RECIPROCITY THEOREMS FOR HOLOMORPHIC REPRESENTATIONS OF SOME INFINITE-DIMENSIONAL

GROUPS

QUELQUES THÉORÈMES DE RÉCIPROCITÉ POUR LES REPRÉSENTATIONS HOLOMORPHES IRRÉDUCTIBLES DE

CERTAINS GROUPES DE DIMENSION INFINIE

TUONG TON-THAT THE UNIVERSITY OF IOWA IOWA CITY, IA 52242-1419 USA

                      tonthat@math.uiowa.edu

PACS codes. 02.20.Tw, 02.20.Qs, 03.65.Fd.

Abstract. Let μ denote the Gaussian measure on ℂ^{n×k} defined by dμ (Z) = π^{−nk} exp [− Tr (ZZ^{†})] dZ, where Tr denotes the trace function, Z^{†} = Z̄^{T}, and dZ denotes the Lebesgue measure on ℂ^{n×k}. Let F_{n×k} denote the Bargmann–Segal–Fock space of holomorphic entire functions on ℂ^{n×k} which are also square-integrable with respect to μ. Fix n and let F_{n×∞} denote the Hilbert-space completion of the inductive limit lim_{k→∞} F_{n×k}. Let G_{k} and H_{k} be compact groups such that H_{k} ⊂ G_{k} ⊂ GL_{k} (ℂ). Let G_{∞} (resp. H_{∞}) denote the inductive limit ⋃_{k=1}^{∞} G_{k} (resp. ⋃_{k=1}^{∞} H_{k}). Then the representation R_{G_{∞}} (resp. R_{H_{∞}}) of G_{∞} (resp. H_{∞}), obtained by right translation on F_{n×∞}, is a holomorphic representation of G_{∞} (resp. H_{∞}) in the sense defined by Ol’shanskii. Then R_{G_{∞}} and R_{H_{∞}} give rise to the dual representations R′_{G′_{n}} and R′_{H′_{n}} of the dual pairs (G′_{n}, G_{∞}) and (H′_{n}, H_{∞}), respectively. The generalized Bargmann–Segal–Fock space F_{n×∞} can be considered as both a (G′_{n}, G_{∞})-dual module and an (H′_{n}, H_{∞})-dual module. It is shown that the following multiplicity-free decompositions of F_{n×∞} into isotypic components F_{n×∞} = ∑_{(λ)}⊕ I_{n×∞}^{(λ)} = ∑_{(μ)}⊕ I_{n×∞}^{(μ)} hold, where (λ) is a common irreducible

signature of the pair (G′_{n}, G_{∞}) and (μ) a common irreducible signature of the pair (H′_{n}, H_{∞}), and I_{n×∞}^{(λ)} (resp. I_{n×∞}^{(μ)}) is both the isotypic component of the equivalence classes (λ)_{G_{∞}} (resp. (μ)_{H_{∞}}) and (λ′)_{G′_{n}} (resp. (μ′)_{H′_{n}}). A reciprocity theorem, giving the multiplicity of (μ)_{H_{∞}} in the restriction to H_{∞} of (λ)_{G_{∞}} in terms of the multiplicity of (λ′)_{G′_{n}} in the restriction to G′_{n} of (μ′)_{H′_{n}}, constitutes the main result of this paper. Several applications of this theorem to Physics are also discussed.

Résumé. Soit μ la mesure de Gauss definie sur l’espace vectoriel ℂ^{n×k} par la formule

dμ (Z) = π^{−nk} exp [− Tr (ZZ^{†})] dZ, z ∈ ℂ^{n×k},

où l’on désigne par Tr la trace d’une matrice, Z^{†} = Z̄^{T}, et par dZ la mesure de Lebesgue sur ℂ^{n×k}. Soit F_{n×k} l’espace hilbertien de Bargmann–Segal–Fock des fonctions entières holomorphes f: ℂ^{n×k} → ℂ telles que f soient de carréintegrable par rapport à la mesure μ. On fixe n et l’on désigne par F_{n×∞} le complété de la limite inductive par rapport à k des espaces F_{n×k}. Pour chaque k soient G_{k} et H_{k} deux groupes compacts tels que H_{k} ⊂ G_{k} ⊂ GL_{k} (ℂ), et l’on suppose aussi que H_{k−1} ⊂ H_{k} ⊂ H_{k+1} ⊂ ⋯ et G_{k−1} ⊂ G_{k} ⊂ G_{k+1} ⊂ · · ·. Soit G_{∞} (resp. H_{∞}) la limite inductive de la chaine {G_{k}} (resp. {H_{k}}). Alors la représentation R_{G_{∞}} (resp. R_{H_{∞}}) de G_{∞} (resp. H_{∞}), obtenue par translation à droite sur F_{n×∞}, est holomorphe dans le sens de Ol’shanskii. Les représentations R_{G_{∞}} et R_{H_{∞}} donnent lieu aux représentations R′_{G′_{n}} et R′_{H′_{n}}, respectivement, des paires duales (G′_{n}, G_{∞}) et (H′_{n}, H_{∞}). L’espace hilbertien generalisé de Bargmann–Segal–Fock F_{n×∞} peut être consideré en même temps comme un (G′_{n}, G_{∞})-module et un (H′_{n}, H_{∞})-module. On montre que l’on a les décompositions suivantes de F_{n×∞} en uniques composantes isotypiques

F_{n×∞} = ∑_{(λ)}⊕ I_{n×∞}^{(λ)} = ∑_{(μ)}⊕ I_{n×∞}^{(μ)},

où (λ) est une signature irréductible commune de la paire (G′_{n}, G_{∞}) et (μ) celle de la paire (H′_{n}, H_{∞}), et où I_{n×∞}^{(λ)} (resp. I_{n×∞}^{(μ)}) est à la fois la composante isotypique de la classe d’équivalence de (λ)_{G_{∞}} (resp. (μ)_{H_{∞}}) et celle de (λ′)_{G′_{n}} (resp. (μ′)_{H′_{n}}). On donne une démonstration d’un théorème de réciprocité, donnant la multiplicité de (μ)_{H_{∞}} dans la restriction à H_{∞} de (λ)_{G_{∞}}, en fonction de la multiplicité de (λ′)_{G′_{n}} dans la restriction à G′_{n} de (μ′)_{H′_{n}}. L’article se termine par une discussion de plusieurs applications en Physique du théorème précédant.

1. Introduction

In recent years there is great interest, both in Physics and in Mathematics, in the theory of unitary representations of infinite-dimensional groups and their Lie algebras (see, for example, [Ka1], and the literature cited therein). Starting with the seminal work of I. Segal in [Se] the representation theory of U (∞) and other classical infinite-dimensional groups was thoroughly investigated by Kirillov in [Ki], Stratila and Voiculescu in [S&V], Pickrell in [Pi], Ol’shanskii in [Ol1], Gelfand and Graev in [Ge&Gr], Kac in [Ka2], to cite just a few. A more complete list of references can be found in the comprehensive and important work of Ol’shanskii in [Ol2].

In [Ol2] Ol’shanski generalized Howe’s theory of dual pairs to some infinitedimensional dual pairs of groups. Recently in [TT1] and [TT2] we investigated the generalized Casimir invariants of these infinite-dimensional dual pairs. In [TT3] we gave a general reciprocity theorem for finite-dimensional dual pairs of groups which generalized our previous results in [KT1] and [LT1]. In this article we give a generalization of this reciprocity theorem to the case of dual pairs where one member is infinite-dimensional and the other is finite-dimensional, and discuss the general case where both members are infinite-dimensional. If Section 2 we will review the reciprocity theorem given in [TT3] which serves as the necessary background for the generalized theorem, and more importantly, discuss several interesting applications of this theorem. Section 3 deals with our main theorem, and the paper ends with a short conclusion in Section 4.

2. The Reciprocity Theorem for Finite-Dimensional Pairs of Groups and Its Applications

In [TT3] our reciprocity theorem can be applied to the more general context of dual representations but for this paper we shall restrict ourself to the case of the oscillator dual representations and where one of the members is a compact group.

Let ℂ^{n×k} denote the vector space of all n × k complex matrices. Let μ denote the Gaussian measure on ℂ^{n×k} defined by

dμ (Z) = π^{−nk} exp [− Tr (ZZ^{†})] dZ, Z ∈ ℂ^{n×k}, (2.1)

where in Eq. (2.1) Z^{†} denotes the adjoint of the matrix Z and dZ denotes the Lebesgue measure on ℂ^{n×N}. Let F_{n×k} ≡ F (ℂ^{n×k}) denote the Bargmann–Segal– Fock space of all holomorphic entire functions on ℂ^{n×k} which are also squareintegrable with respect to dμ. Endowed with the inner product

(f g) = ∫_{ℂ^{n×k}} f (Z) \overline{g (Z)} dμ (Z) ; f, g ∈ F_{n×k}, (2.2)

F_{n×k} has a Hilbert-space structure. It can be easily verified that the inner product

⟨f g⟩ = f (D) \overline{g \binom{}{Z̄}}|_{Z=0} (2.3)

where f (D) denotes the formal power series obtained by replacing Z_{αj} by the partial derivative ∂/∂Z_{αj} (1 ≤ α ≤ n, 1 ≤ j ≤ k). In fact if (r) = (r_{11}, …, r_{nk}) is a multi-index of integers r_{αj} ≥ 0 let Z^{(r)} ≡ Z_{11}^{r_{11}} ⋯ Z_{nk}^{r_{nk}} and (r)! = r_{11}! ⋯ r_{nk}! then it is easy to verify that

(\frac{Z^{(r)}}{[(r)!]^{\frac{1}{2}}} \frac{Z^{(r′)}}{[(r′)!]^{\frac{1}{2}}} ) = ⟨\frac{Z^{(r)}}{[(r)!]^{\frac{1}{2}}} \frac{Z^{(r′)}}{[(r′)!]^{\frac{1}{2}}} ⟩ = δ_{(r)}_{(r′)}. (2.4)

It follows immediately from Eq. (2.4) that \{Z^{(r)} / [(r)!]^{\frac{1}{2}} \}_{(r)} forms an orthonormal

basis for F_{n×k} when (r) ranges over all multi-indices; moreover P_{n×k} ≡ P (ℂ^{n×k}), the subspace of all polynomial functions on ℂ^{n×k}, is dense in F_{n×k}.

Let G and G′ be two topological groups. Let R_{G} and R′_{G′} be continuous unitary and completely (discretely) reducible representations of G and G′ on F_{n×k} such that R_{G} and R′_{G′} commute. Then we have the following definition of dual representations (for the definition of dual representations in a more general context see [TT3]).

Definition 2.1. The representations R_{G} and R′_{G′} are said to be dual if the G′ ×Gmodule F_{n×k} is decomposed into a multiplicity-free orthogonal direct sum of the form

F_{n×k} = ∑_{(λ)}⊕ I_{n×k}^{(λ)}, (2.5)

where in Eq. (2.5) the label (λ) characterizes both an equivalence class of an irreducible unitary representation λ_{G} of G and an equivalence class of an irreducible representation λ′_{G′}, and I_{n×k}^{(λ)} denotes the (λ)-isotypic component, i.e., the direct sum (not canonical) of all irreducible subrepresentations of R_{G} (resp. R′_{G′}) that belong to the equivalence class λ_{G} (resp. λ′_{G′}). Moreover the G′ × G-submodule I_{n×k}^{(λ)} is irreducible for all signatures (λ); i.e., I_{n×k}^{(λ)} ≈ V^{(λ_{G})} ˆ⊗ W^{(λ′_{G′})}, where V^{(λ_{G})}

(resp. W^{(λ′_{G′})}) is an irreducible G-module of class (λ_{G}) (resp. G′-module of class (λ′_{G′})).

We refer to the decomposition (2.5) as the canonical decomposition of the G′×Gmodule F_{n×k}.

In this context we have the following theorem which is a special case of Theorem 4.1 in [TT3].

Theorem 2.2. Let G be a compact group. Let R_{G} and R′_{G′} be given dual representations on F_{n×k}. Let H be a compact subgroup of G and let R_{H} be the representation of H on F_{n×k} obtained by restricting R_{G} to H. If there exists a group H′ ⊃ G′ and a representation R′_{H′} on F_{n×k} such that R′_{H′} is dual to R_{H} and R′_{G′} is the restriction of R′_{H′} to the subgroup G′ of H′ then we have the following multiplicity-free decompositions of F_{n×k} into isotypic components

F_{n×k} = ∑_{(λ)}⊕ I_{n×k}^{(λ)} = ∑_{(μ)}⊕ I_{n×k}^{(μ)} (2.6)

where (λ) is a common irreducible signature of the pair (G′, G) and (μ) is a common irreducible signature of the pair (H′, H).

