Journal of Lie Theory Volume 10 (2000) 157–170 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

Ladder representation norms for hermitian symmetric groups

Mark G. Davidson and Ronald J. Stanke^{∗}

Communicated by B. Ørsted

Abstract. Let G be a connected noncompact simple Hermitian symmetric group with finite center. Let H(λ) denote the geometric realization of an irreducible unitary highest weight representation with highest weight λ. Then H(λ) consists of vector-valued holomorphic functions on G/K and the action of G on H(λ) is given in terms of a factor of automorphy. For highest weights λ corresponding to ladder representations, we obtain the G-invariant inner product on H(λ). This inner product arises as the pullback of an isometry Φ_{λ} : H(λ)→ H(λ̃) ⊗ Y_{λ}, where Y_{λ} is finite dimensional and the weight λ̃ corresponds to a scalar valued representation. In all but finitely many cases the G-invariant inner product on H(λ̃) is known and is used to express the G-invariant inner product on H(λ). Explicit examples are given for families of ladder representations of SU(p, q) and SO^{∗}(2n). Finally, inversion formulas for unitary intertwining operators between H(λ) and any equivalent realization are exhibited.

1. Introduction

The ladder representations are among the most reductive of the unitary highest weight representations. To be more specific, we refer to the Enright, Howe and Wallach classification [5] where the unitary highest weight modules associated to a K -type are parametrized by the union of a half-line and a set of isolated points. The endpoint of the half line and the isolated points are called reduction points because the corresponding Verma modules are reducible. The discrete series representations and the generalized limits of discrete series [8] lie within the half-line on the left and their unitary structures are well known. The ladder representations, on the other hand, appear among the right most isolated points and their unitary structures have been recently studied. [12], [13], [14].

In this article, we describe the invariant unitary norm of highest weight ladder representations, realized in the Harish-Chandra setting [6] where the relevant Hermitian symmetric space is a bounded domain. The unitary norm is obtained by constructing an equivariant isometry which maps from the ladder representation to a tensor product that has a uniquely determined scalar-valued unitary

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag representation as one of it factors. The form of the unitary norm, being the pullback of this isometry, fundamentally depends on the norm of the corresponding scalar-valued representation. For all but finitely many ladder representations, the scalar-valued representation lies in the discrete series or limits thereof. The upshot of this construction is explicit formulas for norms of ladder representations. We now give a more detailed description of the results.

∗ Second author supported in part by a research grant from Baylor University.

Let G be a connected noncompact simple Hermitian symmetric group with finite center. Let K denote a maximal compact subgroup of G. Highest weight representations of G are realized here as holomorphically induced multiplier representations. In this setting, the representation space H(λ) with highest weight λ consists of holomorphic, vector-valued functions on the Hermitian symmetric space G/K, viewed as a bounded domain. Ladder representations are characterized by the property that the K-types occur as a multiplicity-free chain.

Let Λ_{l}(G) (resp. Λ_{0}(G)) denote the set of highest weights corresponding to unitary highest weight ladder (resp. scalar-valued) representations of G. Let H(λ)^{K} denote the K -finite vecotrs in H(λ). For λ ∈ Λ_{l}(G) and μ ∈ Λ_{0}(G), we define in section five a K -map ϕ : H(λ)^{K} → H(μ) ⊗ V_{λ−μ}, where V_{λ−μ} is K - irreducible with highest weight λ − μ. For each λ ∈ Λ_{l}(G), we uniquely determine a weight μ = λ̃ such that if λ̃ ∈ Λ_{0}(G), the above map extends to a G-equivariant isometry (cf Theorem 6.1). The invariant inner product on H(λ) arises as the pullback of the inner product on H(λ̃) ⊗ V_{λ−λ̃}.

As the explicit examples for G = SO^{∗}(2n) in section seven show, it is not always the case that λ̃ ∈ Λ_{0}(G). However, if one has λ̃ ∈ Λ_{0}(G̃), where G̃ is a finite covering group of G, then one can proceed as indicated above to obtain a G̃-invariant, and thus G-invariant, inner product on H(λ).

For all but finitely many λ ∈ Λ_{l}(G), the associated weight λ̃ corresponds to a scalar-valued discrete series or generalized limit of discrete series representation. Inner products for such representations H(λ̃) are explicitly known and are given by integration over G/K or one of its boundary components. For such λ, the form of the inner product for H(λ) in turn inherits this feature (cf. Theorem 7.1).

In order to provide some concrete examples, we consider families of ladder representations of SU(p, q) and SO^{∗}(2n) whose highest weights are found in Table 11.1 of [2]. For example, if G = SU(p, q) then λ = mω_{p−1} − (m + 1)ω_{p} ∈ Λ_{l}(G). If m + 1 ≥ p + q, then the inner product on H(λ) has the explicit form

(f, g)_{λ} = d_{λ} ∫_{D}(f(z), g(z))det(I − zz^{∗})^{m+1−p−q}dz,

where D ⊂ ℂ^{p×q} denotes the bounded realization of G/K, dz is Lebesgue measure and d_{λ} is a constant. The constraint m + 1 ≥ p + q is precisely the condition for which H(λ̃) is a discrete series representation.

As an application of the explicit form of the invariant inner products given in section seven, we compute in section eight the form of the inverse of intertwining operators Ξ : ℍ_{λ} → H(λ), where ℍ_{λ} is any equivalent realization of H(λ). In the special case where G = SU(p, q), SO^{∗}(2n) or Sp(n, ℝ) and ℍ_{λ} is the harmonic realization of H(λ), we express the inverse for the unitary intertwining operator found in section eight of [3] as an integral over G/K. This expression lacks the differentiation and limits found in [12], [13], and [14], where G = SU(p, q) is considered.

2. Preliminaries

Let G be a connected noncompact simple Lie group with finite center. Let K denote a maximal compact subgroup of G and assume that the space G/K is Hermitian symmetric. Choose a Cartan subgroup T in K. Let 𝔤_{0}, 𝔨_{0} and 𝔱_{0} denote the Lie algebras of G, K and T, respectively. Fix a Cartan decomposition 𝔤_{0} = 𝔨_{0} ⊕ 𝔭_{0}. By notational convention, the removal of the subscript _{0} denotes complexification. Then there exists a decomposition 𝔤 = 𝔭_{+} ⊕ 𝔨 ⊕ 𝔭_{−}, where 𝔭_{+} and 𝔭_{−} are abelian subalgebras of 𝔭 satisfying

[𝔨, 𝔭_{±}] ⊂ 𝔭_{±}, [𝔭_{+}, 𝔭_{−}] ⊂ 𝔨 and \overline{𝔭}_{±} = 𝔭_{∓},  (2.1)

where the bar denotes conjugation on 𝔤 with respect to 𝔤_{0}.

