Spaces of bounded spherical functions on Heisenberg groups: part II

Chal Benson¹ &
Gail Ratcliff¹

410 Accesses
2 Citations
Explore all metrics

Abstract

Consider a linear multiplicity free action by a compact Lie group $K$ on a finite dimensional Hermitian vector space $V$. Letting $K$ act on the Heisenberg group $H_V=V\times \mathbb {R}$ yields a Gelfand pair. The condition that $K:V$ be “well-behaved” establishes a relationship between the associated moment mapping and highest weight vectors occurring in the polynomial ring ${\mathbb {C}}[V]$. Under this condition, an application of the Orbit Method produces a topological embedding of the space of bounded spherical functions for $(K,H_V)$ in the space of $K$-orbits in the dual of the Lie algebra for $H_V$. In part I of this work, it was shown that every irreducible multiplicity free action is well-behaved. Here we extend this result to encompass all multiplicity free actions. Our proof uses case-by-case analysis of multiplicity free actions which are indecomposable but not irreducible.

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

Article Open access 01 October 2016

AN ORBIT MODEL FOR THE SPECTRA OF NILPOTENT GELFAND PAIRS

Article 02 August 2019

On the Korányi spherical maximal function on Heisenberg groups

Article Open access 05 December 2022

Use our pre-submission checklist

Avoid common mistakes on your manuscript.

1 Introduction

Let $V\cong {\mathbb {C}}^n$ be a finite dimensional complex vector space with Hermitian inner product $\left\langle \cdot ,\cdot \right\rangle $ and $K$ be a compact Lie group acting on $(V,\left\langle \cdot ,\cdot \right\rangle )$ by some unitary representation. The group $K$ acts by automorphisms on the associated Heisenberg group

$$\begin{aligned} H_V=V\times \mathbb {R}\quad \text {with product}\quad (z,t)(z',t')=\left( z+z',\ t+t'-\frac{1}{2}\hbox {Im}\left\langle z,z'\right\rangle \right) \end{aligned}$$

via

$$\begin{aligned} k\cdot (z,t)=(k\cdot z, t). \end{aligned}$$

$(K,H_V)$ is said to be a Gelfand pair if the convolution algebra $L^1_K(H_V)$ of integrable $K$-invariant functions on $H_V$ is commutative. As is well known, this is the case if and only if $K:V$ is a linear multiplicity free action [8]. That is, if and only if the representation of $K$ in the space ${\mathbb {C}}[V]$ of holomorphic polynomial functions on $V$,

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

is multiplicity free.

In this context, the spectrum, or Gelfand space, for $L^1_K(H_V)$ can be identified, via integration, with the set $\Delta (K,H_V)$ of bounded $K$-spherical functions on $H_V$ endowed with the compact-open topology. An application of the Orbit Method, given in [5], produces an injective mapping $\Psi :\Delta (K,H_V)\rightarrow {\mathfrak {h}}_V^*/K$ from $\Delta (K,H_V)$ to the set of $K$-orbits in the dual of the Lie algebra for $H_V$. Giving ${\mathfrak {h}}_V^*/K$ the quotient topology our main result here is that

Theorem 1.1

$\Psi :\Delta (K,H_V)\rightarrow {\mathfrak {h}}_V^*/K$ is a homeomorphism onto its image.

Theorem 1.1 was conjectured, in more general form, in [5] and is the focus of [6] and [2]. This paper is a continuation of these works. We proved in [6] that $\Psi $ is indeed a homeomorphism whenever $K:V$ is a well-behaved multiplicity free action. (See Definition 2.3 below.) Thus Theorem 1.1 is a direct consequence of the following.

Theorem 1.2

Every linear multiplicity free action is well-behaved.

If $K_1:V_1$ and $K_2:V_2$ are multiplicity free actions, then so is the product action $(K_1\times K_2):(V_1\oplus V_2)$. Lemma 3.1 below shows, moreover, that if both $K_1:V_1$ and $K_2:V_2$ are well-behaved then so is $(K_1\times K_2):(V_1\oplus V_2)$. So to prove Theorem 1.2, it suffices to verify that every indecomposable multiplicity free action is well-behaved. These are the multiplicity free actions that do not decompose as product actions.

The papers [3, 13, 15] classify indecomposable multiplicity free actions up to geometric equivalence. Kac’s paper [13] gives all multiplicity free actions $K:V$ in which $K$ acts irreducibly on $V$. In [2], we applied this classification to show that each irreducible multiplicity free action is well-behaved. Here we complete the proof of Theorem 1.2 by analyzing the reducible but indecomposable actions given in [3, 15]. As explained in [2, Section 3.7], a byproduct of our calculations is that the orbital model for $\Delta (K,H_V)$, provided by Theorem 1.1, becomes relatively explicit in each case.

2 Preliminaries and background results

Let $K:V$ denote a fixed multiplicity free action. We summarize below some results from [2], retaining the notational conventions established there. In particular

$\mathfrak {k}$ is the Lie algebra for $K$, $T\subset K$ a maximal torus, $\mathfrak {t}\subset \mathfrak {k}$ its Lie algebra and ${\mathfrak {h}}:=\mathfrak {t}_{\mathbb {C}}$. Choosing a system of positive roots, we decompose $\mathfrak {k}_{{\mathbb {C}}}={\mathfrak {h}}\oplus \mathfrak {n}_+\oplus \mathfrak {n}_-$ and let $B:=HN=HN_+$ be the resulting Borel subgroup in $K_{{\mathbb {C}}}$.
$\Lambda \subset {\mathfrak {h}}^*$ is the set of $B$-highest weights for irreducible representations of $K_{{\mathbb {C}}}$ occurring in ${\mathbb {C}}[V]$. For each $\alpha \in \Lambda $ choose a $B$-highest weight vector $h_\alpha \in {\mathbb {C}}[V]$ with weight $\alpha $ (unique modulo ${\mathbb {C}}^\times $) and
let $\alpha _\mathfrak {k}\in \mathfrak {k}^*$ be the (real-valued) linear functional on $\mathfrak {k}$ with $\alpha _\mathfrak {k}|_{\mathfrak {t}}=-i\alpha $ and $\alpha _\mathfrak {k}\equiv 0$ on the orthogonal complement to $\mathfrak {t}$ in $\mathfrak {k}$ with respect to an $Ad(K)$-invariant inner product.
$\tau :V\rightarrow \mathfrak {k}^*$ is the (unnormalized) moment map for $K:V$, namely $\tau (v)(A):=i\left\langle A\cdot v,v\right\rangle .$

2.1 Spherical points and well-behaved multiplicity free actions

Definition 2.1

[6] A point $v_\alpha \in V$ is said to be a spherical point for the highest weight $\alpha \in \Lambda $ if $\tau (v_\alpha )=\alpha _\mathfrak {k}$.

Results from [1, 9] ensure that a spherical point $v_\alpha $ exists for each $\alpha \in \Lambda $ and that the $K$-orbit $\mathcal {K}_\alpha =K\cdot v_\alpha $ is uniquely determined. The following result facilitates the calculation of spherical points in examples.

Lemma 2.2

[2, Lemma 3.1] $v_\alpha \in V$ is a spherical point for $\alpha \in \Lambda $ if and only if

$$\begin{aligned} \left\{ \begin{array}{ll} \left\langle X\cdot v_\alpha , v_\alpha \right\rangle =-\alpha (X) &{} \text {for all}\,X\in {\mathfrak {h}}\,\text {and}\\ \left\langle X\cdot v_\alpha , v_\alpha \right\rangle =0 &{} \text {for all} X\in \mathfrak {n}_+\oplus \mathfrak {n}_- \end{array}\right\} \end{aligned}$$

(2.1)

As (2.1) is linear in $X$, this amounts to a system of $\dim (\mathfrak {k})$ quadratic equations whose solutions give all spherical points for $\alpha $.

Definition 2.3

[6] Given $\alpha \in \Lambda $, we say that a spherical point $v_\alpha $ for $\alpha $ is well-adapted to $h_\alpha $ when the following conditions hold.

(i)
$h_\alpha (v_\alpha )\ne 0$, and
(ii)
$\bigl (\partial _wh_\alpha \bigr )(v_\alpha )=\left\langle w,v_\alpha \right\rangle h_\alpha (v_\alpha )$ for all $w\in V$.

The multiplicity free action $K:V$ is said to be well-behaved if for every $\alpha \in \Lambda $, one can choose a spherical point $v_\alpha $ well-adapted to $h_\alpha $.

Lemma 2.4

[2, Lemma 3.7] Let $K_1:V$ be a multiplicity free action obtained by restricting a multiplicity free action $K_2:V$ to a closed Lie subgroup $K_1\subset K_2$. Assume, moreover, that ${\mathbb {C}}[V]$ shares a common decomposition under the associated representations of $K_1$ and $K_2$. Then $K_1:V$ is well-behaved if and only if $K_2:V$ is well-behaved.

2.2 A limiting procedure

Recall that $\alpha \in \Lambda $ is a fundamental highest weight for $K:V$ when $h_\alpha $ is an irreducible polynomial. The fundamental highest weights are finite in number and freely generate $\Lambda $ as an additive semigroup [12]. The number of fundamental highest weights for $K:V$ is its rank. A technical Lemma from [2] is used to study examples.

Lemma 2.5

[2, Lemma 3.5] Let $K:V$ be a rank $r$ multiplicity free action with fundamental highest weights $\{\alpha _1,\ldots ,\alpha _r\}$ and associated fundamental highest weight vectors $h_j=h_{\alpha _j}$. Suppose that for all positive real numbers $x_1,\ldots ,x_r>0$, there is a point $v(\mathbf{x})=v(x_1,\ldots ,x_r)$ in $V$ which satisfies the following four conditions:

(1)
(2)
$h_i(v(\mathbf{x}))\ne 0$ for $1\le i\le r$.
(3)
For each $1\le k<r$ and indices $1\le j_1<j_2<\cdots <j_k\le r$ the limit
$$\begin{aligned} \lim _{x_{j_k}\rightarrow 0^+}\cdots \lim _{x_{j_1}\rightarrow 0^+}v(x_1,\ldots ,x_r) \end{aligned}$$
exists in $V$, and
(4)
$\lim _{x_{j_k}\rightarrow 0^+}\cdots \lim _{x_{j_1}\rightarrow 0^+}h_i\bigl (v(x_1,\ldots ,x_r)\bigr )\ne 0$ for each $i\in \{1,\ldots ,r\}\backslash \{j_1,\ldots ,j_k\}$.

Then $K:V$ is a well-behaved multiplicity free action.

Definition 2.6

We call a point $v(\mathbf{x})=v(x_1,\ldots ,x_r)\in V$ satisfying condition (1) in Lemma 2.5 a generalized spherical point. Moreover, $v(\mathbf{x})$ is said to be a generic generalized spherical point when each parameter $x_j$ is nonzero.

As $\Lambda =\bigl \{x_1\alpha _1+\cdots +x_r\alpha _r~:~x_j\in \mathbb {Z}, x_j\ge 0\bigr \}$, Lemma 2.2 shows $v(\mathbf{x})$ to be a spherical point for weight $(\alpha =x_1\alpha _1+\cdots +x_r\alpha _r)\in \Lambda $ whenever each $x_j$ in a nonnegative integer. If each $x_j$ is a positive integer, then we call the weight $\alpha $ generic.

3 Product actions

A routine Lemma enables us to reduce the proof of Theorem 1.2 to the study of indecomposable multiplicity free actions.

Lemma 3.1

Products of well-behaved multiplicity free actions are well-behaved.

Proof

Consider a product action $(K_1\times K_2):(V_1\oplus V_2)$ where $K_1:V_1$ and $K_2:V_2$ are well-behaved multiplicity free actions in Hermitian vector spaces $(V_j,\left\langle \cdot ,\cdot \right\rangle _j)$. Equipping $V_1\oplus V_2$ with the direct sum Hermitian inner product, it follows that the moment mapping $\tau :V_1\oplus V_2\rightarrow (\mathfrak {k}_1\times \mathfrak {k}_2)^*=\mathfrak {k}_1^*\times \mathfrak {k}_2^*$ is just $\tau (v_1,v_2)=\bigl (\tau _1(v_1),\tau _2(v_2)\bigr )$ where $\tau _j:V_j\rightarrow \mathfrak {k}_j$ is the moment mapping for $K_j:V_j$.

Let $\Lambda _j\subset {\mathfrak {h}}_j^*$ be the set of $B_j$-highest weights for representations of $K_j$ occurring in ${\mathbb {C}}[V_j]$. The set of $(B_1\times B_2)$-highest weights for representations of $K_1\times K_2$ occurring in ${\mathbb {C}}[V_1\oplus V_2]\cong {\mathbb {C}}[V_1]\otimes {\mathbb {C}}[V_2]$ is $\Lambda =\Lambda _1\times \Lambda _2\subset \bigl (({\mathfrak {h}}_1\times {\mathfrak {h}}_2)^*={\mathfrak {h}}_1^*\times {\mathfrak {h}}_2^*\bigr ).$ If $h_{\alpha _j}\in {\mathbb {C}}[V_j]$ are highest weight vectors with weights $\alpha _j\in \Lambda _j$, then $h_{\alpha _1}\otimes h_{\alpha _2}$ is a highest weight vector in ${\mathbb {C}}[V_1\oplus V_2]$ with weight $(\alpha _1,\alpha _2)\in \Lambda $. Let $v_{\alpha _j}\in {\mathbb {C}}[V_j]$ be a spherical point for $\alpha _j$ well-adapted to $h_{\alpha _j}$. We claim that the spherical point $(v_{\alpha _1},v_{\alpha _2})\in V_1\oplus V_2$ for $(\alpha _1,\alpha _2)\in \Lambda $ is well-adapted to $h_{\alpha _1}\otimes h_{\alpha _2}$. Indeed $\bigl (h_{\alpha _1}\otimes h_{\alpha _2}\bigr )(v_{\alpha _1},v_{\alpha _2})=h_{\alpha _1}(v_{\alpha _1})h_{\alpha _2}(v_{\alpha _2})\ne 0$ and by linearity, it suffices to check Definition 2.3 condition (ii) for derivatives in directions lying in $V_1 \cup V_2\subset V_1\oplus V_2$. If say $w\in V_1$, then one has

$$\begin{aligned} \bigl (\partial _{(w,0)}(h_{\alpha _1}\otimes h_{\alpha _2})\bigr )(v_{\alpha _1},v_{\alpha _2})&= \bigl (\partial _w h_{\alpha _1}\bigr )(v_{\alpha _1})h_{\alpha _2}(v_{\alpha _2}) =\left\langle w,v_{\alpha _1}\right\rangle _1h_{\alpha _1}(v_{\alpha _1})h_{\alpha _2}(v_{\alpha _2})\\&= \left\langle (w,0),(v_{\alpha _1},v_{\alpha _2})\right\rangle \bigl (h_{\alpha _1}\otimes h_{\alpha _2}\bigr )(v_{\alpha _1},v_{\alpha _2}). \end{aligned}$$

$\square $

4 Reducible but indecomposable multiplicity free actions

Each multiplicity free action splits as a product of indecomposable multiplicity free actions. The indecomposable but non-irreducible multiplicity free actions are classified in [3, 15]. See also [14]. Scalar actions somewhat complicate the classification. Lemma 2.4 ensures, however, that a multiplicity free action obtained by adding or removing a copy of the scalars $\mathbb {T}$ from a well-behaved multiplicity free action remains well-behaved. So for our purposes, it suffices to describe the multiplicity free actions, which are fully saturated. That is, actions $K:V$ which include a full copy of the scalars acting on each irreducible subspace of $V$.

Up to geometric equivalence, there are twelve saturated indecomposable non-irreducible multiplicity free actions $K:V$. In each case, $V=V_1\oplus V_2$ is the direct sum of two $K$-irreducible subspaces, $K=K'\times \mathbb {T}\times \mathbb {T}$ with $K'$ compact semisimple and $K:V_1$, $K:V_2$ are irreducible multiplicity free actions. The possibilities for $K':V$ are listed in Table 1, which follows notational conventions from [3].^{Footnote 1} To complete the proof of Theorem 1.2, we will verify that each of these actions is well-behaved. Numbers in the first column of the table refer to subsections treating each example in turn.