If λ_{G} (resp. λ′_{G′}) denotes an irreducible unitary representation of class (λ) and μ_{H} (resp. μ′_{H′}) denotes an irreducible unitary representation of class (μ) then the multiplicity dim [Hom_{H} (μ_{H} : λ_{G}|_{H})] of the irreducible representation μ_{H} in the restriction to H of the representation λ_{G} is equal to the multiplicity dim [Hom_{G′} (λ′_{G′} : μ′_{H′}|_{G′})] of the irreducible representation λ′_{G′} in the restriction to G′ of the representation μ′_{H′}.

Remarks. In many cases Hom_{H} (μ : λ_{G}|_{H}) and Hom_{G} (λ′_{G} : μ′_{H′}|_{G}) are shown to be isomorphic and can be explicitly constructed in terms of generalized Casimir operators as given in [KT2] and [LT2].

To illustrate this theorem we devote the rest of this section to some typical examples and discuss their generalization.

Examples 2.3. 1) Consider F_{1×k} with k ≥ 2; then F_{1×k} is the classical Bargmann space first considered by V. Bargmann in [Ba]. Then P_{1×k} is the algebra of all polynomial functions in k variables (Z_{1}, …, Z_{k}) = Z. Let G = U (k) and G′ = U (1); then the complexification of U (k) (resp. U (1)) is G_{ℂ} = GL_{k} (ℂ) (resp. G′_{ℂ} = GL_{1} (ℂ)). An element f of F_{1×k} is of the form

f (Z) = ∑_{|(r)|=0}^{∞} c_{(r)}Z^{(r)} (2.7)

with (r) = (r_{1}, …, r_{k}), |(r)| = r_{1} + ⋯ + r_{k}, and Z^{(r)} = Z_{1}^{r_{1}} ⋯ Z_{k}^{r_{k}}, c_{(r)} ∈ ℂ such

that ∑_{|(r)|=0}^{∞} |c_{(r)}|^{2} (r)! < ∞, where (r)! = r_{1}! ⋯ r_{k}!. The system \{Z^{(r)} / [(r)!]^{\frac{1}{2}} \}, where (r) ranges over all multi-indices, forms an orthonormal basis for F_{1×k}. R_{G_{ℂ}} and R_{G} are defined by

\begin{cases}[R_{G_{ℂ}} (g) f] (Z) = f (Zg) , & g ∈ GL_{k} (ℂ) , \\ [R_{G} (u) f] (Z) = f (Zu) , & u ∈ U (k) . \end{cases} (2.8)

R′_{G′_{ℂ}} and R′_{G′} are defined by

\begin{cases}[R′_{G′_{ℂ}} (g′) f] (Z) = f ((g′)^{t} Z) , & g′ ∈ GL_{1} (ℂ) , \\ [R′_{G′} (u′) f] (Z) = f ((u′)^{t} Z) , & u′ ∈ U (1) . \end{cases} (2.9)

The infinitesimal action of R_{G_{ℂ}} is given by

R_{ij} = Z_{i} \frac{∂}{∂Z_{j}} , 1 ≤ i, j ≤ k, (2.10)

which form a basis for a Lie algebra isomorphic to gl_{k} (ℂ).

The infinitesimal action of R′_{G′_{ℂ}} is given by

L = ∑_{i=1}^{k} Z_{i} \frac{∂}{∂Z_{i}} , (2.11)

which forms a basis for a Lie algebra isomorphic to gl_{1} (ℂ). If p, q ∈ P_{1×k} then from Eq. (2.1) of [TT4] we have

R_{G_{ℂ}} (g) p (D) R_{G_{ℂ}} (g^{−1}) = [R_{G_{ℂ}} (g^{✓}) p] (D) ,  g ∈ GL_{k} (ℂ) , g^{✓} = (g^{−1})^{t} , (2.12)

so that if u ∈ U (k) then

(2.13)  ⟨R_{G} (u) p R_{G} (u) q⟩ = [R_{G} (u) p] (D) \overline{(R_{G} (u) q) \binom{}{Z̄}}|_{Z=0}

= R_{G} (u^{✓}) p (D) R_{G} (u^{t}) R \binom{}{ū} \overline{q \binom{}{Z̄}}|_{Z=0}

= p (D) R_{G} (u^{t}ū) q (Z̄u^{✓}) |_{Z=0}

= ⟨p q⟩ , since u^{t}ū = 1. A similar computation shows that

R′_{G′_{ℂ}} (g′) p (D) R′_{G′_{ℂ}} ((g′)^{−1}) = [R ((g′)^{✓})] (D) , g′ ∈ GL_{1} (ℂ) , (2.14)

so that if u ∈ U (1) then

⟨R′_{G′} (u′) p R′_{G′} (u′) q⟩ = ⟨p q⟩ . (2.15)

Note that all equations above from (2.12) to (2.15) remain valid if we replace ℂ^{1×k} by ℂ^{n×k} and GL_{1} (ℂ) (resp. U (1)) by GL_{n} (ℂ) (resp. U (n)).

It follows that R_{G}, G = U (k) (resp. R′_{G′}, G′ = U (n)) is a continuous unitary representation of G (resp. G′) on F_{n×k}.

Let P_{1×k}^{(m)} denote the subspace (of F_{1×k}) of all homogeneous polynomial functions of degree m ≥ 0. Then by the Borel–Weil theorem (see, e.g., [TT4]) the restriction of R_{G_{ℂ}} to P_{1×k}^{(m)} is an irreducible subrepresentation of R_{G_{ℂ}} with highest weight (\underbrace{m, 0, …, 0}_{k}) and highest weight vector cZ_{1}^{m}, c ∈ ℂ^{∗}. In fact, by letting

the infinitesimal operators R_{ij} act on P_{1×k}^{(m)} one can easily show that P_{1×k}^{(m)} is an irreducible subrepresentation of R_{G_{ℂ}}. By “Weyl’s unitarian trick” the restriction of this irreducible subrepresentation to G gives an irreducible unitary representation of G.

Let 0 ≠ p ∈ P_{1×k}^{(m)}. Then (R′_{G′} (g′) p) (Z) = p ((g′)^{t} Z) = p (g′Z) = (g′)^{m} p (Z) for all g′ ∈ GL_{1} (ℂ). So the one-dimensional subspace of F_{1×k} spanned by p is an irreducible G′_{ℂ}-submodule with highest weight (m) and its restriction to G′ is an irreducible unitary G′-submodule. In fact, Euler’s formula implies that

Lp = mp, for all p ∈ P_{1×k}^{(m)}. (2.16)

Thus the canonical decomposition of the G′ × G-module F_{1×k} is simply

F_{1×k} = ∑_{m=0}^{∞}⊕ P_{1×k}^{(m)}. (2.17)

Let H denote the special orthogonal subgroup SO (k). Then H_{ℂ} = SO_{k} (ℂ). Then the ring of all H (or H_{ℂ})-invariant polynomials in P_{1×k} is generated by the constants and p_{0} (Z) = ∑_{1≤i≤k} Z_{i}^{2}. The ring of all H (or H_{ℂ})-invariant differential operators with constant coefficients is generated by the constants and the Laplacian △ = p_{0} (D) = ∑_{1≤i≤k} ∂^{2}/∂Z_{i}^{2}. To find the dual representation of R_{H} we follow the method given in [TT3] by setting

X^{+} = \frac{1}{2}p_{0},  X^{−} = \frac{1}{2}p_{0} (D) = \frac{1}{2} △,  and E = \frac{k}{2} + L. (2.18)

Then X^{+} (resp. X^{−}) acts on F_{1×k} as a creation (resp. annihilation) operator and E acts on F_{1×k} as a number operator. In fact, if p ∈ P_{1×k}^{(m)} then X^{+}p = \frac{1}{2}p_{0}p, X^{−}p = \frac{1}{2} △ p, and Ep = ((k/2) + m) p, so that X^{+} raises P_{1×k}^{(m)} to P_{1×k}^{(m+2)}, X^{−} lowers P_{1×k}^{(m)} to P_{1×k}^{(m−2)} and H multiplies (elementwise) P_{1×k}^{(m)} by the number (k/2) + m. An easy computation shows that

[E, X^{+}] = 2X^{+},  [E, X^{−}] = −2X^{−},  [X^{−}, X^{+}] = E. (2.19)

Eq. (2.19) gives a faithful representation of the Lie algebra sl_{2} (ℝ). Thus the dual action of H is given by this representation. The integrated form of this Lie algebra representation is more subtle to describe: it is the metaplectic representation of the two-sheeted covering group \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} of SL_{2} (ℝ) (or Sp_{2} (ℝ)), and this group is not a matrix group. Its concrete description can be obtained by applying the Bargmann– Segal transform which sends the Schrödinger representation of this group to its Fock representation F_{1×k}. However, for our purpose, its infinitesimal action (2.19) together with the action of its maximal compact group G′ = U (1), which is particularly simple, will suffice. Indeed, it is easy to show that we have the following decomposition of P_{1×k}^{(m)}:

P_{1×k}^{(m)} = ∑_{i=0,...,[m/2]} ⊕ p_{0}^{i}H_{1×k}^{(m−2i)}, (2.20)

where [m/2] denotes the integral part of m/2, and H_{1×k}^{(m−2i)} denotes the subspace of all harmonic homogeneous polynomials of degree (m − 2i), i.e., all functions p ∈ P_{1×k}^{(m−2i)} such that △p = 0. For an integer r ≥ 0 then it can be easily shown that the restriction R_{H}^{(r)} of R_{H} to H_{1×k}^{(r)} is an irreducible representation of H with signature (\underbrace{r, 0, …, 0}_{[k/2]}) and highest weight vector

f^{(r)^{H}} (Z) = \begin{cases}(Z_{1} + iZ_{s+1})^{r} , & if k = 2s, \\ (Z_{1} + iZ_{s+2})^{r} , & if k = 2s + 1, i = \sqrt{−1}. \end{cases} (2.21)

For each integer j ≥ 0, the restriction of R_{H} to the subspace p_{0}^{j}H^{(r)} is equivalent to R_{H}^{(r)} since p_{0}^{j} is H-invariant. Set

I_{1×k}^{(r)} = ∑_{j=0}^{∞}⊕ p_{0}^{j}H_{1×k}^{(r)}; (2.22)

then I_{1×k}^{(r)} is the (\underbrace{r, 0, …, 0}_{k})-isotypic component of R_{H}^{(r)}. From (2.20) and (2.22)

we see that

F_{1×k} = ∑_{r=0}^{∞}⊕ I_{1×k}^{(r)}. (2.23)

Obviously, R′_{H′} (u) = R′_{G} (u), u ∈ G′, leaves each one-dimensional subspace cp_{0}^{j}h, c ∈ ℂ, invariant, since R′_{G} (u) (p_{0}^{j}h) = u^{r+2j} (p_{0}^{j}h) (alternatively, E (p_{0}^{j}h) =

((k/2) + r + 2j) (p_{0}^{j}h)), for all h ∈ H_{1×k}^{(r)}. Clearly, X^{+} (p_{0}^{j}h) = \frac{1}{2}p_{0}^{(j+1)}h, h ∈

H_{1×k}^{(r)}. Finally from the equation

X^{−} (p_{0}f) = (k + 2s) f + \frac{1}{2}p_{0} △ f (2.24)

if f is a polynomial function of degree s, we deduce by induction on the integer j ≥ 1 that

X^{−} (p_{0}^{j}h) = j (k + 2 (r + j − 1)) p_{0}^{j−1}h, h ∈ H_{1×k}^{(r)}. (2.25)

For each fixed h ∈ H_{1×k}^{(r)} let Jh denote the subspace of I_{1×k}^{(r)} spanned by the set

{p_{0}^{j}h | j = 0, 1, 2, … }. Then it follows from the previous discussion that the subrepresentation of the Lie algebra sl_{2} (ℝ) on Jh is irreducible, and thus the metaplectic subrepresentation of \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} on Jh is irreducible as well. As a U (1)-module Jh is reducible, and for this special case each one-dimensional subspace cp_{0}^{j}h, c ∈ ℂ, is an irreducible submodule, and the lowest one is ch which has weight r (or (k/2)+r) since

R′_{G} (u) h = u^{r}h,  u ∈ U (1) ,  or  Eh = (\frac{k}{2} + r) h. (2.26)

In general, if a holomorphic discrete series of a noncompact semisimple Lie group such as \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} considered as a K-module, where K is its maximal compact subgroup, decomposes into a discrete sum of irreducible submodules, each one of them can be characterized by a signature (highest weight, for example) and the one with the lowest highest weight (under the lexicographic ordering) is unique. This lowest K-type highest weight which corresponds to the Harish Chandra’s or Blattner’s parameter, can be used to label the given holomorphic discrete series. We shall call this label its signature. In our example, the holomorphic discrete series Jh of

\begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} has signature r. If dim (H_{1×r}^{(r)}) = d (actually, d = \binom{k+r−1}{r} − \binom{k+r−3}{r−2}) I_{1×k}^{(r)}

is the r-isotypic component (of the metaplectic representation of \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}) which contains d isomorphic copies of signature r.