The subalgebra 𝔱 is a Cartan subalgebra of 𝔤 and we let Φ denote the roots corresponding to the pair (𝔤, 𝔱). We let Φ_{c} and Φ_{n} denote the set of compact and noncompact roots, respectively. The root space corresponding to α ∈ Φ is denoted by 𝔤_{α}. We choose a positive system of roots Φ^{+} so that, if Φ_{n}^{+} = Φ^{+} ∩ Φ_{n} and Φ_{n}^{−} = (−Φ^{+}) ∩ Φ_{n}, then

𝔭_{+} = ⊕_{α∈Φ_{𝔫}^{+}}𝔤_{α} and 𝔭_{−} = ⊕_{α∈Φ_{𝔫}^{−}}𝔤_{α}.  (2.2)

Let B denote the Killing form and θ denote Cartan involution on 𝔤. We choose root vectors E_{α} ∈ 𝔤_{α}, α ∈ Φ, so that B(E_{α}, E_{−α}) = 2/(α, α) and θ(\overline{E}_{α}) = −E_{−α}, where (·, ·) is the standard inner product on the real space of linear functionals on 𝔱 taking purely imaginary values on 𝔱_{0}. If we set H_{α} = [E_{α}, E_{−α}], then

[H_{α}, E_{α}] = 2E_{α} and [H_{α}, E_{−α}] = −2E_{−α}.  (2.3)

3. Holomorphically induced multiplier representations

In this article, we shall work exclusively with the geometric realization of unitary highest weight representations of G. In this realization, the representation space consists of vector-valued functions that are holomorphic on G/K. We adopt the Harish-Chandra realization [6] of G/K as a bounded domain of 𝔭_{+}. The representation of G is a multiplier representation that is expressed in terms of a factor of automorphy. We now discuss some basic facts of this realization. For additional details, see [6], [10].

Let (τ, V_{τ}) denote an irreducible unitary representation of K with representation space V_{τ}. Then τ extends to a holomorphic representation of K_{ℂ} on V_{τ}. Let j : G × \overline{D} → K_{ℂ} be the continuous map in Proposition (4.7) of [4]. Define the factor of automorphy J_{τ} by J_{τ}(g, z) = τ(j(g, z)) for (g, z) ∈ G × \overline{D}. Let O(D, V_{τ}) denote the space of V_{τ} -valued holomorphic functions on D. We define the associated multiplier representation T = T_{τ} of G on O(D, V_{τ}) by

(T (g)F)(z) = J_{τ}(g^{−1}, z)^{−1}F (g^{−1} · z),  (3.1)

for F ∈ O(D, V_{τ}), g ∈ G and z ∈ D. Here g · z denotes the action of G on \overline{D}.

It is well-known that the K -finite vectors in O(D, V_{τ}) are the polynomial functions (cf. Lemma 1.6 of [2]). Such functions have unique extensions to 𝔭_{+} since D is open in 𝔭_{+}. Let 𝕍_{τ} denote the V_{τ}−valued polynomial functions on 𝔭_{+}.

Let 1_{v} denote the constant function on 𝔭_{+} whose value is v ∈ V_{τ} and let 𝕍_{τ}^{0} denote the linear span of these constant functions. Let L(τ) denote the 𝔤-module generated by 𝕍_{τ}^{0} . It is known that L(τ) lies in each 𝔤-invariant submodule of 𝕍_{τ}. Consequently, L(τ) is irreducible. Moreover, if v_{τ} ∈ V_{τ} is a highest weight vector for 𝔨, then 1_{v_{τ}} is a highest weight vector for 𝔤.

We conclude this section with an elementary fact regarding particular weight spaces of 𝕍_{τ}. Write Φ_{n}^{+} as the set {α_{0}, α_{1}, …, α_{s}}, where α_{0} is the simple noncompact root, and let E_{α} ∈ 𝔤_{α} be chosen as in section two. Let γ = (γ_{0}, …, γ_{s}) ∈ ℕ_{0}^{s+1}, where ℕ_{0} = {0, 1, 2, ⋯ }, denote a multindex and (·, ·) denote the standard Hermitian inner product on 𝔤. By (6.3) of [4], for each weight vector v_{μ} of V_{τ} with weight μ, one observes that

f_{γ,μ}(z) = (z, E_{α_{0}})^{γ_{0}} ⋯ (z, E_{α_{s}})^{γ_{s}}v_{μ}, z ∈ 𝔭_{+},

is a nonzero weight vector with weight − ∑_{i=0}^{s} γ_{i}α_{i} +μ. Denote by W_{ω} the weight space in 𝕍_{τ} with weight ω. Since 𝕍_{τ} is spanned by the monomials f_{γ,μ}, we have 𝕍_{τ} = ⊕_{ω}W_{ω} where W_{ω} = span_{ℂ}{f_{γ,μ} : ω = − ∑_{i=0}^{s} γ_{i}α_{i} + μ}.

Proposition 3.1. Let (τ, V_{τ}) be an irreducible unitary representation of K with highest weight λ. If α_{0} denotes the simple noncompact root, then, for each integer m ≥ 0, the weight space W_{λ−mα_{0}} is spanned by the vector (·, E_{α_{0}})^{m}v_{λ}, where v_{λ} is a unit highest weight vector for V_{τ}.

Proof. By the above discussion, it suffices to show that if − ∑_{i=0}^{s} γ_{i}α_{i} + μ = λ − mα_{0}, then μ = λ, γ_{0} = m and γ_{1} = … = γ_{s} = 0. Recall that μ = λ − Q where Q is a sum of elements from Φ_{c}^{+}. If − ∑_{i=0}^{s} γ_{i}α_{i} + μ = λ − mα_{0}, then ∑_{i=0}^{s} γ_{i}α_{i} + Q = mα_{0}. By writing each α_{i}, 1 ≤ i ≤ s, and Q as the sum of simple roots, we conclude γ_{1} = … = γ_{s} = 0 and Q = 0. Consequently, we have μ = λ and γ_{0} = m.

