Eigenvalues of K-invariant Toeplitz Operators on Bounded Symmetric Domains

Abstract

We determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains $D=G/K,$ for the irreducible K-types indexed by all partitions of length $r={\mathrm {rank}}(D)$.

1 Introduction

The well-known Toeplitz–Berezin calculus, acting on the Bergman space $H^2(D)$ of a bounded domain $D\subset {{\mathbf {C}}}^d,$ is covariant under the biholomorphic automorphism group G of D. Actually, Berezin [3] considered two kinds of symbolic calculus (contravariant and covariant symbols) which are related by the Berezin transform. For a bounded symmetric domain $D=G/K$ of rank r,  where G acts transitively on D and K is a maximal compact subgroup of G,  one has a more general covariant Toeplitz–Berezin calculus acting on the weighted Bergman spaces $H^2_\nu (D)$ over D. Here $\nu$ is a scalar parameter for the (scalar) holomorphic discrete series of G and its analytic continuation. Since G acts irreducibly on $H_\nu ^2(D),$ there are no non-trivial G-invariant operators in the $C^*$-algebra generated by Toeplitz operators. On the other hand, there exist interesting K-invariant Toeplitz type operators, which have been studied in relation to complex and harmonic analysis [2, 6]. These operators are uniquely determined by a sequence of eigenvalues indexed over all partitions of length r. In this paper, we determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators, both for the covariant and contravarient symbol. While the covariant symbol is treated as a direct generalization of [2], the contravariant symbol eigenvalue formula requires more effort. Here a crucial ingredient is the dimension formula for the irreducible K-types.

2 K-invariant Toeplitz Operators

In the following we use the Jordan theoretical description of bounded symmetric domains. For more details, see [1, 5, 10, 11, 16]. Each irreducible bounded symmetric domain D of rank r and dimension d can be realized as the (spectral) open unit ball of a hermitian Jordan triple $Z\approx {{\mathbf {C}}}^d.$ Let G be the identity component of the biholomorphic automorphism group of D,  and let K be the stabilizer subgroup at $0\in D.$ Then K is a compact linear group consisting of Jordan triple automorphisms of Z. The Shilov boundary S of D consists of all tripotents in Z of maximal rank r. Let (z|w) denote the unique K-invariant inner product on Z such that $(e|e)=r$ for any $e\in S.$ For any maximal tripotent $e\in Z$ the so-called Peirce 2-space $Z_2(e)$ [10] is a hermitian Jordan triple of tube type. Put

$$ d_e:=\dim Z_2(e). $$

Let $e_1,\ldots ,e_r$ be a frame of orthogonal minimal tripotents, and put $e=e_1+\ldots +e_r.$ The joint Peirce decomposition [10] gives rise to two numerical invariants a, b such that

$$\begin{aligned}&\tfrac{d_e}{r}=1+\tfrac{a}{2}(r-1), \\&\tfrac{d}{r}=1+\tfrac{a}{2}(r-1)+b=\tfrac{d_e}{r}+b. \end{aligned}$$

The tube type case is characterized by $d_e=d$ or, equivalently, $b=0.$ The triple (r, a, b) characterizes D up to isomorphism. As an important special case, the spin factor Z of dimension $d\geqslant 3$ has the invariants $r=2,\ a=d-2$ and $b=0.$ Here the normalized inner product on $Z\approx {{\mathbf {C}}}^d$ is $(z|w)=2z\cdot {\overline{w}},$ and the unit element $e=(1,0,\ldots ,0).$

Let ${\mathcal {P}}(Z)$ denote the algebra of all (holomorphic) polynomials on Z. The natural action

$$\begin{aligned} (k\cdot f)(z):=f(k^{-1}z) \end{aligned}$$

(2.1)

of K on functions f on Z (or D), induces a Peter–Weyl decomposition [12]

$$\begin{aligned} {\mathcal {P}}(Z)=\sum \limits _{{\varvec{m}}}{\mathcal {P}}^{{\varvec{m}}}(Z) \end{aligned}$$

(2.2)

ranging over all integer partitions ${{\varvec{m}}}=(m_1\geqslant \ldots \geqslant m_r\geqslant 0)$ of length r. Here ${\mathcal {P}}^{{\varvec{m}}}(Z)$ denotes the irreducible K-module of all polynomials on Z of type ${{\varvec{m}}}$ [14]. By irreducibility, any two K-invariant inner products are proportional on each submodule ${\mathcal {P}}^{{\varvec{m}}}(Z).$

Let $H^2(Z)$ denote the Fock space of all entire functions on Z,  with Fischer-Fock inner product

$$\begin{aligned} (\phi |\psi )=\int \limits _Z{\frac{dz}{\pi ^d}}\ e^{-(z|z)}\ {\overline{\phi (z)}}\psi (z) \end{aligned}$$

(2.3)

and reproducing kernel

$$ {\mathcal {K}}(z,w)=e^{-(z|w)}. $$

Note that for function spaces we use inner products which are conjugate-linear in the first variable.

Consider the classical Pochhammer symbol

$$ (\lambda )_k:={\frac{\Gamma (\lambda +k)}{\Gamma (\lambda )}} $$

which equals $\lambda (\lambda +1)\cdots (\lambda +k-1)$ if $k\in {{\mathbf {N}}}.$ For any scalar parameter $\nu$ and r-tuple $\mu =(\mu _1,\ldots ,\mu _r)$ we define the multi-variable Pochhammer symbol

$$\begin{aligned} (\nu )_\mu :=\prod \limits _{j=1}^r(\nu -\tfrac{a}{2}(j-1))_{\mu _j}. \end{aligned}$$

(2.4)

The numerical invariant $p:={\frac{1}{r}}(d_e+d)=2+a(r-1)+b$ is called the genus of D. For a scalar parameter $\nu >p-1$ the weighted Bergman space $H_\nu ^2(D)$ consists of all holomorphic functions $\phi$ on D which are square-integrable for the inner product

$$\begin{aligned} (\phi |\psi )_\nu =c_\nu \int \limits _D{\frac{dz}{\pi ^d}}\ \Delta (z,z)^{\nu -p}\ {\overline{\phi (z)}}\psi (z). \end{aligned}$$

(2.5)

Here $\Delta (z,w)$ denotes the Jordan triple determinant, which is a sesqui-polynomial uniquely determined by the property

$$\begin{aligned} \Delta (z,z)=\prod \limits _{i=1}^r(1-t_i^2) \end{aligned}$$

for all $z=\sum \limits _{i=1}^r t_ie_i\in Z,$ where $e_1,\ldots ,e_r$ is any frame of minimal orthogonal tripotents and $t_1,\ldots ,t_r$ are the (non-negative) “singular values” of z. For a Lie theoretic definition see [5, p. 262] (for tube type domains) and [4, (3.4)-(3.6)] (general case). The normalizing constant $c_\nu$, giving rise to a probability measure, is

$$ c_\nu =(\nu -\tfrac{d}{r})_{(d/r,\ldots ,d/r)}=\prod \limits _{j=1}^r{\frac{\Gamma (\nu -\tfrac{a}{2}(j-1))}{\Gamma (\nu -\tfrac{d}{r}-\tfrac{a}{2}(j-1))}}. $$

The reproducing kernel is

$$\begin{aligned} {\mathcal {K}}^\nu (z,w)=\Delta (z,w)^{-\nu }. \end{aligned}$$

(2.6)

For the continuous part of the so-called Wallach set, explicitly given by the condition

$$ \nu >\tfrac{a}{2}(r-1) $$

[4, 8, 9], the kernel (2.6) is strictly positive on $D\times D,$ and the associated reproducing kernel Hilbert space, still denoted by $H_\nu ^2(D),$ contains ${\mathcal {P}}(Z)$ as a dense subspace. As a special case, the parameter $\nu =\tfrac{d}{r}=1+b+\tfrac{a}{2}(r-1)$ corresponds to the Hardy space

$$\begin{aligned} H_{d/r}^2(D)=H^2(S) \end{aligned}$$

on the Shilov boundary S of D.

