Covariant Symbolic Calculi on Real Symmetric Domains

Jonathan Arazy, Harald Upmeier

We introduce the concept of ”covariant symbolic calculus” on real and complex symmetric domains, prove a general product formula for the link transform (generalized Berezin transform) between two such calculi, and describe a basic example (Toeplitz calculus) in more detail.

1. Introduction

The complex hermitian spaces of non-compact type, realized as bounded symmetric domains D ⊂ ℂ^{n}, are a fundamental class of non-compact Kähler manifolds whose quantization, e.g. by the well-known Berezin-Toeplitz operators, has been studied intensively [BLU], [UU]. Writing D = G/K for a semi-simple Lie group G of hermitian type and its maximal compact subgroup K, the quantization map

(1.1)  A : C^{∞}(D) → L(H)
f ↦ A_{f}

realized by (possibly unbounded) operators on a complex Hilbert space H should satisfy the covariance condition

A_{f∘g^{−1}} = U(g) A_{f} U(g^{−1})

for all g ∈ G, where U denotes an irreducible (projective) representation of G acting on H. In [AU1] a general theory concerning such ”covariant quantizations” on complex symmetric domains (including the flat case D = ℂ^{n}) is developed for the weighted Bergman spaces H = H_{ν}^{2}(D) of holomorphic functions on D, belonging to the scalar holomorphic discrete series of G. The Toeplitz calculus and also the Weyl calculus are natural examples of covariant quantizations, and the main result of [AU1] gives a ”product formula” for the link transform f ↦ A^{∗}(B_{f}) on C^{∞}(D) in terms of the spherical Fourier transform of certain K-invariant ”characteristic” functions associated with the covariant quantizations A and B.

While the setting of complex symmetric domains is still an active research area (e.g., concerning vector-valued representations or the Weyl calculus for domains of higher dimension), a promising new direction is to apply ideas and methods from quantization theory to real symmetric domains G_{ℝ}/K_{ℝ}, for a semi-simple or reductive Lie group G_{ℝ} not necessarily of hermitian type. Besides dealing with a much wider geometric framework compared to the complex case, a major application of this more general approach is to harmonic analysis of G_{ℝ}, yielding an explicit decomposition of certain (reducible) G_{ℝ}-representations into irreducible components. The complex case described above corresponds to the special case of a tensor product representation.

2. Real Symmetric Domains

Let

D = G/K

be a hermitian symmetric space of non-compact type, realized as the (spectral) unit ball

D = {z ∈ Z : ∥z∥ < 1}

of a complex JB^{∗}-triple Z of finite dimension. Then

G = Aut(D)^{∘}

is the identity component of the holomorphic automorphism group of D, and K = Aut(Z)^{∘}. Let

z ↦ \overline{z}

be a conjugation of Z preserving the triple product, and define

Z_{ℝ} := \{z ∈ Z : \overline{z} = z\},
D_{ℝ} := D ∩ Z_{ℝ} = {z ∈ Z_{ℝ} : ∥z∥ < 1},
G_{ℝ} := \{g ∈ G : \overline{g(z)} = g(\overline{z}) ∀ z ∈ Z\} = \{g ∈ G : g(D_{ℝ}) = D_{ℝ}\},
K_{ℝ} := K ∩ G_{ℝ}.

Then Z_{ℝ} is a real JB^{∗}-triple and

D_{ℝ} = G_{ℝ}/K_{ℝ}

is a real Riemannian symmetric space of non-compact type, called a real bounded symmetric domain. Up to a few low dimensional exceptions, all Riemannian symmetric spaces of non-compact type can be realized in this way. In the following we assume that D_{ℝ} is irreducible. The domain D is called the complexification of D_{ℝ}. It is not necessarily irreducible.

Example 2.1. Let Z_{ℝ} be a complex (irreducible) JB^{∗}-triple, considered as a real JB^{∗}-triple. Then its unit ball D_{ℝ} is a complex bounded symmetric domain, considered as a real bounded symmetric domain. Put

Z := Z_{ℝ} × \overline{Z}_{ℝ}, D := D_{ℝ} × \overline{D}_{ℝ}

endowed with the flip conjugation

\overline{(z_{1}, \overline{z}_{2})} := (z_{2}, \overline{z}_{1}) (2.1)

for all z_{1}, z_{2} ∈ Z_{ℝ}. The corresponding real form is the diagonal

\{(z_{1}, \overline{z}_{2}) ∈ Z : z_{1} = z_{2}\} = \{(z, \overline{z}) : z ∈ Z_{ℝ}\} ≡ Z_{ℝ},
\{(z_{1}, \overline{z}_{2}) ∈ D : z_{1} = z_{2}\} = \{(z, \overline{z}) : z ∈ D_{ℝ}\} ≡ D_{ℝ}.

Thus the complex case gives rise to a product domain in the complexification. Putting

(g_{1}, \overline{g}_{2})(z_{1}, \overline{z}_{2}) := (g_{1}(z_{1}), \overline{g_{2}(z_{2})})

for z_{1}, z_{2} ∈ D_{ℝ} and g_{1}, g_{2} ∈ G_{ℝ} := Aut(D_{ℝ})^{∘}, we have

G := Aut(D)^{∘} = \{(g_{1}, \overline{g}_{2}) : g_{1}, g_{2} ∈ G_{ℝ}\} ≈ G_{ℝ} × G_{ℝ}.

Furthermore, an automorphism (g_{1}, \overline{g}_{2}) ∈ G commutes with (2.1) if and only if g_{1} = g_{2}, since we have for z_{1}, z_{2} ∈ D_{ℝ}

(g_{1}, \overline{g}_{2})\overline{(z_{1}, \overline{z}_{2})} = (g_{1}, \overline{g}_{2})(z_{2}, \overline{z}_{1}) = (g_{1}(z_{2}), \overline{g_{2}(z_{1})})

and

\overline{(g_{1}, \overline{g}_{2})(z_{1}, \overline{z}_{2})} = \overline{(g_{1}(z_{1}), \overline{g_{2}(z_{2})})} = (g_{2}(z_{2}), g_{1}(z_{1})).

Example 2.2. The complex matrix ball

D = {z ∈ ℂ^{p×q} : z^{∗} z < I}

in Z := ℂ^{p×q}, endowed with the usual conjugation z ↦ \overline{z}, gives rise to the real matrix ball

D_{ℝ} = {z ∈ ℝ^{p×q} : z^{∗} z < I}

in Z_{ℝ} = ℝ^{p×q}. In particular, the interval D_{ℝ} = (−1, 1) ⊂ ℝ has the unit disk D = {z ∈ ℂ : |z| < 1} as complexification.

Example 2.3. The complex matrix ball

D = {z ∈ ℂ^{2×2} : z^{∗} z < I}

in Z = ℂ^{2×2}, endowed with the conjugation

( \begin{matrix}a & b \\ c & d \end{matrix} )^{−} := ( \begin{matrix}\overline{d} & −\overline{c} \\ −\overline{b} & \overline{a} \end{matrix} ) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}\overline{a} & \overline{b} \\ \overline{c} & \overline{d} \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ,

has the real form

D_{ℝ} = {z ∈ ℍ : z z^{∗} < I}

in Z_{ℝ} = ℍ. Here ℍ denotes the real division algebra of quaternions.

Example 2.4. Every (irreducible) symmetric cone Ω in a euclidean Jordan algebra X is a real symmetric domain (realized as the unit ball of X via a Cayley transform) whose complexification can be realized as the tube domain

T (Ω) := {x + iy : x ∈ Ω, y ∈ X}

in Z := X^{ℂ}.

Generalizing Example 2.4 every irreducible real bounded symmetric domain can be realized as a real Siegel domain as follows:

Let Z be an irreducible complex JB^{∗}-triple with conjugation z ↦ \overline{z}, and consider the Peirce decomposition

Z = U ⊕ V, U = Z_{1}(e), V = Z_{1/2}(e)

with respect to a maximal tripotent e = \overline{e} ∈ Z. Then the respective real forms satisfy Z_{ℝ} = U_{ℝ} ⊕ V_{ℝ} (2.2)

and U_{ℝ} is a semi-simple (not necessarily euclidean) real Jordan algebra with a decomposition U_{ℝ} = X ⊕ Y, (2.3)

where

X := {x ∈ U_{ℝ} : x^{∗} = x}

is an irreducible euclidean Jordan algebra, and we put

Y := {y ∈ U_{ℝ} : y^{∗} = −y}.

Combining (2.2) and (2.3) we obtain

Z_{ℝ} = X ⊕ Y ⊕ V_{ℝ}. (2.4)

The self-adjoint part X ⊕ iY of U is also a euclidean Jordan algebra. According to [L2, 10.1] or [U, Section 21], the unit ball of Z is equivalent (via a Cayley transformation) to a complex Siegel domain

D = \{u + v ∈ Z : \frac{u + u^{∗}}{2} − \{e v^{∗} v\} ∈ Ω_{X⊕iY} \}

where Ω_{X⊕iY} is the positive cone of X ⊕ iY. It follows that

D_{ℝ} = {x + y + v ∈ Z_{ℝ} : x − {e v^{∗} v} ∈ Ω}, (2.5)

where Ω = X ∩Ω_{X⊕iY} is the positive cone of X. This is the unbounded realization of D_{ℝ} as a real Siegel domain.

