## 1. Introduction

Pairs of groups that have dual representations on a Hilbert space, such as those given by the Schur-Weyl duality theorem [

1,

2] and by Howe’s dual reductive pairs [

3], have wide applications in both mathematics [

4] and physics [

5,

6]. They were shown [

7], for example, to expose an intimate relationship between symmetry groups and dynamical groups in the quantum mechanics of many-particle systems. However, this contribution is concerned with a different kind of duality relationship that emerges in vector-coherent-state theory [

8,

9].

VCS theory is a physics version of the mathematical theory of induced representations [

10,

11] which focuses on the construction of irreducible representations. Vector coherent states were introduced [

8] for the purpose of deriving explicit realisations of the holomorphic representations of the non-compact symplectic groups and their Lie algebras, as needed in applications of a symplectic model in nuclear collective dynamics [

12,

13], and were subsequently found to have many other applications [

11,

14,

15,

16,

17],

i.e., induced representations of a more general type [

18,

19,

20,

21,

22]. Partially-coherent state representations, related to VCS representations, were also introduced for this purpose by Deenen and Quesne [

23]. A recent review of the applications of VCS theory in nuclear physics can be found in [

24]. Classes of vector coherent states, for which there is a resolution of the identity, have been considered more recently by Ali and others [

25,

26,

27] but not, as far as we are aware, for the construction of Lie group or Lie algebra representations.

Holomorphic discrete-series representations of connected non-compact simple Lie groups were defined by Harish-Chandra [

28,

29,

30] and further explored by Godement [

31], Gelbart [

32] and others [

33]. However, for computational applications in physics, the evaluation of the integral expressions for their inner products posed problems. Moreover, they applied only to holomorphic representations which are in the discrete series, even though there are others that are sometimes needed. (Discrete series representations of a Lie group are subrepresentations of its regular representation.) Thus, complementary coherent-state methods were developed [

8,

9] for deriving the matrix elements of the infinitesimal generators of many Lie groups in terms of a VCS generalisation of standard coherent-state representations [

34,

35,

36]. Essential ingredients of VCS theory, which enabled it to provide practical computational algorithms, were the so-called K-matrix methods [

8,

37] for calculating inner products.

VCS theory was shown [

38] to reproduce the Harish-Chandra/Godement expressions of the Sp

$(N,\mathbb{R})$ holomorphic discrete-series representations. It was also used to derive explicit representations of numerous Lie algebras, including a graded (super) Lie algebra [

17], as outlined in several reviews [

11,

39,

40]. In this paper, a refined version of K-matrix theory is developed and its underlying structure is exposed in terms of complementary VCS irreps on simply-defined extensions of Bargmann Hilbert spaces. These irreps are not unitary but are equivalent to unitary irreps. They are biorthogonal duals of each other relative to the extended Bargmann inner products and in combination, a complementary pair of irreps define an orthonormal basis for an irreducible unitary representation. This approach provides systematic procedures for calculating the explicit matrices of irreducible Lie algebra representations as needed in applications of dynamical symmetry in physics. Moreover, they are not restricted to discrete-series representations and apply to both compact and non-compact Lie groups.

For the purpose of constructing unitary irreps of a Lie group

${G}_{0}$, coherent states are most usefully defined [

36,

41] as elements of a minimum-dimensional orbit of

${G}_{0}$ within the Hilbert space of an irreducible unitary representation. For simplicity, we here restrict consideration to connected semi-simple Lie groups and irreps with extremal (highest- or lowest-weight) states. A minimum-dimensional orbit for such an irrep is one that contains an extremal state. The Lie algebra of

G, the complex extension of

${G}_{0}$, is then of the form

$\mathfrak{g}={\mathfrak{n}}_{+}\oplus \mathfrak{k}\oplus {\mathfrak{n}}_{-}$, where

$\mathfrak{k}$ is the complex extension of the Lie algebra of the isotropy subgroup

${K}_{0}\subset {G}_{0}$ that leaves the extremal state invariant;

${\mathfrak{n}}_{+}$ and

${\mathfrak{n}}_{-}$ are, respectively, subalgebras of raising and lowering operators.