Table 1 Indecomposable multiplicity free actions

Full size table

5 Case-by-case analysis: fixed rank examples

In this section, we examine the first eight actions in Table 1. Each of these (families of) examples has a fixed rank and one can use brute force calculation. For each action $K:V$, we will apply Lemma 2.5 (the limiting procedure) and proceed as follows.

(a)
Give explicit fundamental highest weights $\alpha _j$ and highest weight vectors $h_j$. These data can be found in [3, 4, 14, 15].
(b)
Use Lemma 2.2 to obtain a system of quadratic equations whose solutions are generic generalized spherical points.
(c)
Produce one such solution, $v(\mathbf{x})$ say.
(d)
Obtain formulas for the $h_j(v(\mathbf{x}))$’s to verify condition (2) in Lemma 2.5.
(e)
Take limits as subsets of the parameters $\mathbf{x}$ approach zero in $v(\mathbf{x})$ and in the formulas for the $h_j(v(\mathbf{x}))$’s, verifying conditions (3) and (4) in the lemma.

We made extensive use of a computer algebra system, Maple, to facilitate the calculations. In most cases, however, Maple was unable to solve the equations from step (b) on general symbolic inputs. Considerable experimentation with numerical examples was used to conjecture the expressions given below for a generic generalized spherical point in each case. It is, however, not difficult to check by hand that these points do solve the equations from step (b). Steps (d) and (e) are straightforward in each case. But for a multiplicity free action or rank $r$, there are $2^r-2$ non-empty proper subsets of the parameters and thus $2^r-2$ limits to examine in all. We used Maple to perform step (e) and, except for the rank 3 actions, will omit the details. The interested reader can find a Maple worksheet concerning the rank 7 example (5.6) at the first author’s web page [7].

5.1 Notational conventions

The first seven actions $K:V$ from Table 1 will each be realized in a suitable space of complex matrices, $V=M_{n,m}({\mathbb {C}})$ say, and the usual Hermitian inner product on $M_{n,m}({\mathbb {C}})$, namely

$$\begin{aligned} \left\langle z,w\right\rangle =\mathrm{tr}(zw^*), \end{aligned}$$

is $K$-invariant. For matrices $z\in V$, the notations $z_{i,\bullet }$ and $z_{\bullet ,j}$ indicate row and column vectors. The row and column spaces carry their standard inner product and norm.

We fix the following notation concerning highest weight theory for the general linear and symplectic groups.

$B_n$ will denote the Borel subgroup of lower triangular matrices in $\mathrm{GL}(n,{\mathbb {C}})$, ${\mathfrak {h}}_n$ the Cartan subalgebra of diagonal matrices in $\mathrm{gl}(n,{\mathbb {C}})$ and $\varepsilon _j\in {\mathfrak {h}}_n^*$ the functional
$$\begin{aligned} \varepsilon _j\bigl (\mathrm{diag}(d_1,\ldots ,d_n)\bigr )=d_j,\quad (1\le j\le n). \end{aligned}$$
(5.1)
The compact symplectic group is $\mathrm{Sp}(2n)=\mathrm{Sp}(2n,{\mathbb {C}})\cap U(2n)$ where $\mathrm{Sp}(2n,{\mathbb {C}})$ is the subgroup of $\mathrm{GL}(2n,{\mathbb {C}})$ preserving the symplectic form
$$\begin{aligned} \omega \bigl ((z_1,\ldots ,z_{2n}), (w_1,\ldots ,w_{2n})\bigr )=\sum _{j=1}^n(z_jw_{n+j}-z_{n+j}w_j). \end{aligned}$$
(5.2)
The group $\mathrm{Sp}(2n,{\mathbb {C}})$ has Lie algebra
$$\begin{aligned} \mathrm{sp}(2n,{\mathbb {C}})=\left\{ \left[ \begin{array}{c|c}A&{}B\\ C&{}-A^t\end{array}\right] ~:~A,B,C\in \mathrm{gl}(n,{\mathbb {C}}),\ B^t=B,\ C^t=C\right\} . \end{aligned}$$
As Borel subgroup $B_{2n}^{\mathrm{Sp}}$ in $\mathrm{Sp}(2n,{\mathbb {C}})$, we choose $B_{2n}^{\mathrm{Sp}}=\exp (\mathfrak {b}_{2n}^{\mathrm{Sp}})$ where $\mathfrak {b}_{2n}^{\mathrm{Sp}}$ is the subalgebra of $\mathrm{sp}(2n,{\mathbb {C}})$ consisting of matrices as above with $B=0$ and $A$ lower triangular. A Cartan subalgebra in $\mathrm{sp}(2n,{\mathbb {C}})$ is given by
$$\begin{aligned} {\mathfrak {h}}_{2n}^{\mathrm{Sp}}=\bigl \{\mathrm{diag}(a_1,\ldots ,a_n,-a_1,\ldots ,-a_n)~:~a_j\in {\mathbb {C}}\bigr \} \end{aligned}$$
and in this context we let $\varepsilon _j\in ({\mathfrak {h}}_{2n}^{\mathrm{Sp}})^*$ ($1\le j\le n$) denote the linear functional
$$\begin{aligned} \varepsilon _j\bigl (\mathrm{diag}(a_1,\ldots ,a_n,-a_1,\ldots ,-a_n)\bigr )=a_j. \end{aligned}$$
(5.3)

5.2 $ \mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\mathbf{SU(n)}\ (n\ge 2)$

Here $K=U(n)\times \mathbb {T}$ acts on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus {\mathbb {C}}^n\cong M_{n,2}({\mathbb {C}})$ via

$$\begin{aligned} (k,c)\cdot z=kz\left[ \begin{array}{l@{\quad }l} c&{}0\\ 0&{}1 \end{array}\right] =\bigl [ckz_{\bullet ,1} \bigl | kz_{\bullet ,2}\bigr ] \end{aligned}$$

for $(k,c)\in K$ and $z\in M_{n,2}({\mathbb {C}})$.

As Borel subgroup in $K_{\mathbb {C}}=\mathrm{GL}(n,{\mathbb {C}})\times {\mathbb {C}}^\times $, we take $B=B_n\times {\mathbb {C}}^\times $ and as Cartan subalgebra ${\mathfrak {h}}={\mathfrak {h}}_n\times {\mathbb {C}}$. Let $T=(0,1)\in {\mathfrak {h}}$ and $\varepsilon _\circ \in {\mathfrak {h}}^*$ be the functional with $\varepsilon _\circ (T)=1$, $\varepsilon _\circ |_{{\mathfrak {h}}_n}=0$. This is a rank 3 multiplicity free action with fundamental $B$-highest weights and associated highest weight vectors

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-(\varepsilon _1+\varepsilon _\circ ) &{} h_1(z)=z_{11}\\ \alpha _2=-\varepsilon _1 &{} h_2(z)=z_{12}\\ \alpha _3=-(\varepsilon _1+\varepsilon _2+\varepsilon _\circ ) &{} h_3(z)={\det }_2(z) \end{array}\right\} . \end{aligned}$$

Here ${\det }_2(z)$ is the determinant of the $2\times 2$ matrix formed by the first two rows in $z$. For nonnegative integer exponents, $h_\alpha =h_1^ah_2^bh_3^c$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha =a\alpha _1+b\alpha _2+c\alpha _3=-\bigl ((a+b+c)\varepsilon _1+c\varepsilon _2+(a+c)\varepsilon _\circ \bigr ). \end{aligned}$$

One has $\left\langle T\cdot z,z\right\rangle =\Vert z_{\bullet ,1}\Vert ^2$ and $\left\langle E_{i,j}\cdot z,z\right\rangle =\left\langle z_{j,\bullet }, z_{i,\bullet }\right\rangle $ for elementary matrices $E_{i,j}\in \mathrm{gl}(n,{\mathbb {C}})$. Thus Lemma 2.2 shows that the matrix entries $z_{ij}$ of a spherical point for $\alpha $ must satisfy

$$\begin{aligned} \Vert z_{\bullet ,1}\Vert ^2=a+c,\quad \Vert z_{1,\bullet }\Vert ^2=a+b+c,\quad \Vert z_{2,\bullet }\Vert ^2=c \end{aligned}$$

and $\left\langle z_{j,\bullet },z_{i,\bullet }\right\rangle =0$ for $i\ne j$.

It is easy to verify that the entries of

$$\begin{aligned} v(a,b,c):=\left[ \begin{array}{l@{\quad }l} \sqrt{ \frac{a(a+b+c)}{a+b}}&{}\sqrt{\frac{b({a+b+c})}{a+b}}\\ -\sqrt{\frac{bc}{a+b}}&{}\sqrt{\frac{ac}{a+b}}\\ 0&{}0\\ \vdots &{}\vdots \\ 0&{}0\end{array} \right] \end{aligned}$$

(5.4)

satisfy the above equations for any given real parameters $a,b,c\ge 0$ provided $a+b\ne 0$. In particular (5.4) is a generic generalized spherical point if $a,b,c>0$. To show that the action $U(n)\times \mathbb {T}:M_{n,2}({\mathbb {C}})$ is well-behaved it remains to check conditions (2)–(4) of Lemma 2.5.

Evaluating the fundamental highest weight vectors at $v=v(a,b,c)$ yields

$$\begin{aligned} h_1(v)=\sqrt{ \frac{a(a+b+c)}{a+b}},\quad h_2(v)=\sqrt{\frac{b({a+b+c})}{a+b}},\quad h_3(v)=\sqrt{c(a+b+c)}.\qquad \end{aligned}$$

(5.5)

As these values are nonzero for positive parameters $(a,b,c)$, condition (2) in Lemma 2.5 holds here. As regards condition (3) in the lemma, we just need to observe that for fixed $c>0$ the limit

$$\begin{aligned} \lim _{b\rightarrow 0^+}\lim _{a\rightarrow 0^+}v(a,b,c) =\lim _{b\rightarrow 0^+}\left[ \begin{array}{l@{\quad }l} 0 &{}\quad \sqrt{b+c}\\ -\sqrt{c} &{}\quad 0 \\ 0&{}\quad 0\\ \vdots &{}\quad \vdots \\ 0&{}\quad 0 \end{array}\right] = \left[ \begin{array}{l@{\quad }l} 0 &{}\quad \sqrt{c} \\ -\sqrt{c}&{}\quad 0 \\ 0&{}\quad 0\\ \vdots &{}\quad \vdots \\ 0&{}\quad 0\end{array}\right] \end{aligned}$$

does exist. Limiting values for $h_j\bigl (v(a,b,c)\bigr )$ ($j=1,2,3$) as one or two parameters approach zero are given by

$$\begin{aligned} \left\{ \begin{array}{l|l|l|l} \text {limit}&{}h_1\bigl (v(a,b,c)\bigr )&{}h_2\bigl (v(a,b,c)\bigr )&{}h_3\bigl (v(a,b,c)\bigr )\\ \hline \lim _{a\rightarrow 0^+} &{} 0 &{} \sqrt{b+c}&{} \sqrt{c(b+c)}\\ \hline \lim _{b\rightarrow 0^+} &{} \sqrt{a+c} &{} 0 &{} \sqrt{c(a+c)}\\ \hline \lim _{c\rightarrow 0^+} &{} \sqrt{a} &{} \sqrt{b} &{} 0 \\ \hline \lim _{b\rightarrow 0^+}\lim _{a\rightarrow 0^+} &{} 0 &{} \sqrt{c} &{} c\\ \hline \lim _{c\rightarrow 0^+}\lim _{a\rightarrow 0^+} &{} 0 &{} \sqrt{b} &{} 0 \\ \hline \lim _{c\rightarrow 0^+}\lim _{b\rightarrow 0^+} &{} \sqrt{a} &{} 0 &{} 0 \end{array}\right\} . \end{aligned}$$

These show, in particular, that condition (4) in Lemma 2.5 holds.

5.3 $\mathbf{SU(n)}\oplus _\mathbf{SU(n)}\mathbf{SU(n)}^*\ ({n}\ge 3)$

This is a twisted variant of Example 5.2 with $K=U(n)\times \mathbb {T}$ acting on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus {\mathbb {C}}^n\cong M_{n,2}({\mathbb {C}})$ via

$$\begin{aligned} (k,c)\cdot z=\bigl [ckz_{\bullet ,1}\ |\ k^{-t}z_{\bullet ,2}\bigr ]\quad \quad \text {where } k^{-t}:=(k^{-1})^t. \end{aligned}$$

Here the action in the second column is contragredient to the standard action in the first column. One takes $n\ge 3$ here as Examples 5.3 and 5.2 are geometrically equivalent when $n=2$. This is so because the standard representation for $SU(2)$ is self-contragredient.

Again $K:V$ has rank 3 with fundamental $B$-highest weights and highest weight vectors

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-(\varepsilon _1+\varepsilon _\circ ) &{} h_1(z)=z_{11}\\ \alpha _2=+\varepsilon _n &{} h_2(z)=z_{n2}\\ \alpha _3=-\varepsilon _\circ &{} h_3(z)=\sum _{i=1}^nz_{i1}z_{i2}=z_{\bullet ,1}\cdot z_{\bullet ,2} \end{array}\right\} . \end{aligned}$$

For nonnegative integer exponents, $h_\alpha =h_1^ah_2^bh_3^c$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha =a\alpha _1+b\alpha _2+c\alpha _3=-\bigl (a\varepsilon _1-b\varepsilon _n+(a+c)\varepsilon _\circ \bigr ). \end{aligned}$$

Now $\left\langle T\cdot z,z\right\rangle =\Vert z_{\bullet ,1}\Vert ^2$ as before but $\left\langle E_{i,j}\cdot z,z\right\rangle =z_{j1}\overline{z_{i1}}-z_{i2}\overline{z_{j2}}$ for elementary matrices $E_{i,j}\in \mathrm{gl}(n,{\mathbb {C}})$. So the matrix entries $z_{ij}$ of a spherical point for $\alpha $ must satisfy

$$\begin{aligned} \Vert z_{\bullet ,1}\Vert ^2=a+c,\quad |z_{11}|^2-|z_{12}|^2=a,\quad |z_{n1}|^2-|z_{n2}|^2=-b \end{aligned}$$

and $z_{j1}\overline{z_{i1}}-z_{i2}\overline{z_{j2}}=0$ for $i\ne j$. The matrix entries in

$$\begin{aligned} v(a,b,c):=\left[ \begin{array}{c@{\quad }c} \sqrt{ \frac{a(a+b+c)}{a+b}}&{}\quad \sqrt{\frac{ac}{a+b}}\\ 0&{}\quad 0\\ \vdots &{}\quad \vdots \\ 0&{}\quad 0\\ \sqrt{\frac{bc}{a+b}}&{}\quad \sqrt{\frac{b({a+b+c})}{a+b}} \end{array} \right] \end{aligned}$$

(5.6)

satisfy these equations for any real values $a,b,c\ge 0$ with $a+b\ne 0$. In particular (5.6) is a generic generalized spherical point when $a,b,c>0$. Evaluating the fundamental highest weight vectors $h_j$ at $v=v(a,b,c)$ again yields Eq. 5.5. Thus, conditions (2)–(4) from Lemma 2.5 hold as verified in Example 5.2.