Now let us verify Theorem 2.2 for this simple example. From Eq. (2.20) we have

(2.27) dim [Hom_{SO(k)} ((\underbrace{r, 0, …, 0}_{[k/2]})_{SO(k)} : (\underbrace{m, 0, …, 0}_{k})_{U(k)}|_{SO(k)})]

= \begin{cases}1, & if r = m − 2i for i = 0, …, [m/2] , \\ 0, & otherwise, \end{cases}

and from Eq. (2.22) and Eq. (2.26) we have

(2.28) dim [Hom_{U(1)} (m_{U(1)} : r\begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}|_{U(1)})] = \begin{cases}1, & if 2j + r = m, \\ 0, & otherwise, \end{cases}

which are obviously identical.

For arbitrary n such that n ≤ k Eq. (2.7) remains valid with (r) = (r_{11}, …, r_{nk}) and Z^{(r)} = Z_{11}^{r_{11}} ⋯ Z_{nk}^{r_{nk}}. Eq. (2.8), (2.12), (2.13), (2.14) remain valid. Eq. (2.10) is replaced by

R_{ij} = ∑_{α=1}^{n} Z_{αi} \frac{∂}{∂Z_{αj}} , 1 ≤ i, j ≤ k. (2.10)′

Eq. (2.11) is replaced by

L_{αβ} = ∑_{i=1}^{k} Z_{αi} \frac{∂}{∂Z_{βi}} , 1 ≤ α, β ≤ n. (2.11)′

Let B′_{n} denote the lower triangular Borel subgroup of G′_{ℂ} = GL_{n} (ℂ), let (λ) be an n-tuple of integers such that λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n} ≥ 0, let λ: B′_{n} → ℂ^{∗} be the holomorphic character defined on B′_{n} by

λ (b′) = (b′_{11})^{λ_{1}} ⋯ (b′_{nn})^{λ_{n}} if b′ = [\begin{matrix}b′_{11} & & & & 0 \\ & . & . & . \\ ∗ & & & & b′_{nn} \end{matrix}] belongs to B′_{n}.

Let P_{n×k}^{(λ)} denote the subspace of all polynomial functions on ℂ^{n×k} which also satisfy the covariant condition

f (b′Z) = λ (b′) f (Z) , (b′, Z) ∈ B′_{n} × ℂ^{n×k}. (2.29)

Let R_{λ} denote the representation of G obtained by right translation on P_{n×k}^{(λ)}. Then by the Borel–Weil theorem (see, e.g., [TT4, Theorem 1.5]) R_{λ} is irreducible with highest weight (λ) and highest weight vector

cf_{λ} (Z) = cΔ_{1}^{λ_{1}−λ_{2}} (Z) Δ_{2}^{λ_{2}−λ_{3}} (Z) ⋯ Δ^{λ_{n}} (Z) , c ∈ ℂ^{∗}, (2.30)

where in Eq. (2.30) Δ_{i} (Z) denotes the i^{th} principal minor of Z.

Similarly let B_{k}^{t} denote the upper triangular Borel subgroup of G_{ℂ} = GL_{k} (ℂ) and let λ′: B_{k}^{t} → ℂ^{∗} be the holomorphic character defined on B_{k}^{t} by

λ′ (b) = b_{11}^{λ_{1}} ⋯ b_{nn}^{λ_{n}} if b = [\begin{matrix}b_{11} & & & & & & & & & ∗ \\ & & . & . & . \\ & & & & & b_{nn} \\ & & & & & & . & . & . \\ & 0 & & & & & & & & b_{kk} \end{matrix}] belongs to B_{k}^{t}.

Let P_{n×k}^{(λ′)} denote the subspace of all polynomial functions on ℂ^{n×k} which also satisfy the covariant condition

f (Zb) = λ′ (b) f (Z) , (b, Z) = B_{k}^{t} × ℂ^{n×k}. (2.31)

Let R′_{λ′} denote the representation of G′ on P_{n×k}^{(λ′)} defined by

[R′_{λ′} (g′) f] (Z) = f ((g′)^{t} Z) , g′ ∈ G′. (2.32)

Then R′_{λ′} is irreducible with highest weight (λ′) and with the same highest weight vector given by Eq. (2.30). By Weyl’s unitarian trick the restriction of R_{λ} (resp. R′_{λ′}) to G = U (k) (resp. G′ = U (n)) remains irreducible with the same signature.

Let I_{n×k}^{(λ)} denote the G′_{ℂ} × G_{ℂ} (or G′ × G)-cyclic module in F_{n×k} generated by the highest vector f_{λ} given by Eq. (2.29); then by Theorem 3, p. 150, of [Ze], I_{n×k}^{(λ)} is irreducible with highest weight (λ′, λ). For the sake of simplicity we say that the

G′_{ℂ} × G_{ℂ}-module I_{n×k}^{(λ)} has signature (λ). To prove that I_{n×k}^{(λ)} ≈ P_{n×k}^{(λ′)} ˆ⊗ P_{n×k}^{(λ)} we

define a map Φ: P_{n×k}^{(λ′)} ˆ⊗ P_{n×k}^{(λ)} → I_{n×k}^{(λ)} as follows:

Let f′ ⊗ f ∈ P_{n×k}^{(λ′)} ˆ⊗ P_{n×k}^{(λ)}. Then f′ and f can be represented in the following form:

f′ = ∑_{i∈I′} c′_{i}R′_{λ′} (g′_{i}) f_{λ}, f = ∑_{j∈I} c_{j}R_{λ} (g_{j}) f_{λ}, (2.33)

where in Eq. (2.33) c′_{i}, c_{j} ∈ ℂ, g′_{i} ∈ G′_{ℂ}, g_{j} ∈ G_{ℂ}, and I′ and I are two finite index sets. Set Φ (f′ ⊗ f) = ∑_{i∈I′, j∈I} c′_{i}c_{j}T (g′_{i}, g_{j}) f_{λ}, where [T (g′_{i}, g_{j}) f_{λ}] (Z) =

f ((g′_{i})^{t} Zg_{j}). Since

R′_{λ′} (g′) f′ = ∑_{i∈I} c′_{i}R′_{λ′} (g′g′_{i}) f_{λ}

and

R_{λ} (g) f = ∑_{j∈I} c_{j}R_{λ} (gg_{j}) f_{λ}

it follows that

Φ [(R′_{λ′} (g′) ⊗ R_{λ} (g)) (f′ ⊗ f)] = ∑_{i∈I, j∈T} c′_{i}c_{j}T (g′g′_{i}, gg_{j}) f_{λ}

= T (g′, g) Φ (f′ ⊗ f)

for all g′ ∈ G′_{ℂ} and g ∈ G_{ℂ}. This means that Φ is an intertwining operator and by Schur’s lemma Φ is either 0 or an isomorphism. Since

Φ (f_{λ} ⊗ f_{λ}) = f_{λ}

it follows that Φ is an isomorphism. Since P_{n×k} is dense in F_{n×k} Theorem 3 (p.

150) of [Ze] (see also [KT1]) implies that we have the Hilbert sum F_{n×k} = ∑_{(λ)}⊕ I_{n×k}^{(λ)} for the pair (U (n) , U (k)).

Now suppose k > 2n and set H = SO (k), H_{ℂ} = SO_{k} (ℂ). Let J_{n×k} denote the ring of all H (or H_{ℂ})-invariant polynomials in P_{n×k}. Then J_{n×k} is generated by the constants and the n (n + 1) /2 algebraically independent polynomials

p_{αβ} (Z) = ∑_{i=1}^{k} Z_{αi}Z_{βi}, 1 ≤ α ≤ β ≤ n. (2.34)

It follows that the ring of all H (or H_{ℂ})-invariant differential operators with constant coefficients is generated by the constants and the Laplacians

△_{αβ} = p_{αβ} (D) = ∑_{i=1}^{k} \frac{∂^{2}}{∂Z_{αi}∂Z_{βi}} , 1 ≤ α ≤ β ≤ n. (2.35)

The infinitesimal action of R′_{G_{ℂ}} is generated by

L_{αβ} = ∑_{i=1}^{k} Z_{αi} \frac{∂}{∂Z_{βi}} , 1 ≤ α, β ≤ n. (2.36)

Set P_{αβ} = −p_{αβ}, E_{αβ} = L_{αβ} + \frac{1}{2}kδ_{αβ}, and D_{αβ} = △_{αβ}; then it follows from [KLT] (see Eq. (3.3)) that {E_{αβ}, P_{αβ}, D_{αβ}} defines a faithful representation of sp_{2n} (ℝ) on F_{n×k}. By construction this representation is dual to the infinitesimal action of R_{H}. The global action R′_{H′} is a unitary metaplectic representation of \begin{matrix} \\ Sp & 2n & (ℝ) \end{matrix}, the two-sheeted covering of Sp_{2n} (ℝ) (see [KLT] for details). As in the case of the pair (U (n) , U (k)) the common highest weight vector (for R′_{H′} the lowest K′-type highest weight vector) of signature (μ) = (μ_{1}, …, μ_{n}) with μ_{1} ≥ ⋯ ≥ μ_{n} ≥ 0 and μ_{i} ∈ ℕ, 1 ≤ i ≤ n, of the pair ({ \atop Sp_{2n} (ℝ) ,}SO (k)) is

f_{μ} (Z) = Δ_{1}^{μ_{1}−μ_{2}} (Zq) Δ_{2}^{μ_{2}−μ_{3}} (Zq) ⋯ Δ_{n}^{μ_{n}} (Zq) , (2.37)

where the k × k matrix q is given by

\frac{1}{\sqrt{2}} [ \begin{matrix}11_{ν} & 11_{ν} \\ i11_{ν} & −i11_{ν} \end{matrix} ] if k = 2ν,

and

\frac{1}{\sqrt{2}} [\begin{matrix}11_{ν} & 0 & 11_{ν} \\ 0 & \sqrt{2} & 0 \\ i11_{ν} & 0 & −i11_{ν} \end{matrix} ] if k = 2ν + 1,

and where 11_{ν} is the unit matrix of order ν.

An element p of P_{n×k} is called H-harmonic if △_{αβ}p = 0 for all α, β = 1, …, n. Let H_{n×k} denote the subspace of all H-harmonic polynomial functions of P_{n×k} and let H_{n×k} (μ) denote the subspace of all elements h of H_{n×k} which also satisfy the covariant condition

h (b′Z) = (b′_{11})^{μ_{1}} ⋯ (b′_{nn})^{μ_{n}} h (Z) , ∀ b′ ∈ B′_{n}. (2.38)

Then according to Theorem 3.1 of [TT4], the representation R_{H} of H which is obtained by right translations on H_{n×k} (μ) is irreducible with signature (μ).