Under the Fischer-Fock inner product (2.3) each finite-dimensional Hilbert space ${\mathcal {P}}^{{\varvec{m}}}(Z)$ has a reproducing kernel

$$ E^{{\varvec{m}}}(z,w)=\sum \limits _\alpha \psi _\alpha ^{{\varvec{m}}}(z)\ {\overline{\psi _\alpha ^{{\varvec{m}}}(w)}} $$

for any orthonormal basis $\psi _\alpha ^{{\varvec{m}}}\in {\mathcal {P}}^{{\varvec{m}}}(Z).$ The Hilbert spaces $H^2(Z)$ and $H_\nu ^2(D)$ (in the continuous Wallach set) are invariant under the action (2.1) of K. The Faraut–Korányi binomial formula [4]

$$\begin{aligned} {\mathcal {K}}^\nu (z,w)=\Delta (z,w)^{-\nu }=\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ E^{{\varvec{m}}}(z,w) \end{aligned}$$

(2.7)

implies that the inner product (2.5) on $H^2_\nu (D)$ and the Fischer-Fock inner product (2.3) are related by

$$\begin{aligned} (p|q)_\nu ={\frac{(p|q)}{(\nu )_{{\varvec{m}}}}} \end{aligned}$$

(2.8)

for each partition ${{\varvec{m}}}$ and all $p,q\in {\mathcal {P}}^{{\varvec{m}}}(Z),$ using the multi-variable Pochhammer symbol (2.4).

We now introduce Toeplitz operators. Let F(z, w) be a sequi-holomorphic symbol function, written as a sum (or series)

$$\begin{aligned} F(z,w)=\sum \limits _i\phi _i(z)\ {\overline{\psi _i(w)}} \end{aligned}$$

for holomorphic functions $\phi _i,\psi _i.$ We are mainly concerned with sesqui-polynomials, where $\phi _i,\psi _i\in {\mathcal {P}}(Z)$ and the sum is finite. We write $F_w(z):=F(z,w)$ for fixed w.

Let $M_\phi$ (resp., $M_\phi ^\nu$) denote the multiplication operator by a polynomial $\phi$ acting on ${\mathcal {P}}(Z)$ or $H_\nu ^2(D),$ respectively. The $\nu$-th Toeplitz operator $T^\nu _F$ on $H_\nu ^2(D)$ with symbol function $F_|$ (where $F_|$ denotes the restriction of F to the diagonal) has the form

$$ T^\nu _F=\sum \limits _i M_{\psi _i}^{\nu *}\ M_{\phi _i}^\nu . $$

We define similarly

$$ T_F=\sum \limits _i M_{\psi _i}^*\ M_{\phi _i} $$

acting on ${\mathcal {P}}(Z),$ or as a densely defined unbounded operator on the Fock space $H^2(Z).$ Here $M_\psi ^*$ is the constant coefficient differential operator associated with a polynomial $\psi$ via the normalized inner product. Thus

$$ M^*_\psi \ e^{(z|w)}={\overline{\psi (w)}}\ e^{(z|w)} $$

for all $z,w\in Z.$ For $p,q,\psi \in {\mathcal {P}}(Z)$ we have

$$ (p|q)=M^*_p q(0) $$

and

$$ (p|M^*_\psi q)=(M_\psi p|q)=(\psi \cdot p|q). $$

Note that $M^*_\psi$ depends in a conjugate-linear way on $\psi .$

The Toeplitz calculus is sometimes called the “anti-Wick” calculus. On the other hand, the “Wick” functional calculus (normal ordering, where annihilation operators are moved to the right) yields the operator

$$ F_{T^\nu }:=\sum \limits _i M_{\phi _i}^\nu \ M_{\psi _i}^{\nu *} $$

on $H_\nu ^2(D),$ and, similarly,

$$ F_T:=\sum \limits _i M_{\phi _i}\ M_{\psi _i}^*, $$

acting on ${\mathcal {P}}(Z)$ or as a densely defined unbounded operator on $H^2(Z).$

If F is K-invariant in the sense that

$$ F(kz,kw)=F(z,w) $$

for all $k\in K,$ then the operators $T_F,\ T_F^\nu$ and $F_T,\ F_{T^\nu }$ commute with the K-action (2.1). Since the decomposition (2.2) is multiplicity free, it follows that K-invariant operators form a commutative algebra, and every such operator T is a block-diagonal operator uniquely determined by its sequence of eigenvalues $T({{\varvec{m}}})$ defined by

$$ Tp=T({{\varvec{m}}})\ p $$

for all $p\in {\mathcal {P}}^{{\varvec{m}}}(Z).$ For $T^\nu _F,$ with F K-invariant, we obtain

$$ T^\nu _F\ p=:T^\nu _F({{\varvec{m}}})\ p $$

for all $p\in {\mathcal {P}}^{{\varvec{m}}}(Z),$ with eigenvalues given by

$$\begin{aligned} T^\nu _F({{\varvec{m}}})\ (p|p)_\nu= & {} (p|T^\nu _F({{\varvec{m}}})p)_\nu =(p|T^\nu _F p)_\nu =(p|F_|\ p)_\nu \\= & {} c_\nu \int \limits _D{\frac{dz}{\pi ^d}}\ {\overline{p(z)}}\ \Delta (z,z)^{\nu -p}\ F(z,z)\ p(z). \end{aligned}$$

The Jordan triple determinant $\Delta (z,w)=\Delta _w(z)$ is K-invariant, and for its powers we obtain

Lemma 1

$$ T^\nu _{\Delta ^\beta }({{\varvec{m}}})={\frac{c_\nu }{c_{\nu +\beta }}}{\frac{(\nu )_{{\varvec{m}}}}{(\nu +\beta )_{{\varvec{m}}}}} $$

Proof

Let $p\in {\mathcal {P}}^{{\varvec{m}}}(Z).$ Then (2.8) implies

$$\begin{aligned} T^\nu _{\Delta ^\beta }({{\varvec{m}}})\ (p|p)_\nu= & {} c_\nu \int \limits _D{\frac{dz}{\pi ^d}}\ {\overline{p(z)}}\ \Delta (z,z)^{\nu -p}\ \Delta (z,z)^\beta \ p(z) \\= & {} {\frac{c_\nu }{c_{\nu +\beta }}}c_{\nu +\beta }\int \limits _D{\frac{dz}{\pi ^d}}\ {\overline{p(z)}}\ \Delta (z,z)^{\nu +\beta -p}\ p(z) \\= & {} {\frac{c_\nu }{c_{\nu +\beta }}}(p|p)_{\nu +\beta }={\frac{c_\nu }{c_{\nu +\beta }}}{\frac{(\nu )_{{\varvec{m}}}}{(\nu +\beta )_{{\varvec{m}}}}}(p|p)_\nu \end{aligned}$$

$\square$

Proposition 2

For a K-invariant sesqui-holomorphic function F, let $F^\vDash$ denote the $\nu$-th Berezin transform (defined via the Berezin symbol of operators on $H_\nu ^2(D)$). Then

$$ T^\nu _F({{\varvec{m}}})=F^\Vdash _{T^\nu }({{\varvec{m}}}). $$

Proof

Using Einstein summation convention, the identity

$$\begin{aligned} F(z,w)\ {\mathcal {K}}^\nu (z,w)=\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ F_{T^\nu }({{\varvec{m}}})\ E^{{\varvec{m}}}(z,w) \end{aligned}$$

(2.9)

follows from the computation

$$\begin{aligned} F(z,w){\mathcal {K}}^\nu (z,w)= & {} \phi _i(z){\overline{\psi _i(w)}}({\mathcal {K}}_z^\nu |{\mathcal {K}}_w^\nu )_\nu =({\overline{\phi _i(z)}}{\mathcal {K}}_z^\nu |{\overline{\psi _i(w)}}{\mathcal {K}}_w^\nu )_\nu \\= & {} (M_{\phi _i}^{\nu *}{\mathcal {K}}_z^\nu |M_{\psi _i}^{\nu *}{\mathcal {K}}_w^\nu )_\nu =({\mathcal {K}}_z^\nu |M_{\phi _i}^\nu M_{\psi _i}^{\nu *}{\mathcal {K}}_w^\nu )_\nu =({\mathcal {K}}_z^\nu |F_{T^\nu }{\mathcal {K}}_w^\nu )_\nu \\= & {} \sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}({\mathcal {K}}_z^\nu |F_{T^\nu }E_w^{{\varvec{m}}})_\nu =\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}({\mathcal {K}}_z^\nu |F_{T^\nu }({{\varvec{m}}}) E_w^{{\varvec{m}}})_\nu \\= & {} \sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ F_{T^\nu }({{\varvec{m}}})\ ({\mathcal {K}}_z^\nu |E_w^{{\varvec{m}}})_\nu =\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ F_{T^\nu }({{\varvec{m}}})\ E_w^{{\varvec{m}}}(z) \\= & {} \sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ F_{T^\nu }({{\varvec{m}}})\ E^{{\varvec{m}}}(z,w). \end{aligned}$$