Harish-Chandra showed that the quotient

${K}_{0}\setminus {G}_{0}$ can be regarded in a natural way as a bounded open subset of the complex vector space

${\mathfrak{n}}_{+}$. This is shown in the VCS context in

Section 2.3. It gives a natural complex structure on the quotient space and makes it possible to discuss holomorphic functions. This underlies the construction of holomorphic discrete series [

28,

29,

30].

Construction of a holomorphic unitary irrep of ${G}_{0}$ by coherent-state methods is then straightforward when ${\mathfrak{n}}_{+}$ and ${\mathfrak{n}}_{-}$ are Abelian. However, this condition imposes an unacceptable restriction on the set of holomorphic representations that can be constructed by coherent state methods. Fortunately, the construction can be extended by considering a larger group ${K}_{0}\subset {G}_{0}$ such that $\mathfrak{g}$ continues to be of the form ${\mathfrak{n}}_{+}\oplus \mathfrak{k}\oplus {\mathfrak{n}}_{-}$ for some subalgebras ${\mathfrak{n}}_{\pm}$ which are Abelian. Consider an irrep of ${G}_{0}$ which contains an irrep of ${K}_{0}$ that is extremal in the sense that it is killed by ${\mathfrak{n}}_{-}$ (or by ${\mathfrak{n}}_{+}$). Then, instead of the orbit of a single extremal state, we consider the orbit of this ${K}_{0}$-irrep. With this extension, it becomes straightforward to construct all the holomorphic discrete series irreps, considered by Harish-Chandra, plus others that are limits of the discrete series.

It is known [

42,

43] that coherent-state orbits are diffeomorphic to coadjoint orbits. It is also known [

44,

45] that coadjoint orbits are classical phase spaces and carry classical representations of the dynamics of the model for which

${G}_{0}$ is a dynamical group. Thus, the construction of the unitary irreps of a Lie group

${G}_{0}$ by VCS methods is, in effect, a partial realisation of Kirillov’s objectives of constructing unitary irreps from coadjoint orbits. As shown by Bartlett

et al. [

46], the VCS construction is also a powerful tool in realising the objectives of geometric quantisation.

This paper starts with consideration of a unitary irrep with lowest weight κ of a connected non-compact semi-simple Lie group ${G}_{0}$ that has a Lie algebra ${\mathfrak{g}}_{0}$ with complex extension $\mathfrak{g}={\mathfrak{n}}_{+}\oplus \mathfrak{k}\oplus {\mathfrak{n}}_{-}$ for which $\mathfrak{k}$ is the complex extension of a compact subalgebra ${\mathfrak{k}}_{0}$ of ${\mathfrak{g}}_{0}$ and ${\mathfrak{n}}_{\pm}$ are, respectively, subalgebras of Abelian raising and lowering operators. Vector coherent states are then defined for this irrep based on a set of lowest-grade states $\{|\kappa \alpha \rangle \}$, i.e., states that are annihilated by the lowering operators of ${\mathfrak{n}}_{-}$ and are a basis for an irrep ${\widehat{\sigma}}_{\kappa}$ of ${\mathfrak{k}}_{0}$.

We adopt conventions standard in physics, in which ${\widehat{X}}^{\u2020}$ denotes the Hermitian adjoint of an operator $\widehat{X}$, $\tilde{M}$ denotes the transpose of a matrix M, and * denotes complex conjugation. Bases of raising and lowering operators will be denoted, respectively, by $\{{A}_{i},i=1,\cdots ,d\}$ and $\{{B}_{i},i=1,\cdots ,d\}$, and the representation of any element $X\in \mathfrak{g}$ in the representation with lowest weight κ will be denoted by $\widehat{X}$. The raising and lowering operators will be defined such that, in a unitary irrep, ${\widehat{B}}_{i}={\widehat{A}}_{i}^{\u2020}$.

Following a derivation in

Section 2 and

Section 3 of the Harish-Chandra holomorphic discrete-series representations from a VCS perspective, the following sections develop the dual VCS theory of holomorphic representation. An early version of the theory was initiated by the authors [

47] in terms of coherent-state triplets and led to many analytical results for scalar coherent-state representations. The present VCS developments are new and apply to a much wider class of representations.