4. Unitary highest weight representations

The purpose of this section is to give some general results on unitary highest weight representations of G. In particular, we point out that the results of [3] extend from the setting of a linear group G to the setting of a Hermitian, connected noncompact simple Lie group G with finite center. As a consequence of this extension, we employ a theorem of Kunze [11] which characterizes the unitary space of functions on G/K on which G acts irreducibly. The representation in this case is the representation T given in section three. However, the unitary norm in Kunze’s result is given in an abstract way. The goal of the remaining sections is to make explicit this norm in specific cases.

Let λ denote the highest weight for the irreducible unitary representation (τ, V_{τ}). We change notation slightly and let L(λ) denote the irreducible 𝔤-module defined in section three. Similarly, v_{λ} will replace v_{τ} and 1_{λ} will replace 1_{v_{τ}}. Let Λ(𝔤) denote the set of highest weights such that the 𝔤-module L(λ) is unitarizable. The set Λ(𝔤) has been completely classified [5], [9]. We are interested in the subset Λ(G) ⊂ Λ(𝔤) of highest weights that correspond to unitary highest weight representations of G and, in particular, their realizations as spaces of vector valued functions on G/K.

We begin with a main result in [3] on positive definite operator-valued kernel functions. If G is linear, λ ∈ Λ(G), and J_{λ} is the factor of automorphy in section three, define Q : D × D → Aut(V_{λ}) by

Q(w, z) = J(g_{2}, 0)J(g_{2}^{−1}, z)^{∗−1},  (4.1)

where g_{1}, g_{2} ∈ G, g_{1} · 0 = z and g_{2} · 0 = w. By Theorem 7.1 and Proposition 7.3 of [3], Q defines a positive definite operator-valued kernel function that is holomorphic in w and antiholomorphic in z.

There are only some minor details to check to see that the proofs in [3] extend to the setting of finite covering groups. For example, the calculations needed to establish the results through Proposition 6.2 are done at the level of the Lie algebra and so trivially extend. Theorem 6.1 extends by utilizing arguments similar to those found in the proof of Lemma (4.15) of [4]. One uses the connectedness of the subgroup A along with the finiteness of the center of G to establish the calculation on p. 26 of [3]. From this, both results from [3] follow by the arguments provided there.

Equipped with the positive definite operator Q in (4.1), one can construct a Hilbert space H(λ) of V_{λ}-valued functions on D for which the representation of G is given by the formula for T in (3.1). The details of the construction of H(λ) are found in [11]. We summarize the essential properties.

Proposition 4.1. If λ ∈ Λ(G), there exists a Hilbert space H(λ) of continuous functions f : D → V_{λ} with inner product (·, ·)_{λ} such that

(1) for each z ∈ D, the map e_{z} : H(λ) → V_{λ} defined by e_{z}(f) = f(z) is continuous,

(2) Q(w, z) = e_{w}e_{z}^{∗} for all w, z ∈ D,

(3) the linear space span_{ℂ}{w → Q(w, z)v : z ∈ D, v ∈ V_{λ}} is dense in H(λ),

(4) (Q(·, z)v, Q(·, w)u)_{λ} = (Q(w, z)v, u) for all z, w ∈ D and v, u ∈ V_{λ},

(5) the formula (3.1) defines a strongly continuous unitary representation of G on H(λ).

Remark 4.2. Since Q is holomorphic in the first variable, it follows that H(λ) consists of holomorphic functions on D. By (4.1), Q(z, 0) is the identity on V_{λ}. Then by proposition 4.1 (3) it follows that H(λ) contains the constant functions 1_{v}, v ∈ V_{λ}. If H(λ)^{K} denotes the K-finite vectors in H(λ), one concludes H(λ)^{K} = L(λ).

For weights μ and α, we let the number 2(μ, α)/(α, α) be denoted by μ_{α}.

Proposition 4.3. Let λ ∈ Λ(G). Then for each integer m ≥ 0, the weight vector T (\overline{E}_{α_{0}})^{m}(1_{λ}) is a nonzero highest weight vector for K satisfying:

(1) ∥ T (\overline{E}_{α_{0}})^{m}(1_{λ}) ∥_{λ}^{2}= (−1)^{m}m! λ_{α_{0}}(λ_{α_{0}} − 1) ⋯ (λ_{α_{0}} − m + 1), (2) T (\overline{E}_{α_{0}})^{m}(1_{λ}) = c_{λ}(m)(·, E_{α_{0}})^{m}v_{λ},

where c_{λ}(m) = \frac{(−1)^{m}}{(E_{α_{0}},E_{α_{0}})^{m}} λ_{α_{0}}(λ_{α_{0}} − 1) ⋯ (λ_{α_{0}} − m + 1).

Proof. We suppress the use of the symbol T and denote the 𝔤-action by juxtaposition. To show (1), recall from Proposition 4.1 that (·, ·)_{λ} is the unitary inner product for H(λ), so we have ∥ \overline{E}_{α_{0}}^{m}(1_{λ}) ∥_{λ}^{2}= (−1)^{m}(E_{α_{0}}^{m}\overline{E}_{α_{0}}^{m}(1_{λ}), 1_{λ})_{λ}. But by Lemma 4.2 (b) of [3], one knows

E_{α_{0}}^{m}\overline{E}_{α_{0}}^{m}(1_{λ}) = m!λ_{α_{0}}(λ_{α_{0}} − 1) ⋯ (λ_{α_{0}} − m + 1)1_{λ}.  (4.2)

Since (1_{λ}, 1_{λ})_{λ} = (v_{λ}, v_{λ}) = 1, we have

∥ \overline{E}_{α_{0}}^{m}(1_{λ}) ∥_{λ}^{2}= (−1)^{m}m!λ_{α_{0}}(λ_{α_{0}} − 1) ⋯ (λ_{α_{0}} − m + 1),

which establishes (1).