Summing up, real symmetric domains D_{ℝ} are of the following types:

\underline{Type 1} Z is irreducible and D_{ℝ} has an unbounded realization

D_{ℝ} ≈ {x + y + v ∈ Z_{ℝ} : x − {e v^{∗} v} ∈ Ω}

as described above.

\underline{Type 2} Z_{ℝ} is a complex irreducible JB^{∗}-triple, with unit ball D_{ℝ} being a complex hermitian symmetric domain, and the complexification is given by

Z = Z_{ℝ} × \overline{Z}_{ℝ} and D = D_{ℝ} × \overline{D}_{ℝ}.

We also include the flat case:

\underline{Type 3} D = Z = ℂ^{n}, endowed with the usual conjugation, so that D_{ℝ} = Z_{ℝ} = ℝ^{n}.

3. Quantization Hilbert spaces

For a real symmetric domain D_{ℝ} = G_{ℝ}/K_{ℝ}, with complexification D = G/K, let H_{ν}^{2}(D) (3.1)

be the ν-th weighted Bergman (resp. Bargmann) space of holomorphic functions on D, endowed with the canonical irreducible (projective) representation

U_{ν} : G → U(H_{ν}^{2}(D)). (3.2)

More precisely, according to the three types of domains (cf. Section 1), (3.1) and (3.2) are defined as follows:

\underline{Case 1} Let Z be an irreducible JB^{∗}-triple, with unit ball D of rank r, dimension n and genus p. Let Δ(z, w) denote the Jordan triple determinant and denote by (z|w) the unique scalar product invariant under K := Aut(Z)^{∘} normalized by the condition (c|c) = 1 for all minimal tripotents c ∈ Z. Let dm(z) be the associated Lebesgue measure. Fix ν > p − 1 and consider the probability measure

dμ_{ν}(z) = \frac{Γ_{Ω}(ν)}{π^{n} Γ_{Ω}(ν − \frac{n}{r})} Δ(z, z)^{ν−p} dm(z) (3.3)

where Γ_{Ω} denotes the so-called Koecher-Gindikin Γ-function of the positive cone Ω associated with Z [FK]. The weighted Bergman space

H_{ν}^{2}(D) := {h ∈ L^{2}(D, μ_{ν}) : h holomorphic}

= {h : D → ℂ holomorphic : ∥h∥_{ν}^{2} := ∫_{D} dμ_{ν}(z) |h(z)|^{2} < +∞}

has the reproducing kernel

K(z, w) = Δ(z, w)^{−ν} (3.4)

for all z, w ∈ D. The irreducible unitary (projective) representation U_{ν} of G on H_{ν}^{2}(D) is defined by (U_{ν}(g^{−1})h)(z) := j(g, z) h(g(z)) (3.5)

for all g ∈ G, h ∈ H_{ν}^{2}(D) and z ∈ D, where

j(g, z) := [Det_{Z} g′(z)]^{ν/p}.

\underline{Case 2} If D_{ℝ} ⊂ Z_{ℝ} is an irreducible complex symmetric domain, we consider the Hilbert space tensor product

H_{ν}^{2}(D) := H_{ν}^{2}(D_{ℝ}) ⊗ \overline{H_{ν}^{2}(D_{ℝ})} (3.6)

for the weighted Bergman space H_{ν}^{2}(D_{ℝ}) over D_{ℝ}, as described in Case 1. Here ν > p_{ℝ}−1 where p_{ℝ} = genus (D_{ℝ}). H_{ν}^{2}(D) consists of sesqui-holomorphic functions h(z_{1}, \overline{z}_{2}) on D := D_{ℝ} × \overline{D}_{ℝ} ⊂ Z := Z_{ℝ} × \overline{Z}_{ℝ} which are square-integrable under the product measure dμ_{ν}(z_{1}, z_{2}) := dμ_{ν}^{ℝ}(z_{1}) dμ_{ν}^{ℝ}(z_{2}). (3.7)

An equivalent realization of (3.6) is via Hilbert-Schmidt integral operators

(hϕ)(z_{1}) = ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(z_{2}) h(z_{1}, \overline{z}_{2}) ϕ(z_{2}) (3.8)

on H_{ν}^{2}(D_{ℝ}). The reproducing kernel of H_{ν}^{2}(D) is the product

K(z_{1}, \overline{z}_{2}, w_{1}, \overline{w}_{2}) := K^{ℝ}(z_{1}, w_{1}) K^{ℝ}(w_{2}, z_{2}), (3.9)

where K^{ℝ}(z_{1}, w_{1}) is the kernel function for H_{ν}^{2}(D_{ℝ}). H_{ν}^{2}(D) carries an irreducible unitary (projective) representation U_{ν} of G = G_{ℝ} × G_{ℝ} defined by

U_{ν}(g_{1}, \overline{g}_{2}) T = U_{ν}^{ℝ}(g_{1}) T U_{ν}^{ℝ}(g_{2})^{∗} (3.10)

for all g_{1}, g_{2} ∈ G_{ℝ} and T ∈ H_{ν}^{2}(D), regarded as a Hilbert-Schmidt operator on H_{ν}^{2}(D_{ℝ}). For the diagonal (g, \overline{g}) ∈ G, with g ∈ G_{ℝ}, we obtain the adjoint action

U_{ν}(g, \overline{g}) T = U_{ν}^{ℝ}(g) T U_{ν}^{ℝ}(g)^{−1}.

We put

j(g_{1}, \overline{g}_{2}; z_{1}, \overline{z}_{2}) := j^{ℝ}(g_{1}, z_{1}) \overline{j^{ℝ}(g_{2}, z_{2})}

Covariant Symbolic Calculi 7

for all g_{1}, g_{2} ∈ G_{ℝ} and z_{1}, z_{2} ∈ D_{ℝ}.

\underline{Case 3} For the flat case D = ℂ^{n}, let (z|w) denote the inner product and let dm(z) be the associated Lebesgue measure. Fix ν > 0 and consider the probability measure

dμ_{ν}(z) := (\frac{ν}{π} )^{n} e^{−ν(z|z)}dm(z). (3.11)

The weighted Bargmann space

H_{ν}^{2}(ℂ^{n}) := {h ∈ L^{2}(ℂ^{n}; dμ_{ν}) : h holomorphic}
= {h : ℂ^{n} → ℂ holomorphic : ∥h∥_{ν}^{2} := ∫_{ℂ^{n}} dμ_{ν}(z) |h(z)|^{2} < ∞}

has the reproducing kernel K(z, w) = e^{ν(z|w)} (3.12) for all z, w ∈ ℂ^{n}. H_{ν}^{2}(ℂ^{n}) carries an irreducible unitary (projective) representation of the semi-direct product G := U(n) ◁ ℂ^{n} via (U_{ν}(g) h)(z) := j(g^{−1}, z) h(g^{−1}(z)) (3.13) for all g ∈ G, h ∈ H_{ν}^{2}(ℂ^{n}) and z ∈ ℂ^{n}, where

j(g^{−1}, z) = \frac{K(z, g(0))}{K(g(0), g(0))} = exp (ν(z|g(0)) − \frac{ν}{2} (g(0)|g(0))).

This completes the definition of H_{ν}^{2}(D). In all three cases we have j(g, z) K(g(z), g(w)) \overline{j(g, w)} = K(z, w) (3.14) for all g ∈ G and z, w ∈ D. Put K_{w}(z) := K(z, w). (3.15) Then K_{w} ∈ H_{ν}^{2}(D) for all w ∈ D, and (3.14) shows

U(g) K_{w} = \overline{j(g, w)} K_{g(w)}

for all g ∈ G and w ∈ D since

\overline{j(g, w)} K_{g(w)}(g(z)) = \overline{j(g, w)} K(g(z), g(w))

= j(g, z)^{−1} K(z, w) = j(g^{−1}, g(z)) K_{w}(z)
= (U(g) K_{w})(g(z))

for all z ∈ D. Let

(h|k)_{ν} := ∫_{D} dμ_{ν}(z) \overline{h(z)} k(z)

denote the scalar product in H_{ν}^{2}(D). Then

h(z) = (K_{z}|h)_{ν} (3.16)

for all h ∈ H_{ν}^{2}(D) by the reproducing kernel property.

Let z ↦ \overline{z} denote the conjugation of D with real form D_{ℝ}. Since D is simply connected I(z) := K(z, \overline{z})^{1/2} (3.17)

defines a holomorphic function on D (not belonging to H_{ν}^{2}(D)).