If

${\mathbb{H}}^{\kappa}$ denotes the Hilbert space of a unitary irrep of the Lie algebra

${\mathfrak{g}}_{0}$ with lowest weight

κ, a state

$|{\psi}_{\nu}\rangle \in {\mathbb{H}}^{\kappa}$ is observed to have two naturally-defined VCS wave functions: one,

${\Phi}_{\nu}$, defined by the expansion of the state

$|{\psi}_{\nu}\rangle $ as a combination of vector coherent states

where

$dv\left(z\right)$ is any conveniently-chosen measure; the other,

${\Psi}_{\nu}$, defined by its overlaps with vector coherent states

(Note that a state in a Hilbert space can be represented by a variety of wave functions. For example, a given eigenstate of a particle in a harmonic oscillator potential can be represented by a function of its position coordinates and by a function of its momentum coordinates. A remarkable property of a dual pair of VCS wave functions for a state is that they are different functions of a common set of variables.)

If one chooses

$dv$ to be the Bargmann measure [

48], for which

one obtains from these equations the powerful results

where

and

with

Systematic procedures are given in the text for constructing the unitary VCS irreps with these dual VCS wave functions. Similar constructions are also given for compact Lie groups defined by highest-weights.

## 2. VCS Construction of Holomorphic Discrete-series Representations

It is shown in this section that VCS theory reproduces the standard results for holomorphic-discrete-series representations.

Let

${G}_{0}$ be a real, simple and connected Lie group with Lie algebra

${\mathfrak{g}}_{0}$ which, if non-compact, has a faithful finite-dimensional representation. (Extension to a reductive Lie group is straightforward but will not be considered.) Without loss of generality, we regard

${G}_{0}$ as a matrix subgroup of GL(

$n,\mathbb{C}$). Let

G with Lie algebra

$\mathfrak{g}$ denote the complex extension of

${G}_{0}$. Then,

${G}_{0}$ has unitary holomorphic representations if its Lie algebra

${\mathfrak{g}}_{0}$ has a compact subalgebra

${\mathfrak{k}}_{0}$ that contains a Cartan subalgebra for

${G}_{0}$ and

$\mathfrak{g}$ is expressible as a direct sum

in which

$\mathfrak{k}$ is the complex extension of

${\mathfrak{k}}_{0}$, and

${\mathfrak{n}}_{\pm}$ are, respectively, Abelian subalgebras of raising and of lowering operators for which

$[\mathfrak{k},{\mathfrak{n}}_{\pm}]\in {\mathfrak{n}}_{\pm}$ and

$[{\mathfrak{n}}_{-},{\mathfrak{n}}_{+}]\in \mathfrak{k}$.

The conditions on

${G}_{0}$ are such that, if

${K}_{0}\subset {G}_{0}$ is a compact subgroup with Lie algebra

${\mathfrak{k}}_{0}$,

${G}_{0}/{K}_{0}$ and

${K}_{0}\setminus {G}_{0}$ are symmetric spaces, and

${G}_{0}$ and

${K}_{0}$ have defining representations of the block-matrix form

with

$a,e\in {M}_{pp}\left(\mathbb{C}\right)$,

$d,f\in {M}_{qq}\left(\mathbb{C}\right)$,

$b\in {M}_{pq}\left(\mathbb{C}\right)$ and

$c\in {M}_{qp}(\mathbb{C}$). Likewise, the subgroups of

G generated by the subalgebras

${\mathfrak{n}}_{\pm}$ can be defined by matrix representations of the form

in which

${I}_{p}\in {M}_{pp}$ and

${I}_{q}\in {M}_{qq}$ are identity matrices.

Substantial use will be made in the following of the Gauss factorisation

where

a,

b,

c, and

d are real or complex matrices, which applies to any matrix of the given block matrix form, provided

$det\left(d\right)\ne 0$.

#### 2.1. Holomorphic VCS Representations of the Group ${G}_{0}$

Let

${\widehat{\mathcal{U}}}^{\kappa}$ denote an irreducible unitary representation of