To prove (2), we observe that the weight vector \overline{E}_{α_{0}}^{m}(1_{λ}) has weight λ − mα_{0}. But \overline{E}_{α_{0}}^{m}(1_{λ}) ∈ H(λ)^{K} = L(λ) ⊂ 𝕍_{λ}, the space of K -finite vectors in O(D, V_{λ}). By Proposition 3.1, the weight space for the weight λ − mα_{0} in 𝕍_{λ} is spanned by the vector (·, E_{α_{0}})^{m}v_{λ}. Thus there exists a constant c such that \overline{E}_{α_{0}}^{m}(1_{λ}) = c(·, E_{α_{0}})^{m}v_{λ}. From (6.3) of [4], we see that E_{α_{0}}^{m} is the differential operator (−1)^{m}δ(E_{α_{0}})^{m}, where δ is the directional derivative. A straightforward computation shows that δ(E_{α_{0}})^{m}((·, E_{α_{0}})^{m}v_{λ}) = m! (E_{α_{0}}, E_{α_{0}})^{m}v_{λ}. Thus, by applying E_{α_{0}}^{m} to the equation \overline{E}_{α_{0}}^{m}(1_{λ}) = c(·, E_{α_{0}})^{m}v_{λ}, one concludes by (4.2) that m! λ_{α_{0}}(λ_{α_{0}} − 1) ⋯ (λ_{α_{0}} − m + 1) = c(−1)^{m}m! (E_{α_{0}}, E_{α_{0}})^{m}. This establishes (2).

Since α_{0} is the simple noncompact root, it follows that for each α ∈ Φ_{c}^{+}, one has [𝔤_{α}, 𝔤_{−α_{0}}] =0. Consequently, for α ∈ Φ_{c}^{+}, E_{α} · \overline{E}_{α_{0}}^{m}(1_{λ}) = \overline{E}_{α_{0}}^{m}E_{α}(1_{λ}) = 0 since 1_{λ} is a highest weight vector. This shows that \overline{E}_{α_{0}}^{m}(1_{λ}) is a highest weight vector for K.

To show that \overline{E}_{α_{0}}^{m}(1_{λ}) ≠ 0, it suffices by (1) to show that λ_{α_{0}} < 0. For this, we refer to the proof of Theorem 5.1 of [3]. There it is shown that if μ is any weight of V_{λ}, γ_{1} is the maximal root of 𝔭_{+} and λ ∈ Λ(G), then μ_{γ_{1}} < 0. We observe that −α_{0} is the highest weight of 𝔭_{−} and that 𝔭_{−} and 𝔭_{+} are dual via the Killing form. Thus, if σ denotes the longest element of W_{K} (the Weyl group for K), then −σ(γ_{1}) = −α_{0}. But then λ_{α_{0}} = 2(λ, α_{0})/(α_{0}, α_{0}) = 2(σ^{−1}(λ), σ^{−1}(α_{0}))/(σ^{−1}(α_{0}), σ^{−1}(α_{0})) = 2(σ^{−1}(λ), γ_{1})/(γ_{1}, γ_{1}) = σ^{−1}(λ)_{γ_{1}} . But W_{K} permutes the weights of V_{λ}. Putting μ = σ^{−1}(λ), we have λ_{α_{0}} = μ_{γ_{1}} < 0.

5. Scalar-valued representations and isometries

Recall that Λ(G) denotes the set of highest weights that correspond to irreducible, unitary highest weight representations of G. Let Λ_{0}(G) ⊂ Λ(G) denote the subset of nonzero highest weights μ for which the corresponding representation (τ_{μ}, V_{μ}) of K satisfies dim(V_{μ}) = 1. We refer to the corresponding representation T_{μ} on H(μ) as a scalar-valued representation of G. The purpose of this section is to associate to λ ∈ Λ(G) a unique λ̃ ∈ Λ_{0}(G) for which an isometry exists between a certain K -invariant subspace of H(λ) and a tensor product H(λ̃) ⊗ Y_{λ}, where Y_{λ} is a finite dimensional Hilbert space.

For μ ∈ Λ_{0}(G), we let ℂ_{μ} denote the corresponding one dimensional K space and let 1_{μ} ∈ ℂ_{μ} denote a unit vector. The dual ℂ_{μ}^{∗} of ℂ_{μ} is a K space with highest weight −σ(μ), where σ is the longest element of the Weyl group W_{K}. Since dim(ℂ_{μ}) = 1, it follows that σ(μ) = μ. The dual of ℂ_{μ} is thus identified with ℂ_{−μ} and K acts on ℂ_{−μ} by multiplication by τ_{μ}(k)^{−1}, k ∈ K. For μ ∈ Λ_{0}(G) and λ ∈ Λ(G), we form the K -space ℂ_{μ} ⊗ V_{λ} ⊗ ℂ_{−μ} and observe that the natural representation of K on this space reduces to 1⊗τ_{λ} ⊗1, where 1 denotes the trivial representation. It is routine to check that for each μ ∈ Λ_{0}(G) and λ ∈ Λ(G), the mapping ψ_{μ,λ} : ℂ_{μ} ⊗ V_{λ} ⊗ ℂ_{−μ} → V_{λ} defined by

ψ_{μ,λ}(Σc_{i} ⊗ v_{i} ⊗ d_{i}) = ∑_{i}(c_{i}, 1_{μ})(d_{i}, 1_{−μ})v_{i}

is a K -equivariant isometry.

For μ ∈ Λ_{0}(G), let P (𝔭_{+}, ℂ_{μ} ⊗ V_{λ} ⊗ ℂ_{−μ}) denote the K -space of ℂ_{μ} ⊗ V_{λ} ⊗ ℂ_{−μ}-valued polynomials on 𝔭_{+}. The K -action here is (k · f)(z) = (1 ⊗ τ_{λ}(k) ⊗ 1)(f(k^{−1}.z)) for k ∈ K and z ∈ 𝔭_{+}. For f ∈ P (𝔭_{+}, ℂ_{μ}⊗V_{λ}⊗ℂ_{−μ}), define R_{μ,λ}(f) by

(R_{μ,λ}(f))(z) = ψ_{μ,λ}(f(z)), z ∈ 𝔭_{+}.  (5.1)

Then R_{μ,λ} defines a K -isomorphism of P (𝔭_{+}, ℂ_{μ}⊗V_{λ}⊗ℂ_{−μ}) onto 𝕍_{λ}, the K -space of V_{λ}-valued polynomials on 𝔭_{+}.