Example 3.1. In the product case (Case 2) we have

(3.18)  I(z_{1}, \overline{z}_{2}) = K(z_{1}, \overline{z}_{2}, \overline{(z_{1}, \overline{z}_{2})})^{1/2} = K(z_{1}, \overline{z}_{2}, z_{2}, \overline{z}_{1})^{1/2}
= [K^{ℝ}(z_{1}, z_{2}) K^{ℝ}(z_{1}, z_{2})]^{1/2} = K^{ℝ}(z_{1}, z_{2})

as a holomorphic function on D = D_{ℝ}×\overline{D}_{ℝ}. Since the reproducing kernel property implies

ϕ(z_{1}) = ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(z_{2}) K^{ℝ}(z_{1}, z_{2}) ϕ(z_{2})

for all ϕ ∈ H_{ν}^{2}(D_{ℝ}), it follows that I corresponds to the identity operator on H_{ν}^{2}(D_{ℝ}). On the other hand, the kernel

K_{w_{1},\overline{w}_{2}}(z_{1}, \overline{z}_{2}) = K(z_{1}, \overline{z}_{2}, w_{1}, \overline{w}_{2})

= K^{ℝ}(z_{1}, w_{1}) K^{ℝ}(w_{2}, z_{2}) = K_{w_{1}}^{ℝ}(z_{1}) \overline{K_{w_{2}}^{ℝ}(z_{2})} (3.19)

for (w_{1}, \overline{w}_{2}) ∈ D corresponds to the rank 1 operator K_{w_{1}}^{ℝ}(K_{w_{2}}^{ℝ})^{∗} acting on H_{ν}^{2}(D_{ℝ}) since for all ϕ ∈ H_{ν}^{2}(D_{ℝ})

∫_{D_{ℝ}} dμ_{ν}^{ℝ}(z_{2}) K_{w_{1},w_{2}}(z_{1}, \overline{z}_{2}) ϕ(z_{2}) =

∫_{D_{ℝ}} dμ_{ν}^{ℝ}(z_{2}) K_{w_{1}}^{ℝ}(z_{1}) K^{ℝ}(w_{2}, z_{2}) ϕ(z_{2}) =

K_{w_{1}}^{ℝ}(z_{1}) ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(z_{2}) K^{ℝ}(w_{2}, z_{2}) ϕ(z_{2}) =

K_{w_{1}}^{ℝ}(z_{1}) ϕ(z_{2}) = K_{w_{1}}^{ℝ}(K_{w_{2}}^{ℝ}|ϕ)_{ν}.

Proposition 3.2. I is the unique (up to a multiplicative constant) holomorphic function on D which is U_{ν}-invariant under G_{ℝ} ⊂ G.

Proof. Let g ∈ G_{ℝ}. In order to show

I(z) = (U_{ν}(g^{−1}) I)(z) = j(g, z) I(g(z)) (3.20) for all z ∈ D, we may assume that z = ζ ∈ D_{ℝ} since both sides of (3.20) are holomorphic. Since I(ζ), j(g, ζ) and I(g(ζ)) are positive, the assertion follows by taking squares

[j(g, ζ) I(g(ζ))]^{2} = j(g, ζ) K(g(ζ), g(ζ)) j(g, ζ) = K(ζ, ζ) = I(ζ)^{2}.
□

As a consequence of (3.20) we have

I(ζ) = I(g_{ζ}(o)) = j(g_{ζ}, o)^{−1} (3.21)

for all ζ ∈ D_{ℝ}, with g_{ζ} ∈ G_{ℝ} satisfying g_{ζ}(o) = ζ. Here o is the origin of D_{ℝ}.

Lemma 3.3. For all z ∈ D we have

∫_{D_{ℝ}} dμ_{0}(ζ) K(z, ζ) I(ζ)^{−1} = I(z) ∫_{D_{ℝ}} dμ_{0}(ζ) K(o, ζ) I(ζ)^{−1}. (3.22)

Proof. Since both sides of (3.22) are holomorphic on D we may assume z ∈ D_{ℝ}. Write z = g(o) for some g ∈ G_{ℝ}. Then G_{ℝ}-invariance of μ_{0} implies

∫_{D_{ℝ}} dμ_{0}(ζ) K(z, ζ) K(ζ, ζ)^{−1/2} =

∫_{D_{ℝ}} dμ_{0}(ζ) K(z, g(ζ)) K(g(ζ), g(ζ))^{−1/2} =

∫_{D_{ℝ}} dμ_{0}(ζ) j(g, o)^{−1} K(o, ζ) j(g, ζ)^{−1} j(g, ζ) K(ζ, ζ)^{−1/2}

= j(g, o)^{−1} ∫_{D_{ℝ}} dμ_{0}(ζ) K(o, ζ) K(ζ, ζ)^{−1/2}.

Since K(z, \overline{z}) = K(g(o), g(o)) = j(g, o)^{−2} K(o, o) = j(g, o)^{−2} the assertion follows. □

Notation. In view of Lemma 3.3 it is natural to normalize the G_{ℝ}-invariant measure μ_{0} on D_{ℝ} by the condition

∫_{D_{ℝ}} dμ_{0}(ζ) K(o, ζ) I(ζ)^{−1} = 1. (3.23)

This normalization (depending on ν) will be chosen in the sequel. Then

I(z) = ∫_{D_{ℝ}} dμ_{0}(ζ) K(z, ζ) I(ζ)^{−1} (3.24)

for all z ∈ D. In the case of complex bounded symmetric domains D_{ℝ} the normalization (3.23) amounts to

dμ_{0}(ζ) = dμ_{ν}^{ℝ}(ζ) K^{ℝ}(ζ, ζ) (3.25)

since K^{ℝ}(0, ζ) = 1.

Proposition 3.4. For holomorphic h ∈ H_{ν}^{2}(D) we have

∫_{D} dμ_{ν}(w) \overline{h(w)} I(w) = ∫_{D_{ℝ}} dμ_{0}(ζ) \overline{h(ζ)} I(ζ)^{−1}. (3.26)

Proof. We may assume h = K_{z} for some z ∈ D. Since I(z) := K(z, \overline{z})^{1/2} is holomorphic, Lemma 3.3 implies

∫_{D_{ℝ}} dμ_{0}(ζ) \overline{K_{z}(ζ)} I(ζ)^{−1} = ∫_{D_{ℝ}} dμ_{0}(ζ) K(z, ζ) I(ζ)^{−1}

= I(z) = (K_{z}|I)_{ν} = ∫_{D} dμ_{ν}(w) \overline{K_{z}(w)} I(w).

□

4. Covariant Symbolic Calculi on Real Symmetric Domains

We will now introduce the new concept of covariant calculus for real symmetric domains, which arises as a natural generalization of the complex case [AU1] replacing operators by Hilbert space vectors.

Definition 4.1. A covariant symbolic calculus is given by a linear ”symbol map”

(4.1)  σ : H_{ν}^{2}(D) → {functions on D_{ℝ}}
h ↦ σh

satisfying the covariance condition

σ(U_{ν}(g) h) = (σh) ∘ g^{−1} (4.2)

for all g ∈ G_{ℝ}. More precisely, σ should have a G_{ℝ}-invariant domain Dom(σ) of holomorphic functions on D containing all reproducing kernel vectors K_{w}, w ∈ D. Then the covariance condition becomes

(σK_{w}) ∘ g^{−1} = σ(U_{ν}(g) K_{w}) = \overline{j(g, w)} σ(K_{gw}) (4.3) for all w ∈ D and g ∈ G_{ℝ}. In addition we require that the holomorphic function I(z) := K(z, \overline{z})^{1/2} on D belongs to Dom(σ). Then σI is a constant function by (4.2) and Lemma 3.3, and we normalize σ by assuming

σI = 1. (4.4)

The function σh is also called the passive (or weak) symbol of h. The main examples of symbolic calculi are ”real” in the sense that σK_{o} is a real-valued function on D_{ℝ}. This will be assumed in the sequel.

Proposition 4.2. For every symbolic calculus σ we have

(σh)(ζ) = (\overline{(σK_{⊔})(ζ)}|h)_{ν} (4.5)

for all h ∈ Dom(σ) and ζ ∈ D_{ℝ}, where for fixed ζ the function

z ↦ (σK_{⊔})(ζ) (z) := (σK_{z})(ζ) (4.6)

is anti-holomorphic on D.

Proof. We may assume h = K_{w} for w ∈ D. Then

\overline{(σK_{w})(ζ)} = \overline{(σK_{⊔})(ζ)} (w) = (K_{w}|\overline{(σK_{⊔})(ζ)})_{ν}

and hence

(σK_{w})(ζ) = \overline{(K_{w}|\overline{(σK_{⊔})(ζ)})_{ν}} = (\overline{(σK_{⊔})(ζ)}|K_{w})_{ν}.
□

Definition 4.3. The adjoint

(4.7) σ^{∗} : {functions on D_{ℝ}} → H_{ν}^{2}(D)

of a covariant symbolic calculus σ is defined by assigning to a function f on D_{ℝ} (belonging to Dom(σ^{∗})) the holomorphic function

(σ^{∗}f)(z) := ∫_{D_{ℝ}} dμ_{0}(ζ) f(ζ) \overline{(σK_{z})(ζ)} (4.8)

on D. Here the invariant measure μ_{0} is normalized by the condition (3.23), depending on ν but not on σ. We call f the active (or strong) symbol of σ^{∗}f. The adjoint σ^{∗}, corresponding to the map A of (1.1), can of course also be taken as the starting point of the theory. The ”dual” view point emphasizing the symbol map σ is closer to Berezin’s original approach.

Proposition 4.4. A covariant symbolic calculus σ and its adjoint σ^{∗} are related by duality

(σ^{∗}f|h)_{ν} = (f|σh) := ∫_{D_{ℝ}} dμ_{0}(ζ) \overline{f(ζ)} (σh)(ζ) (4.9)

for all f ∈ Dom(σ^{∗}) and h ∈ Dom(σ).

Proof. We may assume h = K_{z} for some z ∈ D. Then

(σ^{∗}f|K_{z})_{ν} = \overline{(K_{z}|σ^{∗}f)_{ν}} = \overline{(σ^{∗}f)(z)}

and (4.9) follows from (4.8) by taking conjugates. □

Since (4.9) involves the G_{ℝ}-invariant measure μ_{0} and σ is G_{ℝ}-covariant, it follows that σ^{∗} satisfies the covariance condition

σ^{∗} (f ∘ g^{−1}) = U_{ν}(g)(σ^{∗}f) (4.10)

for all g ∈ G_{ℝ} and f ∈ Dom(σ^{∗}).