The infinitesimal action of R_{H} is given by

R_{ij}^{H} = ∑_{α=1,...,n} (Z_{αi} \frac{∂}{∂Z_{αj}} − Z_{αj} \frac{∂}{∂Z_{αi}} ) , 1 ≤ i < j ≤ k. (2.39)

From [KLT] the dual infinitesimal action of R_{H} is given by the system {E_{αβ}, P_{αβ}, D_{αβ}} which satisfies the commutation relations

\begin{cases}[E_{αβ}, E_{μν}] = δ_{βμ}E_{αν} − δ_{αν}E_{μβ} \\ [E_{αβ}, P_{μν}] = δ_{βμ}P_{αν} + δ_{βν}P_{αμ} \\ [E_{αβ}, D_{μν}] = −δ_{αμ}D_{βν} − δ_{αν}D_{βμ} \\ [P_{αβ}, D_{μν}] = δ_{αμ}E_{νβ} + δ_{αν}E_{μβ} + δ_{βμ}E_{να} + δ_{βν}E_{μα} \\ [P_{αβ}, P_{μν}] = [D_{αβ}, D_{μν}] = 0 \\ P_{αβ}, P_{βα}, D_{αβ} = D_{βα} \\ P_{αβ}^{†} = D_{αβ},  D_{αβ}^{†} = P_{αβ},  E_{αβ}^{†} = E_{βα}, \\ for all α, β, μ, ν = 1, …, n. \end{cases} (2.40)

By Corollary 3.11 of [TT4] the μ-isotypic component in H_{n×k} consists of d_{μ} copies isomorphic to H_{n×k} (μ), where d_{μ} is the degree of an irreducible representation of G′ = U (n) of signature (μ_{1}, …, μ_{n}). Since from Eq. (2.40) and the fact that f_{μ} is H-harmonic

D_{μν}E_{αβ}f_{μ} = [D_{μν}, E_{αβ}] f_{μ} + E_{αβ}D_{μν}f_{μ}
= δ_{αμ}D_{βν}f_{μ} + δ_{αν}D_{βμ}f_{μ}
= 0,

it follows that E_{αβ}f_{μ} is H-harmonic for every α, β = 1, …, n. Since [E_{αβ}, R_{ij}^{H}] = 0 for all α, β = 1, …, n and i, j = 1, …, k it follows that E_{αβ}: H_{n×k} (μ) → H_{n×k} are intertwining operators, and thus are either 0 or isomorphisms. It follows that the 𝔤′- module generated by the cyclic vector f_{μ} is irreducible with signature (μ_{1}, …, μ_{n}). In fact, from Eq. (3.14) of [TT4] this space is a G′-module. Let G′f_{μ} denote this G′-module; then by construction G′f_{μ} ⊂ H_{n×k}.

If h ∈ G′f_{μ} then from Eq. (2.40) we have

D_{μν}P_{αβ}h = [D_{μν}, P_{αβ}] h + P_{αβ}D_{μν}h
= − (δ_{αμ}E_{νβ} + δ_{αν}E_{μβ} + δ_{βμ}E_{να} + δ_{βν}E_{μν}) h,

and therefore D_{μν}P_{αβ}h belongs to G′f_{μ}. It follows that J_{n×k}G′f_{μ} is an irreducible sp_{2n} (ℝ)-module with signature (μ). Let H′_{n×k} (μ) denote this module and let I_{n×k}^{(μ)} be the H′ × H-cyclic module generated by f_{μ}; then a proof similar to the case I_{n×k}^{(λ)} shows that H′_{n×k} (μ) ˆ⊗ H_{n×k} (μ) is isomorphic to I_{n×k}^{(μ)}. By the “separation of variables theorem” 2.5 of [TT4] and from the fact that P_{n×k} is dense in F_{n×k} it follows that the orthogonal direct sum decomposition F_{n×k} = ∑_{(μ)}⊕ I_{n×k}^{(μ)} holds.

Therefore the reciprocity theorem 2.2 holds for these pairs (G′, G) and (H′, H) as well.

2) Let k = 2l and consider again the dual pair (G′ = U (n) , G = U (k)). Let H =Sp(k); then H_{ℂ} =Sp_{k} (ℂ). If l ≥ n ≥ 2 then the theory of symplectic harmonic polynomials in [TT5] implies that the dual representation to the representation R_{H} on F_{n×k} is a representation of the group SO^{∗} (2n) = H′ whose infinitesimal action is given by Eq. (4.2) of [KLT]. Using Theorem 2.1 of [TT5] and the “separation of variables theorem” for this case we can show similarly that F_{n×k} = ∑_{(μ)}⊕ I_{n×k}^{(μ)} for

this dual pair (SO^{∗} (2n) , Sp (k)). Thus the reciprocity theorem 2.2 holds again for these pairs (G′, G) and (H′, H).

3) The case of the dual pairs (G′ = U (p) × U (q) , G = U (k) × U (k)) and (H′ = U (p, q) , H = U (k)) can be treated in a similar fashion using the results of [TT6] and the infinitesimal action of H′ on F_{n×k} is given by Eq. (6.4) of [TT3]. However, its generalization to the case H = U (∞) in Section 3 is quite delicate and requires a quite different embedding that we shall describe in detail below.

Let p and q be positive integers such that p+q = n. Let k be an integer such that k ≥ 2 max (p, q). Let (λ) be a q-tuple of integers such that λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{q} ≥ 0. Let R_{λ} denote the representation of GL_{k} (ℂ) (or U (k)) defined on P_{q×k}^{(λ)} given by Eq. (2.29) and (2.30) with n replaced by q. We define the contragredient (or dual) representation of R_{λ} as follows.

Let s_{r} denote the r × r matrix with ones along the reverse diagonal and zero elsewhere:

( 0 1 ; \begin{aligned}& & & & & & · \\ & & & & & · \\ & & & & · \\ & & & · \\ & & · \\ & · \\ · \end{aligned} ; 1 0) .

If W ∈ ℂ^{q×k} let W̃ = s_{q}W s_{k}. Thus W̃ is of the form

W̃ = [\begin{matrix}W_{q,k} & ⋯ & W_{q,1} \\ ⋮ & & ⋮ \\ W_{1,k} & ⋯ & W_{1,1} \end{matrix}] . (2.41)

Let P_{q×k}^{(λ^{✓})} denote the subspace of all polynomial functions in W̃ which also satisfy the covariant condition

f (b̃′W̃ ) = λ (b′) f \binom{}{W̃} (2.42)

for all b′ ∈ B′_{q}, where B′_{q} is the lower triangular Borel subgroup of GL_{q} (ℂ), and b̃′ = s_{q}b′s_{q}.

Define the representation R_{λ^{✓}} of GL_{k} (ℂ) (or U (k)) on P_{q×k}^{(λ^{✓})} by

[R_{λ^{✓}} (g) f] \binom{}{W̃} = f (W̃ s_{k}gs_{k}) , g ∈ GL_{k} (ℂ) . (2.43)

Then R_{λ^{✓}} is irreducible with signature \binom{\underbrace{0, …, 0, −λ_{q}, −λ_{q−1}, …, −λ_{1}}}{k} and lowest weight vector

cf_{λ^{✓}} \binom{}{W̃} = Δ_{1}^{λ_{1}−λ_{2}} \binom{}{w̃} Δ_{2}^{λ_{2}−λ_{3}} \binom{}{w̃} ⋯ Δ^{λ_{q}} \binom{}{w̃} , c ∈ ℂ^{∗}, (2.44)

of weight (−λ_{1}, −λ_{2}, …, −λ_{q}, 0, …, 0).

Let P_{q×k}^{(λ^{✓})′} denote the subspace of all polynomial functions in W̃ which also satisfy the covariant condition

f (W̃ b̃) = λ (b) f \binom{}{W̃} , (2.45)

where b̃ = s_{k}bs_{k}, b ∈ B_{k}^{t} (it follows that b̃ is a lower triangular matrix of the

form b̃ = ( ..._{∗ b_{11}}^{b_{kk} 0} )). Let R′_{(λ^{✓})′} denote the representation of GL_{q} (ℂ) (or of

G′ = U (q)) on P_{q×k}^{(λ^{✓})′} defined by

[R′_{(λ^{✓})′} (g′) f] \binom{}{W̃} = f (s_{q} (g′)^{−1} s_{q}W̃ ) , g′ ∈ GL_{q} (ℂ) . (2.46)

Then R′_{(λ^{✓})′} is irreducible with highest weight (λ^{✓})′ and with lowest weight vector given by cf_{λ^{✓}}, c ∈ ℂ^{∗}, of weight (−λ_{1}, −λ_{2}, …, −λ_{q}).

As in the case (λ′) ⊗ (λ) it can be shown that P_{q×k}^{(λ^{✓})′} ˆ⊗ P_{q×k}^{(λ^{✓})} is isomorphic to

I_{q×k}^{(λ^{✓})} and we have the Hilbert sum decomposition F_{q×k} = ∑_{(λ^{✓})}⊕ I_{q×k}^{(λ^{✓})} for the pair (U (q) , U (k)).

Now let G = U (k) × U (k) act on F_{n×k} via the outer tensor product

[R_{U(k)} ˆ⊗ R_{U(k)^{✓}}] (g_{1}, g_{2}) f ([ Z ; W̃ ]) = f ([ \begin{matrix}Zg_{1} \\ W̃ s_{k}g_{2}^{✓}s_{k} \end{matrix} ]) , (2.47)

where Z ∈ ℂ^{p×k}, W̃ ∈ ℂ^{q×k}, p + q = n, g_{1}, g_{2} ∈ U (k). Then G′ = U (p) × U (q) acts on F_{n×k} via the outer tensor product

[R′_{U(p)} ˆ⊗ R′_{U(q)^{✓}}] (g′_{1}, g′_{2}) f ([ Z ; W̃ ]) = f ([\begin{matrix}(g′_{1})^{t} Z \\ s_{q} (g′_{2})^{−1} s_{q}W̃ \end{matrix} ]) , (2.48)

where (g′_{1}, g′_{2}) ∈ U (p) × U (q).

It follows that we have the isotypic decomposition for the dual pairs (G′, G)

F_{n×k} = ∑_{(ν)⊗(λ^{✓})}⊕ I_{n×k}^{(ν)⊗(λ^{✓})}, (2.49)

where I_{n×k}^{(ν)⊗(λ^{✓})} is isomorphic to I_{p×k}^{(ν)} ⊗ I_{q×k}^{(λ^{✓})}.

Let H = {(g, g) : g ∈ U (k)}; then H is isomorphic to U (k) and H acts on F_{n×q} via the inner (or Kronecker) tensor product R_{H} = R_{U(k)}⊗R_{U(k)^{✓}}. Let J_{n×k} denote the ring of all H (or H_{ℂ} ≈ GL_{k} (ℂ))-invariant polynomials in P_{n×k}. Then from [TT6] and [TT3] J_{n×k} is generated by the constants and the p × q algebraically independent polynomials

p_{αβ} ([ Z ; W̃ ]) = (Zs_{k}W̃^{t})_{αβ} = ∑_{i=1}^{k} Z_{αi}W_{βi}, 1 ≤ α ≤ p, 1 ≤ β ≤ q. (2.50)

It follows that the ring of all H or (H_{ℂ})-invariant differential operators with constant coefficients is generated by the constants and the Laplacians

△_{αβ} = p_{αβ} (D) = ∑_{i=1}^{k} \frac{∂^{2}}{∂Z_{αi}∂W_{βi}} , 1 ≤ α ≤ p, 1 ≤ β ≤ q. (2.51)

Together with the infinitesimal action of GL_{n} (ℂ) on F_{n×k} the p_{αβ}’s and △_{αβ}’s generate a Lie algebra isomorphic to su(p, q) with commutation relations given by Eq. (6.4) in [TT3]. The global action of this infinitesimal action defines a representation R′_{H′} of H′ = SU (p, q) on F_{n×k} which is dual to the representation R_{H}.