For λ ∈ Λ(G), we let {v_{i}} denote an orthonormal basis of V_{λ}. For μ ∈ Λ_{0}(G), we define a map ϕ_{μ,λ} : H(λ)^{K} → P (𝔭_{+}, ℂ_{μ}⊗V_{λ}⊗ℂ_{−μ}) by the expression

ϕ_{μ,λ}(f)(z) = ∑_{i}(f(z), v_{i})1_{μ} ⊗ v_{i} ⊗ 1_{−μ}.  (5.2)

By a standard calculation, ϕ_{μ,λ} is independent of the choice of orthonormal basis {v_{i}}. But then, for k ∈ K, one has

ϕ_{μ,λ}(T (k)f)(z) = ∑_{i}((T (k)f)(z), v_{i})1_{μ} ⊗ v_{i} ⊗ 1_{−μ}
= ∑_{i}(τ_{λ}(k)f(k^{−1} · z), v_{i})1_{μ} ⊗ v_{i} ⊗ 1_{−μ}
= ∑_{i}(f(k^{−1} · z), τ_{λ}(k^{−1})v_{i})1_{μ} ⊗ v_{i} ⊗ 1_{−v}
= (1 ⊗ τ_{λ}(k) ⊗ 1)(ϕ_{μ,λ}(f)(k^{−1}.z))
= (k · ϕ_{μ,λ}f)(z).

Thus ϕ_{μ,λ} is a K -map. In addition, it is clear by the definition of R_{μ,λ} that

R_{μ,λ} ∘ ϕ_{μ,λ} = Id_{H(λ)^{K}}.  (5.3)

In particular, ϕ_{μ,λ} is injective.

For λ ∈ Λ(G), let V_{λ−mα_{0}} ⊂ H(λ)^{K} denote the K -irreducible space generated by the K -highest weight vector T (\overline{E}_{α_{0}})^{m}(1_{λ}) in Proposition 4.3. If μ ∈ Λ_{0}(G), note that H(μ)^{K} is a K -invariant space of ℂ_{μ}-valued polynomials on 𝔭_{+}. Consequently, H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ} can be identified with a subspace of P (𝔭_{+}, ℂ_{μ}⊗V_{λ}⊗ℂ_{−μ}).

Finally, note that μ ∈ Λ_{0}(G) is uniquely determined by the value of (μ, α_{0}). Indeed, μ ∈ Λ_{0}(G) is of the form μ = rω_{0} where r ∈ ℝ and ω_{0} denotes the fundamental weight corresponding to α_{0}. Proposition 5.1. Let λ ∈ Λ(G), μ ∈ Λ_{0}(G) and let B(λ) denote the K -space ⊕_{m≥0}V_{λ−mα_{0}}.

(1) For all integers m ≥ 0, one has ϕ_{μ,λ}(V_{λ−mα_{0}}) ⊂ H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ}.

(2) The restricted map ϕ_{μ,λ} : B(λ) → H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ} is norm-preserving if and only if μ satisfies μ_{α_{0}} = λ_{α_{0}} .

Proof. To prove (1), we choose an orthonormal basis of V_{λ} by adjoining v_{λ} to an orthonormal basis of (ℂv_{λ})^{⊥}. Recall from Proposition 4.3 (2) that T_{λ}(\overline{E}_{α_{0}})^{m}(1_{λ}) = c_{λ}(m)(·, E_{α_{0}})^{m}v_{λ}. Using the above basis in the definition of ϕ_{μ,λ}, one has

ϕ_{μ,λ}(T_{λ}(\overline{E}_{α_{0}})^{m}(1_{λ})) = c_{λ}(m)(·, E_{α_{0}})^{m}1_{μ} ⊗ v_{λ} ⊗ 1_{−μ}.

Again from Proposition 4.3 (2), one knows that T_{μ}(\overline{E}_{α_{0}})^{m}(1_{μ}) = c_{μ}(m)(·, E_{α_{0}})^{m}1_{μ}. Consequently, since T_{μ}(\overline{E}_{α_{0}})^{m}(1_{μ}) ∈ H(μ)^{K}, one concludes that

ϕ_{μ,λ}(T (\overline{E}_{α_{0}})^{m}(1_{λ})) ∈ H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ}.

But since ϕ_{μ,λ} is K -equivariant and H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ} is K -invariant, one has ϕ_{μ,λ}(V_{λ−mα_{0}}) ⊂ H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ} as well. This establishes (1).

To prove (2), first observe the center of K acts on 𝔭_{+} by scalars. By (6.3) of [4] it follows that the K -finite vectors in H(λ) of different homogeneous degree must be orthogonal. It follows that ⊕_{m≥0}V_{λ−mα_{0}} is an orthogonal direct sum in H(λ) and, likewise, ⊕_{m≥0}ϕ_{μ,λ}(V_{λ−mα_{0}}) is an orthogonal direct sum in H(μ)^{K} ⊗ V_{λ} ⊗ ℂ_{−μ}. Let (·, ·)_{⊗} denote the canonical inner product on the Hilbert space H(μ) ⊗ V_{λ} ⊗ ℂ_{−μ}. Since V_{λ−mα_{0}} is an irreducible K-space, there exists a constant b_{m} > 0 such that

(ϕ_{μ,λ}(f), ϕ_{μ,λ}(g))_{⊗} = b_{m}(f, g)_{λ}, for all f, g ∈ V_{λ−mα_{0}}.  (5.4)

The above calculation shows that

ϕ_{μ,λ}(T_{λ}(\overline{E}_{α_{0}})^{m}(1_{λ})) = \frac{c_{λ}(m)}{c_{μ}(m)}T_{μ}(\overline{E}_{α_{0}})^{m}(1_{μ}) ⊗ v_{λ} ⊗ 1_{−μ}.

Letting f = g = T_{λ}(\overline{E}_{α_{0}})^{m}(1_{λ}) in (5.4), we obtain

(\frac{c_{λ}(m)}{c_{μ}(m)})^{2}(T_{μ}(\overline{E_{α_{0}}})^{m}(1_{μ}), T_{μ}(\overline{E_{α_{0}}})^{m}(1_{μ}))_{μ}

= b_{m}(T_{λ}(\overline{E_{α_{0}}})^{m}(1_{λ}), T_{λ}(\overline{E_{α_{0}}})^{m}(1_{λ}))_{λ}.  (5.5)

By Proposition 4.3 (1), we know that

∥ T_{λ}(\overline{E}_{α_{0}})^{m}(1_{λ}) ∥_{λ}^{2}= m!(E_{α_{0}}, E_{α_{0}})^{m}c_{λ}(m)

and

∥ T_{μ}(\overline{E}_{α_{0}})^{m}(1_{μ}) ∥_{μ}^{2}= m!(E_{α_{0}}, E_{α_{0}})^{m}c_{μ}(m).