${G}_{0}$ and its Lie algebra

${\mathfrak{g}}_{0}$ on a Hilbert space

${\mathbb{H}}^{\kappa}$ and let

${\widehat{T}}^{\kappa}$ denote the extension of

${\widehat{\mathcal{U}}}^{\kappa}$ to the complex Lie algebra

$\mathfrak{g}$. We also let

${\widehat{T}}^{\kappa}$ denote the extension of

${\widehat{\mathcal{U}}}^{\kappa}$ from the compact subgroup

${K}_{0}\subset {G}_{0}$ to the corresponding complex group

$K\subset G$. The matrix

then has the property that the subalgebra

$\mathfrak{k}$ is an eigenspace of ad(

${c}_{0}$) with zero eigenvalue,

i.e.,

and the subalgebras

${\mathfrak{n}}_{\pm}$ are eigenspaces of ad(

${c}_{0}$) with respective eigenvalues

$\pm 2$, given by

We consider holomorphic representations for which the spectrum of the self-adjoint operator ${\widehat{c}}_{0}={\widehat{T}}^{\kappa}\left({c}_{0}\right)$ on ${\mathbb{H}}^{\kappa}$ is bounded from below (if the required spectrum is bounded from above, replace ${c}_{0}$ with -${c}_{0}$). The eigenspace belonging to the least ${\widehat{c}}_{0}$ eigenvalue is then called the lowest-grade subspace. It is the subspace ${\mathbb{H}}_{0}^{\kappa}$ of states in ${\mathbb{H}}^{\kappa}$ that are annihilated by the lowering operators representing elements of ${\mathfrak{n}}_{-}$.

Each eigenspace of

${\widehat{c}}_{0}$ is a

${K}_{0}$-invariant subspace of

${\mathbb{H}}^{\kappa}$ and is a direct sum of subspaces for unitary irreps of

${K}_{0}$. In particular, the lowest-grade subspace,

${\mathbb{H}}_{0}^{\kappa}$, carries a single irrep of

$\mathfrak{k}$, which we denote by

${\widehat{\sigma}}^{\kappa}$. As will be seen, this irrep

${\widehat{\sigma}}^{\kappa}$ uniquely determines, up to unitary equivalence, the irrep

${\widehat{\mathcal{U}}}^{\kappa}$ of

${G}_{0}$. Thus, the irrep

${\widehat{\mathcal{U}}}^{\kappa}$ and its Hilbert space

${\mathbb{H}}^{\kappa}$ are appropriately labelled by the highest weight

κ of the irrep

${\widehat{\sigma}}^{\kappa}$. Thus, if

${\widehat{\Pi}}^{\kappa}$ denotes the projection operator

to the lowest-grade subspace of the Hilbert space

${\mathbb{H}}^{\kappa}$ that we wish to construct, the representation

${\widehat{\sigma}}^{\kappa}$ is related to the representation

${\widehat{T}}^{\kappa}$ by the intertwining relationship

The objective is now to induce the representation ${\widehat{\mathcal{U}}}^{\kappa}$ of ${G}_{0}$ from the irrep ${\widehat{\sigma}}^{\kappa}$ of its subgroup ${K}_{0}$ by VCS methods. If $\widehat{Z}\left(z\right)$ represents an element $Z\left(z\right)\in {\mathfrak{n}}_{-}$, its Hermitian adjoint ${\widehat{Z}}^{\u2020}\left(z\right)$ represents an element of ${\mathfrak{n}}_{+}$ and is a raising operator for the representation under construction. A set of coherent states $\{{e}^{{\widehat{Z}}^{\u2020}\left(z\right)}|\alpha \rangle ,Z\left(z\right)\in {\mathfrak{n}}_{-}\}$ is then defined for every vector $|\alpha \rangle $ in the space of lowest-grade states.

Holomorphic wave functions for states of the Hilbert space

${\mathbb{H}}^{\kappa}$ can now be defined. Observe that an element

$Z\left(z\right)\in {\mathfrak{n}}_{-}$ can be expanded

$Z\left(z\right)={\sum}_{i}{z}_{i}{B}_{i}$, in a basis

$\{{B}_{i},i=1,\cdots ,d\}$ for the

d-dimensional subalgebra

${\mathfrak{n}}_{-}$. A vector

z, with components

$\{{z}_{i}\}$, is then an element of a complex vector space

$\mathcal{Z}$ that is isomorphic, as a vector space, to

${\mathfrak{n}}_{-}$, and the Hilbert space

${\mathbb{H}}^{\kappa}$ is spanned by a subset of the coherent states

$\left\{{e}^{{\widehat{Z}}^{\u2020}\left(z\right)}|\alpha \rangle ,|\alpha \rangle \in {\mathbb{H}}_{0}^{\kappa},z\in D\right\}$, where