An element p of P_{n×k} is called H-harmonic if △_{αβ}p = 0 for all α = 1, …, p, and β = 1, …, q. Let H_{n×k} denote the subspace of all H-harmonic polynomial functions of P_{n×k} and let H_{n×k} (μ) denote the subspace of H_{n×k} generated by the

elements f ∈ P_{p×k}^{(ν)} ⊗ P_{q×k}^{(λ^{✓})} which also satisfy the condition △_{αβ} = 0, 1 ≤ α ≤ p,

1 ≤ β ≤ q. Let R_{H}^{(μ)}, μ = (ν) ⊗ (λ^{✓}), denote the representation of H on H_{n×k} (μ) defined by

[R_{H}^{(μ)} (g) f] ([ Z ; W̃ ]) = f ([ \begin{matrix}Zg \\ W̃ s_{k}g^{✓}s_{k} \end{matrix} ]) , (2.52)

for all g ∈ H. Then Theorem 5.2 of [TT3] implies that:

The representation R_{H}^{(μ)} of H ≈ U (k) on H_{n×k} (μ) is an irreducible unitary representation of class (μ) which has signature

(μ) = (\underbrace{ν_{1}, …, ν_{p}, 0, …, 0, −λ_{q}, …, −λ_{1}}_{k}), (2.53)

where in Eq. (2.53) ν_{α}, 1 ≤ α ≤ p, and λ_{β}, 1 ≤ β ≤ q, are integers such that

ν_{1} ≥ ⋯ ≥ ν_{p} ≥ 0 and λ_{1} ≥ ⋯ ≥ λ_{q} ≥ 0. Let f_{μ} ([ Z ; W̃ ]) = f_{ν} (Z) f_{λ^{✓}} \binom{}{W̃},

where f_{ν} is given by Eq. (2.30) with ν replacing λ and f_{λ^{✓}} is given by Eq. (2.44). Let I_{n×k}^{(μ)} be the H′ × H-cyclic module generated by f_{μ}; then a proof similar to the previous cases shows that H′_{n×k} (μ) ˆ⊗ H_{n×k} (μ) is isomorphic to I_{n×k}^{(μ)}. By the “separation of variables theorem” 1.5 of [TT6] and Theorem 5.1 of [TT3] it follows that the orthogonal direct sum decomposition F_{n×k} = ∑_{(μ)}⊕ I_{n×k}^{(μ)} holds. Therefore

the reciprocity theorem 2.2 also holds for these pairs (G′, G) and (H′, H).

4) This example is a generalization of the previous example. Consider r copies of one of the following groups: U (k), SO (k), or Sp(k), with k even for the last, and let each of them act on a Bargmann–Segal–Fock space F_{p_{i}×k}, 1 ≤ i ≤ r, by right translations. Let p_{1} + p_{2} + ⋯ + p_{r} = n, and let G denote the direct product of r copies of each type of group. In the case of U (k) we allow the r^{th} copy to act on F_{p_{2}×k} either directly or contragrediently; for the other cases it is not necessary to consider the contragredient representations since they are identical to the direct representations.

On each F_{p_{i}×k} for the U (k) action we have the dual action of U (p_{i}) by left translations, and with possibly the dual (left) contragredient representation in the case i = r. For SO (k) we have the metaplectic representation of \begin{matrix} \\ Sp & 2p_{i} \end{matrix} (ℝ), and for Sp(k) we have the corresponding representation of SO^{∗} (2p_{i}). Let G′ denote the dual group of G thus obtained. Let H denote the diagonal subgroup of G; then in the case of U (k) an element of H is of the form (\underbrace{u, u, …, u}_{r}) or (\underbrace{u, …, u}, ū_{r−1}), u ∈ U (k), and

in other cases an element of H is of the form (\underbrace{u, u, …, u}_{r}), u ∈ SO (k) or u ∈Sp(k).

Let H′ denote the dual group of H thus obtained. Then H′ is isomorphic in each case to U (n), \begin{matrix} \\ Sp & 2n \end{matrix} (ℝ), or SO^{∗} (2n). As in previous examples it is straightforward to verify that the reciprocity theorem 2.2 holds for these pairs (G′, G) and (H′, H).

3. Reciprocity Theorems for Finite-Infinite Dimensional Dual Pairs of Groups

Let H be an infinite-dimensional separable complex Hilbert space with a fixed basis {e_{1}, e_{2}, …, e_{k}, … }. Let GL_{k} (ℂ) denote the group of all invertible bounded linear operators on H which leave the vectors e_{n}, n > k, fixed. We define GL_{∞} (ℂ) as the inductive limit of the ascending chain of subgroups

GL_{1} (ℂ) ⊂ ⋯ ⊂ GL_{k} (ℂ) ⊂ · · · .

Thus

GL_{∞} (ℂ) = {A = (a_{ij}) , i, j ∈ ℕ | A is invertible

and all but a finite number of a_{ij} − δ_{ij} are 0}.

If for each k we have a Lie subgroup G_{k} of GL_{k} (ℂ) such that G_{k} is naturally embedded in G_{k+1}, k = 1, …, n, …, then we can define the inductive limit G_{∞} = \underrightarrow{lim} G_{k} = ⋃_{k=1}^{∞} G_{k}. For example, U (∞) = \{u ∈ GL_{∞} (ℂ) : u^{∗} = u^{−1}\}, and thus U (∞) is the inductive limit of the groups U_{k} of all unitary operators of H which leave the vectors e_{n}, n > k, fixed.

Following Ol’shanskii we call a unitary representation of G_{∞} tame if it is continuous in the group topology in which the ascending chain of subgroups of type

\{(\begin{matrix}1_{k} & 0 \\ 0 & ∗ \end{matrix})\} , k = 1, 2, 3, …, constitutes a fundamental system of neighborhoods

of the identity 1_{∞}. Assume that for each k a continuous unitary representation (R_{k}, H_{k}) is given and an isomorphic embedding i_{k+1}^{k}: H_{k} → H_{k+1} commuting with the action of G_{k} (i.e., i_{k+1}^{k} ∘ R_{k} (g) = R_{k+1} (g) ∘ i_{k+1}^{k}) is given. For j ≤ k define the connecting map φ_{jk}: G_{j} × H_{j} → G_{k} × H_{k} by

φ_{jk} (g_{j}, x_{j}) = (g_{k}, x_{k}) , (g_{j}, x_{j}) ∈ G_{j} × H_{j}, (3.1)

where in Eq. (3.1) g_{k} (resp. x_{k}) denotes the natural embedding of g_{j} (resp. x_{j}) in G_{k} (resp. H_{k}). Then obviously the diagram

\begin{matrix}G_{j} × H_{j} & ⟶^{R_{j}} & H_{j} \\ φ_{jk}↓ & & ↓i_{k}^{j}=i_{k}^{k−1}∘···∘i_{j+2}^{j+1}∘i_{j+1}^{j} \\ G_{k} × H_{k} & ⟶^{R_{k}} & H_{k} \end{matrix} (3.2)

is commutative. Let H_{∞} denote the Hilbert-space completion of ⋃_{k=1}^{∞} H_{k} and define a representation R_{∞} of G_{∞} on H_{k} by

R_{∞} (g) x = R_{k} (g) x if g ∈ G_{k} and x ∈ H_{k}. (3.3)

Then obviously R_{∞} is a unique continuous unitary representation of G_{∞} on ⋃_{k=1}^{∞} H_{k} which can be extended to a unique continuous unitary representation of G_{∞} on H_{∞}. Let φ_{k} denote the canonical map of (G_{k}, H_{k}) into (G_{∞}, H_{∞}) and i_{k} denote the canonical map of H_{k} into H_{∞}; then obviously the diagram

\begin{matrix}G_{k} × H_{k} & ⟶^{R_{k}} & H_{k} \\ φ_{k}↓ & & ↓i_{k} \\ G_{∞} × H_{∞} & ⟶^{R_{∞}} & H_{∞} \end{matrix} (3.4)

is commutative.

The following theorem, which is well-known when i_{k+1}^{k} is an isometric embedding (see, e.g., [Ol2]), is crucial for what follows.

Theorem 3.1. If the representations (R_{k}, H_{k}) are all irreducible then the inductive limit representation (R_{∞}, H_{∞}) is also irreducible.

Proof. Let A be a bounded operator on H_{∞} which belongs to the commutant of the algebra of operators generated by the set {R_{∞} (g) , g ∈ G_{∞}}. Since ⋃_{k=1}^{∞} H_{k} is dense in H_{∞} and all the linear operators involved are continuous we can without loss of generality consider them as operating on ⋃_{k=1}^{∞} H_{k} and satisfying A (i_{l}^{k} (x)) = Ax for k ≤ l and for all x ∈ H_{k}. Let P_{k} denote the projection of ⋃_{k=1}^{∞} H_{k} onto H_{k}. Let A_{k} denote the restriction of A to H_{k}; then A_{k} is a bounded linear operator of H_{k} into ⋃_{n=1}^{∞} H_{n}. It follows immediately that P_{k}A_{k}: H_{k} → H_{k} is a bounded linear operator on H_{k}. Let x ∈ H_{k} and suppose A_{k}x = Ax belongs to H_{l}. If l ≤ k we may use the isomorphic embedding i_{k}^{l} = i_{k}^{k−1} ∘ ⋯ ∘ i_{l+1}^{l}: H_{l} → H_{k} to identify Ax with an element of H_{k} so that P_{k}A_{k}x = A_{k}x = Ax, and thus

R_{k} (g_{k}) P_{k}A_{k}x = R_{∞} (g_{k}) Ax = AR_{∞} (g_{k}) x = P_{k}R_{k} (g_{k}) x, ∀ g_{k} ∈ G_{k}.

If l > k then use i_{l}^{k} to identify H_{k} with a subspace of H_{l}. Write Ax = y + z where y belongs to the identified subspace of H_{k} and z belongs to its orthogonal complement in H_{l}. Since all representations are unitary and for g_{k} ∈ G_{k} we have i_{l}^{k} ∘ R_{k} (g_{k}) = R_{l} (g_{k}) ∘ i_{l}^{k} it follows that

P_{k}R_{∞} (g_{k}) A_{k}x = P_{k}R_{k} (g_{k}) y = R_{∞} (g_{k}) P_{k}Ax.

By assumption R_{∞} (g_{k}) Ax = AR_{∞} (g_{k}) x, therefore

R_{k} (g_{k}) P_{k}A_{k}x = R_{∞} (g_{k}) P_{k}Ax

= P_{k}R_{∞} (g_{k}) A_{k}x = P_{k}A_{k}R_{∞} (g_{k}) x = P_{k}A_{k}R_{k} (g_{k}) x.

Since this relation holds for all x ∈ H_{k} and g_{k} ∈ G_{k} it follows that P_{k}A_{k} belongs to the commutant of the algebra of operators on H_{k} generated by the set {R_{k} (g_{k}) , g_{k} ∈ G_{k}}. Schur’s lemma for operator algebras (see, e.g., [Di, Proposition 2.3.1, p. 39]) implies that P_{k}A_{k} = λ_{k}I_{k}, where λ_{k} is a scalar depending on k and I_{k} is the identity operator on H_{k}. Now A is a map of inductive limit sets such that P_{k}A_{k}: H_{k} → H_{k}, and it follows from the definition of an inductive limit map that λ_{k} = λ_{l} for sufficiently large k, l with k < l. Indeed, if x ∈ H_{k} and A_{k}x = Ax ∈ H_{j} with j ≤ k then P_{k}A_{k}x = i_{k}^{j} (Ax) = λ_{k}x. For l > k we then have

λ_{l}i_{l}^{k} (x) = P_{l}A_{l} (i_{l}^{k} (x)) = P_{l}A (i_{l}^{k} (x)) = P_{l}Ax

= i_{l}^{j} (Ax) = i_{l}^{k} (i_{k}^{j} (Ax)) = i_{l}^{k} (P_{k}A_{k} (x)) = λ_{k}i_{l}^{k} (x) .

On the other hand, if Ax ∈ H_{j} with j > k then for all l ≥ j we have

P_{l}A_{l} (i_{l}^{k} (x)) = P_{l}A (i_{l}^{k} (x)) = P_{l}Ax = P_{l}A_{j} (i_{j}^{k} (x))

= P_{l}P_{j}A_{j} (i_{j}^{k} (x)) = P_{l} (λ_{j}i_{j}^{k} (x)) = i_{l}^{j} (λ_{j}i_{j}^{k} (x)) = λ_{j}i_{l}^{j} (i_{j}^{k} (x)) = λ_{j}i_{l}^{k} (x) .

Since P_{l}A_{l} (i_{l}^{k} (x)) = λ_{l}i_{l}^{k} (x), we must have λ_{j} = λ_{l} for all l ≥ j. This implies that A = λI_{∞} where λ ∈ ℂ is a constant and I_{∞} is the identity on H_{∞}. By the same Schur’s lemma quoted above the representation R_{∞} on H_{∞} must be irreducible.

Now fix n and consider the chain of Hilbert spaces F_{n×k} from Section 2 with k > 2n. Let (G′_{n}, G_{k}) denote a dual pair of groups with dual representations (R′_{n}, R_{k}) acting on F_{n×k} as in Theorem 2.2. Then we have the chain of embedded subgroups G_{k} ⊂ G_{k+1} ⊂ ⋯; for example, U (k) is naturally embedded in U (k + 1) via the embedding u → ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ), u ∈ U (k). Therefore we can define the inductive limit G_{∞} = \underrightarrow{lim} G_{k} = ⋃_{k>2n}^{∞} G_{k}. We also have an isometric embedding i_{k+1}^{k}: F_{n×k} → F_{n×(k+1)} such that

i_{k+1}^{k} ∘ R_{k} (g) = R_{k+1} (g) ∘ i_{k+1}^{k}.