But then (5.5) reduces to c_{λ}(m) = b_{m}c_{μ}(m). Consequently b_{m} = 1 for all m ≥ 0 if and only if λ_{α_{0}} = μ_{α_{0}} . This completes the proof of (2).

For λ ∈ Λ(G), we define the weight λ̃ by

λ̃ = (λ_{α_{0}})ω_{0},  (5.6)

where ω_{0} is the fundamental dominant weight corresponding to α_{0}. Corollary 5.2. If λ, λ̃ ∈ Λ(G), then the restriction of ϕ_{λ̃,λ} to B(λ) uniquely extends to an isometry Φ_{λ} : \overline{B(λ)} → H(λ̃) ⊗ V_{λ} ⊗ ℂ_{−λ̃} such that Φ_{λ}(f)(z) = ∑_{i}(f(z), v_{i})1λ̃ ⊗ v_{i} ⊗ 1_{−λ̃}, where {v_{i}} is an orthonormal basis of V_{λ}. Proof. The unique extension to an isometry Φ_{λ} on the closure of B(λ) follows from Proposition 5.1 (2). To establish the formula for Φ_{λ}, recall from Proposition 4.1(1) that convergence in H(λ) implies pointwise convergence. Consequently, if \{f_{n}\} ⊂ B(λ) converges to f in \overline{B(λ)}, then (f_{n}(z), v) converges to (f(z), v) for each z ∈ D and v ∈ V_{λ}. Since both Φ_{λ} and point evaluation e_{z} are continuous, one has

Φ_{λ}f(z) = e_{z}Φ_{λ}(f)
= lim_{n→∞} e_{z}Φ_{λ}(f_{n})
= lim_{n→∞} ∑_{i}(f_{n}(z), v_{i})1λ̃ ⊗ v_{i} ⊗ 1_{−λ̃}
= ∑_{i}(f(z), v_{i})1λ̃ ⊗ v_{i} ⊗ 1_{−λ̃}.

Remark 5.3. In [13], certain orthogonal families of scalar-valued polynomials on D, G = U(p, q), are studied. These polynomials are used to invert an intertwining operator which is subsequently used to define a G-invariant norm for ladder representations of G. However, these polynomials are easily seen to be an orthogonal basis of B(λ̃). The proof of Proposition 5.1 (1) shows that the image of Φ_{λ} is contained in B(λ̃) ⊗ V_{λ} ⊗ ℂ_{−λ̃}. We can therefore see the relevance of these orthogonal families in describing the image of Φ_{λ}.

6. Ladder Representations

In this section we restrict our attention to highest weights λ ∈ Λ(G) that correspond to a ladder representation of G. We define ladder representations and show how the unitary inner product (·, ·)_{λ} of H(λ), discussed in section four, is completely determined by the unitary inner product of H(λ̃) (with λ̃ defined as in (5.6)) and the isometry in Corollary 5.2.

Recall by Proposition 4.3 that for any λ ∈ Λ(G), each of the K -types V_{λ−mα_{0}}, m ≥ 0, appears in H(λ). We define (T_{λ}, H(λ)) to be a ladder representation of G if these are the only K -types that appear. Let Λ_{l}(G) denote the subset of Λ(G) such that (T_{λ}, H(λ)) is a ladder representation. Then λ ∈ Λ_{l}(G) if and only if

(\overline{⊕_{m≥0}V_{λ−mα_{0}}}) = H(λ).  (6.1)

We define a representation S_{λ} of G on O(D, ℂλ̃ ⊗V_{λ} ⊗ℂ_{−λ̃}) by the formula

(S_{λ}(g)f)(z) = (1 ⊗ J_{λ}(g^{−1}, z)^{−1} ⊗ 1)f(g^{−1} · z),  (6.2)

for z ∈ D. Here, 1 denotes the identity map on ℂλ̃ as well as ℂ_{−λ̃}.

Theorem 6.1. Let λ ∈ Λ_{l}(G). If λ̃ ∈ Λ_{0}(G) then Φ_{λ} intertwines T_{λ} and S_{λ}. Moreover, the invariant unitary inner product (·, ·)_{λ} on H(λ) is given by the formula

(f_{1}, f_{2})_{λ} = ∑_{i}((f_{1}(·), v_{i})1λ̃, (f_{2}(·), v_{i})1λ̃)λ̃,

where f_{1}, f_{2} ∈ H(λ), {v_{i}} is an orthonormal basis of V_{λ} and (·, ·)λ̃ denotes the invariant unitary inner product on H(λ̃). Proof. The intertwining property is a consequence of the form of Φ_{λ} in Corollary 5.2. Now let f_{1}, f_{2} ∈ H(λ) and let (·, ·)_{⊗} denote the inner product on the Hilbert space H(λ̃) ⊗ V_{λ} ⊗ ℂ_{−λ̃}. Since Φ_{λ} is an isometry, we have

(f_{1}, f_{2})_{λ} = (Φ_{λ}(f_{1}), Φ_{λ}(f_{2}))_{⊗}
= (∑_{i}(f_{1}(·), v_{i})1λ̃ ⊗ v_{i} ⊗ 1_{−λ̃}, ∑_{j}(f_{2}(·), v_{j})1λ̃ ⊗ v_{j} ⊗ 1_{−λ̃})_{⊗}
= ∑_{i,j}((f_{1}(·), v_{i})1λ̃, (f_{2}(·), v_{j})1λ̃)λ̃(v_{i}, v_{j})
= ∑_{i}((f_{1}(·), v_{i})1λ̃, (f_{2}(·), v_{i})1λ̃)λ̃.

Remark 6.2. As seen in the next section, it can happen that λ̃ ∉ Λ_{0}(G) but λ̃ ∈ Λ_{0}(G̃) for some covering group G̃ of G. The representation T = T_{λ} of G lifts to a unitary representation of G̃ on the same Hilbert space H(λ). Theorem 6.1 then applies and gives the unitary inner product on H(λ), which is G̃-invariant. However, this inner product is also G-invariant via the covering map G̃ → G.