D is an open neighbourhood of

$z=0$ in

$\mathcal{Z}$. Thus, in this paper, the same symbol

z is used, without ambiguity, to denote both an element of

$\mathcal{Z}$ and the corresponding matrix representing an element of

${\mathfrak{n}}_{-}$ in Equation (12). It follows that a state vector

$|\psi \rangle \in {\mathbb{H}}^{\kappa}$ can be represented by a VCS wave function Ψ which is a holomorphic function of the complex variables

$\{{z}_{i}\}$ that takes values in

${\mathbb{H}}_{0}^{\kappa}$, given by

The corresponding holomorphic VCS representation

$\widehat{\Gamma}$, isomorphic to the desired irrep

${\widehat{\mathcal{U}}}^{\kappa}$, is then defined by

In fact, because the VCS wave functions are holomorphic functions of the variables

$\{{z}_{i}\}$, their values are defined at all

$z\in \mathcal{Z}$ by their values in any open neighbourhood

D of

$z=0$ in

$\mathcal{Z}$. Let

${\mathcal{F}}^{\kappa}\left(D\right)$ denote the space of holomorphic functions on

D with values in

${\mathbb{H}}_{0}^{\kappa}$, defined by Equation (

19). The VCS linear mapping

then intertwines the actions

${\widehat{\mathcal{U}}}^{\kappa}\left(g\right)$ and

$\widehat{\Gamma}\left(g\right)$ for all

$g\in {G}_{0}$, i.e.,

Consider the action

$\widehat{\Gamma}\left(g\right)$ defined by Equation (

20) of a group element

$g=g(a,b,c,d)\in {G}_{0}$ starting from the observation that the product

${e}^{\widehat{Z}\left(z\right)}{\widehat{\mathcal{U}}}^{\kappa}\left(g\right)$ is a representation of the element

Thus, provided

$det(d+zb)\ne 0$, the product

${e}^{Z\left(z\right)}g$ has the Gauss factorisation given, according to Equation (

13), by

It follows that, for

$g\in {G}_{0}$ and

$z\in D$,

where

$k\in K$ is the element given in the defining representation by

The domain

D on which the VCS representation

$\widehat{\Gamma}$ acts can now be defined as a convenient subset of

$\mathcal{Z}$ that is invariant under the transformations

for all the group elements

$g(a,b,c,d)\in {G}_{0}$. The domain

D must also exclude any

z for which

$det(d+zb)$ could vanish, or be such that all such points are of zero measure. Such a domain is identified with a subset of

${K}_{0}\setminus {G}_{0}$ as follows. Let

$P\subset G$ denote the parabolic subgroup with Lie algebra

The action given by Equation (

27) allows us to canonically identify

${K}_{0}\setminus {G}_{0}$ with an open submanifold of

$P\setminus G$ [

29,

49,

50], and by regarding

$exp\left(Z\left(z\right)\right)$ as a representative of a coset

$Pexp\left(Z\left(z\right)\right)\in P\setminus G$, the complex numbers

$z=\{{z}_{i}\}$ become coordinates for

$P\setminus G$. This is discussed further in

Section 2.3.

Note that the above results have not been restricted to representations of the discrete series.

#### 2.2. Holomorphic Representation of a Lie Algebra $\mathfrak{g}$

The VCS derived representation

$\widehat{\Gamma}\left(X\right)$ of an element

X in the Lie algebra

$\mathfrak{g}$ is defined as for the Lie group

${G}_{0}$ by

where

$\{{B}_{i}\}$ is a basis for

${\mathfrak{n}}_{-}$. Explicit expressions are obtained from the expansion

and the identities

where

${\widehat{A}}_{i}$ is the Hermitian adjoint of

${B}_{i}^{\u2020}$,

${\widehat{\mathfrak{C}}}_{p}={\widehat{\sigma}}^{\kappa}\left({C}_{p}\right)$ is defined, in accordance with Equation (

18), by

and

$\{{C}_{p}\}$ is a basis for

$\mathfrak{k}$, the Lie algebra of

K. From these definitions, it follows that

where

${\widehat{z}}_{i}$ is the multiplicative operator for which

${\widehat{z}}_{i}\Psi \left(z\right)={z}_{i}\Psi \left(z\right)$. The first of these equations is obtained immediately. The second is obtained from the expansion

$[{B}_{i},{C}_{p}]={\sum}_{j}{C}_{p}^{ij}{B}_{j}$ and

The third is obtained from the expansion

$[{B}_{i},{A}_{j}]={\sum}_{p}{D}_{p}^{ij}{C}_{p}$ followed by Equation (

38).