Proposition 4.5. For any covariant symbolic calculus σ the adjoint σ^{∗} satisfies

σ^{∗}1 = I. (4.11)

Proof. For ζ ∈ D_{ℝ} and g_{ζ} ∈ G_{ℝ} satisfying ζ = g_{ζ}(o) the covariance of σ implies

(σK_{o})(g_{ζ}^{−1}(o)) = σ(U_{ν}(g_{ζ})K_{o})(o) = j(g_{ζ}, o)(σK_{ζ})(o) = I(ζ)^{−1}(σK_{ζ})(o).

Since G_{ℝ} is unimodular and (σK_{ξ})(η) is real for ξ, η ∈ D_{ℝ}, Proposition 3.4 and Proposition 4.2 imply

(σ^{∗}1)(o) = ∫_{D_{ℝ}} dμ_{0}(ζ)(σK_{o})(ζ) = ∫_{D_{ℝ}} dμ_{0}(ζ)(σK_{o})(g_{ζ}(o))

= ∫_{D_{ℝ}} dμ_{0}(ζ)(σK_{o})(g_{ζ}^{−1}(o)) = ∫_{D_{ℝ}} dμ_{0}(ζ) I(ζ)^{−1}(σK_{ζ})(o)

= ∫_{D} dμ_{ν}(z) I(z)(σK_{z})(o) = (\overline{(σK_{⊔})(o)} | I)_{ν} = (σI)(o) = 1.

Since σ^{∗}1 = (σ^{∗}1)(o) I by covariance, the assertion follows. □

Definition 4.6. Let σ be a covariant symbolic calculus with adjoint σ^{∗}. The composite map

f ↦ (σσ^{∗}) f = σ(σ^{∗}f)

acting on functions on D_{ℝ} is called the link transform associated with σ. Note that σσ^{∗} maps the active symbol to the passive symbol. By our assumptions, σσ^{∗} has a G_{ℝ}-invariant domain and commutes with the G_{ℝ}-action

(σσ^{∗})(f ∘ g) = (σσ^{∗}f) ∘ g (4.12)

for all functions f ∈ Dom(σσ^{∗}) and g ∈ G_{ℝ}. More generally, one may consider transforms

f ↦ (σ_{1}σ_{2}^{∗}) f = σ_{1}(σ_{2}^{∗}f)

linking two covariant symbolic calculi σ_{1}, σ_{2} on H_{ν}^{2}(D), and the invariance property (4.12) still holds. In view of (4.4) and (4.11) these transforms are ”stochastic” operators: (σ_{1}σ_{2}^{∗}) 1 = σ_{1}(σ_{2}^{∗}1) = σ_{1}I = 1.

Proposition 4.7. Let σ_{1}, σ_{2} be covariant symbolic calculi. Then the link transform σ_{1}σ_{2}^{∗} has the integral kernel

(\overline{(σ_{1}K_{⊔})(ξ)} | \overline{(σ_{2}K_{⊔})(η)})_{ν} = ∫_{D} dμ_{ν}(z)(σ_{1}K_{z})(ξ) \overline{(σ_{2}K_{z})(η)} (4.13)

with respect to μ_{0} (normalized by (3.23)).

Proof. Using (4.5) and (4.8) we obtain

(σ_{1}σ_{2}^{∗}f)(ξ) = (\overline{(σ_{1}K_{⊔})(ξ)} | σ_{2}^{∗}f)_{ν} =

∫_{D} dμ_{ν}(z)(σ_{1}K_{z})(ξ)(σ_{2}^{∗}f)(z) =

∫_{D} dμ_{ν}(z)(σ_{1}K_{z})(ξ) ∫_{D_{ℝ}} dμ_{0}(η) f(η) \overline{(σ_{2}K_{z})(η)} =

∫_{D_{ℝ}} dμ_{0}(η) f(η) ∫_{D} dμ_{ν}(z)(σ_{1}K_{z})(ξ) \overline{(σ_{2}K_{z})(η)}.

□

It is well known [H] that the L^{2}-space L^{2}(D_{ℝ}, μ_{0}) has a multiplicity-free decomposition

L^{2}(D_{ℝ}, μ_{0}) ≡ ∫_{𝔞^{#}} |c(λ)|^{−2}dλ ⟨G_{ℝ}⟩_{λ} (4.14)

under the group G_{ℝ}, where ⟨G_{ℝ}⟩_{λ} denotes the principal series representation with spectral parameter λ ∈ 𝔞^{#}, c(λ) is Harish-Chandra’s c-function and we choose an Iwasawa decomposition G_{ℝ} = K_{ℝ} A N (4.15) of G_{ℝ}, with 𝔞 = Lie(A). By G_{ℝ}-invariance it follows that the link transform σ_{1}σ_{2}^{∗} is diagonalized under the G_{ℝ}-action. More precisely, let ϕ_{λ} ∈ C^{∞}(D_{ℝ}) be the normalized spherical function associated with λ. Then

σ_{1}σ_{2}^{∗}ϕ_{λ} = σ̃_{1}σ_{2}^{∗}(λ) ϕ_{λ} (4.16)

for all ϕ_{λ} ∈ Dom(σ_{1}σ_{2}^{∗}), and the eigenvalues σ̃_{1}σ_{2}^{∗}(λ) completely characterize the link transform. For every λ ∈ 𝔞^{#} the Iwasawa decomposition (4.15) defines ”exponential functions” [H]

e_{λ}(ζ) = exp ⟨A(g)|λ + ρ⟩, g ∈ G_{ℝ}, g(o) = ζ

on D_{ℝ} satisfying e_{λ}(o) = 1 and

e_{λ}(g(ζ)) = e_{λ}(g(o)) e_{λ}(ζ)

for all g ∈ NA, such that

ϕ_{λ}(ζ) = ∫_{K_{ℝ}} dk e_{λ}(kζ)

is the spherical function of type λ. In the flat case e_{λ} is expressed in terms of the exponential function [AU1]; in the curved setting an explicit description of e_{λ} can be given by realizing D_{ℝ} as a real Siegel domain (cf. Section 4).

Proposition 4.8. For every covariant symbolic calculus σ we have

(σ^{∗}e_{λ})(z) = (σ^{∗}e_{λ})(o) e_{λ}(z) I(z) (4.17)

where e_{λ}(z) denotes the unique extension of e_{λ} to a holomorphic function on D.

Proof. Since both sides of (4.17) are holomorphic it suffices to let z = ζ ∈ D_{ℝ}. Let g_{ζ} ∈ NA satisfy g_{ζ}(o) = ζ. Then

(σ^{∗}e_{λ})(ζ) = (σ^{∗}e_{λ})(g_{ζ}(o)) = j(g_{ζ}, o)^{−1}(U_{ν}(g_{ζ}^{−1})(σ^{∗}e_{λ}))(o)
= j(g_{ζ}, o)^{−1}(σ^{∗}(e_{λ} ∘ g_{ζ}))(o) = j(g_{ζ}, o)^{−1}e_{λ}(ζ) (σ^{∗}e_{λ})(o)
= I(ζ) e_{λ}(ζ) (σ^{∗}e_{λ})(o).
□

Our main result is a ”product formula” for the link transform for any pair of covariant symbolic calculi.

Theorem 4.9. Let σ_{1}, σ_{2} be covariant symbolic calculi on an irreducible real symmetric domain D_{ℝ}. Then the G_{ℝ}-invariant link transform σ_{1}, σ_{2}^{∗} has the eigenvalues

σ̃_{1}σ_{2}^{∗}(λ) = \frac{\overline{(σ_{1}^{∗}e_{λ})(o)} (σ_{2}^{∗}e_{λ})(o)}{c_{λ}} (4.18)

for all λ ∈ Dom(σ̃_{1}σ_{2}^{∗}) ⊂ 𝔞^{#}, where c_{λ} is a positive constant independent of σ_{1}, σ_{2} which will be computed explicitly in Section 5. In particular,

σ̃σ^{∗}(λ) = \frac{1}{c_{λ}} |(σ^{∗}e_{λ})(o)|^{2}. (4.19)

Proof. By Proposition 4.7, σ_{1}σ_{2}^{∗} is the adjoint of σ_{2}σ_{1}^{∗} with respect to μ_{0}. Using Proposition 4.8 and Proposition 4.2 it follows that

(σ_{2}^{∗}e_{λ})(o) \overline{(e_{λ}I|\overline{(σ_{1}K_{⊔})(o)})_{ν}} =

(σ_{2}^{∗}e_{λ})(o) (\overline{(σ_{1}K_{⊔})(o)} | e_{λ}I)_{ν} = (\overline{(σ_{1}K_{⊔})(o)} | σ_{2}^{∗}e_{λ})_{ν} (4.20)

= σ_{1}(σ_{2}^{∗}e_{λ})(o) = σ̃_{1}σ_{2}^{∗}(λ) = \overline{σ̃_{2}σ_{1}^{∗}(λ)}
= \overline{(σ_{1}^{∗}e_{λ})(o)} (e_{λ}I|\overline{(σ_{2}K_{⊔})(o)})_{ν}.

Therefore

c_{λ} := \frac{(σ_{2}^{∗}e_{λ})(o)}{(e_{λ}I|\overline{(σ_{2}K_{⊔})(o)})_{ν}} = \overline{[\frac{(σ_{1}^{∗}e_{λ})(o)}{(e_{λ}I|\overline{(σ_{1}K_{⊔})(o)})_{ν}} ]}.