To see this we take the case n = 1: then an element f of F_{n×k} is a function of Z = (Z_{1}, …, Z_{k}) of the form given by Eq. (2.7), and the verification of the equation above is straightforward. Let F_{n×∞} denote the Hilbert-space completion on ⋃_{k>2n}^{∞} F_{n×k}. Then it is clear that the inductive limit representation R_{∞} of G_{∞} on F_{n×∞} is tame and satisfies the relations (3.2), (3.3), and (3.4).

If G_{k} is a compact group then every irreducible unitary representation of G_{k} is of the form (ρ_{λ_{k}}, V_{λ_{k}}) with highest weight (λ_{k}) = (m_{1}, m_{2}, …, m_{i}, …), where m_{1}, m_{2}, … are nonnegative integers satisfying m_{1} ≥ m_{2} ≥ ⋯ and the numbers m_{i} are equal to 0 for sufficiently large i. Consider the decomposition (2.5) of Definition 2.1 of the dual module F_{n×k} into isotypic components

F_{n×k} = ∑_{(λ_{k})}⊕ I_{n×k}^{(λ_{k})}

where the signatures (λ_{k}) actually depend essentially on n, but since n is fixed, to alleviate the notation we just tacitly assume this dependence. Also for k sufficiently large if (λ_{k}) = (m_{1}, …, m_{i}, …) then (λ_{k+1}) = (m_{1}, …, m_{i}, …, …) and we write succinctly (λ_{k}) ⊂ (λ_{k+1}).

For sufficiently large k we can exhibit an isomorphic embedding i_{k+1}^{k}: I_{n×k}^{(λ_{k})} → I_{n×(k+1)}^{(λ_{k+1})}. If H_{k} is a subgroup of G_{k} such that H′_{n} contains G′_{n} and (H′_{n}, H_{k}) forms a dual pair then the same process can be repeated for the chain (H′_{n}, H_{k}) ⊂

(H′_{n}, H_{k+1}) ⊂ · · ·. If G_{k} (or H_{k}) is of the type \underbrace{U (k) × ⋯ × U (k)}_{r} then each

i_{k+1}^{k} is an isometric embedding; for other types of G_{k} (or H_{k}) the definition of i_{k+1}^{k} is more subtle. This can be examined case by case although the process is very tedious. To illustrate this we consider the case F_{1×k} with H_{k} = SO (k) and G_{1} = \begin{matrix} \\ Sp & 2 & (ℝ) \end{matrix} = \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}. Then Eq. (2.22) and Eq. (2.23) imply that

F_{1×k} = ∑_{r=0}^{∞}⊕ I_{1×k}^{(r)_{k}}  with  I_{1×k}^{(r)_{k}} = ∑_{j=0}^{∞}⊕ p_{0,k}^{j}H_{1×k}^{(r)_{k}}

where p_{0,k} (Z) = Z_{1}^{2} + ⋯ + Z_{k}^{2}, (r)_{k} = (\underbrace{r, 0, …, 0}_{k}), and H_{1×k}^{(r)} are the subspace of

all harmonic homogeneous polynomials of degree r. Obviously a harmonic homogeneous polynomial h of degree r in k variables can be considered as a harmonic homogeneous polynomial of r in k + 1 variables. So we can define an isomorphic embedding i_{k+1}^{k}: I_{1×k}^{(r)_{k}} → I_{1×(k+1)}^{(r)_{k+1}} by sending p_{0,k}^{j}h into p_{0,(k+1)}^{j}h, and clearly

R_{H} (u_{k}) (p_{0,(k+1)}^{j}h) = p_{0,(k+1)}^{j}R_{H} (u_{k}) h = i_{k+1}^{k}p_{0,k}^{j}R_{H} (u_{k}) h

= i_{k+1}^{k} ((R_{H} (u_{k}) p_{0,k}^{j}) (R_{H} (u_{k}) h)) = i_{k+1}^{k} (R_{H} (u_{k}) (p_{0,k}^{j}h))

for all u_{k} ∈ H_{k}. Thus, R_{H} (u_{k}) ∘ i_{k+1}^{k} = i_{k+1}^{k} ∘ R_{H} (u_{k}) for all u_{k} ∈ H_{k}. It follows that i_{k+1}^{k} can be extended to the whole space F_{1×k} and that i_{k+1}^{k} (F_{1×k}) = ∑_{r=0}^{∞}⊕ i_{k+1}^{k} (I_{1×k}^{(r)_{k}}) is an isomorphic embedding of F_{1×k} into F_{1×(k+1)}. Also note

in this very special case (r)_{k} ⊂ (r)_{k+1} for all k > 2 and that no other signatures (r)_{k+1} occur in F_{1×(k+1)} without (r)_{k} occurring in F_{1×k}; this fact is an exception and almost never happens in the general case (e.g., n ≥ 2). By Theorem 3.1

the tensor product representations R_{G′_{n}}^{(λ′)} ⊗ R_{G_{∞}}^{(λ)} and R_{H′_{n}}^{(μ′)} ⊗ R_{H_{∞}}^{(μ)} of G′_{n} × G_{∞}

and H′_{n} × H_{∞} on I_{n×∞}^{(λ)} and I_{n×∞}^{(μ)}, respectively, are irreducible with signature (λ)_{∞} and (μ)_{∞}, respectively, where if (λ_{k}) = (m_{1}, m_{2}, …, m_{i}, …) then (λ)_{∞} = (\underbrace{m_{1}, m_{2}, …, m_{i}, …, 0}_{∞}, …, 0) and similarly for (μ)_{∞}. Note that as n is fixed, the

group G′_{n} remains fixed; however, its representation R′_{G′_{n}} on F_{n×k} does depend

on k, and should be written as (R′_{G′_{n}})_{k}, and as k → ∞, (R′_{G′_{n}})_{∞} has to be considered as an inductive limit of representations, although for k sufficiently large

all the representations (R_{G′_{n}}^{(λ′)})_{k} are equivalent. The same observations apply to

(R′_{H′_{n}})_{k} and (R_{H′_{n}}^{(μ′)})_{k}. To illustrate this let us consider again the case U (1)×U (k)

and \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} × SO (k) acting on F_{1×k}. Indeed, the infinitesimal action of R′_{G′_{ℂ}} is

given by Eq. (2.11) as L_{k} = ∑_{i=1}^{k} Z_{i}∂/∂Z_{i} and L_{k+1} = ∑_{i=1}^{k+1} Z_{i}∂/∂Z_{i}, and for p ∈ P_{1×k}^{(m)} ⊂ P_{1×(k+1)}^{(m)} Eq. (2.16) implies that

L_{k}p = L_{k+1}p = mp.

By Eq. (2.18) the infinitesimal actions of R′_{H′_{1}} on F_{1×k} and F_{1×(k+1)} are given, respectively, by

\{ \begin{matrix}E_{k} & = \frac{k}{2} + L_{k}, & X_{k}^{+} & = \frac{1}{2} ∑_{i=1}^{k} Z_{i}^{2}, & X_{k}^{−} & = \frac{1}{2} ∑_{i=1}^{k} \frac{∂^{2}}{∂Z_{i}^{2}} , & and \\ E_{k+1} & = \frac{k + 1}{2} + L_{k+1}, & X_{k+1}^{+} & = \frac{1}{2} ∑_{i=1}^{k+1} Z_{i}^{2}, & X_{k+1}^{−} & = \frac{1}{2} ∑_{i=1}^{k+1} \frac{∂^{2}}{∂Z_{i}^{2}} . \end{matrix} (3.5)

If h_{k} ∈ H_{1×k}^{(r)} then Eqs. (2.24), (2.25) applied to {E_{k}, X_{k}^{+}, X_{k}^{−}} show that J_{k}h_{k} is an irreducible representation of sl_{2} (ℝ) with signature (r). Similarly if h_{k+1} ∈ H_{1×(k+1)}^{(r)} then J_{k+1}h_{k+1} is also an irreducible representation of sl_{2} (ℝ) with signature (r).

Let F_{n×∞} denote the Hilbert-space completion of ⋃_{k} F_{n×k}; then F_{n×∞} = \underrightarrow{lim} F_{n×k} is the inductive limit of the chain \{F_{n×k}\}.

After this necessary preparatory work we can now state and prove the main theorem of this paper.

Theorem 3.2. Let G_{∞} denote the inductive limit of a chain G_{k} ⊂ G_{k+1} ⊂ ⋯ of compact groups. Let R_{G_{∞}} and R′_{(G′_{n})_{∞}} be given dual representations on F_{n×∞}. Let H_{∞} denote the inductive limit of a chain of compact subgroups H_{k} ⊂ H_{k+1} ⊂ ⋯ such that H_{k} ⊂ G_{k} for all k. Let R_{H_{∞}} be the representation of H_{∞} on F_{n×∞} obtained by restricting R_{G_{∞}} to H_{∞}. If there exists a group H′_{n} ⊃ G′_{n} and a representation R′_{(H′_{n})_{∞}} on F_{n×∞} such that R′_{(H′_{n})_{∞}} is dual to R_{H_{∞}} and R′_{(G′_{n})_{∞}} is the restriction of R′_{(H′_{n})_{∞}} to the subgroup G′_{n} of H′_{n} then we have the following multiplicity-free decompositions of F_{n×∞} into isotypic components:

F_{n×∞} = ∑_{(λ)}⊕ I_{n×∞}^{(λ)} = ∑_{(μ)}⊕ I_{n×∞}^{(μ)} (3.6)

where (λ) is a common irreducible signature of the pair (G′_{n}, G_{∞}) and (μ) is a common irreducible signature of the pair (H′_{n}, H_{∞}).

If λ_{G_{∞}} (resp. λ′_{(G′_{n})_{∞}}) denotes an irreducible unitary representation of class (λ) and μ_{H_{∞}} (resp. μ′_{(H′_{n})_{∞}}) denotes an irreducible unitary representation of class (μ) then the multiplicity dim [Hom_{H_{∞}} (μ_{H_{∞}} : λ_{G_{∞}}|_{H_{∞}})] of the irreducible representation μ_{H_{∞}} in the restriction to H_{∞} of the representation λ_{G_{∞}} is equal to the

multiplicity dim [Hom_{G′_{n}} (λ′_{(G′_{n})_{∞}} : μ′_{(H′_{n})_{∞}} |_{G′_{n}})] of the irreducible representation

λ′_{(G′_{n})_{∞}} in the restriction to G′_{n} of the representation μ′_{(H′_{n})_{∞}}.

Proof. As remarked above, the dual (G′_{n}, G_{∞})-module I_{n×∞}^{(λ)} is irreducible (by Theorem 3.1) with signature (λ), and isotypic components of different signatures are mutually orthogonal since their projections I_{n×k}^{(λ)_{k}} are mutually orthogonal. Finally if a vector in F_{n×∞}, which we may assume to belong to F_{n×k} for some k, is orthogonal to I_{n×∞}^{(λ)} for all (λ), it must therefore be orthogonal to I_{n×k}^{(λ)_{k}} for all (λ)_{k}, and hence must be the zero vector in F_{n×k}, and thus zero in F_{n×∞}. A similar argument applies to the isotypic components I_{n×∞}^{(μ)}, and thus Eq. (3.6) holds.

Now fix (λ) and (μ). Then the restriction of R_{G_{∞}} to I_{n×∞}^{(λ)} decomposes into a (non-canonical) orthogonal direct sum of equivalent irreducible unitary representations of signature (λ)_{∞}. A representative of this representation may be obtained by applying Theorem 3.1 to get the inductive limit (G_{∞}, R_{(λ)_{∞}}) of the chain (G_{k}, R_{λ_{k}}); for example, when G_{k} = U (k), the representation R_{λ_{k}} is given by Eq. (2.29) on P_{n×k}^{(λ)_{k}}. Considered as a G′_{n}-module I_{n×∞}^{(λ)} decomposes into a (non-canonical) orthogonal direct sum of equivalent irreducible unitary representations of signature (λ′)_{n}. A representative of this representation may be obtained by applying Theorem 3.1 to get the inductive limit (G′_{n}, R′_{(λ′_{n})_{∞}}) (note that although G′_{n} is a stationary chain at n, the representations R′_{(λ′_{n})_{k}} depend on k even though they are all equivalent and belong to the class (λ′)_{n}); for example, when G_{n} = U (n) the

representation R′_{λ′_{n}} is given by Eq. (2.32) on P_{n×k}^{(λ′)_{n}} which is defined by Eq. (2.31).