Similarly, the identity

$$\begin{aligned} F^\vDash (z,w)\ {\mathcal {K}}^\nu (z,w)=\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ T^\nu _F({{\varvec{m}}})\ E^{{\varvec{m}}}(z,w) \end{aligned}$$

(2.10)

follows from the computation

$$\begin{aligned} F^\vDash (z,w)\ {\mathcal {K}}^\nu (z,w)= & {} ({\mathcal {K}}_z^\nu |T^\nu _F{\mathcal {K}}_w^\nu )_\nu =\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}({\mathcal {K}}_z^\nu |T^\nu _F E_w^{{\varvec{m}}})_\nu \\= & {} \sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}({\mathcal {K}}_z^\nu |T^\nu _F({{\varvec{m}}})E_w^{{\varvec{m}}})_\nu =\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ T^\nu _F({{\varvec{m}}})\ ({\mathcal {K}}_z^\nu |E_w^{{\varvec{m}}})_\nu \\= & {} \sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ T^\nu _F({{\varvec{m}}})\ E_w^{{\varvec{m}}}(z)=\sum \limits _{{\varvec{m}}}(\nu )_{{\varvec{m}}}\ T^\nu _F({{\varvec{m}}})\ E^{{\varvec{m}}}(z,w). \end{aligned}$$

Comparing coefficients in (2.9) and (2.10), the assertion follows. $\square$

In general, the Berezin transform of a K-invariant function is difficult to compute. For powers of $\Delta$ we obtain with Lemma 1

Corollary 3

$$ (\Delta ^\beta )^\vDash (z,w)\ {\mathcal {K}}^\nu (z,w)={\frac{c_\nu }{c_{\nu +\beta }}}\sum \limits _{{\varvec{m}}}{\frac{(\nu )_{{\varvec{m}}}^2}{(\nu +\beta )_{{\varvec{m}}}}}\ E^{{\varvec{m}}}(z,w) $$

3 The First Eigenvalue Formula

The Jordan triple determinant has a decomposition

$$ \Delta (z,w)=\sum \limits _{\ell =0}^r(-1)^\ell \ \Delta ^{(\ell )}(z,w) $$

into sesqui-polynomials $\Delta ^{(\ell )}$ which are homogeneous of bi-degree $(\ell ,\ell ).$ For $\ell =1$ we obtain the normalized K-invariant inner product

$$ \Delta ^{(1)}(z,w)=(z|w). $$

If Z is of tube type with unit element e, then

$$\begin{aligned} \Delta ^{(r)}(z,w)=N(z)\ {\overline{N(w)}}, \end{aligned}$$

(3.1)

where N is the Jordan algebra determinant normalized by $N(e)=1.$ The first eigenvalue formula gives the eigenvalues of the K-invariant operators $\Delta ^{(\ell )}_{T^\nu }$ and $\Delta ^{(\ell )}_T$ for $0\leqslant \ell \leqslant r.$ This generalizes the approach in [2] for $\ell =1.$

Consider the fundamental partitions

$$\begin{aligned} (\ell ):=(1,\ldots ,1,0,\ldots ,0) \end{aligned}$$

(3.2)

with $\ell$ ones. Let $\psi _\alpha ^{(\ell )}$ be an orthonormal basis of ${\mathcal {P}}^{(\ell )}(Z)$ and consider the Fischer-Fock reproducing kernel

$$\begin{aligned} E^{(\ell )}(z,w)=\sum \limits _\alpha \psi _\alpha ^{(\ell )}(z)\ {\overline{\psi _\alpha ^{(\ell )}(w)}} \end{aligned}$$

(3.3)

of ${\mathcal {P}}^{(\ell )}(Z).$ Then ${\mathcal {P}}^{(0)}(Z)={{\mathbf {C}}}$ consists of constant functions and $\Delta ^{(0)}=E^{(0)}=1$. In the first interesting case $\ell =1$ we obtain the dual space

$$ {\mathcal {P}}^{(1)}(Z)=Z^* $$

of all linear forms on Z,  and

$$ \Delta ^{(1)}(z,w)=E^{(1)}(z,w)=(z|w)=\sum \limits _i(z|u_i)(u_i|w) $$

for any orthonormal basis $u_i\in Z.$

Lemma 4

$$ \Delta ^{(\ell )}(z,w)=E^{(\ell )}(z,w)\ \prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1)). $$

Proof

The Faraut–Korányi formula (2.7) applied to the parameter $-1$ yields

$$ \Delta ^{(\ell )}(z,w)=(-1)^\ell \ (-1)_{(\ell )}\ E^{(\ell )}(z,w) $$

with (positive) constant given by

$$\begin{aligned} (-1)^\ell \ (-1)_{(\ell )}= & {} (-1)^\ell \prod \limits _{j=1}^r(-1-\tfrac{a}{2}(j-1))_{(\ell )_j}\\= & {} (-1)^\ell \prod \limits _{j=1}^\ell (-1-\tfrac{a}{2}(j-1))=\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1)). \end{aligned}$$

$\square$

Lemma 4 implies that

$$ \Delta ^{(\ell )}_{T^\nu }=\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1))\ E^{(\ell )}_{T^\nu } =\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1))\sum \limits _\alpha M^\nu _{\psi _\alpha ^{(\ell )}}M^{\nu *}_{\psi _\alpha ^{(\ell )}} $$

and

$$ \Delta ^{(\ell )}_T=\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1))\ E^{(\ell )}_T=\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1))\sum \limits _\alpha M_{\psi _\alpha ^{(\ell )}}M^*_{\psi _\alpha ^{(\ell )}}. $$

In particular, for $\ell =1,$

$$ \Delta ^{(1)}_{T^\nu }=E^{(1)}_{T^\nu }=\sum \limits _i M_{(z|u_i)}^\nu M_{(z|u_i)}^{\nu *} $$

and

$$ \Delta ^{(1)}_T=E^{(1)}_T=\sum \limits _i M_{(z|u_i)} M_{(z|u_i)}^*. $$

For any partition ${{\varvec{m}}}$ and $1\leqslant k\leqslant r$ we put

$$\begin{aligned} m_k':=m_k-\tfrac{a}{2}(k-1). \end{aligned}$$

(3.4)

This notation (unrelated to the ’dual’ partition sometimes also denoted by ${{\varvec{m}}}'$) will be used throughout the paper. For any $\ell$-element subset $L\subset \{1,\ldots ,r\},$ with characteristic function

$$ \chi _i^L:={\left\{ \begin{array}{ll}1&{}i\in L\\ 0&{}i\notin L\end{array}\right. }, $$

we define

(3.5)

whenever ${{\varvec{m}}}+\chi ^L$ is also a partition, and put $\alpha _{{\varvec{m}}}^L:=0$ otherwise. Similarly, we define

(3.6)

whenever ${{\varvec{m}}}-\chi ^L$ is also a partition, and put $\beta _{{\varvec{m}}}^L:=0$ otherwise.

Proposition 5

Assume that Z is of tube type, with unit element e. Then the spherical polynomial $\Phi ^{{\varvec{m}}}$ of type ${{\varvec{m}}}$ satisfies

$$\begin{aligned} \Delta _e^{(\ell )}\ \Phi ^{{\varvec{m}}}=\sum \limits _{|L|=\ell }\Phi ^{{{\varvec{m}}}+\chi ^L}\ \alpha _{{\varvec{m}}}^L \end{aligned}$$

(3.7)

for $0\leqslant \ell \leqslant r,$ with summation over all $\ell$-element subsets $L\subset \{1,\ldots ,r\}$ such that ${{\varvec{m}}}+\chi ^L$ is a partition.