5.4 $(\mathbf{SU(n)}\otimes \mathbf{SU}(2))\oplus _\mathbf{SU}(2)(\mathbf{SU}(2)\otimes \mathbf{SU(m)}) \ (n,m\ge 2)$

Here $K=U(n)\times U(m)\times U(2)$ acts on $V=V_1\oplus V_2=({\mathbb {C}}^n\otimes {\mathbb {C}}^2)\oplus ({\mathbb {C}}^m\otimes {\mathbb {C}}^2)\cong M_{n+m,2}({\mathbb {C}})$ via

$$\begin{aligned} (k_1,k_2,k_3)\cdot z=\left[ \begin{array}{l|l}k_1 &{} 0 \\ \hline 0 &{} k_2\end{array}\right] z k_3^t. \end{aligned}$$

(5.7)

We take Borel subgroup $B=B_n\times B_m\times B_2$ in $K_{\mathbb {C}}=\mathrm{GL}(n,{\mathbb {C}})\times \mathrm{GL}(m,{\mathbb {C}})\times \mathrm{GL}(2,{\mathbb {C}})$ and let $\varepsilon _j$, $\varepsilon _j'$, $\varepsilon _j''$ denote functionals on ${\mathfrak {h}}={\mathfrak {h}}_n\times {\mathfrak {h}}_m\times {\mathfrak {h}}_2$ as in (5.1) supported on each of the three factors. This is a rank 5 multiplicity free action with fundamental $B$-highest weights and highest weight vectors

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-(\varepsilon _1+\varepsilon _1'') &{} h_1(z)=z_{11}\\ \alpha _2=-(\varepsilon _1'+\varepsilon _1'') &{} h_2(z)=z_{n+1,1}\\ \alpha _3=-(\varepsilon _1+\varepsilon _2+\varepsilon _1''+\varepsilon _2'') &{} h_3(z)={\det }_2(z)\\ \alpha _4=-(\varepsilon _1'+\varepsilon _2'+\varepsilon _1''+\varepsilon _2'') &{} h_4(z)=\left| \begin{array}{ll}z_{n+1,1}&{}z_{n+1,2}\\ z_{n+2,1}&{}z_{n+2,2}\end{array}\right| \\ \\ \alpha _5=-(\varepsilon _1+\varepsilon _1'+\varepsilon _1''+\varepsilon _2'') &{} h_5(z)=\left| \begin{array}{ll}z_{11}&{}z_{12}\\ z_{n+1,1}&{}z_{n+1,2}\end{array}\right| \end{array}\right\} . \end{aligned}$$

(5.8)

For nonnegative integer exponents $h_\alpha =h_1^ah_2^bh_3^ch_4^dh_5^e$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+c\alpha _3+d\alpha _4+e\alpha _5\\&= -\bigl ((a+c+e)\varepsilon _1+c\varepsilon _2+(b+d+e)\varepsilon _1'+d\varepsilon _2'+(a+b+c+d+e)\varepsilon _1''\\&+(c+d+e)\varepsilon _2''\bigr ). \end{aligned}$$

Letting $E_{i,j}$, $E_{i,j}'$ and $E_{i,j}''$ denote elementary matrices in $\mathrm{gl}(n,{\mathbb {C}})$, $\mathrm{gl}(m,{\mathbb {C}})$ and $\mathrm{gl}(2,{\mathbb {C}})$, respectively, one has

$$\begin{aligned} \left\langle E_{i,j}\cdot z,z\right\rangle =\left\langle z_{j,\bullet },z_{i,\bullet }\right\rangle ,\quad \left\langle E_{i,j}'\cdot z,z\right\rangle =\left\langle z_{n+j,\bullet },z_{n+i,\bullet }\right\rangle ,\quad \left\langle E_{i,j}''\cdot z,z\right\rangle =\left\langle z_{\bullet ,j},z_{\bullet ,i}\right\rangle \end{aligned}$$

for $z\in M_{n+m,2}({\mathbb {C}})$. Thus a spherical point $z$ for weight $\alpha $ must have

orthogonal columns,
rows $1\ldots n$ pairwise orthogonal and rows $n+1\ldots n+m$ pairwise orthogonal,
$\Vert z_{1\bullet }\Vert ^2=a+c+e$, $\Vert z_{2,\bullet }\Vert ^2=c$, $\Vert z_{n+1,\bullet }\Vert ^2=b+d+e$, $\Vert z_{n+2,\bullet }\Vert ^2=d$,
$\Vert z_{\bullet ,1}\Vert ^2=a+b+c+d+e$, $\Vert z_{\bullet ,2}\Vert ^2=c+d+e$.

To solve these equations, we may set rows other than 1, 2, $n+1$ and $n+2$ to zero and reduce to the case $n=m=2$. So now $K=U(2)\times U(2)\times U(2)$, $V=M_{4,2}({\mathbb {C}})$. One can check that the matrix entries of

$$\begin{aligned} v\bigl (a,b,c,d,e):=\left[ \begin{array}{l@{\quad }l} -\sqrt{{\frac{ a\left( a+b+e \right) \left( a+c+e \right) }{ \left( a+b \right) \left( a+e \right) }}}&{} \sqrt{{\frac{ be\left( a+c+e \right) }{ \left( a+b \right) \left( a+e \right) }}}\\ \sqrt{{\frac{bce}{ \left( a+b \right) \left( a+e \right) }}}&{} \sqrt{{\frac{ ac\left( a+b+e\right) }{ \left( a+b\right) \left( a+e \right) }}}\\ \sqrt{{\frac{ b\left( a+b+e \right) \left( b+d+e \right) }{ \left( a+b \right) \left( b+e \right) }}}&{} \sqrt{{\frac{ae \left( b+d+e \right) }{ \left( a+b \right) \left( b+e\right) }}}\\ -\sqrt{{\frac{ade}{ \left( a+b \right) \left( b+e \right) }}}&{} \sqrt{{\frac{b d\left( a+b+e \right) }{ \left( a+b \right) \left( b+e \right) }}}\end{array} \right] \end{aligned}$$

satisfy each of the above conditions for arbitrary positive real parameters. This is a generic generalized spherical point for this example. Evaluating the fundamental highest weight vectors at $v=v(a,b,c,d,e)$ gives

$$\begin{aligned} \left\{ \begin{array}{l} \begin{array}{ll} h_1(v)=-\sqrt{{\frac{ a\left( a+b+e \right) \left( a+c+e \right) }{ \left( a+b \right) \left( a+e \right) }}} &{} h_2(v)=\sqrt{{\frac{ b\left( a+b+e \right) \left( b+d+e \right) }{ \left( a+b \right) \left( b+e \right) }}} \\ h_3(v)=-\sqrt{c(a+c+e)} &{} h_4(v)=\sqrt{d(b+d+e)} \end{array} \\ h_5(v)=-\sqrt{\frac{e(a+b+e)(a+c+e)(b+d+e)}{(a+e)(b+e)}} \end{array}\right\} . \end{aligned}$$

These values are nonzero as required by Lemma 2.5 condition (2). To check conditions (3) and (4) in the Lemma, one needs to take limits as one or more parameters approach zero in succession and compute the limiting values of the $h_j(v)$’s. This is routine but there are $2^5-2=30$ limits to examine in all. We used Maple to perform this task. For example, one finds

$$\begin{aligned} \lim _{c\rightarrow 0^+}\lim _{a\rightarrow 0^+}v(a,b,c,d,e)=\lim _{c\rightarrow 0^+} \left[ \begin{array}{ll} 0&{}\sqrt{c+e} \\ \sqrt{c}&{}0\\ \sqrt{b+d+e}&{}0\\ 0&{}\sqrt{d}\end{array}\right] = \left[ \begin{array}{ll} 0&{}\sqrt{e} \\ 0&{}0\\ \sqrt{b+d+e}&{}0\\ 0&{}\sqrt{d}\end{array}\right] =v_\circ \end{aligned}$$

say and each of the values

$$\begin{aligned} h_2(v_\circ )=\sqrt{b+d+e},\quad h_4(v_\circ )=\sqrt{d(b+d+e)},\quad h_5(v_\circ )=-\sqrt{e(b+d+e)} \end{aligned}$$

are nonzero.

5.5 $(\mathbf{Sp(2n)}\otimes \mathbf{{SU}(2)})\oplus _\mathbf{SU(2)}(\mathbf{{SU}(2)}\otimes \mathbf{SU(m)}) \ (n,m\ge 2)$

Now $K=\mathrm{Sp}(2n)\times U(m)\times U(2)$ acts on $V=V_1\oplus V_2=({\mathbb {C}}^{2n}\otimes {\mathbb {C}}^2)\oplus ({\mathbb {C}}^m\otimes {\mathbb {C}}^2)\cong M_{2n+m,2}({\mathbb {C}})$ as in Eq. 5.7 from the previous example. This is a rank 6 multiplicity free action.

We use Borel subgroup $B=B_{2n}^{\mathrm{Sp}}\times B_m\times B_2$ in $K_{\mathbb {C}}=\mathrm{Sp}(2n,{\mathbb {C}})\times \mathrm{GL}(m,{\mathbb {C}})\times \mathrm{GL}(2,{\mathbb {C}})$ and Cartan subalgebra ${\mathfrak {h}}={\mathfrak {h}}_{2n}^{\mathrm{Sp}}\times {\mathfrak {h}}_m\times {\mathfrak {h}}_2$. Let $\varepsilon _j\in {\mathfrak {h}}^*$ be given by (5.3) on ${\mathfrak {h}}_{2n}^{\mathrm{Sp}}$ and $\varepsilon _j',\varepsilon _j''\in {\mathfrak {h}}^*$ be given by (5.1) on the factors ${\mathfrak {h}}_m$ and ${\mathfrak {h}}_2$, respectively. With these notational conventions, the fundamental highest weights and highest weight vectors are as in Eq. 5.8 above (but with “$n$” replaced by “$2n$” in the formulas for $h_2(z)$, $h_4(z)$, $h_5(z)$) together with

$$\begin{aligned} \alpha _6=-(\varepsilon _1''+\varepsilon _2''),\quad h_6(z)=\omega (z'_{\bullet ,1},z'_{\bullet ,2}), \end{aligned}$$

where $z'\in M_{2n,2}({\mathbb {C}})$ denotes the first $2n$ rows in $z\in M_{2n+m,2}({\mathbb {C}})$ and $\omega $ the symplectic inner product (5.2). For nonnegative integer exponents, $h_\alpha =h_1^ah_2^bh_3^ch_4^dh_5^eh_6^f$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+ c\alpha _3+d\alpha _4+ e\alpha _5 +f\alpha _6\\&= -\bigl ((a+c+e)\varepsilon _1+c\varepsilon _2+(b+d+e)\varepsilon _1'+d\varepsilon _2'+(a+b+c+d+e+f)\varepsilon _1'' \\&\quad +(c+d+e+f)\varepsilon _2''\bigr ). \end{aligned}$$

It will suffice, as in the Example 5.4, to consider the case $n=m=2$. So now $K=\mathrm{Sp}(4)\times U(2)\times U(2)$ and $V=M_{6,2}({\mathbb {C}})$. From the actions of the two copies of $\mathrm{gl}(2,{\mathbb {C}})$, one obtains $\left\langle E_{i,j}'\cdot z,z\right\rangle =\left\langle z_{4+j,\bullet },z_{4+i,\bullet }\right\rangle $ and $\left\langle E_{i,j}''\cdot z,z\right\rangle =\left\langle z_{\bullet ,j},z_{\bullet ,i}\right\rangle $. The action of $\mathrm{sp}(4,{\mathbb {C}})$ gives

$$\begin{aligned} \left\{ \begin{array}{ll} \left\langle (E_{1,1}-E_{3,3})\cdot z,z\right\rangle =\Vert z_{1,\bullet }\Vert ^2-\Vert z_{3,\bullet }\Vert ^2 &{} \left\langle E_{1,3}\cdot z,z\right\rangle =\left\langle z_{3,\bullet },z_{1,\bullet }\right\rangle \\ \left\langle (E_{2,2}-E_{4,4})\cdot z,z\right\rangle =\Vert z_{2,\bullet }\Vert ^2-\Vert z_{4,\bullet }\Vert ^2 &{} \left\langle E_{2,4}\cdot z,z\right\rangle =\left\langle z_{4,\bullet },z_{2,\bullet }\right\rangle \\ \left\langle (E_{1,2}-E_{4,3})\cdot z,z\right\rangle =\left\langle z_{2,\bullet },z_{1,\bullet }\right\rangle -\left\langle z_{3,\bullet },z_{4,\bullet }\right\rangle \\ \left\langle (E_{1,4}+E_{2,3})\cdot z,z\right\rangle =\left\langle z_{4,\bullet },z_{1,\bullet }\right\rangle +\left\langle z_{3,\bullet },z_{2,\bullet }\right\rangle \end{array}\right\} . \end{aligned}$$

Thus a spherical point $z\in M_{6,2}({\mathbb {C}})$ for weight $\alpha $ must have

$\left\langle z_{\bullet ,1},z_{\bullet ,2}\right\rangle =0$,
$\left\langle z_{1,\bullet },z_{3,\bullet }\right\rangle =0=\left\langle z_{2,\bullet },z_{4,\bullet }\right\rangle =\left\langle z_{5,\bullet },z_{6,\bullet }\right\rangle $,
$\left\langle z_{1,\bullet },z_{2,\bullet }\right\rangle =\left\langle z_{4,\bullet },z_{3,\bullet }\right\rangle $, $\left\langle z_{1,\bullet },z_{4,\bullet }\right\rangle =-\left\langle z_{2,\bullet },z_{3,\bullet }\right\rangle $
$\Vert z_{1,\bullet }\Vert ^2-\Vert z_{3,\bullet }\Vert ^2=a+c+e$, $\Vert z_{2,\bullet }\Vert ^2-\Vert z_{4,\bullet }\Vert ^2=c$,
$\Vert z_{5,\bullet }\Vert ^2=b+d+e$, $\Vert z_{6,\bullet }\Vert ^2=d$,
$\Vert z_{\bullet ,1}\Vert =a+b+c+d+e+f$, $\Vert z_{\bullet ,2}\Vert ^2=c+d+e+f$.

A generic generalized spherical point whose matrix entries solve these equations for arbitrary positive values of $a,\ldots , f$ is given below.

$$\begin{aligned} v(a,b,c,d,e,f):=\left[ \begin{array}{l@{\quad }l} \sqrt{{\frac{a \left( a+b+e \right) \left( a+c+e \right) \left( a+2\,c+e+f \right) }{ (a+b)(a+e)(a+2c+e) }}} &{} -\sqrt{{\frac{ be\left( a+c+e \right) \left( a+2\,c+e+f \right) }{ (a+b)(a+e)(a+2c+e) }}} \\ \sqrt{{\frac{bce \left( a+2\,c+e+f \right) }{ (a+b)(a+e)(a+2c+e) }}} &{} \sqrt{{\frac{ ac\left( a+b+e \right) \left( a+2\,c+e+f \right) }{ (a+b)(a+e)(a+2c+e) }}} \\ \sqrt{{\frac{bef \left( a+c+e \right) }{ (a+b)(a+e)(a+2c+e) }}} &{} \sqrt{{\frac{af \left( a+b+e \right) \left( a+c+e \right) }{ (a+b)(a+e)(a+2c+e) }}} \\ -\sqrt{{\frac{acf \left( a+b+e \right) }{ (a+b)(a+e)(a+2c+e) }}} &{} \sqrt{{\frac{bcef}{ (a+b)(a+e)(a+2c+e) }}} \\ -\sqrt{{\frac{ b\left( a+b+e \right) \left( b+d+e \right) }{ (a+b)(b+e) }}} &{} -\sqrt{{\frac{ ae\left( b+d+e \right) }{ (a+b)(b+e) }}} \\ -\sqrt{{\frac{ade}{ (a+b)(b+e) }}} &{} \sqrt{{\frac{bd \left( a+b+e \right) }{ (a+b)(b+e) }}} \end{array} \right] \end{aligned}$$

Evaluating the $h_j's$ at $v=v(a,b,c,d,e,f)$ yields

$$\begin{aligned} \left\{ \begin{array}{ll} h_1(v)=\sqrt{{\frac{a \left( a+b+e \right) \left( a+c+e \right) \left( a+2\,c+e+f \right) }{ (a+b)(a+e)(a+2c+e) }}} &{} h_2(v)=-\sqrt{{\frac{ b\left( a+b+e \right) \left( b+d+e \right) }{ (a+b)(b+e) }}} \\ \\ h_3(v)=\frac{(a+2c+e+f)\sqrt{c(a+c+e)}}{a+2c+e} &{} h_4(v)=-\sqrt{d(b+d+e)} \\ \\ h_5(v)=-\sqrt{\frac{e(a+b+e)(a+c+e)(b+d+e)(a+2c+e+f)}{(a+e)(b+e)(a+2c+e)}} &{} h_6(v)=\sqrt{f(a+2c+e+f)} \end{array}\right\} . \end{aligned}$$

upon simplification using a computer algebra system. These show, in particular, that condition (2) in Lemma 2.5 holds. We also used Maple to check conditions (3) and (4) from the lemma, completing the verification for this example. This entails routine examination of $2^6-2=62$ limits.