By an analogous argument we infer that the same conclusions hold for (μ), I_{n×∞}^{(μ)},

(H_{∞}, R_{(μ)_{∞}}), (H′_{n}, R′_{(μ′_{n})_{∞}}).

Now consider the decomposition of the restriction to H_{k} of the representation R_{λ_{k}} of G_{k}. The multiplicity of (μ)_{k} in (λ)_{k} |_{H_{k}} is the dimension of Hom_{H_{k}} (R_{μ_{k}} : R_{λ_{k}}|_{H_{k}}), where Hom_{H_{k}} (R_{μ_{k}} : R_{λ_{k}}|_{H_{k}}) is the vector space of linear homomorphisms intertwining R_{μ_{k}} and R_{λ_{k}}|_{H_{k}}. Since G_{k} and H_{k} are, by assumption, compact, this dimension is finite. If T_{k}: H_{μ_{k}} → H_{λ_{k}} is an element of Hom_{H_{k}} (R_{μ_{k}} : R_{λ_{k}}|_{H_{k}}), where H_{μ_{k}} (resp. H_{λ_{k}}) denotes the representation space

of R_{μ_{k}} (resp. R_{λ_{k}}), then since H_{μ_{k}} ⊂ I_{n×k}^{(μ)_{k}} and H_{λ_{k}} ⊂ I_{n×k}^{(λ)_{k}} it follows that we have an inductive chain of homomorphisms {T_{k}: H_{μ_{k}} → H_{λ_{k}}}. Let H_{μ_{∞}} (resp. H_{λ_{∞}}) denote the inductive limit of H_{μ_{k}} (resp. H_{λ_{k}}); then there exists a unique homomorphism T_{∞}: H_{μ_{∞}} → H_{λ_{∞}} (see, for example, [Du, Theorem 2.5, p. 430], or [Ro, p. 44]). Again by Theorem 3.1, R_{λ_{∞}} = \underrightarrow{lim} R_{λ_{k}} (resp. R_{μ_{∞}} = \underrightarrow{lim} R_{μ_{k}}) is irreducible with signature (λ)_{∞} (resp. (μ)_{∞}), and it is easy to show that T_{∞} is an intertwining homomorphism. Conversely, all homomorphisms of inductive limits arise that way. Consequently, the chain Hom_{H_{k}} (R_{μ_{k}} : R_{λ_{k}}|_{H_{k}}) induces the inductive limit Hom_{H_{∞}} (R_{μ_{∞}} : R_{λ_{∞}}|_{H_{∞}}). Obviously for sufficiently large k, dim [Hom_{H_{k}} (R_{μ_{k}} : R_{λ_{k}}|_{H_{k}})] = dim [Hom_{H_{∞}} (R_{μ_{∞}} : R_{λ_{∞}}|_{H_{∞}})]. By duality, we

obtain in the same way the inductive limit [Hom_{G′_{n}} (R′_{(λ′_{n})_{∞}} : R′_{(μ′_{n})_{∞}} |_{G′_{n}})]; ac-

tually this chain stabilizes for k sufficiently large. It follows from Theorem 2.2 (see also the proof of Theorem 4.1 in [TT3]) that dim [Hom_{H_{∞}} (μ_{H_{∞}} : λ_{G_{∞}}|_{H_{∞}})] =

dim [Hom_{G′_{n}} (λ′_{(G′_{n})_{∞}} : μ′_{(H′_{n})_{∞}} |_{G′_{n}})].

As an example we again consider the case F_{1×∞} with G_{∞} = U (∞), G′_{1} = U (1), H_{∞} = SO (∞), and H′_{1} = \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}. Then from Eq. (2.17), (λ)_{k} = (\underbrace{m, 0, …, 0}_{k}),

λ′_{1} = (m), and I_{1×k}^{(λ)_{k}} = P_{1×k}^{(m)}. It follows that (λ)_{∞} = (m, 0, 0,0) and I_{1×∞}^{(λ)_{∞}} =

P_{1×∞}^{(m)_{∞}}, the vector space of all homogeneous polynomials of degree m in infinitely many variables Z_{1}, Z_{2}, etc. The infinitesimal action of R′_{(U(1))_{k}} is given by Eq.

(2.10), L_{k} = ∑_{i=1}^{k} Z_{i}∂/∂Z_{i}, so the infinitesimal action L_{(m)_{∞}} is given the formal

series ∑_{i=1}^{∞} Z_{i}∂/∂Z_{i}. For H_{∞} = SO (∞) and H′_{1} = \begin{matrix} \\ SL & 2 & (ℝ) \end{matrix} the actions are more delicate to describe. From Eq. (2.22), (μ)_{k} = (\underbrace{r, 0, …, 0}_{[k/2]}), where r is an integer

≥ 0, and therefore (μ)_{∞} = (r, 0, 0,0). Let H_{1×k}^{(r)_{k}} denote the space of all harmonic homogeneous polynomials of degree r in k variables Z_{1}, …, Z_{k} then from Eq. (2.22)

I_{1×k}^{(r)_{k}} = ∑_{j=0}^{∞}⊕ p_{0,k}^{j}H_{1×k}^{(r)_{k}}, where p_{0,k} (Z) = ∑_{i=1}^{k} Z_{i}^{2}. We define the actions R_{SO(∞)}

and R′_{\binom{}{SL_{2}(ℝ)}_{∞}} as follows:

Consider the algebras (sl_{2} (ℝ))_{k} with the bases {E_{k}, X_{k}^{+}, X_{k}^{−}} given by Eq. (3.5);

define the projective or inverse limit of the family {(sl_{2} (ℝ))_{k} , I_{1×k}^{(r)_{k}}} as follows: For

each pair of indices l, k with l ≤ k a continuous homomorphism ϕ_{l}^{k}: (sl_{2} (ℝ))_{k} → (sl_{2} (ℝ))_{l} by sending E_{k} to E_{l}, X_{k}^{+} to X_{l}^{+}, X_{k}^{−} to X^{−l}, and extends by linearity to (sl_{2} (ℝ))_{k} → (sl_{2} (ℝ))_{l}. Clearly ϕ_{l}^{k} satisfies fhe following:

a) ϕ_{k}^{k} is the identity map for all k,

b) if i ≤ l ≤ k then ϕ_{i}^{k} = ϕ_{i}^{l} ∘ ϕ_{l}^{k}.

The inverse limit of the system {sl_{2} (ℝ)_{k}} is denoted by

(3.7) sl_{2} (ℝ)_{∞} = \underleftarrow{lim} sl_{2} (ℝ)_{k} = ⟨E_{∞}, X_{∞}^{+}, X_{∞}^{−}⟩ ,

where E_{∞} = \frac{1}{2}1_{∞} + L_{∞}, X_{∞}^{+} = \frac{1}{2} ∑_{i=1}^{∞} Z_{i}^{2}, X_{∞}^{−} = \frac{1}{2} ∑_{i=1}^{∞} \frac{∂}{∂Z_{i}^{2}} .

Then {E_{∞}, X_{∞}^{+}, X_{∞}^{−}} acts on F_{1×∞} as follows: If f ∈ F_{1×∞} then we may assume that f ∈ F_{1×k} for some k and

E_{∞}f = E_{k}f,  X_{∞}^{+}f = X_{k}^{+}f  and  X_{∞}^{−}f = X_{k}^{−}f. (3.8)

If H_{1×∞}^{(r)_{∞}} denotes the subspace (of P_{1×∞}^{(r)_{∞}}) of all harmonic homogeneous polynomials of infinitely many variables Z_{1}, Z_{2}, etc. (i.e., h ∈ H_{1×∞} if and only if h ∈ P_{1×∞}^{(r)_{∞}} and X_{∞}^{−}h = 0) then

I_{1×∞}^{(r)_{∞}} = ∑_{j=0}^{∞}⊕ (2X_{∞}^{+})^{j} H_{1×∞}^{(r)_{∞}}, (3.9)

where in Eq. (3.9) 2X_{∞}^{+} = (p_{0})_{∞} = ∑_{i=1}^{∞} Z_{i}^{2}. Note that H_{1×∞}^{(r)_{∞}} corresponds to

the inductive limit of the chain \{H_{1×k}^{(r)_{k}}\}. Let R\begin{matrix}(r) & _{∞} \\ SO & (∞) \end{matrix} denote the inductive limit

representation of the chain R\begin{matrix}(r) & _{l} \\ SO & (k) \end{matrix}; then R\begin{matrix}(r) & _{∞} \\ SO & (∞) \end{matrix} together with Eq. (3.8) describes

completely the action of the dual pair (\begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}, SO (∞)) on the isotypic component

I_{1×∞}^{(r)_{∞}} and thus we have the isotypic decompositions for the dual pairs (U (1) , U (∞))

and (\begin{matrix} \\ SL & 2 & (ℝ) \end{matrix}, SO (∞)),

F_{1×∞} = ∑_{m=0}^{∞}⊕ I_{1×∞}^{(m)_{∞}} = ∑_{r=0}^{∞}⊕ I_{1×∞}^{(r)_{∞}},

and thus Theorem 3.2 is verified for this example.

Since the next two examples are very important by their applications to Physics we shall state them as corollaries to Theorem 3.2.

Corollary 3.3. Let G_{∞} denote the direct product of r copies of H_{∞} where H_{∞} = U (∞), SO (∞), or Sp (∞). If G_{∞} acts as the exterior tensor product representation V^{(λ_{1})^{∞}} ⊗⋯⊗V^{(λ_{r})^{∞}}, where each V^{(λ_{i})^{∞}}, 1 ≤ i ≤ r, is an irreducible unitary H_{∞}- module, then H_{∞} acts as the inner (or Kronecker) tensor product representation on V^{(λ_{1})^{∞}} ˆ⊗ ⋯ ˆ⊗ V^{(λ_{r})^{∞}}. If λ_{G_{∞}} denotes an irreducible unitary representation of class (λ_{1})_{G_{∞}} ⊗ ⋯ ⊗ (λ_{r})_{G_{∞}} and μ_{H_{∞}} denotes an irreducible unitary representation of class (μ)_{H_{∞}} then the multiplicity dim [Hom_{H_{∞}} (μ_{H_{∞}} : λ_{G_{∞}}|_{H_{∞}})] of the representation (μ)_{H_{∞}} in the inner tensor product (λ_{1})_{∞} ˆ⊗ ⋯ ˆ⊗ (λ_{r})_{∞} is equal to the multiplicity of (μ)_{H_{k}} in the inner tensor product (λ_{1})_{k} ˆ⊗ ⋯ ˆ⊗ (λ_{r})_{k} for sufficiently large k.

Proof. If (λ_{i})_{∞} = (λ_{i}^{1}, λ_{i}^{2}, …, λ_{i}^{j}, …) where λ_{i}^{j} are integers such that λ_{i}^{1} ≥ λ_{i}^{2} ≥

⋯ and λ_{i}^{j} = 0 for all but a finite number of j, let n denote the total number of all nonzero entries λ_{i}^{j}, 1 ≤ i ≤ r; then V^{(λ_{1})^{∞}} ⊗ ⋯ ⊗ V^{(λ_{r})^{∞}} can be realized as a subspace of the Bargmann–Segal–Fock space F_{n×∞}. From Theorem 3.2 it follows that V^{(λ_{1})^{∞}} ⊗⋯⊗V^{(λ_{r})^{∞}} belongs to the isotypic component I_{n×∞}^{(λ)_{G∞}} of F_{n×∞}, thus V^{(λ_{1})^{∞}} ⊗ ⋯ ⊗ V^{(λ_{r})^{∞}} is the inductive limit of the chain {V^{(λ_{1})^{k}} ⊗ ⋯ ⊗ V^{(λ_{r})^{k}}}. If μ_{H_{∞}} is an irreducible unitary representation of class (μ)_{H_{∞}} then by Theorem 3.2

dim [Hom_{H_{∞}} (μ_{H_{∞}} : λ_{G_{∞}}|_{H_{∞}})] = dim [Hom_{G′_{n}} (λ′_{(G′_{n})_{∞}} : μ′_{(H′_{n})_{∞}} |_{G′_{n}})] ,

where λ′_{(G′_{n})_{∞}} (resp. μ′_{(H′_{n})_{∞}}) is the representation of G′_{n} (resp. H_{n}) dual to λ_{G_{∞}}(resp. μ_{H_{∞}}). For sufficiently large k every μ_{H_{∞}} is the inductive limit of a chain μ_{H_{k}} and for such a k Theorem 2.2 implies that

dim [Hom_{H_{k}} (μ_{H_{k}} : λ_{G_{k}}|_{H_{k}})] = dim [Hom_{G′_{n}} (λ′_{(G′_{n})_{k}} : μ′_{(H′_{n})_{k}} |_{G′_{n}})]

= dim [Hom_{G′_{n}} (λ′_{(G′_{n})_{∞}} : μ′_{(H′_{n})_{∞}} |_{G′_{n}})] ,

and this achieves the proof of Corollary 3.3.