Proof

Choose a frame $e_1,\ldots ,e_r$ with $e=e_1+\ldots +e_r.$ For $t=(t_1,\ldots ,t_r)\in {{\mathbf {C}}}^r$ we put $t\cdot e:=t_1e_1+\ldots +t_re_r.$ Then

$$ \Delta _e(t\cdot e)=\Delta (t\cdot e,e)=\prod \limits _{i=1}^r(1-t_i). $$

It follows that

$$\begin{aligned} \Delta _e^{(\ell )}(t\cdot e)=\sigma _\ell (t) \end{aligned}$$

(3.8)

is the $\ell$-th elementary symmetric polynomial. Now consider the ’Selberg-Jack’ symmetric functions $s_{{\varvec{m}}}^{a/2},$ for any partition ${{\varvec{m}}}$ and parameter $k=a/2,$ as defined in [7] (cf. also [13]). Putting

$$ f_r^{a/2}({{\varvec{m}}}):=\prod \limits _{i<j}(m_i'-m_j')_{a/2} $$

as in [7, (1.2)], it is shown in [7, identity (S) on p. 69] that

$$ s_{{\varvec{m}}}^{a/2}(1^r)={\frac{f_r^{a/2}({{\varvec{m}}})}{f_r^{a/2}((0))}}. $$

Therefore

$$\begin{aligned} \Phi ^{{\varvec{m}}}(t\cdot e)={\frac{1}{s_{{\varvec{m}}}^{a/2}(1^r)}}\ s_{{\varvec{m}}}^{a/2}(t)={\frac{f_r^{a/2}((0))}{f_r^{a/2}({{\varvec{m}}})}}\ s_{{\varvec{m}}}^{a/2}(t). \end{aligned}$$

(3.9)

By [7, identity (U) on p. 44 and (5.2)] we have a Pieri formula

$$ \sigma _\ell (t)\ s_{{\varvec{m}}}^{a/2}(t)=\sum \limits _{|L|=\ell }U^{a/2}({{\varvec{m}}}+\chi ^L/{{\varvec{m}}})\ s_{{{\varvec{m}}}+\chi ^L}^{a/2}(t) $$

where, according to [7, (3.15)], the coefficients are given by

$$ U^{a/2}({{\varvec{m}}}+\chi ^L/{{\varvec{m}}})={\frac{f_r^{a/2}({{\varvec{m}}})}{f_r^{a/2}({{\varvec{m}}}+\chi ^L)}}\ \prod \limits _{i<j}\Big (1+\tfrac{a}{2}\ {\frac{\chi _i^L-\chi _j^L}{m_i'-m_j'}}\Big ). $$

In view of (3.9) this implies

$$ \Delta ^{(\ell )}(t\cdot e)\ \Phi ^{{\varvec{m}}}(t\cdot e)=\sum \limits _{|L|=\ell }\Phi ^{{{\varvec{m}}}+\chi ^L}(t\cdot e)\prod \limits _{i<j}\Big (1+\tfrac{a}{2}\ {\frac{\chi _i^L-\chi _j^L}{m_i'-m_j'}}\Big ). $$

For any $i\ne j$ we have $\chi ^L_i-\chi ^L_j=0$ whenever both i, j belong to L or belong to its complement. On the other hand

$$ \chi ^L_i-\chi ^L_j={\left\{ \begin{array}{ll}1&{}i\in L,\ j\notin L\\ -1&{}i\notin L,\ j\in L\end{array}\right. }. $$

It follows that

(3.10)

Thus (3.7) holds for all $t\cdot e\in Z.$ Since the spherical polynomials are uniquely determined by their values on the “diagonal” $t\cdot e$, the assertion follows. $\square$

For $\ell =1$ we use singletons $L=\{k\}$ and obtain

$$ (z|e)\ \Phi ^{{\varvec{m}}}(z)=\sum \limits _{k=1}^r\Phi ^{{{\varvec{m}}}+\chi ^{\{k\}}}(z)\ \alpha _{{\varvec{m}}}^{\{k\}} =\sum \limits _{k=1}^r\Phi ^{{{\varvec{m}}}+\chi ^{\{k\}}}(z)\ \prod \limits _{h\ne k}{\frac{m_k'-m_h'+\tfrac{a}{2}}{m_k'-m_h'}} $$

in accordance with [2, Lemma 4.2].

Example 6

For the spin factor of rank $r=2$ and $m\in {{\mathbf {N}}},\ m\geqslant 1,$ we obtain

$$ (z|e)\ \Phi ^{(m,0)}(z)=\Phi ^{(m+1,0)}(z)\ {\frac{m+a}{m+\tfrac{a}{2}}}+\Phi ^{(m,1)}(z)\ {\frac{m}{m+\tfrac{a}{2}}}. $$

Remark 7

Evaluating (3.7) at e yields the non-obvious identity

$$\begin{aligned} \left( {\begin{array}{c}r\\ \ell \end{array}}\right) =\sum \limits _{|L|=\ell }\alpha _{{\varvec{m}}}^L \end{aligned}$$

(3.11)

valid for any partition ${{\varvec{m}}}$ and any parameter a. This is explicitly stated (for $k={\frac{a}{2}}$) in [7, (6.23)] in the form

$$ \left( {\begin{array}{c}r\\ \ell \end{array}}\right) \prod \limits _{1\leqslant i<j\leqslant r}(m_i-m_j+(j-i)\tfrac{a}{2})=\sum \limits _{|L|=\ell }\prod \limits _{1\leqslant i<j\leqslant r}(m_i+\tfrac{a}{2}\chi _i^L-m_j-\tfrac{a}{2}\chi _j^L +(j-i)\tfrac{a}{2}). $$

With (3.4) this yields

$$ \left( {\begin{array}{c}r\\ \ell \end{array}}\right) =\sum \limits _{|L|=\ell }\prod \limits _{1\leqslant i<j\leqslant r}{\frac{m_i'+{\frac{a}{2}}\chi _i^L-m_j'-{\frac{a}{2}}\chi _j^L}{m_i'-m_j'}} $$

which, by (3.10), implies (3.11).

For any parameter $\nu$ define

$$\begin{aligned} (\nu )_{{\varvec{m}}}^L:=\prod \limits _{k\in L}(\nu +m_k'). \end{aligned}$$

(3.12)

Lemma 8

$$ {\frac{(\nu )_{{{\varvec{m}}}+\chi ^L}}{(\nu )_{{\varvec{m}}}}}=(\nu )_{{\varvec{m}}}^L. $$

In particular, for a singleton $L=\{k\}$

$$ {\frac{(\nu )_{{{\varvec{m}}}+\chi ^{\{k\}}}}{(\nu )_{{\varvec{m}}}}}=(\nu )_{{\varvec{m}}}^{\{k\}}=\nu +m_k' $$

Proof

$$\begin{aligned} {\frac{(\nu )_{{{\varvec{m}}}+\chi ^L}}{(\nu )_{{\varvec{m}}}}}= & {} \prod \limits _{k=1}^r{\frac{(\nu -\tfrac{a}{2}(k-1))_{m_k+\chi _k^L}}{(\nu -\tfrac{a}{2}(k-1))_{m_k}}}\\= & {} \prod \limits _{k\in L}{\frac{(\nu -\tfrac{a}{2}(k-1))_{m_k+1}}{(\nu -\tfrac{a}{2}(k-1))_{m_k}}} \\= & {} \prod \limits _{k\in L}(\nu -\tfrac{a}{2}(k-1)+m_k)=\prod \limits _{k\in L}(\nu +m_k') \end{aligned}$$

$\square$

The dimension $d_{{\varvec{m}}}:=\dim {\mathcal {P}}^{{\varvec{m}}}(Z)$ has been computed in [15, Lemma 2.6] (for tube domains) and [15, Lemma 2.7] (general case). It satisfies

$$\begin{aligned} C\ {\frac{(d_e/r)_{{\varvec{m}}}}{(d/r)_{{\varvec{m}}}}}\ d_{{\varvec{m}}}= & {} \prod \limits _{i<j}(m_i'-m_j')\ (m_i'-m_j'+1-\tfrac{a}{2})_{a-1} \nonumber \\= & {} \prod \limits _{i<j}(m_i'-m_j')\ {\frac{\Gamma (m_i'-m_j'+\tfrac{a}{2})}{\Gamma (m_i'-m_j'+1-\tfrac{a}{2})}} \nonumber \\= & {} \prod \limits _{i<j}{\frac{m_i'-m_j'}{m_i'-m_j'-\tfrac{a}{2}}}\ {\frac{\Gamma (m_i'-m_j'+\tfrac{a}{2})}{\Gamma (m_i'-m_j'-\tfrac{a}{2})}}. \end{aligned}$$

(3.13)

Here the constant

$$\begin{aligned} C= & {} \prod \limits _{i<j}(0_i'-0_j')\ (0_i'-0_j'+1-\tfrac{a}{2})_{a-1} \nonumber \\= & {} \prod \limits _{i<j}(\tfrac{a}{2}(j-i))\ (1+\tfrac{a}{2}(j-i-1))_{a-1} \end{aligned}$$

(3.14)

is determined by the condition $d_{(0)}=1,$ corresponding to ${\mathcal {P}}^{(0)}(Z)={{\mathbf {C}}}.$

Lemma 9

For the partitions $(\ell )$ the dimension is given by

$$ d_{(\ell )}=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) {\frac{(d/r)_{(\ell )}}{\prod \limits _{j=1}^\ell (1+\tfrac{a}{2}(j-1))}}=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) \prod \limits _{j=1}^\ell {\frac{1+b+\tfrac{a}{2}(r-j)}{1+\tfrac{a}{2}(j-1)}} $$