5.6 $(\mathbf{Sp(2n)}\otimes \mathbf{SU(2)})\oplus _{\mathbf{SU(2)}}(\mathbf{SU(2)}\otimes \mathbf{Sp(2m)}) \ (n,m\ge 2)$

Next $K=\mathrm{Sp}(2n)\times \mathrm{Sp}(2m)\times U(2)\times \mathbb {T}$ acts on $V=V_1\oplus V_2=({\mathbb {C}}^{2n}\otimes {\mathbb {C}}^2)\oplus ({\mathbb {C}}^{2m}\otimes {\mathbb {C}}^2)\cong M_{2(n+m),2}({\mathbb {C}})$. Letting $z'$ and $z''$ denote the first $2n$ and last $2m$ rows of $z\in M_{2(n+m),2}({\mathbb {C}})$, we have

$$\begin{aligned} (k_1,k_2,k_3,c)\cdot z=\left[ \begin{array}{l|l} cI_{2n}&{}0\\ \hline 0&{}I_{2m}\end{array}\right] \left[ \begin{array}{l|l}k_1 &{} 0 \\ \hline 0 &{} k_2\end{array}\right] z k_3^t =\left[ \begin{array}{l|l}ck_1z_{\bullet ,1}' &{} ck_1z_{\bullet ,2}' \\ \hline k_2z_{\bullet ,1}'' &{} k_2z_{\bullet ,2}''\end{array}\right] k_3^t. \end{aligned}$$

The factor $\mathbb {T}$ is required to fully saturate this example. In fact this action fails to be multiplicity free if the circle is removed [3, Theorem 6].

We use Borel subgroup $B=B_{2n}^{\mathrm{Sp}}\times B_{2m}^{\mathrm{Sp}}\times B_2\times {\mathbb {C}}^\times $ in $K_{\mathbb {C}}$ and Cartan subalgebra ${\mathfrak {h}}={\mathfrak {h}}_{2n}^{\mathrm{Sp}}\times {\mathfrak {h}}_{2m}^{\mathrm{Sp}}\times {\mathfrak {h}}_2\times {\mathbb {C}}$. Let

$\varepsilon _j,\varepsilon _j'\in {\mathfrak {h}}^*$ be as in (5.3) on the symplectic factors ${\mathfrak {h}}_{2n}^{\mathrm{Sp}}$ and ${\mathfrak {h}}_{2m}^{\mathrm{Sp}}$,
$\varepsilon _j''\in {\mathfrak {h}}^*$ be as in (5.1) on the ${\mathfrak {h}}_2$ factor and
$\varepsilon _\circ \in {\mathfrak {h}}^*$ be dual to $T=(0,0,0,1)\in {\mathfrak {h}}$.

The action $K:V$ has rank 7 with fundamental highest weights

$$\begin{aligned} \left\{ \begin{array}{l} \begin{array}{ll} \alpha _1=-(\varepsilon _1+\varepsilon _1''+\varepsilon _\circ ) &{} \alpha _4=-(\varepsilon _1'+\varepsilon _2'+\varepsilon _1''+\varepsilon _2'') \\ \alpha _2=-(\varepsilon _1'+\varepsilon _1'') &{} \alpha _5=-(\varepsilon _1+\varepsilon _1'+\varepsilon _1''+\varepsilon _2''+\varepsilon _\circ ) \\ \alpha _3=-(\varepsilon _1+\varepsilon _2+\varepsilon _1''+\varepsilon _2''+2\varepsilon _\circ ) &{} \alpha _6=-(\varepsilon _1''+\varepsilon _2''+2\varepsilon _\circ ) \end{array} \\ \alpha _7=-(\varepsilon _1''+\varepsilon _2'') \end{array} \right\} . \end{aligned}$$

Fundamental highest weight vectors $h_1(z),\ldots ,h_6(z)$ with weights $\alpha _1,\ldots ,\alpha _6$ are as in Example 5.5. A highest weight vector for $\alpha _7$ is

$$\begin{aligned} h_7(z)=\omega (z''_{\bullet ,1},z''_{\bullet ,2}). \end{aligned}$$

For nonnegative integer exponents $h_\alpha =h_1^ah_2^bh_3^ch_4^dh_5^eh_6^fh_7^g$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+ c\alpha _3+d\alpha _4+ e\alpha _5 +f\alpha _6+g\alpha _7\\&= -\bigl ((a+c+e)\varepsilon _1+c\varepsilon _2+(b+d+e)\varepsilon _1'+d\varepsilon _2'+(a+b+c+d+e+f+g)\varepsilon _1'' \\&\quad +(c+d+e+f+g)\varepsilon _2''+(a+2c+e+2f)\varepsilon _\circ \bigr ). \end{aligned}$$

As in Examples 5.4, 5.5, we need only consider the case $n=m=2$. So now $K=\mathrm{Sp}(4)\times \mathrm{Sp}(4)\times U(2)\times \mathbb {T}$ and $V=M_{8,2}({\mathbb {C}})$. Lemma 2.2 yields a system of equations for the matrix entries of a spherical point $z\in M_{8,2}({\mathbb {C}})$ for weight $\alpha $, just as in Example 5.5, namely

$\left\langle z_{\bullet ,1},z_{\bullet ,2}\right\rangle =0$,
$\left\langle z_{1,\bullet },z_{3,\bullet }\right\rangle =0=\left\langle z_{2,\bullet },z_{4,\bullet }\right\rangle =\left\langle z_{5,\bullet },z_{7,\bullet }\right\rangle =\left\langle z_{6,\bullet },z_{8,\bullet }\right\rangle $,
$\left\langle z_{1,\bullet },z_{2,\bullet }\right\rangle =\left\langle z_{4,\bullet },z_{3,\bullet }\right\rangle $, $\left\langle z_{1,\bullet },z_{4,\bullet }\right\rangle =-\left\langle z_{2,\bullet },z_{3,\bullet }\right\rangle $
$\left\langle z_{5,\bullet },z_{6,\bullet }\right\rangle =\left\langle z_{8,\bullet },z_{7,\bullet }\right\rangle $, $\left\langle z_{5,\bullet },z_{8,\bullet }\right\rangle =-\left\langle z_{6,\bullet },z_{7,\bullet }\right\rangle $
$\Vert z_{1,\bullet }\Vert ^2-\Vert z_{3,\bullet }\Vert ^2=a+c+e$, $\Vert z_{2,\bullet }\Vert ^2-\Vert z_{4,\bullet }\Vert ^2=c$,
$\Vert z_{5,\bullet }\Vert ^2-\Vert z_{7,\bullet }\Vert =b+d+e$, $\Vert z_{6,\bullet }\Vert ^2-\Vert z_{8,\bullet }\Vert =d$,
$\Vert z_{\bullet ,1}\Vert =a+b+c+d+e+f+g$, $\Vert z_{\bullet ,2}\Vert ^2=c+d+e+f+g$,
$\left\langle z',z'\right\rangle =\Vert z_{\bullet ,1}'\Vert ^2+\Vert z_{\bullet ,2}'\Vert ^2=a+2c+e+2f$.

A generic generalized spherical point whose matrix entries solve these equations for arbitrary positive values of $a,\ldots , g$ is given below.

$$\begin{aligned} v(a,b,c,d,e,f,g):= \left[ \begin{array}{l@{\quad }l} -\sqrt{{\frac{a (a+b+e) (a+c+e)\left( a+2\,c+e+f \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}} &{} \sqrt{{\frac{be \left( a+c+e \right) \left( a+2\,c+e+f \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}}\\ -\sqrt{{\frac{bce \left( a+2\,c+e+f \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}} &{} -\sqrt{{\frac{ac \left( a+b+e \right) \left( a+2\,c+e+f \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}}\\ \sqrt{{\frac{bef \left( a+c+e \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}} &{} \sqrt{{\frac{af \left( a+b+e \right) \left( a+c+e \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}}\\ -\sqrt{{\frac{acf \left( a+b+e \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}} &{} \sqrt{{\frac{bcef}{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}}\\ -\sqrt{{\frac{ b(a+b+e)\left( b+d+e \right) \left( b+2\,d+e+g \right) }{ (a+b)(b+e)(b+2d+e) }}} &{} -\sqrt{{\frac{ ae \left( b+d+e \right) \left( b+2\,d+e+g \right) }{ (a+b)(b+e)(b+2d+e) }}} \\ -\sqrt{{\frac{ ade\left( b+2\,d+e+g \right) }{ (a+b)(b+e)(b+2d+e) }}} &{} \sqrt{{\frac{ bd\left( a+b+e \right) \left( b+2\,d+e+g \right) }{ (a+b)(b+e)(b+2d+e) }}} \\ -\sqrt{{\frac{ aeg\left( b+d+e \right) }{ (a+b)(b+e)(b+2d+e) }}} &{} \sqrt{{\frac{ bg\left( a+b+e \right) \left( b+d+e \right) }{ (a+b)(b+e)(b+2d+e) }}}\\ \sqrt{{\frac{ bdg\left( a+b+e \right) }{ (a+b)(b+e)(b+2d+e) }}} &{} \sqrt{{\frac{adeg}{ (a+b)(b+e)(b+2d+e) }}} \end{array} \right] . \end{aligned}$$

The fundamental highest weight vectors take the following nonzero values at $v=v(a,b,c,d,e,f,g)$.

$$\begin{aligned} \left\{ \begin{array}{l} \begin{array}{l@{\quad }l} h_1(v)=-\sqrt{{\frac{a (a+b+e) (a+c+e)\left( a+2\,c+e+f \right) }{ \left( a+b \right) \left( a+e \right) \left( a+2\,c+e \right) }}} &{} h_2(v)=-\sqrt{{\frac{ b(a+b+e)\left( b+d+e \right) \left( b+2\,d+e+g \right) }{ (a+b)(b+e)(b+2d+e) }}} \\ h_3(v)=\frac{(a+2c+e+f)\sqrt{c(a+c+e)}}{a+2c+e} &{} h_4(v)=-\frac{(b+2d+e+g)\sqrt{d(b+d+e)}}{b+2d+e} \end{array}\\ h_5(v)=\sqrt{\frac{e(a+b+e)(a+c+e)(b+d+e)(a+2c+e+f)(b+2d+e+g)}{(a+e)(b+e)(a+2c+e)(b+2d+e)}} \\ \begin{array}{l@{\quad }l} h_6(v)=-\sqrt{f(a+2c+e+f)} &{} h_7(v)=-\sqrt{g(b+2d+e+g)} \end{array} \end{array}\right\} . \end{aligned}$$

To complete the verification that $K:V$ is well-behaved via Lemma 2.5, we used computer calculations to check each of $2^7-2=126$ relevant limits. A Maple worksheet giving full details can be found at [7].

5.7 $\mathbf{SU(2)}\oplus _{\mathbf{SU(2)}}(\mathbf{SU(2)}\otimes \mathbf{Sp(2n)}) \ (n\ge 2)$

Next consider $K=\mathrm{Sp}(2n)\times U(2)\times \mathbb {T}$ acting on $V=V_1\oplus V_2=({\mathbb {C}}^{2n}\otimes {\mathbb {C}}^2)\oplus {\mathbb {C}}^2\cong M_{2n+1,2}({\mathbb {C}})$ via

$$\begin{aligned} (k_1,k_2,c)\cdot z=\left[ \begin{array}{l|l}c&{}0\\ \hline 0&{}k_1\end{array}\right] zk_2^t. \end{aligned}$$

As Borel subgroup in $K_{\mathbb {C}}=\mathrm{Sp}(2n,{\mathbb {C}})\times \mathrm{GL}(2,{\mathbb {C}})\times {\mathbb {C}}^\times $, we take $B=B_{2n}^{\mathrm{Sp}}\times B_2\times {\mathbb {C}}^\times $ and as Cartan subalgebra ${\mathfrak {h}}={\mathfrak {h}}_{2n}^{\mathrm{Sp}}\times {\mathfrak {h}}_2\times {\mathbb {C}}$. Let $\varepsilon _j\in {\mathfrak {h}}^*$ be as in (5.3) on the symplectic factor ${\mathfrak {h}}_{2n}^{\mathrm{Sp}}$, $\varepsilon _j'$ be as in (5.1) on the ${\mathfrak {h}}_2$ factor and $\varepsilon _\circ $ dual to $T=(0,0,1)$.

This is a rank 5 multiplicity free action with the following fundamental $B$-highest weights and associated highest weight vectors. Here we let $z'\in M_{2n,2}({\mathbb {C}})$ be the matrix obtained by removing the first row from $z\in M_{2n+1,2}({\mathbb {C}})$.

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-(\varepsilon _1'+\varepsilon _\circ ) &{} h_1(z)=z_{11} \\ \alpha _2=-(\varepsilon _1+\varepsilon _1') &{} h_2(z)=z_{21}\\ \alpha _3=-(\varepsilon _1+\varepsilon _1'+ \varepsilon _2'+\varepsilon _\circ ) &{} h_3(z)={\det }_2(z) \\ \alpha _4=-(\varepsilon _1+\varepsilon _2+\varepsilon _1'+\varepsilon _2') &{} h_4(z)={\det _2}(z') \\ \alpha _5=-(\varepsilon _1'+\varepsilon _2') &{} h_5(z)=\omega (z_{\bullet ,1}', z_{\bullet ,2}') \end{array}\right\} . \end{aligned}$$

For nonnegative integer exponents $h_\alpha =h_1^ah_2^bh_3^ch_4^dh_5^e$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+c\alpha _3+d\alpha _4+e\alpha _5\\&= -\bigl ((b+c+d)\varepsilon _1+d\varepsilon _2+(a+b+c+d+e)\varepsilon _1'+(c+d+e)\varepsilon _2'+(a+c)\varepsilon _\circ \bigr ). \end{aligned}$$

As in prior examples, it suffices to examine the case $n=2$. So henceforth $K=\mathrm{Sp}(4)\times U(2)\times \mathbb {T}$ and $V=M_{5,2}({\mathbb {C}})$. Applying Lemma 2.2, one obtains the following equations for the matrix entries of a spherical point $z\in M_{5,2}({\mathbb {C}})$ for weight $\alpha $.

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \Vert z_{\bullet ,1}\Vert ^2=a+b+c+d+e &{} \left\langle z_{\bullet ,1},z_{\bullet ,2}\right\rangle =0 \\ \Vert z_{\bullet ,2}\Vert ^2=c+d+e &{} \left\langle z_{2,\bullet },z_{4,\bullet }\right\rangle =0 \\ \Vert z_{1,\bullet }\Vert ^2=a+c &{} \left\langle z_{3,\bullet },z_{5,\bullet }\right\rangle =0 \\ \Vert z_{2,\bullet }\Vert ^2-\Vert z_{4,\bullet }\Vert ^2=b+c+d &{} \left\langle z_{2,\bullet },z_{3,\bullet }\right\rangle =\left\langle z_{5,\bullet },z_{4,\bullet }\right\rangle \\ \Vert z_{3,\bullet }\Vert ^2-\Vert z_{5,\bullet }\Vert ^2=d &{} \left\langle z_{2,\bullet },z_{5,\bullet }\right\rangle =-\left\langle z_{3,\bullet },z_{4,\bullet }\right\rangle \end{array}\right\} . \end{aligned}$$

One can check that the entries of

$$\begin{aligned} v\bigl (a,b,c,d,e):= \left[ \begin{array}{l@{\quad }l} \sqrt{{\frac{a \left( a+b+c \right) }{a+b}}} &{} -\sqrt{{\frac{bc}{a+b}}} \\ -\sqrt{{\frac{ b(a+b+c)(b+c+d)\left( b+c+2\,d+e \right) }{ \left( b+c+2\,d \right) \left( b+c \right) \left( a+b \right) }}}&{} -\sqrt{{\frac{ ac\left( b+c+d \right) \left( b+c+2\,d+e \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} \\ \sqrt{{\frac{ acd\left( b+c+2\,d+e \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} &{} -\sqrt{{\frac{ bd(a+b+c)\left( b+c+2\,d+e \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} \\ \sqrt{{\frac{ace \left( b+c+d \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} &{} -\sqrt{{\frac{be \left( a+b+c \right) \left( b+c+d \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} \\ \sqrt{{\frac{bde \left( a+b+c \right) }{ (a+b)(b+c)\left( b+c+2\,d \right) }}} &{} \sqrt{{\frac{acde}{ (a+b)(b+c)\left( b+c+2\,d \right) }}} \end{array} \right] \end{aligned}$$

solve these equations for arbitrary positive parameters. This is our generic generalized spherical point. Evaluating the fundamental highest weight vectors at $v=v(a,b,c,d,e)$ gives the values

$$\begin{aligned} \left\{ \begin{array}{l} \begin{array}{l@{\quad }l} h_1(v)=\sqrt{{\frac{a \left( a+b+c \right) }{a+b}}} &{} h_2(v)=-\sqrt{{\frac{ b(a+b+c)(b+c+d)\left( b+c+2\,d+e \right) }{ \left( b+c+2\,d \right) \left( b+c \right) \left( a+b \right) }}} \\ h_3(v)=-\sqrt{\frac{c(a+b+c)(b+c+d)(b+c+2d+e)}{(b+c)(b+c+2d)}} &{} h_4(v)=\frac{(b+c+2d+e)\sqrt{d(b+c+d)}}{b+c+2d} \end{array} \\ h_5(v)=\sqrt{e(b+c+2d+e)} \end{array}\right\} . \end{aligned}$$

which are, in particular, nonzero. A computer algebra system was used to check conditions (3) and (4) from Lemma 2.5. The action $K:V$ is indeed well-behaved.