7. The role of scalar-valued representations

In all but finitely many cases, the associated scalar-valued representation H(λ̃), where λ̃ is defined in (5.6), is a discrete series or a generalized limit of discrete series representation [8]. Since the unitary inner product for such scalar-valued representations is well-known, we can describe the unitary inner product on H(λ) using the formula for (·, ·)_{λ} found in Theorem 6.1. Explicit examples of the unitary inner product (·, ·)_{λ} for families of ladder representations of the groups SU(p, q) and SO^{∗}(2n) appear at the end of this section.

We begin with some basic facts about the unitary inner product for H(μ) where μ corresponds to a scalar-valued discrete series representation. We define the subset Λ_{0}^{d}(G) of Λ_{0}(G) to be the set of highest weights μ satisfying (μ + ρ, β) < 0, where β ∈ Φ_{n}^{+} is the maximal noncompact root and ρ denotes half the sum of the roots in Φ^{+}. Then each μ ∈ Λ_{0}^{d}(G) corresponds to a discrete series representation of G.

Let dm(z) denote the G-invariant measure on D and let M_{μ}, μ ∈ Λ_{0}^{d}(G), denote the function on D determined by the function g → |J_{μ}(g, 0)^{−1}|^{2}. Denote by dμ(z) the measure M_{μ}(z) dm(z), normalized so that ∫_{D} dμ(z) = 1. Then the inner product on H(μ) is given by

(f, h)_{μ} = ∫_{D}(f(z), h(z))_{ℂ_{μ}}dμ(z).  (7.1)

Put r = rank(D) and let μ_{i} ∈ Λ_{0}(G) satisfy (μ_{i}+ρ)_{β} = c(i−1), 1 ≤ i ≤ r, where c is given on page 115 of [5]. These weights, Λ_{0}^{lim}(G), have corresponding unitary structures explicitly given in [8]. We summarize the results found there. Let dm_{i}( z) denote the quasi-invariant measure on the i^{th} boundary component B_{i} of D. The ℝ-valued function on K × G_{i} given by (k, g) → |J_{μ_{i}}(kgc_{i}, 0)^{−1}|^{2} (cf. (6.14) [8]) determines a function M_{i} on B_{i}. Let dμ_{i}(z) denote the normalized measure on B_{i} corresponding to M_{i}(z) dm_{i} (z). Then the (densely defined) inner product on H(μ_{i}), 1 ≤ i ≤ r, is given by

(f, h)_{μ_{i}} = ∫_{B_{i}}(f(z), h(z))_{ℂ_{μ}}dμ_{i}(z),  for f, h ∈ H(μ_{i})^{K}.  (7.2)

Theorem 7.1. Let λ ∈ Λ_{l}(G).

(1) If λ̃ ∈ Λ_{0}^{d}(G) then

(f, h)_{λ} = ∫_{D}(f(z), h(z))dμ(z), for f, g ∈ H(λ).

(2) If λ̃ ∈ Λ_{0}^{lim}(G) then

(f, h)_{λ} = ∫_{B_{i}}(f(z), h(z))dμ_{i}(z), for f, h ∈ H(λ)^{K}.

Proof. To show (1), assume λ̃ ∈ Λ_{0}^{d}(G) . Let {v_{i}} be an orthonormal basis of V_{λ}. Then by Theorem 6.1, we obtain

(f, h)_{λ} = ∑_{i}((f(·), v_{i})1λ̃, (h(·), v_{i})1λ̃)λ̃
= ∑_{i} ∫_{D}(f(z), v_{i})(\overline{h(z), v_{i}})dμ(z) by (7.1)
= ∫_{D}(f(z), h(z))dμ(z).

This proves (1). The proof of (2) follows in the same way by (7.2) and the fact that Φ_{λ} maps K -finite vectors to K -finite vectors.

Explicit examples for SU(p, q)

We now let G = SU(p, q) and utilize Theorem 7.1 to explicitly describe the unitary norm for a family of ladder representations of G. This family includes all but finitely many of the ladder representations of SU(p, q) discussed in section eleven of [2].

Let m ≥ 1 be an integer. Let {ω_{i}} denote the set of standard fundamental dominant weights found in [2]. Let λ^{m} = −(m + 1)ω_{p} + mω_{p+1} and λ_{m} = mw_{p−1}−(m+1)w_{p}. Since α_{p} is the simple noncompact root, we see from (5.6) that λ̃^{m} = λ̃_{m} = −(m+1)ω_{p}. Note that the weight −(m+1)ω_{p} always lies in Λ_{0}(G) and corresponds to a highest weight of a discrete series representation of SU(p, q) if and only if m + 1 ≥ p + q. From [7], p. 84, the invariant measure dm(z) on D is given (up to a positive multiple) by det(I − zz^{∗})^{−(p+q)}dz where dz = Π_{r,s}dx_{rs}dy_{rs} with z_{rs} = x_{rs} + iy_{rs}. Moreover, by [1], one has M_{λ̃m}(z) = Mλ̃^{m}(z) = det(I − zz^{∗})^{m+1}. If m + 1 ≥ p + q and f, h ∈ H(λ_{m}), then we have by Theorem 7.1 the formula

(f, h)_{λ_{m}} = d_{λ_{m}} ∫_{D}(f(z), h(z)) det(I − zz^{∗})^{m+1−p−q}dz,  (7.3)

where d_{λ_{m}}^{−1} = ∫_{D} det(I − zz^{∗})^{m+1−p−q}dz. By replacing the inner product on V_{λ_{m}} in (7.3) with the inner product on V_{λ^{m}}, one obtains the formula for (·, ·)_{λ^{m}} on H(λ^{m}).