Proof

This follows, with (3.13) and (3.14), from the computation

$$\begin{aligned}&\prod \limits _{i<j}{\frac{(\ell )_i'-(\ell )_j'}{\tfrac{a}{2}(j-i)}}\ {\frac{((\ell )_i'-(\ell )_j'+1-\tfrac{a}{2})_{a-1}}{(1+\tfrac{a}{2}(j-i-1))_{a-1}}} \\&\quad =\prod \limits _{i<j}{\frac{(\ell )_i-(\ell )_j+\tfrac{a}{2}(j-i)}{\tfrac{a}{2}(j-i)}}\ {\frac{((\ell )_i-(\ell )_j+1+\tfrac{a}{2}(j-i-1))_{a-1}}{(1+\tfrac{a}{2}(j-i-1))_{a-1}}} \\&\quad =\prod \limits _{i\leqslant \ell<j}{\frac{1+\tfrac{a}{2}(j-i)}{\tfrac{a}{2}(j-i)}}\ {\frac{(2+\tfrac{a}{2}(j-i-1))_{a-1}}{(1+\tfrac{a}{2}(j-i-1))_{a-1}}} \\&\quad =\prod \limits _{i\leqslant \ell <j}{\frac{1+\tfrac{a}{2}(j-i)}{\tfrac{a}{2}(j-i)}}\ {\frac{\tfrac{a}{2}(j-i+1)}{1+\tfrac{a}{2}(j-i-1)}} \\&\quad =\prod \limits _{i=1}^\ell \prod \limits _{j=\ell +1}^r{\frac{1+\tfrac{a}{2}(j-i)}{1+\tfrac{a}{2}(j-i-1)}}\ {\frac{\tfrac{a}{2}(j-i+1)}{\tfrac{a}{2}(j-i)}} \\&\quad =\prod \limits _{i=1}^\ell {\frac{1+\tfrac{a}{2}(r-i)}{1+\tfrac{a}{2}(\ell -i)}}\ {\frac{\tfrac{a}{2}(r+1-i)}{\tfrac{a}{2}(\ell +1-i)}}\\&\quad ={\frac{(d_e/r)_{(\ell )}}{\prod \limits _{k=1}^\ell (1+\tfrac{a}{2}(k-1))}}\ \left( {\begin{array}{c}r\\ \ell \end{array}}\right) \end{aligned}$$

using

$$ \prod \limits _{i=1}^\ell {\frac{\tfrac{a}{2}(r+1-i)}{\tfrac{a}{2}(\ell +1-i)}}=\prod \limits _{i=1}^\ell {\frac{r+1-i}{\ell +1-i}}={\frac{r(r-1)\cdots (r-\ell )}{\ell !}}=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) $$

and

$$ \prod \limits _{i=1}^\ell (1+\tfrac{a}{2}(r-i))=\prod \limits _{i=1}^\ell (1+\tfrac{a}{2}(r-1)-\tfrac{a}{2}(i-1))=(d_e/r)_{(\ell )}. $$

$\square$

For $\ell =1$ we obtain

$$ d_{(1,0\ldots ,0)}=d_{(1)}=\left( {\begin{array}{c}r\\ 1\end{array}}\right) \ (d/r)_{(1)}=r\ \tfrac{d}{r}=d $$

for ${\mathcal {P}}^{(1)}(Z)=Z^*.$ Here the above computation (say, for tube type domains) simplifies to

$$\begin{aligned} d_{(1,0\ldots ,0)}= & {} \prod \limits _{j=2}^r{\frac{1+\tfrac{a}{2}(j-1)}{\tfrac{a}{2}(j-1)}}\cdot {\frac{\tfrac{a}{2} j}{1+\tfrac{a}{2}(j-2)}} =\prod \limits _{j=2}^r{\frac{1+\tfrac{a}{2}(j-1)}{1+\tfrac{a}{2}(j-2)}}\cdot {\frac{\tfrac{a}{2} j}{\tfrac{a}{2}(j-1)}}\\= & {} {\frac{1+\tfrac{a}{2}(r-1)}{1}}\cdot {\frac{\tfrac{a}{2} r}{\tfrac{a}{2}}}=(1+\tfrac{a}{2}(r-1))\cdot r=d. \end{aligned}$$

Example 10

For the spin factor Z and $m\in {{\mathbf {N}}},$ ${\mathcal {P}}^{(m,0)}(Z)$ is the space of all m-homogeneous harmonic polynomials in d variables. Since $a=d-2$ and $b=0$ in this case, we obtain the well-known dimension formula

$$\begin{aligned} d_{(m,0)}= & {} {\frac{m+\tfrac{a}{2}}{\tfrac{a}{2}}}\cdot {\frac{\Gamma (m+a)}{\Gamma (m+1)}}\cdot {\frac{\Gamma (1)}{\Gamma (a)}}\\= & {} {\frac{2m+a}{a}}\cdot {\frac{(m+a-1)!}{m!(a-1)!}}={\frac{(2m+d-2)(m+d-3)!}{m!(d-2)!}} \end{aligned}$$

Proposition 11

Suppose ${{\varvec{m}}}$ and ${{\varvec{m}}}+\chi ^L$ are partitions. Then

$$ {\frac{d_{{\varvec{m}}}}{d_{{{\varvec{m}}}+\chi ^L}}}={\frac{(d_e/r)_{{\varvec{m}}}^L}{(d/r)_{{\varvec{m}}}^L}}\ {\frac{\beta _{{{\varvec{m}}}+\chi ^L}^L}{\alpha _{{\varvec{m}}}^L}}. $$

Proof

For any $i\ne j$ we have $({{\varvec{m}}}+\chi ^L)_i-({{\varvec{m}}}+\chi ^L)_j=m_i-m_j$ whenever both i, j belong to L or belong to its complement. On the other hand

$$ ({{\varvec{m}}}+\chi ^L)_i-({{\varvec{m}}}+\chi ^L)_j={\left\{ \begin{array}{ll}m_i+1-m_j&{}i\in L,\ j\notin L\\ m_i-m_j-1&{}i\notin L,\ j\in L\end{array}\right. }. $$

By (3.13) we have

$\square$

Proposition 12

Let $0\leqslant \ell \leqslant r$ and $w\in Z.$ Then

$$\begin{aligned} \Delta _w^{(\ell )}\ E_w^{{\varvec{m}}}=\sum \limits _{|L|=\ell }(d_e/r)_{{\varvec{m}}}^L\ \beta _{{{\varvec{m}}}+\chi ^L}^L\ E_w^{{{\varvec{m}}}+\chi ^L}. \end{aligned}$$

(3.15)

Proof

We first show that (3.15) holds for any maximal tripotent $w=e\in Z.$ Assume first that Z is of tube type. Since

$$ \Delta _e^{(\ell )}=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) \Phi ^{(\ell )} $$

by (3.8) and

$$ E_e^{{\varvec{m}}}={\frac{d_{{\varvec{m}}}}{(d/r)_{{\varvec{m}}}}}\Phi ^{{\varvec{m}}}$$

as a consequence of Schur orthogonality [5], it follows that

$$\begin{aligned} \Delta _e^{(\ell )}\ E_e^{{\varvec{m}}}= & {} {\frac{d_{{\varvec{m}}}}{(d/r)_{{\varvec{m}}}}}\ \Delta _e^{(\ell )}\ \Phi ^{{\varvec{m}}}={\frac{d_{{\varvec{m}}}}{(d/r)_{{\varvec{m}}}}}\sum \limits _{|L|=\ell }\alpha _{{\varvec{m}}}^L\ \Phi ^{{{\varvec{m}}}+\chi ^L} \\= & {} {\frac{d_{{\varvec{m}}}}{(d/r)_{{\varvec{m}}}}}\sum \limits _{|L|=\ell }{\frac{(d/r)_{{{\varvec{m}}}+\chi ^L}}{d_{{{\varvec{m}}}+\chi ^L}}}\ \alpha _{{\varvec{m}}}^L\ E_e^{{{\varvec{m}}}+\chi ^L} \\= & {} \sum \limits _{|L|=\ell }{\frac{(d/r)_{{{\varvec{m}}}+\chi ^L}}{(d/r)_{{\varvec{m}}}}}\ {\frac{d_{{\varvec{m}}}}{d_{{{\varvec{m}}}+\chi ^L}}}\ \alpha _{{\varvec{m}}}^L\ E_e^{{{\varvec{m}}}+\chi ^L}\\= & {} \sum \limits _{|L|=\ell }(d/r)_{{\varvec{m}}}^L\cdot {\frac{(d_e/r)_{{\varvec{m}}}^L}{(d/r)_{{\varvec{m}}}^L}}\ {\frac{\beta _{{{\varvec{m}}}+\chi ^L}^L}{\alpha _{{\varvec{m}}}^L}}\cdot \alpha _{{\varvec{m}}}^L\ E_e^{{{\varvec{m}}}+\chi ^L}\\= & {} \sum \limits _{|L|=\ell }(d_e/r)_{{\varvec{m}}}^L\ \beta _{{{\varvec{m}}}+\chi ^L}^L\ E_e^{{{\varvec{m}}}+\chi ^L}. \end{aligned}$$