#### 2.3. The Inner Product for Discrete-series Representations

This section confirms that, when the VCS construction is applied to a discrete-series representation, it reproduces the familiar expression for the inner products of its wave functions as obtained by the standard methods [

28,

29,

30,

31,

51].

Let

$\{|\kappa \alpha \rangle \}$ denote an orthonormal basis for the lowest-grade subspace

${\mathbb{H}}_{0}^{\kappa}$ of the Hilbert space

${\mathbb{H}}^{\kappa}$ for an irreducible discrete-series representation

${\widehat{\mathcal{U}}}^{\kappa}$ of

${G}_{0}$. The inner product for the Hilbert space of standard coherent-state wave functions for this representation is then obtained from the resolution of the identity on

${\mathbb{H}}^{\kappa}$ [

34,

35],

where

$d\nu \left(g\right)$ is the suitably normalised

${G}_{0}$-invariant measure. If the integral in this expression is restricted to the compact subgroup

${K}_{0}\subset {G}_{0}$, then the modified operator continues to be the identity operator on the subspace

${\mathbb{H}}_{0}^{\kappa}\subset {\mathbb{H}}^{\kappa}$ but on

${\mathbb{H}}^{\kappa}$ it becomes the projection operator

This implies that the identity operator (

39) can be expressed

where

$dim{\widehat{\sigma}}^{\kappa}$ is the dimension of the irrep

κ of

${K}_{0}$. However, because of the

${K}_{0}$ invariance of

${\widehat{\Pi}}^{\kappa}$,

the integrand

${\widehat{\mathcal{U}}}^{\kappa \u2020}\left(g\right){\widehat{\Pi}}^{\kappa}{\widehat{\mathcal{U}}}^{\kappa}\left(g\right)$ in (

41) is unchanged if

g is replaced by any

${g}^{\prime}$ in the

${K}_{0}\setminus {G}_{0}$ coset

${K}_{0}g$. Thus, the integral in Equation (

41) effectively reduces to an integral over the symmetric space

${K}_{0}\setminus {G}_{0}$.

To derive the Harish-Chandra expression for the inner product of a discrete series holomorphic irrep from this resolution of the identity for the irrep, we need to express the identity operator of Equation (

41) as an integral over

${K}_{0}\setminus {G}_{0}$ coset representatives.

The Gauss factorisation, given by Equation (

13), defines a

$G\to \mathcal{Z}$ map in which

The set of elements of

G that map to a single

$z\in \mathcal{Z}$ is then a

$P\setminus G$ coset, of the form

It has been shown by Harish-Chandra [

29] that, for

z in an open subset

$D\subset \mathcal{Z}$, the subset of elements of the corresponding

$P\setminus G$ coset that lie in

${G}_{0}$ form a

${K}_{0}\setminus {G}_{0}$ coset with a representative element that can be expressed in the form

The expression (

41) of the identity

$\widehat{I}$ can now be replaced by

where

$d\mu \left(z\right)$ is the

${G}_{0}$-invariant measure on the domain

D diffeomorphic to

${K}_{0}\setminus {G}_{0}$. The inner product of state vectors

$|\psi \rangle $ and

$|{\psi}^{\prime}\rangle $ in

${\mathbb{H}}^{\kappa}$ is then expressed in terms of their VCS wave functions, Ψ and

${\Psi}^{\prime}$ by

where

$\langle {\widehat{\sigma}}^{\kappa}\left(h\left(z\right)\right)\Psi \left(z\right)|{\widehat{\sigma}}^{\kappa}\left(h\left(z\right)\right){\Psi}^{\prime}\left(z\right)\rangle $ is an inner product of

${\widehat{\sigma}}^{\kappa}\left(h\left(z\right)\right)\Psi \left(z\right)$ and

${\widehat{\sigma}}^{\kappa}\left(h\left(z\right)\right){\Psi}^{\prime}\left(z\right)$ as vectors in

${\mathbb{H}}_{0}^{\kappa}$.