Explicit examples for SO^{∗}(2n)

We now let G = SO^{∗}(2n) where n ≥ 4. For each integer m ≥ 1, the weight λ = mω_{n−1}−(m+2)ω_{n} corresponds to a ladder representation of SO^{∗}(2n). Following the notation in [2], since α_{n} = e_{n−1} + e_{n} is the simple noncompact root, we have λ̃ = −(m + 2)ω_{n}. Now by Section 7.22 of [2] the highest weights of the unitary scalar-valued representations of SO^{∗}(2n) are all of the form −2kω_{n} where k ≥ 0 is an integer. In particular, if m is odd, then λ̃ ∉ Λ_{0}(SO^{∗}(2n)). We shall return to the case where m is odd. For the moment, assume m is even. From [7], p. 85, the invariant measure on D is given (up to a multiple) by det(I +z\overline{z})^{−n+1}dz, where dz = Π_{r<s}dx_{rs}dy_{rs} with z_{rs} = x_{rs} + iy_{rs}. The function Mλ̃ in this case is Mλ̃(z) = det(I + z\overline{z})^{\frac{m}{2} +1}. The weight λ̃ corresponds to a highest weight for a discrete series representation when \frac{m}{2} − n + 2 ≥ 0.

Then for f, h ∈ H(λ), we have by Theorem 7.1 that

(f, h)_{λ} = d_{λ} ∫_{D}(f(z), h(z)) det(I + z\overline{z})^{\frac{m}{2} −n+2}dz,  (7.4)

where d_{λ}^{−1} = ∫_{D} det(I + z\overline{z})^{\frac{m}{2} −n+2}dz. We observed above that for odd m, the weight λ̃ = −(m + 2)ω_{n} is not the highest weight for a unitary highest weight presentation of SO^{∗}(2n). However, for odd m such that the square integrable condition \frac{m}{2} −n+2 ≥ 0 holds, λ̃ does correspond to a discrete series representation of the two-fold covering group SO^{∗}(2n)^{∼} of SO^{∗}(2n). Remark 6.2 applies in this case and we conclude that the formula for (·, ·)_{λ} appearing in (7.4) remains valid for the m odd case.

8. Applications to Intertwining Operators

In Theorem 7.1 we observed that for λ ∈ Λ_{l}(G), the unitary inner product (·, ·)_{λ} for H(λ) can inherit the intrinsic simplicity of the unitary norm of an associated discrete representation of G on H(λ̃). We now show that this simplicity is also manifested in the form of the inverse of a unitary intertwining operator Ξ : ℍ_{λ} → H(λ), where ℍ_{λ} denotes any equivalent realization of H(λ).

If λ ∈ Λ(G), we let (ω_{λ}, ℍ_{λ}) denote an irreducible unitary highest weight representation of G with highest weight λ. Let v_{λ} denote a unit highest weight vector and V_{λ} its K -invariant span. For z ∈ 𝔭_{+}, define the operator q_{z} : V_{λ} → ℍ_{λ} by the formula

q_{z}(v) = ∑_{n=0}^{∞} \frac{dω_{λ}(\overline{z})^{n}}{n!} (v),  (8.1)

where v ∈ V_{λ} and dω_{λ} denotes the derived action of ω_{λ}. By Theorem 5.1 of [3] the series (8.1) converges in ℍ_{λ} if and only if z ∈ D. Moreover, by Theorem 7.2 of [3], the expression

(Ξf)(z) = q_{z}^{∗}(f), f ∈ ℍ_{λ}, z ∈ D,  (8.2)

defines a unitary intertwining operator Ξ : ℍ_{λ} → H(λ). Here, q_{z}^{∗} : ℍ_{λ} → V_{λ} denotes the adjoint of the operator q_{z}, z ∈ D. Proposition 8.1. Let λ ∈ Λ_{l}(G). If λ̃ ∈ Λ_{0}^{d}(G), then the inverse Ξ^{−1} of Ξ is given by

Ξ^{−1}F = ∫_{D} q_{z}(F (z))dμ(z), for F ∈ H(λ).

Proof. Let < ·, · > denote the unitary inner product on ℍ_{λ}. Using the form of (·, ·)_{λ} in Theorem 7.1 (1), we have for f ∈ ℍ_{λ} and F ∈ H(λ)

(Ξf, F)_{λ} = ∫_{D}(q_{z}^{∗}(f), F (z))dμ(z) = ∫_{D} < f, q_{z}(F (z)) > dμ(z).

Since Ξ is unitary, the proposition now follows.

In order to give a concrete example of the inversion formula in Proposition 8.1, we turn to the harmonic realization of unitary highest weight representations of the linear groups SU(p, q), SO^{∗}(2n) and Sp(n, ℝ). We follow the notation found in section seven of [2]. Let M be the complex space of matrices in Table 7.2 of [2]. The relevant Hilbert space is the Fock space 𝔽 consisting of ℂ-valued holomorphic functions on M which are square integrable with respect to the Gaussian measure on M. We let θ : M → 𝔭_{+} denote the map on page 55 of [2] and let 𝕀 denote the ideal generated by the matrix entries of θ and the constants. Let V_{λ} denote an irreducible K space of harmonic polynomials (cf. (7.8), [2]). Then the closure of 𝕀V_{λ}, denoted by ℍ_{λ}, is an irreducible subspace for ω_{λ}, the harmonic representation of G. By (7.3) of [2], the action of 𝔭_{−} on ℍ_{λ} is given by dω_{λ}(x)f(y) = (θ(y), \overline{x})f(y), for x ∈ 𝔭_{−}, y ∈ M and f ∈ ℍ_{λ}. Consequently, one finds by (8.1) that for z ∈ D

q_{z}h(y) = e^{(θ(y),z)}h(y), h ∈ V_{λ}, y ∈ M.  (8.3)

For λ ∈ Λ(G) (G = SU(p, q), SO^{∗}(2n) or Sp(n, ℝ)), we have by (8.3) of [3] that the intertwining operator Ξ has the form

(Ξf)(z, y) = ∫_{M} f(w)e^{(z,θ(w))}K_{λ}(y, w)dμ(w),

where K_{λ} denotes the reproducing kernel of V_{λ}.

If λ ∈ Λ_{l}(G) and λ̃ ∈ Λ_{0}^{d}(G), then we have by Proposition 8.1 and (8.3) that Ξ^{−1} has the explicit form

(Ξ^{−1}F)(y) = ∫_{D} e^{(θ(y),z)}F (z, y)dμ(z),  (8.4)

where F ∈ H(λ) and y ∈ M.

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Mathematics Department LSU Baton Rouge, LA 70809 davidson@math.lsu.edu Mathematics Department Baylor University Waco, TX 76798 Ronald Stanke@Baylor.edu Received December 2, 1998 and in final form May 23, 1999