If Z is not of tube type, we have $\Delta _e^{(\ell )}(z)=\Lambda _e^{(\ell )}(Pz)$ and $E^{{\varvec{m}}}_e(z)=E^{{\varvec{m}}}_e(Pz),$ where $P:Z\rightarrow Z_2(e)$ is the Peirce 2-projection. Thus (3.15) for $w=e$ holds for Z. Since both sides of (3.15) are K-invariant and the orbit $S=K\cdot e$ is a set of uniqueness for (anti)-holomorphic functions, the assertion follows for all $w\in Z.$ $\square$

Lemma 13

For $0\leqslant \ell \leqslant r$ we have

$$ \Delta ^{(\ell )}={\frac{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) \ (d/r)_{(\ell )}}{d_{(\ell )}}}\ E^{(\ell )}. $$

Proof

This follows from $\Delta ^{(\ell )}_{e,e}=\left( {\begin{array}{c}r\\ \ell \end{array}}\right)$ and $E^{(\ell )}_{e,e}={\frac{d_{(\ell )}}{(d/r)_{(\ell )}}}.$ As a double check, the same result is obtained by combining Lemmas 9 and  4. $\square$

For any polynomial $p\in {\mathcal {P}}(Z)$ we denote by $p_{{\varvec{m}}}\in {\mathcal {P}}^{{\varvec{m}}}(Z)$ its ${{\varvec{m}}}$-th component under the Peter–Weyl decomposition (2.2).

Proposition 14

Let $p\in {\mathcal {P}}^{{\varvec{m}}}(Z).$ Then for each $\ell$-element subset L such that ${{\varvec{m}}}-\chi ^L$ is a partition, we have (on the Fock space)

$$ \sum \limits _\alpha \psi _\alpha ^{(\ell )}(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}=p\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L $$

and

$$ \sum \limits _\alpha \Vert (M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}\Vert ^2=\Vert p\Vert ^2\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L $$

Proof

We follow the argument, for $\ell =1,$ contained in [2]. For fixed w we have

$$ E_w^{(\ell )}=\sum \limits _\alpha \psi _\alpha ^{(\ell )}\ {\overline{\psi _\alpha ^{(\ell )}(w)}} $$

by (3.3). With Lemma 13 and Proposition 12 it follows that

$$\begin{aligned} \sum \limits _\alpha \psi _\alpha ^{(\ell )}(w)(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}(w)= & {} \psi _\alpha ^{(\ell )}(w)(E_w^{{{\varvec{m}}}-\chi ^L}|M^*_{\psi _\alpha ^{(\ell )}}p) \\= & {} \sum \limits _\alpha \psi _\alpha ^{(\ell )}(w)(\psi _\alpha ^{(\ell )}E_w^{{{\varvec{m}}}-\chi ^L}|p) \\= & {} \sum \limits _\alpha \Big ({\overline{\psi _\alpha ^{(\ell )}(w)}}\psi _\alpha ^{(\ell )}E_w^{{{\varvec{m}}}-\chi ^L}|p\Big ) \\= & {} (E_w^{(\ell )}E_w^{{{\varvec{m}}}-\chi ^L}|p)=\Big ((E_w^{(\ell )}E_w^{{{\varvec{m}}}-\chi ^L})_{{\varvec{m}}}|p\Big ) \\= & {} {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L\ (E_w^{{\varvec{m}}}|p) \\= & {} {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L\ p(w). \end{aligned}$$

For the second assertion,

$$\begin{aligned} \sum \limits _\alpha \Vert (M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}\Vert ^2= & {} \sum \limits _\alpha ((M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}|(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L})\\= & {} \sum \limits _\alpha (M^*_{\psi _\alpha ^{(\ell )}}p|(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}) \\= & {} \sum \limits _\alpha (p|\psi _\alpha ^{(\ell )}(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L})\\= & {} {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L\ \Vert p\Vert ^2 \end{aligned}$$

$\square$

For the weighted Bergman spaces we obtain

Lemma 15

Let $p\in {\mathcal {P}}^{{\varvec{m}}}(Z)$ and $\phi \in {\mathcal {P}}^{(\ell )}(Z).$ Then

$$ (M_\phi ^{\nu *}p)_{{{\varvec{m}}}-\chi ^L}={\frac{(\nu )_{{{\varvec{m}}}-\chi ^L}}{(\nu )_{{\varvec{m}}}}}\ (M^*_\phi p)_{{{\varvec{m}}}-\chi ^L}={\frac{1}{(\nu )_{{{\varvec{m}}}-\chi ^L}^L}}\ (M^*_\phi p)_{{{\varvec{m}}}-\chi ^L}. $$

Proof

Let $q\in {\mathcal {P}}^{{{\varvec{m}}}-\chi ^L}(Z).$ Then, with (2.8),

$$\begin{aligned} \Big ((M_\phi ^{\nu *}p)_{{{\varvec{m}}}-\chi ^L}|q\Big )_\nu= & {} (M_\phi ^{\nu *}p|q)_\nu =(p|\phi q)_\nu ={\frac{1}{(\nu )_{{\varvec{m}}}}}(p|\phi q)={\frac{1}{(\nu )_{{\varvec{m}}}}}(M^*_\phi p|q) \\= & {} {\frac{(\nu )_{{{\varvec{m}}}-\chi ^L}}{(\nu )_{{\varvec{m}}}}}(M^*_\phi p|q)_\nu ={\frac{(\nu )_{{{\varvec{m}}}-\chi ^L}}{(\nu )_{{\varvec{m}}}}}((M^*_\phi p)_{{{\varvec{m}}}-\chi ^L}|q)_\nu . \end{aligned}$$

Since q is arbitrary, the assertion follows. $\square$

The first eigenvalue formula is the following:

Theorem 16

For $0\leqslant \ell \leqslant r$ the K-invariant operators $\Delta ^{(\ell )}_{T}$ and $\Delta ^{(\ell )}_{T^\nu }$ have the eigenvalues

$$ \Delta ^{(\ell )}_T({{\varvec{m}}})=\sum \limits _{|L|=\ell }(d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L $$

and

$$ \Delta ^{(\ell )}_{T^\nu }({{\varvec{m}}})=\sum \limits _{|L|=\ell }\ {\frac{(d_e/r)_{{{\varvec{m}}}-\chi ^L}^L}{(\nu )_{{{\varvec{m}}}-\chi ^L}^L}}\ \beta _{{\varvec{m}}}^L, $$

with $\beta _{{\varvec{m}}}^L$ defined in (3.6).

Proof

If $p\in {\mathcal {P}}^{{\varvec{m}}}(Z)$ then

$$ E^{(\ell )}_T p=\sum \limits _\alpha \psi _\alpha ^{(\ell )}(M^*_{\psi _\alpha ^{(\ell )}}p)=\sum \limits _{|L|=\ell }\sum \limits _\alpha \psi _\alpha ^{(\ell )}(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}. $$

In view of Lemma 13, the first assertion for the Fock space follows from Proposition 14. For the second assertion, we use Lemma 15 and obtain, for each subset L such that ${{\varvec{m}}}-\chi ^L$ is a partition

$$\begin{aligned} \sum \limits _\alpha M^\nu _{\psi _\alpha ^{(\ell )}}(M^{\nu *}_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}= & {} \sum \limits _\alpha \psi _\alpha ^{(\ell )}(M^{\nu *}_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}\\= & {} {\frac{1}{(\nu )_{{{\varvec{m}}}-\chi ^L}^L}}\ \sum \limits _\alpha \psi _\alpha ^{(\ell )}(M^*_{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}-\chi ^L}\\= & {} {\frac{1}{(\nu )_{{{\varvec{m}}}-\chi ^L}^L}}\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d_e/r)_{{{\varvec{m}}}-\chi ^L}^L\ \beta _{{\varvec{m}}}^L\ p. \end{aligned}$$

Now the assertion follows from

$$ \Delta ^{(\ell )}_{T^\nu }=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) {\frac{(d/r)_{(\ell )}}{d_{(\ell )}}}\ E^{(\ell )}_{T^\nu }=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) {\frac{(d/r)_{(\ell )}}{d_{(\ell )}}} \ \sum \limits _\alpha M^\nu _{\psi _\alpha ^{(\ell )}}M^{\nu *}_{\psi _\alpha ^{(\ell )}}. $$

$\square$

Example 17

For $\ell =1$ we obtain the formula

$$\begin{aligned} \Delta ^{(1)}_{T^\nu }({{\varvec{m}}})= & {} \Big (\sum \limits _i M^\nu _{(z|u_i)}M^{\nu *}_{(z|u_i)}\Big )({{\varvec{m}}})=\sum \limits _{k=1}^r{\frac{m_k'+{\frac{d_e}{r}}-1}{m_k'+\nu -1}}\ \beta _{{\varvec{m}}}^{\{k\}} \\= & {} \sum \limits _{k=1}^r{\frac{m_k'+{\frac{d_e}{r}}-1}{m_k'+\nu -1}}\ \prod \limits _{h\ne k}{\frac{m_k'-m_h'-\tfrac{a}{2}}{m_k'-m_h'}} \end{aligned}$$

previously obtained in [2, Proposition 4.4].