Remark. The reason that this corollary only holds for sufficiently large k can be seen in the following example. Let G_{k} = \underbrace{U (k) × ⋯ × U (k)}_{4 times} and H_{k} = U (k) and

consider the tensor product (\underbrace{1, 0, …, 0}_{k}) ⊗ (\underbrace{2, 0, …, 0}_{k}) ⊗ (\underbrace{2, 0, …, 0}_{k}) ⊗ (\underbrace{3, 0, …, 0}_{k});

then for k = 2 we have the spectral decomposition

(1, 0) ⊗ (2, 0) ⊗ (2, 0) ⊗ (3, 0) = (8, 0) + 3 (7, 1) + 5 (6, 2) + 5 (5, 3) + 2 (4, 4) ,

for k = 3 we have

(1, 0, 0) ⊗ (2, 0, 0) ⊗ (2, 0, 0) ⊗ (3, 0, 0)

= (8, 0, 0) + 3 (7, 1, 0) + 5 (6, 2, 0) + 5 (5, 3, 0) + 2 (4, 4, 0)
+ 3 (6, 1, 1) + 6 (5, 2, 1) + 5 (4, 3, 1) + 3 (4, 2, 2) + 2 (3, 3, 2) ,

for k ≥ 4 we have

(\underbrace{1, 0, …, 0}_{k}) ⊗ (\underbrace{2, 0, …, 0}_{k}) ⊗ (\underbrace{2, 0, …, 0}_{k}) ⊗ (\underbrace{3, 0, …, 0}_{k})

= (8, 0, …, 0) + 3 (7, 1, 0, …, 0) + 5 (6, 2, 0, …, 0) + 5 (5, 3, 0, …, 0)

+ 2 (4, 4, 0, …, 0) + 3 (6, 1, 1, 0, …, 0) + 6 (5, 2, 1, 0, …, 0) + 5 (4, 3, 1, 0, …, 0)

+ 3 (4, 2, 2, 0, …, 0) + 2 (3, 3, 2, 0, …, 0) + (5, 1, 1, 1, 0, …, 0)

+ 2 (4, 2, 1, 1, 0, …, 0) + (3, 3, 1, 1, 0, …, 0) + (3, 2, 2, 1, 0, …, 0) .

Thus we can see that the spectral decomposition of (1,0)_{∞} ⊗ (2,0)_{∞} ⊗ (2,0)_{∞} ⊗

(3,0)_{∞} is the same as that of order k for k ≥ 4, with infinitely many zeroes at the end of each signature.

Note also that this corollary applied to the tensor product \underbrace{(1,0)_{∞} ⊗ ⋯ ⊗ (1,0)_{∞}}_{r times}

together with the Schur–Weyl Duality Theorem for U (r) implies the generalized Schur–Weyl Duality Theorem proved by Kirillov for U (∞) in [Ki].

Corollary 3.4. Let V^{(λ_{1})^{∞}}, …, V^{(λ_{r})^{∞}} and V^{(μ)^{∞}} be irreducible unitary representation of H_{∞}. Let V^{(μ^{✓})^{∞}} be the representation (of H_{∞}) contragredient to V^{(μ)^{∞}}. Let I^{∞} denote the equivalence class of the identity representation of H_{∞}. Then the multiplicity of (μ)_{∞} in the tensor product (λ_{1})_{∞} ˆ⊗ ⋯ ˆ⊗ (λ_{r})_{∞} is equal to the multiplicity of I^{∞} in the tensor product (λ_{1})_{∞} ˆ⊗ ⋯ ˆ⊗ (λ_{r})_{∞} ˆ⊗ (μ^{✓})_{∞}.

Proof. To prove this corollary we apply Corollary 3.3 to G_{∞} = \underbrace{H_{∞} × ⋯ × H_{∞}}_{r}

and G_{k} = \underbrace{H_{k} × ⋯ × H_{k}}_{r}, then apply Theorem 3.2 to G_{∞} = \underbrace{H_{∞} × ⋯ × H_{∞}}_{r} × H_{∞}^{✓}

and G_{k} = \underbrace{H_{k} × ⋯ × H_{k}}_{r} ⊗ H_{k}^{✓}, and finally apply Theorem 2.1 of [KT3] to obtain

the desired result at order k. The main difficulty resides with the definition of the identity representation on V^{(λ_{1})^{∞}} ˆ⊗ ⋯ ˆ⊗ V^{(λ_{r})^{∞}} ˆ⊗ V^{(μ^{✓})^{∞}}, which we will construct below.

For each k let I^{k} denote the identity representation of H_{k} on V^{(λ_{1})^{k}} ˆ⊗ ⋯ ˆ⊗ V^{(λ_{r})^{k}} ˆ⊗ V^{(μ^{✓})^{k}}. This means that if I^{k} occurs with multiplicity d in V^{(λ_{1})^{k}} ˆ⊗ ⋯ ˆ⊗ V^{(λ_{r})^{k}} ˆ⊗ V^{(μ^{✓})^{k}} then there exist d nonzero vectors f_{i,k}, i = 1, …, d, such that R_{H_{k}} (u) f_{i,k} = f_{i,k} for all u ∈ H_{k}. By construction each f_{i,k} is a polynomial function in F_{n×k} for some n. Thus f_{i,k} is an H_{k}-invariant polynomial in F_{n×k}. If J_{i,k} denotes the one-dimensional subspace spanned by f_{i,k}, then for sufficiently large k and for each fixed i = 1, …, d we have a chain of irreducible unitary representations {H_{k}, I^{k}, J_{i,k}}_{k}. We can define the isomorphism ψ_{k+1}^{k}: J_{i,k} → J_{i,k+1} by ψ_{k+1}^{k} (cf_{i,k}) = cf_{i,k+1}, c ∈ ℂ; then obviously

ψ_{k+1}^{k} (R_{H_{k}} (u) f_{i,k}) = R_{H_{k+1}} (u) f_{i,k+1} = R_{H_{k+1}} (u) ψ_{k+1}^{k} (f_{i,k}) ,

for all u ∈ H_{k}. Also for all k, l, m with k ≤ l ≤ m we have ψ_{m}^{k} = ψ_{m}^{l} ∘ ψ_{l}^{k}. Thus we can define the inductive limit representation {H_{∞}, I^{∞}, J_{i,∞}}, where the action of H_{∞} on J_{i,∞} is defined as follows:

Let u ∈ H_{∞}; then u ∈ H_{k} for some k. If f ∈ J_{i,l} for some l then

R_{H_{∞}} (u) f_{l} = R_{H_{k}} (u) ψ_{k}^{l}f for l < k,

and

R_{H_{∞}} (u) f_{l} = R_{H_{k}} (u) ψ_{l}^{k}f for k ≤ l.

Then it follows from Theorem 3.1 that {H_{∞}, I^{∞}, J_{i,∞}} is irreducible with signature \binom{}{0}_{∞}. The only problem with this approach is that the isomorphism embedding

ψ_{k+1}^{k} is not the isomorphic embedding i_{k+1}^{k}: F_{n×k} → F_{n×(k+1)}. To circumvent this difficulty we define the inverse or projective limit of the family {H_{k}, I^{k}, J_{k}} where

J_{k} denotes the subspace of all H_{k}-invariants in V^{(λ_{1})^{k}} ˆ⊗ ⋯ ˆ⊗ V^{(λ_{r})^{k}} ˆ⊗ V^{(μ^{✓})^{k}}, as follows: For each pair of indices l, k with l ≤ k define a continuous homomorphism ϕ_{l}^{k}: J_{k} → J_{l} such that

i) ϕ_{k}^{k} is the identity map on J_{k}, ii) if i ≤ l ≤ k then ϕ_{i}^{k} = ϕ_{i}^{l} ∘ ϕ_{l}^{k}.

Here we can take ϕ_{l}^{k} as the truncation homomorphism, i.e., ϕ_{l}^{k} is defined on the generators f_{i,k} by

ϕ_{l}^{k} (f_{i,k}) = f_{i,l}.

The projective limit of the system {H_{k}, J_{k}, ϕ_{l}^{k}} is then formally defined by

J_{∞_{←}} := \underleftarrow{lim} J_{k} = \{(f_{k}) ∈ ∏_{k} J_{k} : f_{l} = ϕ_{l}^{k} (f_{k}) , ∀ l ≤ k\} .

Let π_{k}: J_{∞_{←}} → J_{k} denote the projection of J_{∞_{←}} onto J_{k}. Let I^{∞_{←}} denote the representation of H_{∞} on J_{∞_{←}}; then π_{k} (I^{∞_{←}}f) = π_{k} (f). Recall that if P_{n×k} denotes the subspace of all polynomial functions on ℂ^{n×k} the P_{n×k} is dense in F_{n×k}. Let P_{n×∞} = ⋃_{k=1}^{∞} P_{n×k} denote the inductive limit of P_{n×k}; then clearly P_{n×∞} is dense in F_{n×∞}. Let P_{n×∞}^{∗} (resp. F_{n×∞}^{∗}) denote the dual or adjoint space of P_{n×∞} (resp. F_{n×∞}). Then since P_{n×∞} is dense in F_{n×∞}, F_{n×k}^{∗} is dense in P_{n×∞}^{∗}. By the Riesz representation theorem for Hilbert spaces, every element f^{∗} ∈ F_{n×∞}^{∗} is of the form ⟨ · f⟩ for some f ∈ F_{n×∞}, and the map f^{∗} → f is an anti-linear (or conjugate-linear) isomorphism. Thus we can identify F_{∞}^{∗} with F_{∞} and obtain the rigged Hilbert space as the triple P_{n×∞} ⊂ F_{n×∞} ⊂ P_{n×∞}^{∗} (see [G&V] for the definition of rigged Hilbert spaces). However, generally an element of J_{∞_{←}} does not belong to P_{n×∞}^{∗}, but can still be considered as a linear functional (not necessarily continuous) on P_{n×∞}, and furthermore, in this context the identity representation I^{∞_{←}} will respect the isomorphic embedding i_{k+1}^{k}: F_{n×k} → F_{n×(k+1)}.

4. Conclusion

We have studied thoroughly several reciprocity theorems for some dual pairs of groups (G′_{n}, G_{∞}) and (H′_{n}, H_{∞}), where G_{∞} is the inductive limit of a chain {G_{k}} of compact groups, H_{∞} is the inductive limit of a chain {H_{k}} such that for each k, H_{k} is a compact subgroup of G_{k}, and G′_{n} ⊂ H′_{n} are finite-dimensional Lie groups. These theorems show, in particular, that the multiplicity of an irreducible unitary representation of H_{∞} with signature (μ)_{H_{∞}} in the restriction to H_{∞} of an irreducible unitary representation of G_{∞} with signature (λ)_{G_{∞}} is always finite. This is extremely important in the problem of spectral decompositions of tensor products of irreducible unitary representations of inductive limits of compact classical groups. This type of problems arises naturally in Physics (cf. [K&R]), and in [H&T] tensor product decompositions of tame representations of U (∞) are investigated. In [Ol2] Ol’shanskii generalized Howe’s theory of dual pairs to some infinite-dimensional dual pairs of groups. This is the right context to generalize the reciprocity theorem 3.2 for the infinite-dimensional dual pairs (G′_{∞}, G_{∞}) and (H′_{∞}, H_{∞}) which will be part of our work in a forthcoming publication.

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