Taking σ_{1} = σ_{2} it follows that c_{λ} is real. Therefore c_{λ} is independent of σ_{1}, σ_{2} and (4.18) follows from (4.20). □

Remark 4.10. In terms of harmonic analysis

(σ^{∗}e_{λ})(o) = ∫_{D_{ℝ}} dμ_{0}(ζ) e_{λ}(ζ) (σK_{o})(ζ)

= ∫_{D_{ℝ}} dμ_{0}(ζ) ϕ_{λ}(ζ) (σK_{o})(ζ) = (σK_{o})^{∼}(λ)

can be identified with the spherical Fourier transform [H] of the real K_{ℝ}-invariant function σK_{o} on D_{ℝ}. Viewed as a function of λ, we call

a_{o}(λ) := (σ^{∗}e_{λ})(o)

the fundamental function of σ.

5. The Toeplitz-Berezin Calculus

Up to now the discussion of covariant symbolic calculi was quite general. In this section we describe a basic example, the Toeplitz-Berezin calculus, and show that it determines the value of the constant c_{λ} in an explicit way. Let D_{ℝ} = G_{ℝ}/K_{ℝ} be an irreducible real symmetric domain, with complexification D and quantization Hilbert space H_{ν}^{2}(D), as introduced in Section 2. Let o ∈ D_{ℝ} ⊂ D be the origin.

Definition 5.1. The Toeplitz-Berezin symbol

τ : H_{ν}^{2}(D) → C^{∞}(D_{ℝ})

is defined by (τh)(ζ) := K(ζ, ζ)^{−1/2}h(ζ) = I(ζ)^{−1} h(ζ) (5.1)

for all h ∈ H_{ν}^{2}(D) and ζ ∈ D_{ℝ}. In particular

(τK_{z})(ζ) = I(ζ)^{−1} K(ζ, z) (5.2)

for all z ∈ D and ζ ∈ D_{ℝ}.

Note that τ is well-defined (and injective) for all holomorphic functions h on D since D_{ℝ} ⊂ D is a set of uniqueness. The normalization τI = 1 made in (4.4) is trivially satisfied.

By Definition 4.3 the adjoint τ^{∗} of τ (called the Toeplitz-Berezin calculus) is defined by

(τ^{∗}f)(z) = ∫_{D_{ℝ}} dμ_{0}(ζ) f(ζ) I(ζ)^{−1} K(z, ζ) (5.3)

for all functions f ∈ L^{∞}(D_{ℝ}), since dμ_{0}(ζ) I(ζ)^{−1} is a finite measure for ν large enough. Here μ_{0} is normalized by (3.23) so that (3.24) implies τ^{∗}1 = I. According to Proposition 4.7, the link transform ττ^{∗} (called the Berezin transform in the complex case) has the integral kernel

∫_{D} dμ_{ν}(z)(τK_{z})(ξ) \overline{(τK_{z})(η)} =

∫_{D} dμ_{ν}(z) I(ξ)^{−1} K(ξ, z) I(η)^{−1} \overline{K(η, z)} = (5.4)

I(ξ)^{−1} I(η)^{−1} K(ξ, η) = \frac{K(ξ, η)}{K(ξ, ξ)^{1/2} K(η, η)^{1/2}}

for all ξ, η ∈ D_{ℝ}.

Example 5.2. In case D_{ℝ} is complex (Case 2) H_{ν}^{2}(D) = H_{ν}^{2}(D_{ℝ}) ⊗ \overline{H_{ν}^{2}(D_{ℝ})} is the space of Hilbert-Schmidt operators via the identification (3.8), where h is a sesqui-holomorphic function on D_{ℝ} × D_{ℝ}. For fixed z ∈ D_{ℝ} we have

(K_{z}^{ℝ} | hK_{z}^{ℝ})_{ν} = (hK_{z}^{ℝ})(z) = ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(w) h(z, \overline{w}) K_{z}^{ℝ}(w) = h(z, \overline{z})

since w ↦ \overline{h(z, \overline{w})} is holomorphic. Therefore (5.1) amounts to

(τh)(z, \overline{z}) = I(z, \overline{z})^{−1} h(z, \overline{z}) = K^{ℝ}(z, z)^{−1} h(z, \overline{z})
= \frac{(K_{z}^{ℝ}|hK_{z}^{ℝ})_{ν}}{(K_{z}^{ℝ}|K_{z}^{ℝ})_{ν}}

Covariant Symbolic Calculi 17

for all z ∈ D_{ℝ}. This is the classical Berezin symbol of the operator h.

On the other hand, the sesqui-holomorphic function

(τ^{∗}f)(z, \overline{w}) = ∫_{D_{ℝ}} dμ_{0}(ζ) f(ζ) I(ζ, \overline{ζ})^{−1} K(z, \overline{w}, ζ, \overline{ζ})

= ∫_{D_{ℝ}} dμ_{0}(ζ) f(ζ) K^{ℝ}(ζ, ζ)^{−1} K^{ℝ}(z, ζ) K^{ℝ}(ζ, w)

= ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(ζ) f(ζ) K^{ℝ}(z, ζ) K^{ℝ}(ζ, w)

on D_{ℝ} × D_{ℝ}, defined via (5.3), gives rise to the integral operator

((τ^{∗}f)ϕ)(z) = ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(w) (τ^{∗}f)(z, \overline{w}) ϕ(w)

= ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(w) ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(ζ) f(ζ) K^{ℝ}(z, ζ) K^{ℝ}(ζ, w) ϕ(w)

= ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(ζ) f(ζ) K^{ℝ}(z, ζ) ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(w) K^{ℝ}(ζ, w) ϕ(w)

= ∫_{D_{ℝ}} dμ_{ν}^{ℝ}(ζ) f(ζ) K^{ℝ}(z, ζ) ϕ(ζ) = E(fϕ)(z)

where E denotes the orthogonal projection onto H_{ν}^{2}(D_{ℝ}). Thus

(τ^{∗}f)ϕ = E(fϕ) =: T_{f}ϕ

gives the Toeplitz operator with symbol f acting on H_{ν}^{2}(D_{ℝ}). By (5.4) the Berezin transform in the complex case has the kernel

\frac{K(ξ, \overline{ξ}, η, \overline{η})}{K(ξ, \overline{ξ}, ξ, \overline{ξ})^{1/2} K(η, \overline{η}, η, \overline{η})^{1/2}} = \frac{K^{ℝ}(ξ, η) K^{ℝ}(η, ξ)}{K^{ℝ}(ξ, ξ) K^{ℝ}(η, η)} = \frac{|K^{ℝ}(ξ, η)|^{2}}{K^{ℝ}(ξ, ξ) K^{ℝ}(η, η)}

for all ξ, η ∈ D_{ℝ}. Proposition 5.3. For all λ ∈ 𝔞^{#} we have τ̃τ^{∗}(λ) = (τe_{λ})(o) = c_{λ}. (5.5) Proof. Since

\overline{(τK_{z})(o)} = \overline{K(o, o)^{−1/2} K_{z}(o)} = K_{o}(z)

for all z ∈ D, Proposition 4.2 implies

τ̃τ^{∗}(λ) = τ(τ^{∗}e_{λ})(o) = (\overline{(τK_{⊔})(o)} | τ^{∗}e_{λ})_{ν} = (K_{o}|τ^{∗}e_{λ})_{ν} = (τ^{∗}e_{λ})(o).

This shows that (τ^{∗}e_{λ})(o) is real, and the product formula (4.19) implies

(τ^{∗}e_{λ})(o) = τ̃τ^{∗}(λ) = \frac{|(τ^{∗}e_{λ})(o)|^{2}}{c_{λ}} = \frac{(τ^{∗}e_{λ})(o)^{2}}{c_{λ}} .

□

Combining Proposition 5.3 with Theorem 4.9 and Remark 4.10. we obtain

Corollary 5.4. For covariant symbolic calculi σ_{1}, σ_{2} on H_{ν}^{2}(D) the link transform σ_{1}σ_{2}^{∗} has the eigenvalues

σ̃_{1}σ_{2}^{∗}(λ) = \frac{(σ_{2}^{∗}e_{λ})(o) (\overline{σ_{1}^{∗}e_{λ})(o)}}{(τ^{∗}e_{λ})(o)} = \frac{(σ_{2}K_{o})^{∼}(λ) \overline{(σ_{1}K_{o})^{∼}(λ)}}{(τK_{o})^{∼}(λ)}

where τ is the Toeplitz-Berezin calculus and ∼ denotes spherical Fourier transforms.

In view of Corollary 5.4 it is important to compute the integral

(τ^{∗}e_{λ})(o) = ∫_{D_{ℝ}} dμ_{0}(ζ) e_{λ}(ζ) (τK_{o})(ζ)

= ∫_{D_{ℝ}} dμ_{0}(ζ) e_{λ}(ζ) I(ζ)^{−1} K(o, ζ)

= ∫_{D_{ℝ}} dμ_{0}(ζ) e_{λ}(ζ) K(ζ, ζ)^{−1/2} K(o, ζ)

explicitly. In the complex case this has been carried out in [UU], verifying a longstanding conjecture of Berezin. For details on the complex case and also the flat case we refer to [AU1]. In the sequel we determine (τ^{∗}e_{λ})(o) for real symmetric domains. The Toeplitz-Berezin calculus has also been studied in [DP], [N], [Z]; we avoid a case-by-case separation according to the various root systems of D_{ℝ}, thus giving a more uniform treatment.