Example 18

For $\ell =r,\ L=\{1,\ldots ,r\}$ we have $\beta _{{\varvec{m}}}^L=1$ (empty product). If Z is of tube type then $d_e=d.$ Using (3.1) Theorem 16 simplifies to

$$\begin{aligned} (M_NM_N^*)({{\varvec{m}}})= & {} \Delta ^{(r)}_T({{\varvec{m}}})={\frac{(d/r)_{{\varvec{m}}}}{(d/r)_{{{\varvec{m}}}-1}}},\\ (M_N^\nu M_N^{\nu *})({{\varvec{m}}})= & {} \Delta ^{(r)}_{T^\nu }({{\varvec{m}}})={\frac{(d/r)_{{{\varvec{m}}}}}{(d/r)_{{{\varvec{m}}}-1}}}{\frac{(\nu )_{{{\varvec{m}}}-1}}{(\nu )_{{\varvec{m}}}}}. \end{aligned}$$

4 The Second Eigenvalue Formula

The second eigenvalue formula gives the eigenvalues of the K-invariant operators $T^\nu _{\Delta ^{(\ell )}}$ and $T_{\Delta ^{(\ell )}},$ for $0\leqslant \ell \leqslant r.$ In this case the previous arguments, based on reproducing kernel identities, do not apply immediately.

Lemma 19

Under the K-action (2.1) on polynomials, we have

$$ M^*_p(k^{-1}\cdot q)=k^{-1}\cdot (M^*_{k\cdot p}q). $$

Proof

It suffices to check for linear polynomials $p(z)=(z|u),$ where $u\in Z.$ We have

$$\begin{aligned} (M^*_{(z|u)}(k^{-1}\cdot q))(z)= & {} (k^{-1}\cdot q)'(z)u=(q\circ k)'(z)u\\= & {} q'(kz)ku=(M^*_{(z|ku)}q)(kz)=k^{-1}\cdot (M^*_{(z|ku)}q)(z). \end{aligned}$$

Since $(k\cdot p)(z)=p(k^{-1}z)=(k^{-1}z|u)=(z|ku),$ the assertion follows. $\square$

Lemma 20

Let $\phi _\alpha$ and $\psi _\beta$ be orthonormal bases of ${\mathcal {P}}^{(\ell )}(Z).$ Then for any sesqui-linear form $\langle \phi |\psi \rangle$ on ${\mathcal {P}}^{(\ell )}(Z)$ we have

$$ \sum \limits _\alpha \langle \phi _\alpha |\phi _\alpha \rangle =\sum \limits _\beta \langle \psi _\beta |\psi _\beta \rangle $$

Proof

Using Einstein summation convention to simplify notation, we have $\phi _\alpha =\Lambda _\alpha ^\beta \psi _\beta$ for a unitary ’matrix’ $\Lambda .$ Then

$$\begin{aligned} \langle \phi _\alpha |\phi _\alpha \rangle= & {} \langle \Lambda _\alpha ^\sigma \psi _\sigma |\Lambda _\alpha ^\tau \psi _\tau \rangle ={\overline{\Lambda _\alpha ^\sigma }}\ \langle \psi _\sigma |\psi _\tau \rangle \ \Lambda _\alpha ^\tau \\= & {} \langle \psi _\sigma |\psi _\tau \rangle \ (\Lambda ^*\Lambda )_\sigma ^\tau =\langle \psi _\sigma |\psi _\tau \rangle \ \delta _\sigma ^\tau =\langle \psi _\sigma |\psi _\sigma \rangle . \end{aligned}$$

$\square$

For any $p,q\in {\mathcal {P}}(Z)$ the map $(\phi ,\psi )\mapsto \langle \phi |\psi \rangle :=(p|M^*_\phi q)(M^*_\psi q|p)$ is sesqui-linear. Hence Lemma 20 implies for each $k\in K$

$$\begin{aligned} \sum \limits _\alpha (p|M^*_{k\cdot \psi _\alpha ^{(\ell )}}q)(M^*_{k\cdot \psi _\alpha ^{(\ell )}}q|p)= & {} \sum \limits _\alpha \langle k\cdot \psi _\alpha ^{(\ell )}|k\cdot \psi _\alpha ^{(\ell )}\rangle \nonumber \\= & {} \sum \limits _\alpha \langle \psi _\alpha ^{(\ell )}|\psi _\alpha ^{(\ell )}\rangle \nonumber \\= & {} \sum \limits _\alpha (p|M^*_{\psi _\alpha ^{(\ell )}}q)(M^*_{\psi _\alpha ^{(\ell )}}q|p), \end{aligned}$$

(4.1)

since $k\cdot \psi _\alpha ^{(\ell )}$ is also an orthonormal basis.

Proposition 21

Let ${{\varvec{m}}}$ and ${{\varvec{m}}}+\chi ^L$ be partitions. Then we have (on the Fock space)

$$ \sum \limits _\alpha \Vert (\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Vert ^2=\Vert p\Vert ^2\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d/r)_{{\varvec{m}}}^L\ \alpha _{{\varvec{m}}}^L $$

and

$$ \sum \limits _\alpha M^*_{\psi _\alpha ^{(\ell )}}(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}=p\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d/r)_{{\varvec{m}}}^L\ \alpha _{{\varvec{m}}}^L $$

for all $p\in {\mathcal {P}}^{{\varvec{m}}}(Z).$

Proof

Let $q\in {\mathcal {P}}^{{{\varvec{m}}}+\chi ^L}(Z).$ Schur orthogonality applied to ${\mathcal {P}}^{{{\varvec{m}}}+\chi ^L}(Z)$ yields for each $\alpha$

$$\begin{aligned}&{\frac{\Vert q\Vert ^2}{d_{{{\varvec{m}}}+\chi ^L}}}\ \Vert (\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Vert ^2\\&\quad =\int \limits _K dk\ \Big ((\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}|k^{-1}\cdot q\Big )\ \Big (k^{-1}\cdot q|(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Big )\\&\quad =\int \limits _K dk\ \Big (\psi _\alpha ^{(\ell )}p|k^{-1}\cdot q\Big )\ \Big (k^{-1}\cdot q|\psi _\alpha ^{(\ell )}p\Big ) \\&\quad =\int \limits _K dk\ \Big (p|M^*_{\psi _\alpha ^{(\ell )}}(k^{-1}\cdot q)\Big )\ \Big (M^*_{\psi _\alpha ^{(\ell )}}(k^{-1}\cdot q)|p\Big ) \\&\quad =\int \limits _K dk\ \Big (p|k^{-1}\cdot (M^*_{k\cdot \psi _\alpha ^{(\ell )}}q)\Big )\ \Big (k^{-1}\cdot (M^*_{k\cdot \psi _\alpha ^{(\ell )}}q)|p\Big )\\&\quad =\int \limits _K dk\ \Big (k\cdot p|M^*_{k\cdot \psi _\alpha ^{(\ell )}}q\Big )\ \Big (M^*_{k\cdot \psi _\alpha ^{(\ell )}}q|k\cdot p\Big ). \end{aligned}$$