5.8 $\mathbf{Sp(2n)}\oplus _{\mathbf{Sp(2n)}}\mathbf{Sp(2n)} \ (n\ge 2)$

Next $K=\mathrm{Sp}(2n)\times \mathbb {T}\times \mathbb {T}$ acts on $V=V_1\oplus V_2={\mathbb {C}}^{2n}\oplus {\mathbb {C}}^{2n}\cong M_{2n,2}({\mathbb {C}})$ via

$$\begin{aligned} (k,c_1,c_2)\cdot z=kz\left[ \begin{array}{l@{\quad }l}c_1&{}0\\ 0&{}c_2\end{array}\right] =\bigl [c_1 kz_{\bullet ,1}\bigl | c_2kz_{\bullet ,2}\bigr ]. \end{aligned}$$

We take $B=B_{2n}^{\mathrm{Sp}}\times {\mathbb {C}}^\times \times {\mathbb {C}}^\times $, ${\mathfrak {h}}={\mathfrak {h}}_{2n}^{\mathrm{Sp}}\times {\mathbb {C}}\times {\mathbb {C}}$, let $\varepsilon _j\in {\mathfrak {h}}^*$ be as in (5.3) supported on the ${\mathfrak {h}}_{2n}^{\mathrm{Sp}}$ factor, $\varepsilon _\circ \in {\mathfrak {h}}^*$ dual to $T_1=(0,1,0)\in {\mathfrak {h}}$ and $\varepsilon _{\circ \circ }\in {\mathfrak {h}}^*$ dual to $T_2=(0,0,1)\in {\mathfrak {h}}$. This is a rank 4 multiplicity free action with fundamental highest weights and highest weight vectors

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-(\varepsilon _1+\varepsilon _\circ )&{} h_1(z)=z_{11} \\ \alpha _2=-(\varepsilon _1+\varepsilon _{\circ \circ }) &{} h_2(z)=z_{12}\\ \alpha _3=-(\varepsilon _1+\varepsilon _2+\varepsilon _\circ +\varepsilon _{\circ \circ }) &{} h_3(z)={\det }_2(z) \\ \alpha _4=-(\varepsilon _\circ +\varepsilon _{\circ \circ }) &{} h_4(z)=\omega (z_{\bullet ,1},z_{\bullet ,2}) \end{array}\right\} . \end{aligned}$$

For nonnegative integer exponents, $h_\alpha =h_1^ah_2^bh_3^ch_4^d$ has weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+c\alpha _3+d\alpha _4\\&= -\bigl ((a+b+c)\varepsilon _1+c\varepsilon _2+(a+c+d)\varepsilon _\circ +(b+c+d)\varepsilon _{\circ \circ }\bigr ). \end{aligned}$$

We may take $n=2$, $K=\mathrm{Sp}(4)\times \mathbb {T}\times \mathbb {T}$, $V=M_{4,2}({\mathbb {C}})$ here. The Lemma 2.2 equations on the entries in a spherical point $z\in M_{4,2}({\mathbb {C}})$ for weight $\alpha $ read

$$\begin{aligned} \left\{ \begin{array}{l|l|l} \left\langle z_{1,\bullet },z_{3,\bullet }\right\rangle =0=\left\langle z_{2,\bullet },z_{4,\bullet }\right\rangle &{} \left\langle z_{1,\bullet },z_{2,\bullet }\right\rangle =\left\langle z_{4,\bullet },z_{3,\bullet }\right\rangle &{} \left\langle z_{1,\bullet },z_{4,\bullet }\right\rangle =-\left\langle z_{2,\bullet },z_{3,\bullet }\right\rangle \\ \Vert z_{1,\bullet }\Vert ^2-\Vert z_{3,\bullet }\Vert ^2=a+b+c &{} \Vert z_{2,\bullet }\Vert ^2-\Vert z_{4,\bullet }\Vert ^2=c \\ \Vert z_{\bullet ,1}\Vert =a+c+d &{} \Vert z_{\bullet ,2}\Vert ^2=b+c+d \end{array}\right\} . \end{aligned}$$

One generic generalized spherical point whose matrix entries solve these equations for arbitrary positive values of the parameters is

$$\begin{aligned} v(a,b,c,d):= \left[ \begin{array}{l@{\quad }l} -\sqrt{{\frac{a \left( a+b+c \right) \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}&{}-\sqrt{{\frac{b \left( a+b+c \right) \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}\\ \sqrt{{\frac{bc \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}&{}-\sqrt{{\frac{ac \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}\\ -\sqrt{{\frac{bd \left( a+b+c \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}&{}\sqrt{{\frac{ad \left( a+b+c \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}}\\ -\sqrt{{\frac{acd}{ \left( a+b \right) \left( a+b+2\,c \right) }}}&{}-\sqrt{{\frac{bcd}{ \left( a+b \right) \left( a+b+2\,c \right) }}}\end{array} \right] \end{aligned}$$

and the fundamental highest weight vectors take values

$$\begin{aligned} \left\{ \begin{array}{ll} h_1(v)=-\sqrt{{\frac{a \left( a+b+c \right) \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}} &{} h_2(v)=-\sqrt{{\frac{b \left( a+b+c \right) \left( a+b+2\,c+d \right) }{ \left( a+b \right) \left( a+b+2\,c \right) }}} \\ h_3(v)=\frac{(a+b+2c+d)\sqrt{c(a+b+c)}}{a+b+2c} &{} h_4(v)=-\sqrt{d(a+b+2c+d)} \end{array}\right\} \end{aligned}$$

at $v=v(a,b,c,d)$. Thus Lemma 2.5 condition (2) holds and it is not difficult to check conditions (3) and (4) for each of $2^4-2=14$ relevant limits.

5.9 $\mathbf{Spin(8)}\oplus _{\mathbf{Spin(8)}}\mathbf{SO(8)}$

The compact group $\mathrm{Spin}(8)$ has three inequivalent irreducible representations of dimension eight, the natural representation via $\mathrm{SO}(8)$ and two half-spin representations. Consider $\mathrm{Spin}(8)$ acting on a sixteen-dimensional space via the direct sum of two of these representations. As the triality automorphism permutes the eight-dimensional irreducibles (see [10, §20.3]), it makes no difference, up to geometric equivalence, which pair are used. It is convenient to choose the two half-spin representations.

The positive and negative half-spin representations, $\sigma _\pm $ say, can be realized in

$$\begin{aligned} \Lambda ^\mathrm{even}({\mathbb {C}}^4)=\Lambda ^0({\mathbb {C}}^4)\oplus \Lambda ^2({\mathbb {C}}^4)\oplus \Lambda ^4({\mathbb {C}}^4)\quad \text {and}\quad \Lambda ^\mathrm{odd}({\mathbb {C}}^4)=\Lambda ^1({\mathbb {C}}^4)\oplus \Lambda ^3({\mathbb {C}}^4) \end{aligned}$$

respectively. We take $V=V_1\oplus V_2=\Lambda ^\mathrm{even}({\mathbb {C}}^4)\oplus \Lambda ^\mathrm{odd}({\mathbb {C}}^4)=\Lambda ({\mathbb {C}}^4)$ and let $K=\mathrm{Spin}(8)\times \mathbb {T}\times \mathbb {T}$ act via

$$\begin{aligned} (k,c_1,c_2)\cdot (v_1,v_2)=\bigl (c_1\sigma _+(k)(v_1), c_2\sigma _-(k)(v_2)\bigr ) \quad \bigl ((v_1,v_2)\in V_1\oplus V_2\bigr ). \end{aligned}$$

We adopt some notation from [2, Section 4.7]. $V=\Lambda ({\mathbb {C}}^4)=\sum _{j=0}^4\Lambda ^j({\mathbb {C}}^4)$ carries its usual Hermitian inner product and letting $e_{j_1\cdots j_k}=e_{j_1}\wedge \cdots \wedge e_{j_k}$ denotes a wedge product of standard basis vectors in ${\mathbb {C}}^4$,

$$\begin{aligned} \mathcal {B}=\{1,\,e_1,\,e_2,\,e_3,\,e_4,\,e_{12},\,e_{13},\,e_{14},\,e_{23},\,e_{24},\,e_{34},\,e_{234},\,e_{134},\,e_{124},\,e_{123},\,e_{1234}\} \end{aligned}$$

is an orthonormal basis. We write

$$\begin{aligned} (z_\emptyset ,\,z_1,\,z_2,\,z_3,\,z_4,\,z_{12},\,z_{13},\,z_{14},\,z_{23},\,z_{24},\,z_{34},\,z_{234},\,z_{134},\,z_{124},\,z_{123},\,z_{1234}) \end{aligned}$$

for coordinates with respect to $\mathcal {B}$. The compact group $\mathrm{Spin}(8)$ acts unitarily on $V$ via $\sigma _+\oplus \sigma _-$. The image of the derived representation of $so(8)$ in $u(V)$ is given explicitly in [11, Chapter 3] and elsewhere. Complexifying yields a copy of $so(8,{\mathbb {C}})$ inside $\mathrm{gl}(V)$. This is the ${\mathbb {C}}$-span of the 28 operators

$$\begin{aligned} \left\{ \begin{array}{l} H_k=\frac{1}{2}(D_kW_k-W_kD_k)\ (1\le k\le 4),\\ W_kD_\ell \ (1\le k\ne \ell \le 4),\quad W_kW_\ell \ (1\le k<\ell \le 4),\quad D_kD_\ell \ (1\le k<\ell \le 4) \end{array}\right\} . \end{aligned}$$

Here $W_k$ is the operator $W_k(v)=e_k\wedge v$ and $D_k$ its adjoint, contraction by $e_k$. As Cartan subalgebra ${\mathfrak {h}}_8$ and Borel subalgebra $\mathfrak {b}_8={\mathfrak {h}}_8\oplus \mathfrak {n}_{8}$ in this copy of $so(8,{\mathbb {C}})$ we take ${\mathfrak {h}}_8={\mathbb {C}}\text {-Span}\{H_1,\ldots ,H_4\}$,

$$\begin{aligned} \mathfrak {n}_8={\mathbb {C}}\text {-Span}\Bigl (\{W_kD_\ell ~:~1\le k<\ell \le 4\}\cup \{W_kW_\ell ~:~1\le k<\ell \le 4\}\Bigr ). \end{aligned}$$

$K$ has complexified Lie algebra $\mathfrak {k}_{\mathbb {C}}=so(8,{\mathbb {C}})\times {\mathbb {C}}\times {\mathbb {C}}$ with Cartan and Borel subalgebras ${\mathfrak {h}}={\mathfrak {h}}_8\times {\mathbb {C}}\times {\mathbb {C}}$, $\mathfrak {b}=\mathfrak {b}_8\times {\mathbb {C}}\times {\mathbb {C}}$. Let $T_1=(0,1,0)\in {\mathfrak {h}}$, $T_2=(0,0,1)\in {\mathfrak {h}}$ and $\varepsilon _1,\varepsilon _2,\varepsilon _3,\varepsilon _4,\varepsilon _\circ , \varepsilon _{\circ \circ }\in {\mathfrak {h}}^*$ denote the functionals

$$\begin{aligned} \left\{ \begin{array}{l} \varepsilon _j\bigl (a_1H_1+\cdots +a_4H_4+b_1T_1+b_2T_2\bigr )=a_j\\ \varepsilon _\circ \bigl (a_1H_1+\cdots +a_4H_4+b_1T_1+b_2T_2\bigr )=b_1\\ \varepsilon _{\circ \circ }\bigl (a_1H_1+\cdots +a_4H_4+b_1T_1+b_2T_2\bigr )=b_2 \end{array}\right\} . \end{aligned}$$

$K:V$ is a rank 5 multiplicity free action with fundamental highest weights and highest weight vectors

$$\begin{aligned} \left\{ \begin{array}{l|l} \alpha _1=-\frac{1}{2}(\varepsilon _1+\varepsilon _2+\varepsilon _3+\varepsilon _4)-\varepsilon _\circ &{} h_1(z)=z_\emptyset \\ \alpha _2=-\frac{1}{2}(\varepsilon _1+\varepsilon _2+\varepsilon _3-\varepsilon _4)-\varepsilon _{\circ \circ } &{} h_1(z)=z_4\\ \alpha _3=-2\varepsilon _\circ &{} h_3(z)=z_\emptyset z_{234}-z_{12}z_{34}+z_{13}z_{24}-z_{14}z_{23} \\ \alpha _4=-2\varepsilon _{\circ \circ } &{} h_4(z)=z_1z_{234} -z_2z_{134}+z_3z_{124}-z_4z_{123}\\ \alpha _5=-(\varepsilon _1+\varepsilon _\circ +\varepsilon _{\circ \circ }) &{} h_5(z)=z_2z_{34}-z_3z_{24}+z_4z_{23}-z_\emptyset z_{234}\end{array}\right\} . \end{aligned}$$

For nonnegative integer exponents $h_\alpha =h_1^ah_2^bh_3^ch_4^dh_5^e$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha&= a\alpha _1+b\alpha _2+c\alpha _3+d\alpha _4+e\alpha _5\\&= -\Bigg [\left( \frac{1}{2}a+\frac{1}{2}b+e\right) \varepsilon _1+\left( \frac{1}{2}a+\frac{1}{2}b\right) \varepsilon _2+\left( \frac{1}{2}a+\frac{1}{2}b\right) \varepsilon _3+\left( \frac{1}{2}a-\frac{1}{2}b\right) \varepsilon _4 \\&\quad +(a+2c+e)\varepsilon _\circ +(b+2d+e)\varepsilon _{\circ \circ }\Bigg ]. \end{aligned}$$

Lemma 2.2 gives a system of 18 equations for the coordinates $(z_\emptyset ,\ldots ,z_{1234})$ of a spherical point for weight $\alpha $. These are obtained by letting $X$ in (2.1) range over the basis given above for $\mathfrak {b}_8$ together with $T_1$ and $T_2$. Numerical experimentation with a computer algebra system reveals that this system has generic solutions in which eight of the coordinates vanish, namely $z_2, z_3, z_{12}, z_{13}, z_{24}, z_{34}, z_{134}, z_{124}$. Setting these coordinate variables to zero reduces the system to the following eight equations in the remaining eight variables.

$$\begin{aligned} \left\{ \begin{array}{l} z_{{4}}\overline{z_{{1}}}+z_{{234}}\overline{z_{{123}}}=0\\ z_\emptyset \overline{z_{{14}}}+z_{{23}}\overline{z_{{1234}}}=0\\ z_\emptyset \overline{z_{{23}}}+z_{{4}}\overline{z_{{234}}}+z_{{1}}\overline{z_{{123}}}+z_{{14}}\overline{z_{{1234}}} = 0\\ \Vert {z_\emptyset }\Vert ^2-\Vert {z_{{1}}}\Vert ^2+\Vert {z_{{4}}}\Vert ^2-\Vert {z_{{14}}}\Vert ^2+\Vert {z_{{23}}}\Vert ^2+\Vert {z_{{234}}}\Vert ^2-\Vert {z_{{123}}}\Vert ^2-\Vert {z_{{1234}}}\Vert ^2 =a+b+2\,e\\ \Vert {z_\emptyset }\Vert ^2+\Vert {z_{{1}}}\Vert ^2+\Vert {z_{{4}}}\Vert ^2+\Vert {z_{{14}}}\Vert ^2-\Vert {z_{{23}}}\Vert ^2-\Vert {z_{{234}}}\Vert ^2-\Vert {z_{{123}}}\Vert ^2-\Vert {z_{{1234}}}\Vert ^2 =a+b\\ \Vert {z_\emptyset }\Vert ^2+\Vert {z_{{1}}}\Vert ^2-\Vert {z_{{4}}}\Vert ^2-\Vert {z_{{14}}}\Vert ^2+\Vert {z_{{23}}}\Vert ^2-\Vert {z_{{234}}}\Vert ^2+\Vert {z_{{123}}}\Vert ^2-\Vert {z_{{1234}}}\Vert ^2 =a-b\\ \Vert {z_\emptyset }\Vert ^2+\Vert z_{14}\Vert ^2+\Vert z_{23}\Vert ^2+\Vert z_{1234}\Vert ^2 =a+2\,c+e\\ \Vert {z_{{1}}}\Vert ^2+\Vert {z_{{4}}}\Vert ^2+\Vert {z_{{234}}}\Vert ^2+\Vert {z_{{123}}}\Vert ^2 =b+2\,d+e \end{array} \right\} . \end{aligned}$$