As in Section 2 consider the unbounded realization

D_{ℝ} = {x + y + v ∈ Z_{ℝ} : x − {ev^{∗}v} ∈ Ω}

as a real Siegel domain. For the Jordan theoretic concepts used in the sequel we refer to [FK], [L2], [U]. Let e_{1}, …, e_{r} ∈ X be a frame with e = e_{1} + ⋯ + e_{r}. Then we have Peirce decompositions

X = ∑_{1≤i≤j≤r} X_{ij}, Y = ∑_{1≤i≤j≤r} Y_{ij}, V_{ℝ} = ∑_{1≤j≤r} V_{j}^{ℝ}

and we put

1 = dim X_{jj} (1 ≤ j ≤ r)
a := dim X_{ij} (1 ≤ i < j ≤ r).

In case Z_{ℝ} ≠ X we also put

b := dim V_{j}^{ℝ} (1 ≤ j ≤ r)
c := dim Y_{jj} (1 ≤ j ≤ r).

It is known that

dim Y_{ij} = a (1 ≤ i < j ≤ r)

up to one exception (root system D_{2}) which we omit in the sequel. The dimensions of the respective subspaces are then given by

\begin{aligned}n_{X} & := & dim_{ℝ} X & = & r + r(r − 1) a/2 \\ n_{Y} & := & dim_{ℝ} Y & = & cr + r(r − 1) a/2 \\ n_{V} & := & dim_{ℂ} V & = & dim_{ℝ} V_{ℝ} = rb. \end{aligned}

Put n := dim_{ℝ} Z_{ℝ} = n_{X} + n_{Y} + n_{V} . Let P denote the quadratic representation of X and let Δ(x) be the Jordan determinant of X.

Lemma 5.5. For x ∈ Ω we have

Det_{X} P_{x}^{1/2} = Δ(x)^{n_{X}/r}.

Proof. Without loss of generality (due to the spectral decomposition and transitivity of K_{ℝ} on frames) we may assume x = ∑_{j} x_{j}e_{j} diagonal. Putting t′ = P_{x}^{1/2}t

for t ∈ X, the respective Peirce components satisfy t′_{ij} = x_{i}^{1/2} t_{ij} x_{j}^{1/2}. It follows that

Det_{X} P_{x}^{1/2} = ∏_{1≤j≤r} x_{j} ∏_{1≤i<j≤r}(x_{i}x_{j})^{a/2}

= (x_{1} ⋯ x_{r})^{1+(r−1)a/2} = Δ(x)^{1+(r−1)a/2} = Δ(x)^{n_{X}/r}.

□

In the sequel we use the conical functions

Δ\underline{α}(x) = Δ_{1}(x)^{α_{1}−α_{2}}Δ_{2}(x)^{α_{2}−α_{3}} ⋯ Δ_{r}(x)^{α_{r}} (5.6)

on Ω, associated with \underline{α} = (α_{1}, …, α_{r}), where Δ_{1}, …, Δ_{r} are the Jordan theoretic minors. Let Δ\underline{α} ; ∗ denote the corresponding function using the minors in reverse order e_{r}, …, e_{1}. For the following result, cf. [FK, Section VII.1].

20 Arazy, Upmeier

Lemma 5.6. Let x ∈ Ω and Re(α_{j}) > −1 − (r − j) \frac{a}{2}. Then

∫_{Ω} dt Δ\underline{α}(t) e^{−(x|t)} = Γ_{Ω} (\underline{α} + \frac{n_{X}}{r} ) Δ_{∗}^{−\underline{α}^{∗}−n_{X}/r}(x)

where

Γ_{Ω} (\underline{α} + \frac{n_{X}}{r} ) = ∫_{Ω} dt Δ\underline{α}(t) e^{−(e|t)}

is the (multi-variable) Koecher-Gindikin Γ-function and \underline{α}^{∗} := (α_{r}, …, α_{1}). Lemma 5.7. Let Re(β) > 1 + \frac{a}{2} (2r − j − 1) + Re(α_{j}) > \frac{a}{2} (r − 1). Then

Γ_{Ω}(β) ∫_{Ω} dt Δ\underline{α}(t) Δ(e + t)^{−β} = Γ_{Ω} (\underline{α} + \frac{n_{X}}{r} ) Γ_{Ω} (β − \underline{α} + \underline{δ} − \frac{2n_{X}}{r} )

where \underline{δ} = (δ_{1}, …, δ_{r}) is defined by δ_{j} := 1 + (j − 1)a. (5.7) Proof. Applying Lemma 5.6 twice we obtain

Γ_{Ω}(β) ∫_{Ω} dt Δ\underline{α}(t) Δ(e + t)^{−β} =

∫_{Ω} dt Δ\underline{α}(t) ∫_{Ω} ds Δ(s)^{β−n_{X}/r} e^{−(s|e+t)} =

∫_{Ω} ds Δ(s)^{β−n_{X}/r} e^{−(s|e)} ∫_{Ω} dt Δ\underline{α}(t) e^{−(s|t)} =

= Γ_{Ω} (\underline{α} + \frac{n_{X}}{r} ) ∫_{Ω} ds Δ_{∗}^{β−\underline{α}^{∗}−2n_{X}/r}(s) e^{−(s|e)}

= Γ_{Ω} (\underline{α} + \frac{n_{X}}{r} ) Γ_{Ω} (β − \underline{α}^{∗} − \frac{n_{X}}{r} ) .

Now the assertion follows from the well-known identity

Γ_{Ω}(\underline{α}^{∗}) = Γ_{Ω} (\underline{α} + \underline{δ} − \frac{n_{X}}{r} ) . (5.8)

□

Lemma 5.8. [FK, p. 142, Exercise 5] Let 2γ > 1 + (r − 1) a. Then

Γ_{Ω}(γ) ∫_{X} dx Δ(e + x^{2})^{−γ} = π^{n_{X}} 4^{n_{X}−rγ} \frac{Γ_{Ω}(2γ − n_{X}/r)}{Γ_{Ω}(γ)}.

Proof. Let Δ^{ℂ} be the holomorphic extension of Δ to the complexified Jordan algebra X^{ℂ}. By analytic continuation, Lemma 5.6 implies for every x ∈ X

Γ_{Ω}(γ) Δ^{ℂ}(e + ix)^{−γ} = ∫_{Ω} ds Δ(s)^{γ−n_{X}/r} e^{−(e+ix|s)}

= ∫_{Ω} ds Δ(s)^{γ−n_{X}/r} e^{−(e|s)} e^{−i(x|s)}.

It follows that Γ_{Ω}(γ) Δ^{ℂ}(e + ix)^{−γ} is the Fourier transform of the function

f(s) := χ_{Ω}(s) Δ(s)^{γ−n_{X}/r} e^{−(e|s)}

on X, where χ_{Ω} is the characteristic function of Ω ⊂ X. Therefore Parseval’s formula implies

Γ_{Ω}(γ)^{2} ∫_{X} dx |Δ^{ℂ}(e + ix)|^{−2γ} = (2π)^{n_{X}} ∫_{X} ds |f(s)|^{2}

= (2π)^{n_{X}} ∫ ds Δ(s)^{2γ−2n_{X}/r} e^{−2(e|s)}

= (2π)^{n_{X}} Γ_{Ω} (2γ − \frac{n_{X}}{r} ) Δ(2e)^{−2γ+n_{X}/r}

= π^{n_{X}} 4^{n_{X}−rγ} Γ_{Ω} (2γ − \frac{n_{X}}{r} ) .

Now the assertion follows from

|Δ^{ℂ}(e + ix)|^{2} = Δ^{ℂ}(e + ix) Δ^{ℂ}(e − ix) = Δ^{ℂ}((e + ix)(e − ix)) = Δ(e + x^{2}).

□

From now on we assume Y ≠ {0}. For t ∈ X the Jordan multiplication operator M_{t} leaves X and Y invariant.

Lemma 5.9. For t ∈ Ω we have

Δ(t)^{n_{X}/r} Det_{Y} M_{t} = Δ(t)^{n_{Y} /r} Det_{X} M_{t}.

Proof. Assuming t = ∑_{j} t_{j} e_{j} diagonal and putting x′ = M_{t} x, y′ = M_{t} y for x ∈ X and y ∈ Y, the respective Peirce components satisfy x′_{ij} = x_{ij}(t_{i} + t_{j})/2, y′_{ij} = y_{ij}(t_{i} + t_{j})/2 for all 1 ≤ i ≤ j ≤ r. It follows that

Det_{X} M_{t} = ∏_{1≤j≤r} t_{j} · ∏_{1≤i<j≤r} (\frac{t_{i} + t_{j}}{2} )^{a} ,

Det_{Y} M_{t} = ∏_{1≤j≤r} t_{j}^{c} · ∏_{1≤i<j≤r} (\frac{t_{i} + t_{j}}{2} )^{a} .

