite Dimensions

We investigate the group HC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_{\mathbb {C}}$$\end{document} of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H. Irreducible representations of the Weyl–Schrödinger type on the space Lχ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2_\chi $$\end{document} of quadratically integrable C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document}-valued functions are described. Integrability is understood with respect to the projective limit χ=lim←χi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi =\varprojlim \chi _i$$\end{document} of probability Haar measures χi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _i$$\end{document} defined on groups of unitary i×i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\times i$$\end{document}-matrices U(i). The measure χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document} is invariant under the infinite-dimensional group U(∞)=⋃U(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(\infty )=\bigcup U(i)$$\end{document} and satisfies the abstract Kolmogorov consistency conditions. The space Lχ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2_\chi $$\end{document} is generated by Schur polynomials on Paley–Wiener maps. The Fourier-image of Lχ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2_\chi $$\end{document} coincides with the Hardy space Hβ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}^2_\beta $$\end{document} of Hilbert–Schmidt analytic functions on H generated by the correspondingly weighted Fock space Γβ(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _\beta (H)$$\end{document}. An application to heat equation over HC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_{\mathbb {C}}$$\end{document} is considered.


Introduction
An aim of this work is to investigate irreducible Weyl-Schrödinger representations of the complexified Heisenberg group H C (see [17, 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 1 is its identity map.
The group H C has the unit X(0, 0, 0) and inverse elements of the form In what follows, we consider the infinite-dimensional unitary group U (∞) = 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 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 [23,27]).
In [21,24] 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 [5]) on Paley-Wiener 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 [25,34,35].
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 [1,2]. 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 [19] by using the reproducing kernel Hilbert spaces. The Schrödinger representation of such groups using Gaussian measures over a real Hilbert space was described in [3]. Since the group H C 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 (2) 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 (Corollary 3) The space L 2 χ can be generated by two orthonormal bases, consisting of Schur polynomials and power polynomials of variables φ ı = φ ı1 , . . . , φ ıη , respectively, These bases are indexed by tabloids ı λ with strictly ordered ı = (ı 1 , . . . , ı η ) ∈ N η where λ = (λ 1 , . . . , λ η ) ∈ N η is a partition of n ∈ N and η = η(λ) stands for the length of λ. Then we write briefly ı λ n. The orthogonal expansion L 2 χ = L 2,n χ holds (Theorem 1) where L 2,n χ are formed by n-homogeneous polynomials φ λ ı , normed as follows It is also shown that the surjective linear isometry Ψ : where the Taylor expansion on the right-hand side of any analytic function ψ * f ∈ H 2 β on H is uniquely determined by the corresponding element ψ f ∈ Γ β (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 (Theorem 2). 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 (Lemma 7). Using this we show (in Theorem 3) 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 (Theorem 4) that the following Cauchy problem with has the unique solution w(r) = G † r f for any function f from a finite sum 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 , · w0 } 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 [16,17]. The novelty results from the observation that the system of Schur

Invariant Probability Measure
Consider the unitary group U (∞) = U (m) with m ∈ N 0 = N∪{0}, 1 = U (0), irreducibly acting on a separable Hilbert space H, where subgroups U (m) Following to [21,24], we use the Livšic transforms π m+1 with z m ∈ U (m) defined by excluding Their elements u ∈ U are called the virtual unitary matrices. The right action There exists the dense embedding U (∞) U (see [24, 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. [24,Lem.4.8], [27,Thm 2.2], [30,Cor.4.2]), 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 [30,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 Moreover, the measure χ is invariant under the right action, which means that 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.
The uniqueness of the projective limits yieldš χ = χ. So, χ = lim ← − χ m is also well defined and by (9) and (10) we get By the known Portmanteau theorem [14,Thm 13.16] it follows that the limit (6) exists. Whereas, the property (7) is a consequence of the equalities by the Fubini theorem. It yields (8) since the internal integral on the righthand side is independent of g by (7) and d(χ m ⊗ χ m )(g) = 1. The proof is complete.
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 [9,31]), (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 1, we can formulate the following conclusion.

Polynomials on Paley-Wiener Maps
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 η . Taking all ı ∈ I , we conclude that the system indexed by semistandard λ-tabloids forms an orthogonal basis in the symmetric tensor power H n η . The system e ⊗λ : 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) = n≥0 H n with the orthogonal basis e Y := e Yn : n ∈ N 0 of elements ψ = 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 with h ⊗0 = 1, that is convergent, since e λ ı 2 Definition 1. For any h ∈ H and u ∈ U the Paley-Wiener maps are defined to be where projections π i : U u → u i ∈ U (i) are uniquely defined by π i+1 i . These maps satisfy the orthogonal conditions φ ei = φ i and have the natural extension φ h * =φ h onto the adjoint space H * .
This proves that the system φ Y is orthogonal.
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 j ⊥ s λ ı with |μ| = |λ| for any ı, j ∈ I follows from (19), 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 [20, 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 |f − f n | 2 dχ → 0. Hence, the system s Y forms an orthogonal basis in This yields (20).

Lemma 4. The mapping
has the unique isometric conjugate-linear extension As a result, the conjugate-linear isometries Γ β (H) Φ L 2 χ and H n β Φ L 2,n χ hold.
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 [8, I.4.3]) holds,

Fourier Analysis on Virtual Unitary Matrices
Consider the isometry H * n β P P n β (H) (see e.g., [7, 1.6]), where the space P n β (H) of unitarily-weighted n-homogeneous Hilbert-Schmidt polynomials of variable h ∈ H is defined to be a restriction to the diagonal in H × . . . × H of the n-linear forms P • ψ n endowed with the norm ψ * n P n β can be written as The analyticity of H h → ψ * (h) is a result of the composition exp(·) and ψ * (·).

Definition 3.
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 conjugate-linear 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  (26) 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 (13) and (23), we conclude that every ψ * ∈ H 2 β such that ψ = 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 .
Verify the first equality in (27) by substituting (26) into the formula (27). We get If Hence, the second equality in (27)  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.

Theorem 2.
For any f = f n ∈ L 2 χ with f n ∈ L 2,n χ the entire function and its Taylor coefficients at zero d n 0f have the integral representationŝ For every f ∈ D(∂ †m a ) = f ∈ L 2 χ : ∂ †m a f ∈ L 2 χ with m ∈ N 0 , a * m M a * F (f ) = F ∂ †m a T † a f , a ∈ H.