These arise by taking $X=W_1D_4$, $W_1W_4$, $W_2W_3$, $H_1$, $H_2$, $H_4$, $T_1$, $T_2$ in (2.1). One can check that

$$\begin{aligned} v(a,b,c,d,e)&:= \sqrt{{\frac{ a(a+b+e)(a+c+e) }{ (a+b)(a+e) }}}\ 1 +\sqrt{{\frac{ade}{(a+b)(b+e) }}}\ e_1 \\&\quad +\sqrt{{\frac{ b(a+b+e)(b+d+e) }{ (a+b)(b+e)}}}\ e_4 -\sqrt{{\frac{bce}{ (a+b)(a+e) }}}\ e_{14} +\sqrt{{\frac{ be\left( a+c+e \right) }{ (a+b)(a+e)}}}\ e_{23} \\&\quad -\sqrt{{\frac{ ae\left( b+d+e \right) }{ (a+b)(b+e) }}}\ e_{234} +\sqrt{{\frac{ bd\left( a+b+e \right) }{ (a+b)(b+e) }}}\ e_{123} +\sqrt{{\frac{ ac\left( a+b+e \right) }{ (a+b)(a+e) }}}\ e_{1234} \end{aligned}$$

is a generic generalized spherical point solving the above system for arbitrary positive parameter values. Evaluating the fundamental highest weight vectors at $v=v(a,b,c,d,e)$ gives

$$\begin{aligned}\left\{ \begin{array}{l} \begin{array}{l@{\quad }l} h_1(v)=\sqrt{{\frac{ a(a+b+e)(a+c+e) }{ (a+b)(a+e) }}} &{} h_2(v)=\sqrt{{\frac{ b(a+b+e)(b+d+e) }{ (a+b)(b+e)}}} \\ h_3(v)=\sqrt{c(a+c+e)} &{} h_4(v)=-\sqrt{d(b+d+e)} \end{array} \\ h_5(v)=\sqrt{\frac{e(a+b+e)(a+c+e)(b+d+e)}{(a+e)(b+e)}} \end{array}\right\} , \end{aligned}$$

which verifies condition (2) in Lemma 2.5. Maple was used to carry out the routine calculations required to verify the two remaining conditions.

6 Case-by-case analysis: variable rank examples

The last four entries in Table 1 are infinite families of multiplicity free actions with increasing ranks. Lemma 2.5 will be used to show that these are well-behaved. We will discuss these actions together as their spherical points are closely related. Proofs will be provided for the first of these examples. Proofs for the remaining examples are similar, and are omitted for brevity. We begin by tabulating basic data concerning these examples.

6.1 $\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})$ (Table 2)

The group $K=U(n)\times U(m)$ acts on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus M_{n,m}({\mathbb {C}})$ as $(k_1,k_2)\cdot (\xi ,z')=\bigl (k_1\xi ,k_1z'k_2^t\bigr ).$ We identify $V$ with $M_{n,m+1}({\mathbb {C}})$ by adjoining the column vector $\xi $ to $z'$,

$$\begin{aligned} (\xi ,z')\leftrightarrow z=\bigl [\xi \vert z'\bigr ], \end{aligned}$$

and will number the columns of $z\in V$ by $0$ through $m$. By embedding $U(m)$ in $U(m+1)$ as $\left[ \begin{array}{l|l}1&{}0\\ \hline 0&{}U(m)\end{array}\right] $ the action of $K$ is realized by restriction of the usual action of $U(n)\times U(m+1)$ on $M_{n,m+1}({\mathbb {C}})$. The standard Hermitian inner product on $V=M_{n,m+1}({\mathbb {C}})$ is $K$-invariant. Fundamental highest weights, associated highest weight vectors and the set $\Lambda $ of all highest weights that occur in ${\mathbb {C}}[V]$ are listed in Table 2. Here we use Borel subgroup $B=B_n\times B_m$ and let $\varepsilon _j$, $\varepsilon _j'$ denote functionals on ${\mathfrak {h}}={\mathfrak {h}}_n\times {\mathfrak {h}}_m$ as in (5.1) supported on the two factors. We write ${\det }_j$ for the determinant of the first $j$ rows and columns of a matrix. Action $K:V$ has rank $2n$ when $m\ge n$ and rank $2m+1$ when $m<n$. For purposes of verifying that these actions are well-behaved we assume henceforth that either $m=n$ or $m=n-1$. The final entry in Table 2 gives criteria, derived from Lemma 2.2, for a matrix of size $n\times (m+1)$ to be a spherical point for weight $-\left( \sum _j\lambda _j\varepsilon _j+\sum _j\mu _j\varepsilon _j'\right) \in \Lambda $.

Table 2 Data for $\mathbf{SU(n)}\oplus _\mathbf{SU(n)} (\mathbf{SU(n)}\otimes \mathbf{SU(m)})$

Full size table

6.2 $\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})$ (Table 3)

This is a twisted variant of Example 6.1 with $K=U(n)\times U(m)$ acting on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus M_{n,m}({\mathbb {C}})$ via $(k_1,k_2)\cdot (\xi ,z')=\bigl (k_1^{-t}\xi ,k_1z'k_2^t\bigr )$. We identify $V$ with $M_{n,m+1}({\mathbb {C}})$ as in the previous example and number the columns of matrices $z\in V$ by $0$ through $m$. Table 3 lists relevant data for this action. In the formula for highest weight vector $\widetilde{h}_j(z)={\det }_j(\widetilde{z})$, the matrix $\widetilde{z}\in M_{n+1,m}({\mathbb {C}})$ is defined as

$$\begin{aligned} \widetilde{z}:=\left[ \begin{array}{lll} \xi ^tz'_{\bullet ,1}&{}\cdots &{}\xi ^tz'_{\bullet ,m}\\ \hline &{} z' \end{array}\right] \quad \text {for}\quad z=\left[ \begin{array}{l|l}\xi&z'\end{array}\right] . \end{aligned}$$

Entries in the first row of $\widetilde{z}$ are dot products of $\xi $ with the columns of $z'$. Note that $\widetilde{h}_j$ is a polynomial of degree $j+1$. Thus the number and degrees of the fundamental highest weight vectors agree with those in the untwisted example. For purposes of verifying that these actions are well-behaved it suffices to assume that either $m=n$ or $m=n-1$.

Table 3 Data for $\mathbf{SU(n)}^*\oplus _\mathbf{SU(n)}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})$

Full size table

Table 4 Data for $\mathbf{SU(n)}\oplus _\mathbf{SU(n)}\Lambda ^2(\mathbf{SU(n)})$

Full size table

6.3 $\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})$ (Table 4)

Identifying $\Lambda ^2({\mathbb {C}}^n)$ with the space $\mathrm{Skew}(n,{\mathbb {C}})$ of $n\times n$ skew symmetric matrices the group $K=U(n)$ acts on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus \mathrm{Skew}(n,{\mathbb {C}})$ via $k\cdot (\xi ,z')=\bigl (k\xi , kz'k^t\bigr )$. We further identify $V$ with $\mathrm{Skew}(n+1,{\mathbb {C}})$ via the isomorphism $ (\xi ,z')\leftrightarrow \left[ \begin{array}{c|c}0&{}\xi ^t\\ \hline -\xi &{}z'\end{array}\right] $ and number rows and columns as $0$ through $n$. In this model, the action of $U(n)$ on $V_1\oplus V_2$ is realized as a restriction of the usual action of $U(n+1)$ on $\mathrm{Skew}(n+1,{\mathbb {C}})$. The space $V=\mathrm{Skew}(n+1,{\mathbb {C}})$ carries the $K$-invariant Hermitian inner product

$$\begin{aligned} \left\langle z,w\right\rangle =\frac{1}{2}\mathrm{tr}(zw^*)=\sum _{i<j}z_{i,j}\overline{w_{i,j}}. \end{aligned}$$

Table 4 summarizes data for this action. Here $Pf_j$ denotes the Pfaffian of the first $2j$ rows and columns of a skew symmetric matrix and $z'\in \mathrm{Skew}(n,{\mathbb {C}})$ denotes the last $n$ rows and columns of $z$. In our subsequent analysis for this example, we will distinguish the cases $n$ even and $n$ odd.

Remark 6.1

In contrast to all previous examples, this action fails to be fully saturated since the scalars in $U(n)$ act diagonally. This is none-the-less a multiplicity free action [3, Theorem 6]. In view of Lemma 2.4, we choose to work with this non-saturated action. This remark applies equally to Example 6.4 which follows.

6.4 $\mathbf{SU(n)}^{*}\oplus _{\mathbf{SU(n)}}\Lambda ^2\mathbf{(SU(n))}$ (Table 5)

This is the twisted variant of Example 6.3 with $K=U(n)$ acting on $V=V_1\oplus V_2={\mathbb {C}}^n\oplus \mathrm{Skew}(n,{\mathbb {C}})$ via $k\cdot (\xi ,z')=\bigl (k^{-t}\xi , kz'k^t\bigr )$. As in the previous example, we identify $V$ with $\mathrm{Skew}(n+1,{\mathbb {C}})$ and number rows and columns as $0$ through $n$. Explicitly we have

$$\begin{aligned} k\cdot z=\left[ \begin{array}{l|l}0&{}\xi ^tk^{-1}\\ \hline -k^{-t}\xi &{}kz'k^t\end{array}\right] \quad \text {for}\,k\in U(n),\ z=\left[ \begin{array}{l|l}0&{}\xi ^t\\ \hline -\xi &{}z'\end{array}\right] \in \mathrm{Skew}(n+1,{\mathbb {C}}). \end{aligned}$$

In Table 5 for this example, matrix $\widetilde{z}\in \mathrm{Skew}(n+1,{\mathbb {C}})$ is defined as

$$\begin{aligned} \widetilde{z}:=\left[ \begin{array}{l|lll} 0&{}\xi ^tz'_{\bullet ,1}&{}\cdots &{}\xi ^tz'_{\bullet ,n}\\ z'_{1,\bullet }\xi \\ \vdots &{}&{} z' \\ z'_{n,\bullet }\xi \end{array}\right] \quad \text {for}\,\,\,\, z=\left[ \begin{array}{l|l}0&{}\xi ^t\\ \hline -\xi &{}z'\end{array}\right] . \end{aligned}$$

Entries in the first row of $\widetilde{z}$ are dot products of $\xi $ with the columns $z'_{\bullet ,j}$ of $z'$.

Table 5 Data for $\mathbf{SU(n)}^*\oplus _\mathbf{SU(n)}\Lambda ^2(\mathbf{SU(n)})$

Full size table

6.5 Spherical points

Generalized generic spherical points are given below for each of Examples 6.1–6.4. Full justification for the spherical point formulas will be provided for Example 6.1. Given sequences of distinct real parameters $\lambda _1,\ldots , \lambda _N$ and $\mu _1,\ldots \mu _N$ we set, for $1\le i,j\le N$,

$$\begin{aligned} \mathbf{z}_{i,0}=\left| \frac{\prod _{k=1}^n (\lambda _i-\mu _k)}{\prod _{k\ne i} (\lambda _i-\lambda _k)}\right| ^{1/2} \text { and }\quad \mathbf{z}_{i,j}=\left| \frac{\mu _j\prod _{k\ne j}(\lambda _i-\mu _k)\prod _{k\ne i}(\mu _j-\lambda _k)}{ \prod _{k\ne i}(\lambda _i-\lambda _k)\prod _{k\ne j}(\mu _j-\mu _k)} \right| ^{1/2}.\qquad \end{aligned}$$

(6.1)

6.5.1 ${\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})}$ (Table 2)

We must consider the cases $m=n$ and $m=n-1$.

For $m=n,$ generic weights in $\Lambda $ are indexed by (integral) parameters $\varvec{\lambda },\varvec{\mu }$ with $\lambda _1> \mu _1> \lambda _2>\mu _2> \cdots > \lambda _{n-1}> \mu _{n-1}> \lambda _n> \mu _n> 0$. Let $v=v({\varvec{\lambda }},{\varvec{\mu }})\in V$ be the matrix with entries indexed by $(1\le i\le n,\ 0\le j\le n$) given by $\mathbf{z}_{i,0}$ and $sign(i,j)\mathbf{z}_{i,j} $ for $j\ge 1$, where

$$\begin{aligned} \mathrm{sgn}(i,j)=\left\{ \begin{array}{l@{\quad }l} -1 &{} \text {if}\,i>j\\ +1 &{} \text {if}\,i\le j \end{array}\right. , \end{aligned}$$

(6.2)

and $\mathbf{z}_{i,0}$ and $\mathbf{z}_{i,j}$ are as in Eq. 6.1. It is show below that this is a generic generalized spherical point for this example. That is, the rows of $v({\varvec{\lambda }},{\varvec{\mu }})$ are pairwise orthogonal with norms $\lambda _1^{1/2},\ldots ,\lambda _n^{1/2}$ and columns $1$ through $n$ are pairwise orthogonal with norms $\mu _1^{1/2},\ldots ,\mu _n^{1/2}$.

For $m=n-1,$ generic weights in $\Lambda $ are indexed by parameters $\varvec{\lambda },\varvec{\mu }$ with $\lambda _1> \mu _1> \lambda _2> \mu _2> \cdots > \lambda _{n-1}> \mu _{n-1}> \lambda _n> 0$. Let $v(\varvec{\lambda },\varvec{\mu })$ be obtained by setting $\mu _n=0$ in the formulas for the case $m=n$ discussed above and deleting the last column. This gives a generic generalized spherical point for the case $m=n-1$. All of the identities needed to confirm this may also be derived by setting $\mu _n=0$ in the arguments for the case $m=n$. (See Sect. 6.7.) One obtains all lower-dimensional examples by successively setting $\mu _n, \lambda _{n-1},\mu _{n-1},\ldots $ equal to zero and deleting a row or column.

6.5.2 ${\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})}$ (Table 3)

For $m=n$, generic weights in $\Lambda $ are indexed by parameters $\varvec{\lambda },\varvec{\mu }$ with $\mu _1> \lambda _1>\mu _2> \lambda _2>\cdots > \mu _n> \lambda _n$ and $\mu _n\ge 0$. Let $v(\varvec{\lambda },\varvec{\mu })$ be the matrix with entries given by $v_{i,0}=\mathbf{z}_{i,0}$ and $v_{i,j}=sign(i,j)\mathbf{z}_{i,j} $ for $j\ge 1$ as in Eq. 6.1 but where now

$$\begin{aligned} \mathrm{sgn}(i,j)=\left\{ \begin{array}{l@{\quad }l} -1 &{} \text {if}\,i\ge j\\ +1 &{} \text {if}\,i<j \end{array}\right. . \end{aligned}$$

(6.3)

This gives a generic generalized spherical point.

For $m=n-1$, generic weights in $\Lambda $ are indexed by parameters $\varvec{\lambda },\varvec{\mu }$ with $\mu _1> \lambda _1> \mu _2> \lambda _2>\cdots > \mu _{n-1}> \lambda _{n-1}> \lambda _n$ and $\lambda _{n-1}> 0>\lambda _n$. One can obtain a generic generalized spherical point for this data by setting $\mu _n=0$ in the formulas for the case $m=n$ discussed above and deleting the last column. One obtains all lower-dimensional examples by successively setting $\mu _n, \lambda _{n-1},\mu _{n-1},\ldots $ equal to zero and deleting a row or column.