22 Arazy, Upmeier

Since Δ(t) = t_{1} ⋯ t_{r} and c − 1 = \frac{n_{Y} −n_{X}}{r}, the assertion follows. □

Let r_{U} and Δ_{U} be the rank and Jordan determinant of U. We define

ν_{ℝ} := \frac{νr_{U}}{2r} , p_{ℝ} := \frac{pr_{U}}{2r} = \frac{n_{X} + n_{Y} + n_{V} /2}{r}. (5.9)

Lemma 5.10. Let 2ν_{ℝ} > (r − 1) a + c. Then

2^{νr_{U}} ∫_{Y} dy Δ_{U}(e − y)^{−ν} = π^{(n_{X}+n_{Y} )/2} 2^{n_{X}+n_{Y}} \frac{Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y}}{r} )}{Γ_{Ω}(ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )}.

Proof. For t ∈ Ω, x ∈ X and y ∈ Y we have (x^{2}|t) = ({xx^{∗}e}|t) = (x|{xe^{∗}t}) = (x|M_{t}x),

−(y^{2}|t) = ({yy^{∗}e}|t) = (y|{ye^{∗}t}) = (y|M_{t}y).

In view of Lemma 5.9 this implies

π^{−n_{Y} /2} ∫_{Y} dy e^{(y^{2}|t)} = π^{−n_{Y} /2} ∫_{Y} dy e^{−(y|M_{t} y)}

= Det_{Y}^{−1/2} M_{t} = Δ(t)^{(n_{X}−n_{Y} )/2r} Det_{X}^{−1/2} M_{t}

= Δ(t)^{(n_{X}−n_{Y} )/2r} π^{−n_{X}/2} ∫_{X} dx e^{−(x|M_{t} x)}

= Δ(t)^{(n_{X}−n_{Y} )/2r} π^{−n_{X}/2} ∫_{X} dx e^{−(x^{2}|t)}.

Since Δ_{U}(e − y) is real for all y ∈ Y, we have

Δ_{U}(e − y) = \overline{Δ_{U}(e − y)} = Δ_{U}((e − y)^{∗}) = Δ_{U}(e + y)

and therefore Δ_{U}(e−y)^{2} = Δ_{U}(e−y) Δ_{U}(e+y) = Δ_{U}((e−y)(e+y)) = Δ_{U}(e−y^{2}) = Δ(e−y^{2})^{r_{U} /r}. Applying Lemma 5.6 and Lemma 5.8 it follows that

π^{−n_{Y} /2} Γ_{Ω}(ν_{ℝ}) ∫_{Y} dy Δ_{U}(e − y)^{−ν} =

π^{−n_{Y} /2} Γ_{Ω}(ν_{ℝ}) ∫_{Y} dy Δ(e − y^{2})^{−ν_{ℝ}} =

π^{−n_{Y} /2} ∫_{Y} dy ∫_{Ω} dt Δ(t)^{ν_{ℝ}−n_{X}/r} e^{−(e−y^{2}|t)} =

Covariant Symbolic Calculi 23

π^{−n_{Y} /2} ∫_{Ω} dt Δ(t)^{ν_{ℝ}−n_{X}/r} e^{−(e|t)} ∫_{Y} dy e^{(y^{2}|t)} =

π^{−n_{X}/2} ∫_{Ω} dt Δ(t)^{ν_{ℝ}−(n_{X}+n_{Y} )/2r} e^{−(e|t)} ∫_{X} dx e^{−(x^{2}|t)} =

π^{−n_{X}/2} ∫_{X} dx ∫_{Ω} dt Δ(t)^{ν_{ℝ}−(n_{X}+n_{Y} )/2r} e^{−(e+x^{2}|t)} =

π^{−n_{X}/2} Γ_{Ω} (ν_{ℝ} + \frac{n_{X} − n_{Y}}{2r} ) ∫_{X} dx Δ(e + x^{2})^{(n_{Y} −n_{X})/2r−ν_{ℝ}}

= π^{−n_{X}/2} π^{n_{X}} 4^{n_{X}+(n_{Y} −n_{X}−νr_{U})/2} \frac{Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y}}{r} )}{Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )}.

□

For x ∈ X the quadratic representation P_{x} acts also on Y. Lemma 5.11. For x ∈ Ω we have

Det_{Y} P_{x}^{1/2} = Δ(x)^{n_{Y} /r}.

Proof. Assuming x = ∑_{j} x_{j} e_{j} diagonal and putting y′ = P_{x}^{1/2} y for y ∈ Y, the respective Peirce components satisfy y′_{ij} = x_{i}^{1/2} y_{ij} x_{j}^{1/2}. It follows that

Det_{Y} P_{x}^{1/2} = ∏_{1≤j≤r} x_{j}^{c} ∏_{1≤i<j≤r}(x_{i} x_{j})^{a/2}

= (x_{1} ⋯ x_{r})^{c+(r−1) a/2} = Δ(x)^{c+(r−1) a/2} = Δ(x)^{n_{Y} /r}.

□

Lemma 5.12. Let x ∈ Ω and 2ν_{ℝ} > (r − 1) a + c. Then

2^{νr_{U}} ∫_{Y} dy Δ_{U}(x − y)^{−ν} =

Δ(x)^{−2ν_{ℝ}+n_{Y} /r} π^{(n_{X}+n_{Y} )/2} 2^{n_{X}+n_{Y}} \frac{Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y}}{r} )}{Γ_{Ω} (ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )}.

Proof. Putting y′ = P_{x}^{−1/2} y for y ∈ Y, we have

dy = Δ(x)^{n_{Y} /r} dy′

by Lemma 5.8. Since Δ_{U}(x − y) = Δ_{U}(P_{x}^{1/2}(e − y′)) = Δ_{U}(x) Δ_{U}(e − y′) = Δ(x)^{r_{U} /r} Δ_{U}(e − y′)

24 Arazy, Upmeier

it follows that

∫_{Y} dy Δ_{U}(x − y)^{−ν} = Δ(x)^{−2ν_{ℝ}+n_{Y} /r} ∫_{Y} dy′ Δ_{U}(e − y′)^{−ν}.

Now the assertion follows from Lemma 5.10. □

Let R denote the canonical Jordan representation of X on V_{ℝ} [FK], [L1]. Lemma 5.13. For x ∈ Ω we have

Det_{V_{ℝ}} R_{x}^{1/2} = Δ(x)^{n_{V} /2r}.

Proof. Assuming x = ∑_{j} x_{j} e_{j} diagonal and putting v′ = R_{x}^{1/2} v for v ∈ V_{ℝ}, the respective Peirce components satisfy v′_{j} = x_{j}^{1/2} v_{j}. It follows that

Det_{V_{ℝ}} R_{x}^{1/2} = ( ∏_{1≤j≤r} x_{j}^{1/2})^{b} = (x_{1} ⋯ x_{r})^{b/2} = Δ(x)^{b/2} = Δ(x)^{n_{V} /2r}.

□

For v ∈ V_{ℝ} we have (e|{ev^{∗}v}) = ({ee^{∗}v}|v) = (v|v)/2 and therefore

∫_{V_{ℝ}} dv e^{−(e|{ev^{∗}v})} = ∫_{V_{ℝ}} dv e^{−(v|v)/2} = (2π)^{n_{V} /2}. (5.10)

Lemma 5.14. Let x ∈ Ω. Then

∫_{V_{ℝ}} dv e^{−(x|{ev^{∗}v})} = (2π)^{n_{V} /2} Δ(x)^{−n_{V} /2r}.

Proof. Putting v′ = R_{x}^{1/2} v for v ∈ V_{ℝ}, we have

dv = Δ(x)^{−n_{V} /2r} dv′

by Lemma 5.13. Moreover

P_{x}^{1/2} {ev^{∗}v} = {e(R_{x}^{1/2} v)^{∗} (R_{x}^{1/2} v)} = {ev^{∗}′v′}

and hence

(x|{ev^{∗}v}) = (P_{x}^{1/2} e|{ev^{∗}v}) = (e|P_{x}^{1/2} {ev^{∗}v}) = (e|{ev^{∗}′v′}).

Now the assertion follows from (5.10). □ Lemma 5.15. Let β > (r − 1) \frac{a}{2} + \frac{b}{2}. Then

Γ_{Ω}(β) ∫_{V_{ℝ}} dv Δ(e + \{ev^{∗}v\})^{−β} = (2π)^{n_{V} /2} Γ_{Ω} (β − \frac{n_{V}}{2r} ) .

Covariant Symbolic Calculi 25

Proof. Applying Lemma 5.6 and Lemma 5.14 we obtain

Γ_{Ω}(β) ∫_{V_{ℝ}} dv Δ(e + {ev^{∗}v})^{−β} =

∫_{V_{ℝ}} dv ∫_{Ω} dx Δ(x)^{β−n_{X}/r} e^{−(x|e+{ev^{∗}v})} =

∫_{Ω} dx Δ(x)^{β−n_{X}/r} e^{−(x|e)} ∫_{V_{ℝ}} dv e^{−(x|{ev^{∗}v})} =

(2π)^{n_{V} /2} ∫_{Ω} dx Δ(x)^{β−n_{X}/r−n_{V} /2r} e^{−(x|e)} =

(2π)^{n_{V} /2} Γ_{Ω} (β − \frac{n_{V}}{2r} ) .

□

Lemma 5.16. Let x ∈ Ω and β > (r − 1) \frac{a}{2} + \frac{b}{2}. Then

Γ_{Ω}(β) ∫_{V_{ℝ}} dv Δ(x + \{ev^{∗}v\})^{−β} = (2π)^{n_{V} /2} Γ_{Ω} (β − \frac{n_{V}}{2r} ) Δ(x)^{n_{V} /2r−β}.