With (4.1) and Schur orthogonality applied to ${\mathcal {P}}^{{\varvec{m}}}(Z)$ we obtain

$$\begin{aligned}&{\frac{\Vert q\Vert ^2}{d_{{{\varvec{m}}}+\chi ^L}}}\ \sum \limits _\alpha \Vert (\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Vert ^2\\&\quad =\int \limits _K dk\ \sum \limits _\alpha \Big (k\cdot p|M^*_{k\cdot \psi _\alpha ^{(\ell )}}q\Big )\ \Big (M^*_{k\cdot \psi _\alpha ^{(\ell )}}q|k\cdot p\Big ) \\&\quad =\int \limits _K dk\ \sum \limits _\alpha \Big (k\cdot p|M^*_{\psi _\alpha ^{(\ell )}}q\Big )\ \Big (M^*_{\psi _\alpha ^{(\ell )}}q|k\cdot p\Big ) \\&\quad =\sum \limits _\alpha \int \limits _K dk\ \Big (k\cdot p|(M^*_{\psi _\alpha ^{(\ell )}}q)_{{\varvec{m}}}\Big )\ \Big ((M^*_{\psi _\alpha ^{(\ell )}}q)_{{\varvec{m}}}|k\cdot p\Big ) \\&\quad ={\frac{\Vert p\Vert ^2}{d_{{\varvec{m}}}}}\ \sum \limits _\alpha \Vert (M^*_{\psi _\alpha ^{(\ell )}}q)_{{\varvec{m}}}\Vert ^2 \\&\quad ={\frac{\Vert p\Vert ^2}{d_{{\varvec{m}}}}}\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}} (d_e/r)_{{\varvec{m}}}^L\ \beta _{{{\varvec{m}}}+\chi ^L}^L\ \Vert q\Vert ^2, \end{aligned}$$

where in the last step we use Proposition 14. It follows that

$$\begin{aligned} \sum \limits _\alpha \Vert (\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Vert ^2= & {} \Vert p\Vert ^2\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ {\frac{d_{{{\varvec{m}}}+\chi ^L}}{d_{{\varvec{m}}}}}\ (d_e/r)_{{\varvec{m}}}^L\ \beta _{{{\varvec{m}}}+\chi ^L}^L \\= & {} \Vert p\Vert ^2\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ {\frac{(d/r)_{{\varvec{m}}}^L}{(d_e/r)_{{\varvec{m}}}^L}}\ {\frac{\alpha _{{\varvec{m}}}^L}{\beta _{{{\varvec{m}}}+\chi ^L}^L}} \ (d_e/r)_{{\varvec{m}}}^L\ \beta _{{{\varvec{m}}}+\chi ^L}^L \\= & {} \Vert p\Vert ^2\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ (d/r)_{{\varvec{m}}}^L\ \alpha _{{\varvec{m}}}^L. \end{aligned}$$

This proves the first assertion. The second assertion follows, since

$\sum \limits _\alpha M^*_{\psi _\alpha ^{(\ell )}}(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}$ is a multiple of p and

$$\begin{aligned} \sum \limits _\alpha \Big (p|M^*_{\psi _\alpha ^{(\ell )}}(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Big )= & {} \sum \limits _\alpha (\psi _\alpha ^{(\ell )}p|(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}) \\= & {} \sum \limits _\alpha ((\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}|(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}) \\= & {} \sum \limits _\alpha \Vert (\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\Vert ^2. \end{aligned}$$

$\square$

The second eigenvalue formula is the following:

Theorem 22

For $0\leqslant \ell \leqslant r$ the K-invariant operators $T_{\Delta ^{(\ell )}}$ and $T^\nu _{\Delta ^{(\ell )}}$ have the eigenvalues

$$ T_{\Delta ^{(\ell )}}({{\varvec{m}}})=\sum \limits _{|L|=\ell }(d/r)_{{\varvec{m}}}^L\ \alpha _{{\varvec{m}}}^L $$

and

$$ T^\nu _{\Delta ^{(\ell )}}({{\varvec{m}}})=\sum \limits _{|L|=\ell }{\frac{(d/r)_{{\varvec{m}}}^L}{(\nu )_{{\varvec{m}}}^L}}\ \alpha _{{\varvec{m}}}^L $$

with $\alpha _{{\varvec{m}}}^L$ defined in (3.5).

Proof

Let $p\in {\mathcal {P}}^{{\varvec{m}}}(Z).$ By Lemma 15 applied to ${{\varvec{m}}}+\chi ^L$ we have

$$\begin{aligned} \sum \limits _\alpha M^{\nu *}_{\psi _\alpha ^{(\ell )}}(M^\nu _{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}+\chi ^L}= & {} \sum \limits _\alpha M^{\nu *}_{\psi _\alpha ^{(\ell )}}(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L} \\= & {} {\frac{1}{(\nu )_{{\varvec{m}}}^L}}\ \sum \limits _\alpha M^*_{\psi _\alpha ^{(\ell )}}(\psi _\alpha ^{(\ell )}p)_{{{\varvec{m}}}+\chi ^L}\\= & {} p\ {\frac{d_{(\ell )}}{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}}\ {\frac{(d/r)_{{\varvec{m}}}^L}{(\nu )_{{\varvec{m}}}^L}}\ \alpha _{{\varvec{m}}}^L. \end{aligned}$$

Since

$$ T^\nu _{\Delta ^{(\ell )}}p={\frac{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}{d_{(\ell )}}}\ T^\nu _{E^{(\ell )}}p={\frac{\left( {\begin{array}{c}r\\ \ell \end{array}}\right) (d/r)_{(\ell )}}{d_{(\ell )}}} \sum \limits _{|L|=\ell }\sum \limits _\alpha M^{\nu *}_{\psi _\alpha ^{(\ell )}}(M^\nu _{\psi _\alpha ^{(\ell )}}p)_{{{\varvec{m}}}+\chi ^L} $$

the assertion follows by summing over all $\ell$-element subsets L such that ${{\varvec{m}}}+\chi ^L$ is a partition. The proof for the Fock space is similar. $\square$

For the Hardy space $H^2(S)$ over the Shilov boundary S of D,  corresponding to $\nu =\tfrac{d}{r},$ the above formula combined with (3.11) simplifies to

$$ T^{d/r}_{\Delta ^{(\ell )}}({{\varvec{m}}})=\sum \limits _{|L|=\ell }\ \alpha _{{\varvec{m}}}^L=\left( {\begin{array}{c}r\\ \ell \end{array}}\right) . $$

This, however, is trivial since $\Delta ^{(\ell )}(z,z)=\left( {\begin{array}{c}r\\ \ell \end{array}}\right)$ is constant on S.

Example 23

For $\ell =1,$ with $\Delta ^{(1)}(z,z)=(z|z),$ we obtain as a special case

$$ T^\nu _{(z|z)}({{\varvec{m}}})=\sum \limits _{k=1}^r{\frac{(d/r)_{{\varvec{m}}}^{\{k\}}}{(\nu )_{{\varvec{m}}}^{\{k\}}}}\ \alpha _{{\varvec{m}}}^{\{k\}} =\sum \limits _{k=1}^r{\frac{m_k'+\tfrac{d}{r}}{m_k'+\nu }}\ \prod \limits _{h\ne k}{\frac{m_k'-m_h'+\tfrac{a}{2}}{m_k'-m_h'}} $$

for all partitions ${{\varvec{m}}}.$ This formula was conjectured in [6] and proved there, by a different argument, for all bounded symmetric domains of rank $r=2.$

Besides the spin factors, which correspond to the rank 2 domains of tube type, there exist three types of Jordan triples of rank 2 which are not of tube type: (i) the space of all complex $(2\times N)$-matrices with $N>2,$ where $d=2N$ and $a=2,$ (ii) the space of all complex anti-symmetric $(5\times 5)$-matrices, where $d=10$ and $a=4,$ and (iii) the exceptional domain of dimension $d=16,$ where $a=6.$

Example 24

For $\ell =r,\ L=\{1,\ldots ,r\}$ we have $\alpha _{{\varvec{m}}}^L=1$ (empty product). If Z is of tube type, then $d_e=d.$ Using (3.1) Theorem 22 simplifies to

$$\begin{aligned}&(M_N^*M_N)({{\varvec{m}}})=T_{\Delta ^{(r)}}({{\varvec{m}}})={\frac{(d/r)_{{{\varvec{m}}}+1}}{(d/r)_{{\varvec{m}}}}}, \\&(M_N^{\nu *}M_N^{\nu })({{\varvec{m}}})=T^\nu _{\Delta ^{(r)}}({{\varvec{m}}})={\frac{(d/r)_{{{\varvec{m}}}+1}}{(d/r)_{{\varvec{m}}}}}{\frac{(\nu )_{{\varvec{m}}}}{(\nu )_{{{\varvec{m}}}+1}}}. \end{aligned}$$