6.5.3 ${\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})}$ (Table 4)

Generic weights in $\Lambda $ are indexed by strictly decreasing (integral) sequences $c_1>c_2>\cdots >c_n>0$. First suppose that $n$ is even, $n=2m$ say, and let $\lambda _j=c_{2j-1}$, $\mu _j=c_{2j}$, so that $\lambda _1> \mu _1> \lambda _2> \mu _2\cdots > \lambda _m> \mu _m> 0.$ A generic generalized spherical point $v(\varvec{\lambda },\varvec{\mu })$ for such data is the skew symmetric matrix with entries indexed by $0\le i\le n,\ 0\le j\le n$, defined as follows for $i<j$:

$v_{i,j}=0$ if $i$ and $j$ have equal parity.
Nonzero entries on row 0 are
$$\begin{aligned} v_{0,2j-1}=\mathbf{z}_{j,0} \quad \text {for}\,1\le j\le m. \end{aligned}$$
(6.4)
Below row 0 one has
$$\begin{aligned} v_{2i,2j-1}=\mathbf{z}_{j,i} \end{aligned}$$
(6.5)
for $1\le i < j\le m$
and
$$\begin{aligned} v_{2i-1,2j}=\mathbf{z}_{i,j} \end{aligned}$$
(6.6)
for $1\le i\le j\le m$.

Setting $\mu _{m}=0$ in formulas (6.4–6.6) and deleting the last row and column produce the spherical point for the case $n=2m-1$ with data $(\lambda _1,\ldots ,\lambda _m;\mu _1,\ldots ,\mu _{m-1})$.

6.5.4 ${\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})}$ (Table 5)

For $n=2m$, we have parameters $({\varvec{\lambda }},{\varvec{\mu }})$ with $\mu _1>\lambda _1>\cdots >\mu _m>\lambda _m$ and $\mu _m\ge 0$. Let $\ v=v({\varvec{\lambda }},{\varvec{\mu }})\in \mathrm{Skew}(n+1,{\mathbb {C}})$ have entries $z_{i,j}$ defined as follows for $i<j$:

In row 0 we have $v_{0,2j-1}=0$ and
$$\begin{aligned} v_{0,2j}=\mathbf{z}_{j,0} \quad \text {for}\,1\le j\le m. \end{aligned}$$
(6.7)
$z_{i,j}=0$ for $i\ge 2$ if $i$ and $j$ have equal parity and
$$\begin{aligned} v_{2i,2j-1}=\mathbf{z}_{i,j} \end{aligned}$$
(6.8)
for $1\le i< j\le m$
and
$$\begin{aligned} v_{2i-1,2j}=\mathbf{z}_{j,i} \end{aligned}$$
(6.9)
for $1\le i\le j\le m$.

This is a generic generalized spherical point for the data $({\varvec{\lambda }},{\varvec{\mu }})$.

For $n=2m-1$, generic weights are indexed by data $({\varvec{\lambda }},{\varvec{\mu }})$ with $\mu _1> \lambda _1> \cdots > \mu _{m-1}> \lambda _{m-1}> \lambda _m$ and $\lambda _{m-1}> 0>\lambda _m$. Form the generic generalized spherical point for the case $n=2m$ as above with data $(\lambda _1,\ldots ,\lambda _{m};\ \mu _1,\ldots ,\mu _{m-1},0)$ and delete the second last row and column. This gives a generic generalized spherical point for the case $n=2m-1$.

6.6 A combinatorial lemma

To justify the formulas given above for generic generalized spherical points in Examples 6.1–6.4, we make extensive use of the following lemma.

Lemma 6.2

Let $p(x)=\sum _{j=0}^{n-1}p_jx^j$ be a polynomial of degree at most $n-1$ and $a_1,\ldots a_n$ be distinct real numbers. Then

$$\begin{aligned} \sum _{i=1}^n\frac{p(a_i)}{\prod _{k\ne i}(a_i-a_k)}=p_{n-1}. \end{aligned}$$

Proof

For nonnegative integers $j$ let

$$\begin{aligned} Q_j(\mathbf a):= \left| \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} a_1^j&{}a_1^{n-2}&{}\ldots &{}a_1&{}1\\ a_2^j&{}a_2^{n-2}&{}\ldots &{}a_2&{}1\\ \ldots &{}\ldots &{}\ldots &{}\ldots &{}\ldots \\ \ldots &{}\ldots &{}\ldots &{}\ldots &{}\ldots \\ a_n^j&{}a_n^{n-2}&{}\ldots &{}a_n&{}1 \end{array}\right| , \end{aligned}$$

(6.10)

so that $Q_j(\mathbf a)=0$ for $j<n-1$ and $Q_{n-1}(\mathbf a)$ is the Vandermonde determinant,

$$\begin{aligned} Q_{n-1}(\mathbf a)=V_n(\mathbf a)=\prod _{k<\ell }(a_k-a_\ell ). \end{aligned}$$

Expanding along the first column, we obtain

$$\begin{aligned} Q_j(\mathbf a)=\sum _{i=1}^n(-1)^{i-1}a_i^j{V_{n-1}({\widehat{\mathbf{a}}}_i)} =\sum _{i=1}^n(-1)^{i-1}a_i^j{\mathop {\mathop {\prod }\limits _{k<\ell }}\limits _{k,\ell \ne i}}(a_k-a_\ell ) \end{aligned}$$

where $V_{n-1}({\widehat{\mathbf{a}}}_i)$ is the $(n-1)\times (n-1)$ Vandermonde determinant obtained by eliminating $a_i$. Hence also

$$\begin{aligned} \frac{Q_j(\mathbf a)}{V_n(\mathbf a)} =\sum _{i=1}^n(-1)^{i-1}\frac{a_i^j}{\prod _{k<i}(a_k-a_i)\prod _{k>i}(a_i-a_k)} =\sum _{i=1}^n\frac{a_i^j}{\prod _{k\ne i}(a_i-a_k)}. \end{aligned}$$

$$\begin{aligned} \sum _{i=1}^{n}\frac{p(a_i)}{\prod _{k\ne i}(a_i-a_k)}= \sum _{i=1}^{n}\sum _{j=0}^{n-1}\frac{p_j a_i^j}{\prod _{k\ne i}(a_i-a_k)} =\sum _{j=0}^{n-1}p_j\frac{Q_j(\mathbf a)}{V_n(\mathbf a)}=\sum _{j=0}^{n-1}p_j\delta _{j,n-1}=p_{n-1} \end{aligned}$$

as claimed.$\square $

Remark 6.3

Taking $j> n-1$ in (6.10), the quotient $Q_j(\mathbf a)/V_n(\mathbf a)$ becomes the Schur function $s_{(j-n+1,0,\ldots ,0)}(\mathbf a)$. This coincides with a complete symmetric function, explicitly

$$\begin{aligned} \frac{Q_j(\mathbf a)}{V_n(\mathbf a)} =\sum _{|\varvec{k}|=j-n+1}a_1^{k_1}a_2^{k_2}\cdots a_n^{k_n} \end{aligned}$$

for $j> n-1$. (See, for example, [16, §1.15].)

6.7 Justification of the spherical point formulas

We concentrate on Example 6.1 (see Table 2) with $m=n$, as proofs for all other examples are similar. For the matrix $v({\varvec{\lambda }},{\varvec{\mu }})\in M_{n,n+1}({\mathbb {C}})$ given in Sect. 6.5.1, we must verify that

the rows are pairwise orthogonal with norms $\lambda _1^{1/2},\ldots ,\lambda _n^{1/2}$ and
columns $1$ through $n$ are pairwise orthogonal with norms $\mu _1^{1/2},\ldots ,\mu _n^{1/2}$.

6.7.1 Row norms

First consider the polynomial

$$\begin{aligned} E(x):={\prod _k(x-\mu _k)} +\sum _j\frac{\mu _j\prod _{k\ne j}(x-\mu _k)\prod _{k\ge 2}(\mu _j-\lambda _k)}{\prod _{k\ne j}(\mu _j-\mu _k)} \end{aligned}$$

Setting $x=\lambda _2,$ we get

$$\begin{aligned} E(\lambda _2)&= {\prod _k(\lambda _2-\mu _k)} +\sum _j\frac{\mu _j\prod _{k\ne j}(\lambda _2-\mu _k)(\mu _j-\lambda _2)\prod _{k\ge 3}(\mu _j-\lambda _k)}{\prod _{k\ne j}(\mu _j-\mu _k)}\\&= {\prod _k(\lambda _2-\mu _k)} -\sum _j\frac{\mu _j\prod _{k}(\lambda _2-\mu _k)\prod _{k\ge 3}(\mu _j-\mu _k) }{\prod _{k\ne j}(\mu _j-\mu _k)}\\&= \prod _k(\lambda _2-\mu _k)\left[ 1- \sum _j \frac{\mu _j \prod _{k\ge 3}(\mu _j-\lambda _k)}{ \prod _{k\ne j}(\mu _j-\mu _k)}\right] \\&= 0 \end{aligned}$$

by Lemma 6.2. Likewise, by symmetry, $E(\lambda _k)=0$ for all $k\ge 2$. Also,

$$\begin{aligned} E(0)&= {\prod _k(-\mu _k)} +\sum _j\frac{\mu _j\prod _{k\ne j}(- \mu _k)\prod _{k\ge 2}(\mu _j-\lambda _l) }{\prod _{k\ne j}(\mu _j-\mu _k)}\\&= \prod _k(-\mu _k)\left[ 1-\sum _j\frac{\prod _{k\ge 2}(\mu _j-\lambda _k)}{\prod _{k\ne j}(\mu _j-\mu _k)} \right] \\&= 0. \end{aligned}$$

Thus $E(x)$ has zeros at $0, \lambda _2,\ldots \lambda _n$. Since the highest order term is $x^n$, we conclude that in fact

$$\begin{aligned} E(x) =x\prod _{k\ge 2}(x-\lambda _k). \end{aligned}$$

One can check that, for Example 6.1, the expressions inside the absolute values in Eq. 6.1 are positive. Thus the first row of $v=v({\varvec{\lambda }},{\varvec{\mu }})$ satisfies

$$\begin{aligned} \Vert v_{1,\bullet }\Vert ^2&= v_{1,0}^2+\sum _jv_{1,j}^2=\frac{\prod _k(\lambda _1-\mu _k)}{\prod _{k\ge 2}(\lambda _1-\lambda _k)} +\sum _j \frac{\mu _j\prod _{k\ne j}(\lambda _1-\mu _k)\prod _{k\ge 2}(\mu _j-\lambda _k)}{\prod _{k\ge 2}(\lambda _1-\lambda _k)\prod _{k\ne j}(\mu _j-\mu _k)}\\&= \frac{1}{\prod _{k\ge 2}(\lambda _1-\lambda _k)}E(\lambda _1)\\&= \lambda _1. \end{aligned}$$

A similar argument shows that $\Vert v_{i,\bullet }\Vert ^2=\lambda _i$ for $2\le i\le n$. $\square $

6.7.2 Column norms

An argument similar to that used above for the row norms shows that

$$\begin{aligned} E(x):=\sum _i \frac{\prod _{k\ne j }(\lambda _i-\mu _k)\prod _{k\ne i}(x-\lambda _k)}{\prod _{k\ne i}(\lambda _i-\lambda _k)}=\prod _{k\ne j}(x-\mu _k). \end{aligned}$$

So now

$$\begin{aligned} \Vert v_{\bullet ,j}\Vert ^2&= \sum _iv_{i,j}^2=\sum _i\frac{\mu _j\prod _{k\ne j}(\lambda _i-\mu _k)\prod _{k\ne i}(\mu _j-\lambda _k)}{\prod _{k\ne i}(\lambda _i-\lambda _k)\prod _{k\ne j}(\mu _j-\mu _k)}\\&= \frac{\mu _j}{\prod _{k\ne j}(\mu _j-\mu _l)}\sum _i\frac{\prod _{k\ne j}(\lambda _i-\mu _k)\prod _{k\ne i}(\mu _j-\lambda _k)}{\prod _{k\ne i}(\lambda _i-\lambda _k)}\\&= \frac{\mu _j}{\prod _{k\ne j}(\mu _j-\mu _l)}E(\mu _j)\\&= \mu _j \end{aligned}$$

as required. $\square $

6.7.3 Row and column inner products

Next we will show that the rows and columns of $v=v({\varvec{\lambda }},{\varvec{\mu }})$ are pairwise orthogonal. For $1\le a<b\le n$, one has

$$\begin{aligned} \left\langle v_{a,\bullet }, v_{b,\bullet }\right\rangle&= v_{a,0}v_{b,0}+\sum _jv_{a,j}v_{b,j}\\&= \sqrt{\left| \frac{\prod _k(\lambda _a-\mu _k)}{\prod _{k\ne a}(\lambda _a-\lambda _k)} \frac{\prod _k(\lambda _b-\mu _k)}{\prod _{k\ne b}(\lambda _b-\lambda _k)} \right| } \\&\quad +\sum _j \mathrm{sgn}(a,j)\mathrm{sgn}(b,j) \sqrt{\left| \frac{\prod _k(\lambda _a-\mu _k)}{\prod _{k\ne a}(\lambda _a-\lambda _k)} \frac{\prod _k(\lambda _b-\mu _k)}{\prod _{k\ne b}(\lambda _b-\lambda _k)}\right| } \\&\quad \times \left| \frac{\mu _j\prod _{k}(\mu _j-\lambda _k)}{ (\lambda _a-\mu _j)(\lambda _b-\mu _j) \prod _{k\ne j}(\mu _j-\mu _l)}\right| \\&= \sqrt{\left| \frac{\prod _k(\lambda _a-\mu _k)}{\prod _{k\ne a}(\lambda _a-\lambda _k)} \frac{\prod _k(\lambda _b-\mu _k)}{\prod _{k\ne b}(\lambda _b-\lambda _k)} \right| } \\&\quad \times \left( 1+\sum _jsgn(a,j)\mathrm{sgn}(b,j)\left| \frac{\mu _j\prod _{k\ne a,b}(\mu _j-\lambda _k)}{ \prod _{k\ne j}(\mu _j-\mu _l) }\right| \right) .\\ \end{aligned}$$

For $a<b$, the signs have been chosen so that

$$\begin{aligned} \mathrm{sgn}(a,j)\mathrm{sgn}(b,j)\frac{\mu _j\prod _{k\ne a,b}(\mu _j-\lambda _k)}{ \prod _{k\ne j}(\mu _j-\mu _k) }<0. \end{aligned}$$

Hence

$$\begin{aligned} 1+\sum _jsgn(a,j)\mathrm{sgn}(b,j)\left| \frac{\mu _j\prod _{k\ne a,b}(\mu _j-\lambda _k)}{ \prod _{k\ne j}(\mu _j-\mu _l) }\right| =1-\sum _j \frac{\mu _j\prod _{k\ne a,b}(\mu _j-\lambda _l)}{ \prod _{k\ne j}(\mu _j-\mu _l) }=0, \end{aligned}$$

by another application of Lemma 6.2, and thus $\left\langle v_{a,\bullet }, v_{b,\bullet }\right\rangle =0$ as required. The proof that $\left\langle v_{\bullet ,a}, v_{\bullet ,b}\right\rangle =0$ for $1\le a<b\le n$ is similar. $\square $

6.8 Evaluation of the highest weight vectors

Lemma 2.5 will be used to show that Examples 6.1–6.4 are well-behaved. We continue to focus on Example 6.1 as the calculations for the remaining examples are similar. First suppose that $m=n$. The fundamental highest weight vectors are $h_r'(z)={\det }_r(z')$ and $h_r(z)={\det _r}(z)$ ($1\le r\le n$). Below we evaluate these, up to sign, at the generic generalized spherical point $z=v(\varvec{\lambda },\varvec{\mu })$ to show that $h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr )\ne 0\ne h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr )$, as required by Lemma 2.5 condition (2).