Proof. Putting v′ = R_{x}^{−1/2} v for v ∈ V_{ℝ}, we have

dv = Δ(x)^{n_{V} /2r} dv′

by Lemma 5.13. Moreover

P_{x}^{−1/2} {ev^{∗}v} = {e(R_{x}^{−1/2} v)^{∗} (R_{x}^{−1/2} v)} = {ev^{∗}′v′}

and hence

Δ(x + {ev^{∗}v}) = Δ(P_{x}^{1/2} (e + P_{x}^{−1/2} {ev^{∗}v})) = Δ(x) Δ(e + {ev^{∗}′v′}).

Therefore

∫_{V_{ℝ}} dv Δ(x + {ev^{∗}v})^{−β} = Δ(x)^{n_{V} /2r−β} ∫_{V_{ℝ}} dv′ Δ(e + {ev^{∗}′v′})^{−β}

and Lemma 5.15 implies the assertion. □ Theorem 5.17. The link transform ττ^{∗} of the Toeplitz calculus with ”Wallach” parameter ν has the eigenvalues

τ̃τ^{∗}(λ) = \frac{Γ_{Ω} (\underline{λ} + \underline{ρ} + ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r}) Γ_{Ω} (−\underline{λ} + \underline{ρ} + ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r})}{Γ_{Ω} (ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r}) Γ_{Ω}(ν_{ℝ})}.

Here \underline{λ} = (λ_{1}, …, λ_{r}) is the spectral parameter in 𝔞^{#} and \underline{ρ} = (ρ_{1}, …, ρ_{r}) is the half-sum of positive restricted roots [L2], [Z] given by

2ρ_{j} = 1 + (j − 1)a + \frac{n_{Y} − n_{X} + n_{V} /2}{r}. (5.11)

Proof. Writing ζ = x + y + v ∈ X ⊕ Y ⊕ V_{ℝ} according to (2.4), the domain D_{ℝ} is defined by the condition t := x − {ev^{∗}v} ∈ Ω. (5.12)

Using (3.4) and applying Lemma 5.12, Lemma 5.16 and Lemma 5.7 we obtain in case Z_{ℝ} ≠ X

I_{α} := ∫ dx ∫ dv ∫ dy Δ_{U}(x − \{ev^{∗}v\})^{−p/2} Δ\underline{α}(x − \{ev^{∗}v\}) ·

·Δ_{U} (\frac{e + x − y}{2} )^{−ν} Δ_{U}(x − \{ev^{∗}v\})^{ν/2} =

2^{νr_{U}} ∫ dx ∫ dv ∫ dy Δ^{\underline{α}+ν_{ℝ}−p_{ℝ}}(x − \{ev^{∗}v\}) · Δ_{U}(e + x − y)^{−ν} =

2^{νr_{U}} ∫_{Ω} dt Δ^{\underline{α}+ν_{ℝ}−p_{ℝ}}(t) ∫_{V_{ℝ}} dv ∫_{Y} dy · Δ_{U}(e + t + \{ev^{∗}v\} − y)^{−ν} =

2^{n_{X}+n_{Y}} π^{(n_{X}+n_{Y} )/2} \frac{Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y}}{r} )}{Γ_{Ω}(ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )} ·

· ∫_{Ω} dt Δ^{\underline{α}+ν_{ℝ}−p_{ℝ}}(t) ∫_{V_{ℝ}} dv Δ(e + t + \{ev^{∗}v\})^{−2ν_{ℝ}+n_{Y} /r} =

2^{n_{X}+n_{Y}} π^{(n_{X}+n_{Y} )/2} \frac{(2π)^{n_{V} /2} Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r})}{Γ_{Ω}(ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )} ·

· ∫_{Ω} dt Δ^{\underline{α}+ν_{ℝ}−p_{ℝ}}(t) Δ(e + t)^{−2ν_{ℝ}+n_{Y} /r+n_{V} /2r} = 2^{n_{X}+n_{Y} +n_{V} /2} π^{n/2} ·

·\frac{Γ_{Ω} (\underline{α} + ν_{ℝ} − p_{ℝ} + \frac{n_{X}}{r} ) Γ_{Ω} (2ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r} − \underline{α} − ν_{ℝ} + p_{ℝ} + \underline{δ} − \frac{2n_{X}}{r} )}{Γ_{Ω}(ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )} =

2^{n_{X}+n_{Y} +n_{V} /2} π^{n/2} \frac{Γ_{Ω} (\underline{α} + ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r}) Γ_{Ω} (ν_{ℝ} − \underline{α} + \underline{δ} − \frac{n_{X}}{r} )}{Γ_{Ω}(ν_{ℝ}) Γ_{Ω} (ν_{ℝ} + \frac{n_{X}−n_{Y}}{2r} )}

since rp_{ℝ} = n_{X} + n_{Y} + n_{V} /2. In the remaining case Z_{ℝ} = X, Lemma 5.7 yields

I_{α} := ∫_{Ω} dx Δ(x)^{−n_{X}/r} Δ\underline{α}(x) Δ (\frac{e + x}{2})^{−ν} Δ(x)^{ν/2}

= 2^{rν} ∫_{Ω} dx Δ^{\underline{α}−n_{X}/r+ν/2}(x) Δ(e + x)^{−ν}

= 2^{rν} \frac{Γ_{Ω} (\frac{ν}{2} + \underline{α}) Γ_{Ω} (\frac{ν}{2} − \underline{α} + \underline{δ} − \frac{n_{X}}{r} )}{Γ_{Ω}(ν)}.

In general [Z, Lemma 2.3] we have

e_{λ}(x + y + v) = Δ^{\underline{λ}+\underline{ρ}}(x − \{ev^{∗}v\}).

Since τ̃τ^{∗}(−ρ) = (ττ^{∗}1)(e) = 1, it follows that

τ̃τ^{∗}(λ) = I_{λ+ρ}/I_{0} = \frac{Γ_{Ω} (\underline{λ} + \underline{ρ} + ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r}) Γ_{Ω} (ν_{ℝ} − \underline{λ} − \underline{ρ} + \underline{δ} − \frac{n_{X}}{r} )}{Γ_{Ω} (ν_{ℝ} − \frac{n_{Y} +n_{V} /2}{r}) Γ_{Ω} (ν_{ℝ} + \underline{δ} − \frac{n_{X}}{r} )}

in both cases (since ν_{ℝ} = ν/2 if Z_{ℝ} = X). Since

2\underline{ρ} = \underline{δ} + \frac{n_{Y} − n_{X} + n_{V} /2}{r}

by (5.7) and (5.11) and

Γ_{Ω} (ν_{ℝ} + \underline{δ} − \frac{n_{X}}{r} ) = Γ_{Ω} (ν_{ℝ})

by (5.8), the assertion follows. □

Remark 5.18. In case D_{ℝ} is complex, we have ν_{ℝ} = ν and d := dim_{ℂ} D_{ℝ} = n_{Y} + n_{V} /2. Therefore (5.10) simplifies to

τ̃τ^{∗}(λ) = \frac{Γ_{Ω}(\underline{ρ} + ν − \frac{d}{r} + \underline{λ}) Γ_{Ω}(\underline{ρ} + ν − \frac{d}{r} − \underline{λ})}{Γ_{Ω}(ν − \frac{d}{r}) Γ_{Ω}(ν)}

(cf. [AU1]).

Remark 5.19. While the Toeplitz-Berezin calculus is certainly fundamental, it is important to study other covariant symboli calculi such as the Wick calculus and the Weyl calculus and the relationship between them. In [AU1] a detailed investigation is carried out in the complex case, in particular for the Bargmann spaces over ℂ^{n}. In the curved setting the Weyl calculus, which involves the symmetries in a crucial way, poses many open problems, but in [AU2] the eigenvalues for the link transform of the Weyl calculus are determined for all (real and complex) symmetric domains of rank 1. The surprising new feature is the deep role played by hypergeometric functions in this context.

References

[AU1] J. Arazy, H. Upmeier, Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains, Proc. Lund 2000.

[AU2] J. Arazy, H. Upmeier, Weyl calculus on rank 1 symmetric domains, Preprint.

[BLU] D. Borthwick, A. Lesniewski, H. Upmeier, Non-perturbative deformation quantization of Cartan domains, J. Func. Anal 113 (1993), 153-176.

[DP] G. van Dijk, M. Pevzner, Berezin kernels and tube domains, Preprint.

[FK] J. Faraut, A. Korányi, Analysis on Symmetric Cones, Clarendon Press Oxford (1994).

[H] S. Helgason, Groups and Geometric Analysis, Academic Press (1984).

[L1] O. Loos, Jordan Pairs, Springer Lect. Notes 460 (1975).

[L2] O. Loos, Bounded Symmetric Domains and Jordan Pairs, Univ. of California, Irvine (1977).

[N] Y. Neretin, Matrix analogs of Beta-integral and Plancherel formula for Berezin kernel representations, Preprint.

[U] H. Upmeier, Symmetric Banach Manifolds and Jordan C^{∗}-Algebras, North Holland (1985).

[UU] A. Unterberger, H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-597.

[Z] G. Zhang, Berezin transform on real bounded symmetric domains, Preprint.

Department of Mathematics Haifa University Mount Carmel Haifa Israel e-mail address. jarazy@math.haifa.ac.il Fachbereich Mathematik University of Marburg 35032 Marburg Germany e-mail address. upmeier@mathematik.uni-marburg.de

1991 Mathematics Subject Classification. Primary 46L65, 47A60; Secondary 53C35, 17C20