6.8.1 Calculation of ${\det }_r(z')$ for $z=v(\varvec{\lambda },\varvec{\mu })$ in case $m=n$

Pulling common factors from the first $r$ rows and columns of $z'$ gives

$$\begin{aligned} {\det }_r(z')&= {\det }_r\left[ \mathrm{sgn}(i,j) \sqrt{\left| \frac{\mu _j\prod _{k\ne j}(\lambda _i-\mu _k)\prod _{k\ne i}(\mu _j-\lambda _k)}{\prod _{k\ne i}(\lambda _i-\lambda _k)\prod _{k\ne j}(\mu _j-\mu _k)} \right| } \right] \\&= \sqrt{\left| \frac{\prod _{j\le r}\mu _j }{{\mathop {\mathop {\prod }\nolimits _{1\le i\le r,\ 1\le k\le n}}\limits _{k\ne i}}(\lambda _i-\lambda _k) {\mathop {\mathop {\prod }\nolimits _{1\le j\le r,\ 1\le k\le n}}\limits _{k\ne j}}(\mu _j-\mu _k)} \right| }\\&\quad \times {\det }_r\left[ \mathrm{sgn}(i,j) \sqrt{\left| \frac{\prod _{k}(\lambda _i-\mu _k) \prod _{k}(\mu _j-\lambda _k)}{(\lambda _i-\mu _j)(\mu _j-\lambda _i)} \right| } \right] \\&= \frac{ \prod _{j\le r}\sqrt{\mu _j} \prod _{i, k\le r}|\lambda _i-\mu _k|}{\prod _{i<k\le r}|\lambda _i-\lambda _k| \prod _{j<k\le r}|\mu _j-\mu _k|}\\&\quad \times \sqrt{\left| \frac{\prod _{i\le r<k}(\lambda _i-\mu _k) \prod _{j\le r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j\le r<k}(\mu _j-\mu _k) } \right| } {\det }_r\left[ \mathrm{sgn}(i,j) {\left| \frac{1}{\lambda _i-\mu _j} \right| } \right] \end{aligned}$$

Since $\mathrm{sgn}(i,j)$ and $(\lambda _i-\mu _j)$ have opposite signs one has

$$\begin{aligned} {\det }_r\left[ \mathrm{sgn}(i,j) {\left| \frac{1}{\lambda _i-\mu _j} \right| } \right] =(-1)^r{\det }_r\left[ { \frac{1}{\lambda _i-\mu _j}} \right] \end{aligned}$$

where ${\det }_r\bigl [1/(\lambda _i-\mu _j)\bigr ]$ is a Cauchy determinant. As is well-known [16, page 397],

$$\begin{aligned} {\det }_r\left[ \frac{1}{\lambda _i-\mu _j}\right] =\frac{ \prod _{a<b\le r}(\lambda _b-\lambda _a)(\mu _a-\mu _b)}{\prod _{a,b\le r}(\lambda _a-\mu _b)}. \end{aligned}$$

So now

$$\begin{aligned} \left| h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) \right| =\prod _{j\le r}\sqrt{\mu _j}\, \sqrt{\left| \frac{\prod _{i\le r<k}(\lambda _i-\mu _k) \prod _{j\le r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j\le r<k}(\mu _j-\mu _k) } \right| }\,. \end{aligned}$$

(6.11)

Remark 6.4

Thus, in particular, $\bigl |h_n'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr )\bigr |=\sqrt{\mu _1\cdots \mu _n}$. In fact, this is clear since $z'$ is an orthogonal matrix whose $j$th column has norm $\sqrt{\mu _j}$.

6.8.2 Calculation of ${\det }_r(z)$ for $z=v(\varvec{\lambda },\varvec{\mu })$ in case $m=n$

Pulling common factors from the first $r$ rows and columns of $z$ gives

$$\begin{aligned} {\det }_r(z)&= {\det }_r\left[ \begin{array}{l|l} \sqrt{\left| \frac{\prod _{k=1}^n (\lambda _i-\mu _k)}{\prod _{k\ne i} (\lambda _i-\lambda _k)}\right| }&\left[ \mathrm{sgn}(i,j) \sqrt{\left| \frac{\mu _j\prod _{k\ne j}(\lambda _i-\mu _k)\prod _{k\ne i}(\mu _j-\lambda _k)}{\prod _{k\ne i}(\lambda _i-\lambda _k)\prod _{k\ne j}(\mu _j-\mu _k)} \right| }\right] _{\mathop {\mathop {1\le i\le r}}\limits _{1\le j\le r-1}} \end{array} \right] \\&= \frac{ \prod _{j\le r-1}\sqrt{\mu _j} \prod _{1\le i\le r,\, 1\le k\le r-1}|\lambda _i-\mu _k|}{\prod _{i<k\le r}|\lambda _i-\lambda _k| \prod _{j<k\le r-1}|\mu _j-\mu _k|}\\&\quad \times \sqrt{\left| \frac{\prod _{i\le r\le k}(\lambda _i-\mu _k) \prod _{j<r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j< r\le k}(\mu _j-\mu _k) } \right| }\, \det (w) \end{aligned}$$

where $w$ is the $r\times r$ matrix

$$\begin{aligned} w=\left[ \begin{array}{l|l} \begin{array}{l} 1 \\ \vdots \\ 1 \end{array}&\left[ \frac{\mathrm{sgn}(i,j)}{|\lambda _i-\mu _j|}\right] _{\mathop {\mathop {1\le i\le r}} \limits _{1\le j\le r-1}} \end{array}\right] =\left[ \begin{array}{l|l} \begin{array}{l} 1 \\ \vdots \\ 1 \end{array}&\left[ \frac{-1}{\lambda _i-\mu _j}\right] _{\mathop {\mathop {1\le i\le r}}\limits _{1\le j\le r-1}} \end{array}\right] . \end{aligned}$$

Subtracting the $r$’th row of $w$ from the first $r-1$ rows reduces the calculation of $\det (w)$ to that for a Cauchy determinant. Explicitly

$$\begin{aligned} \det (w)&= (-1)^{r-1}\left| \begin{array}{l|l} \begin{array}{l} 1 \\ \vdots \\ 1 \end{array}&\left[ \frac{1}{\lambda _i-\mu _j}\right] _{\mathop {\mathop {1\le i\le r}}\limits _{1\le j\le r-1}} \end{array}\right| =(-1)^{r-1}\left| \begin{array}{c|c} \begin{array}{l}0\\ \vdots \\ 0 \end{array} &{} \left[ \frac{\lambda _r-\lambda _i}{(\lambda _i-\mu _j)(\lambda _r-\mu _j)}\right] _{1\le i,j\le r-1}\\ \hline 1 &{} \begin{array}{lll} \frac{1}{\lambda _r-\mu _1} &{} \cdots &{} \frac{1}{\lambda _r-\mu _{r-1}} \end{array} \end{array}\right| \\&= \det \left( \left[ \frac{\lambda _r-\lambda _i}{(\lambda _i-\mu _j)(\lambda _r-\mu _j)}\right] _{1\le i,j\le r-1}\right) \\&= \frac{\prod _{1\le i\le r-1}(\lambda _r-\lambda _i)}{\prod _{1\le j\le r-1}(\lambda _r-\mu _j)}\, {\det }_{r-1}\left[ \frac{1}{\lambda _i-\mu _j}\right] \\&= \frac{\prod _{1\le i\le r-1}(\lambda _r-\lambda _i)}{\prod _{1\le j\le r-1}(\lambda _r-\mu _j)} \times \frac{ \prod _{a<b\le r-1}(\lambda _b-\lambda _a)(\mu _a-\mu _b)}{\prod _{a,b\le r-1}(\lambda _a-\mu _b)}\\&= \frac{\prod _{a<b\le r}(\lambda _b-\lambda _a) \prod _{a<b\le r-1}(\mu _a-\mu _b)}{\prod _{1\le a\le r,\, 1\le b\le r-1}(\lambda _a-\mu _b)} \end{aligned}$$

Thus we obtain

$$\begin{aligned} \left| h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) \right| =\prod _{j< r}\sqrt{\mu _j}\, \sqrt{\left| \frac{\prod _{i\le r\le k}(\lambda _i-\mu _k) \prod _{j<r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j< r\le k}(\mu _j-\mu _k) } \right| }\,. \end{aligned}$$

(6.12)

6.8.3 Calculation of ${\det }_r(z)$ and ${\det }_r(z')$ for $z=v(\varvec{\lambda },\varvec{\mu })$ in case $m=n-1$

Next consider Example 6.1 with $m=n-1$. We have fundamental highest weight vectors $h_r(z)={\det }_r(z)$ for $1\le r\le n$ and $h_r(z')={\det }_r(z')$ for $1\le r\le n-1$. Values for these at the generic generalized spherical point $z=v(\varvec{\lambda },\varvec{\mu })$ may be obtained by setting $\mu _n=0$ in Eqs. 6.12 and 6.11, respectively. This gives

$$\begin{aligned} \left| h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) \right|&= \prod _{i\le r}\sqrt{\lambda _i}\, \sqrt{\left| \frac{\prod _{i\le r\le k}(\lambda _i-\mu _k) \prod _{j<r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j< r\le k}(\mu _j-\mu _k) } \right| }\,,\end{aligned}$$

(6.13)

$$\begin{aligned} \left| h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) \right|&= \prod _{i\le r}\sqrt{\lambda _i}\, \sqrt{\left| \frac{\prod _{i\le r<k}(\lambda _i-\mu _k) \prod _{j\le r<k}(\mu _j-\lambda _k)}{\prod _{i\le r<k}(\lambda _i-\lambda _k) \prod _{j\le r<k}(\mu _j-\mu _k) } \right| }\,. \end{aligned}$$

(6.14)

6.9 Limit conditions

Finally, we verify conditions (3) and (4) from Lemma 2.5 for Example 6.1 with $m=n$. (See Table 2.) This works by induction on $n$ and $m$. For nonnegative integer exponents $a_j$, $b_j$, the polynomial $h_\alpha =h_1^{a_1}\cdots h_n^{a_n}(h_1')^{b_1}\cdots (h_n')^{b_n}$ is a highest weight vector in ${\mathbb {C}}[V]$ with weight

$$\begin{aligned} \alpha =-\left( \sum _{i=1}^n\lambda _i\varepsilon _{i}+\sum _{i=1}^n\mu _i\varepsilon _{i}'\right) \end{aligned}$$

where $\lambda _1\ge \mu _1\ge \lambda _2\ge \mu _2\cdots \ge \lambda _n\ge \mu _n\ge 0$ are given by

$$\begin{aligned} \lambda _i=\sum _{j\ge i}a_j+\sum _{j\ge i}b_j,\qquad \mu _i=\sum _{j\ge i+1}a_j+\sum _{j\ge i}b_j. \end{aligned}$$

(6.15)

The highest weight vector $h_\alpha $ has all exponents positive if and only if the weight coefficients satisfy $\lambda _1> \mu _1> \lambda _2> \mu _2\cdots > \lambda _n> \mu _n> 0$. Let $v(\varvec{\lambda },\varvec{\mu })$ be the generic generalized spherical point from Sect. 6.5.1 with $(\varvec{\lambda },\varvec{\mu })$ determined by Eq. 6.15 and positive real parameters $(\mathbf a, \mathbf b)$.

Since $\mu _n=b_n$, the limit as $b_n\rightarrow 0$ is just the limit as $\mu _n\rightarrow 0$. As we have already discussed, when the last parameter $\mu _n\rightarrow 0$, the last column of $v(\varvec{\lambda },\varvec{\mu })$ becomes zero and the remaining matrix is a spherical point for the case $m=n-1$. Thus this limit exists. The limiting values for $h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr )$ ($1\le r\le n$) and $h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr )$ ($1\le r\le n-1$) are given, up to sign, by Eqs. 6.13 and 6.14, respectively. In particular, these limiting values are nonzero as required by Lemma 2.5 condition (4).

Other limits as parameters $a_i$, $b_i$ approach zero are equivalent to two adjacent parameters merging in $(\varvec{\lambda },\varvec{\mu })$. Indeed $a_i\rightarrow 0$ ($1\le i\le n$) corresponds to $\lambda _i\rightarrow \mu _i$ and $b_i\rightarrow 0$ ($1\le i\le n-1$) to $\mu _i\rightarrow \lambda _{i+1}$. Taking, for example, the limit as $a_1\rightarrow 0$, one obtains

$$\begin{aligned} \lim _{a_1\rightarrow 0}v(\varvec{\lambda },\varvec{\mu }) =\lim _{\lambda _1\rightarrow \mu _1}v(\varvec{\lambda },\varvec{\mu })= \left[ \begin{array}{l@{\quad }l@{\quad }l} 0&{}\sqrt{\mu _1}&{}0\\ \zeta _{0}&{}0&{}\zeta \\ \end{array} \right] , \end{aligned}$$

where $\left[ \zeta _0|\zeta \right] =v\bigl (\lambda _2,\ldots ,\lambda _n;\, \mu _2,\ldots ,\mu _n\bigr )\in M_{n-1,n}({\mathbb {C}})$ is a generic generalized spherical point for Example 6.1 with $n$ and $m$ reduced by one and data $\lambda _2>\mu _2>\ldots >\mu _n>0$. In particular, this limit exists. Moreover $\lim _{a_1\rightarrow 0}h_1'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =\sqrt{\mu _1}\ne 0$,

$$\begin{aligned} \lim _{a_1\rightarrow 0}h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =\sqrt{\mu _1}\,{\det }_{r-1}(\zeta )=\sqrt{\mu _1}\,h_{r-1}'\bigl (\left[ \zeta _0|\zeta \right] \bigr )\ne 0 \end{aligned}$$

for $2\le r\le n$ and

$$\begin{aligned} \lim _{a_1\rightarrow 0}h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =-\sqrt{\mu _1}\,h_{r-1}\bigl (\left[ \zeta _0|\zeta \right] \bigr )\ne 0 \end{aligned}$$

for $2\le r\le n$, as required. Similarly one has

$$\begin{aligned} \lim _{b_1\rightarrow 0}v(\varvec{\lambda },\varvec{\mu }) =\lim _{\mu _1\rightarrow \lambda _2}v(\varvec{\lambda },\varvec{\mu })= \left[ \begin{array}{c@{\quad }c@{\quad }c} z_{1,0}^\circ &{} 0 &{} \zeta _1^t\\ 0 &{} -\sqrt{\lambda _2} &{} 0 \\ \zeta _0 &{} 0 &{} \zeta \end{array}\right] \end{aligned}$$

where $\left[ \begin{array}{l|l}z_{1,0}^\circ &{} \zeta _1^t\\ \hline \zeta _0 &{} \zeta \end{array}\right] =v\bigl (\lambda _1,\lambda _3,\ldots ,\lambda _n;\, \mu _2,\ldots ,\mu _n\bigr )\in M_{n-1,n}({\mathbb {C}})$ is a generic generalized spherical point for Example 6.1 with $n$ and $m$ reduced by one. Here

$$\begin{aligned} \lim _{b_1\rightarrow 0}h_r'\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =\sqrt{\lambda _2}\,h_{r-1}'\left( \left[ \begin{array}{l|l}z_{1,0}^\circ &{} \zeta _1^t\\ \hline \zeta _0 &{} \zeta \end{array}\right] \right) \ne 0 \end{aligned}$$

for $2\le r\le n$, $\lim _{b_1\rightarrow 0}h_1\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =z_{1,0}^\circ \ne 0$ and

$$\begin{aligned} \lim _{b_1\rightarrow 0}h_r\bigl (v(\varvec{\lambda },\varvec{\mu })\bigr ) =-\sqrt{\lambda _2}\,h_{r-1}'\left( \left[ \begin{array}{l|l}z_{1,0}^\circ &{} \zeta _1^t\\ \hline \zeta _0 &{} \zeta \end{array}\right] \right) \ne 0 \end{aligned}$$

for $2\le r\le n$. Limits as $a_i\rightarrow 0$ for $2\le i\le n$ and $b_i\rightarrow 0$ for $2\le i \le n-1$ behave in a similar manner.

Notes

In fact, references [3, 14, 15] concern actions of complex algebraic groups. The table lists compact forms for these.

References

Benson, C., Jenkins, J., Lipsman, R.L., Ratcliff, G.: A geometric criterion for Gelfand pairs associated with the Heisenberg group. Pac. J. Math. 178(1), 1–36 (1997)
Article MathSciNet Google Scholar
Benson, C., Ratcliff, G:. Spaces of bounded spherical functions on Heisenberg groups: Part I. Ann. Mat. Pura Appl. (4) (to appear)
Benson, C., Ratcliff, G.: A classification of multiplicity free actions. J. Algebra 181(1), 152–186 (1996)
Article MATH MathSciNet Google Scholar
Benson, C., Ratcliff, G.: Rationality of the generalized binomial coefficients for a multiplicity free action. J. Aust. Math. Soc. Ser. A 68(3), 387–410 (2000)
Article MATH MathSciNet Google Scholar
Benson, C., Ratcliff, G.: The space of bounded spherical functions on the free 2-step nilpotent Lie group. Transform. Groups 13(2), 243–281 (2008)
Article MATH MathSciNet Google Scholar
Benson, C., Ratcliff, G.: Geometric models for the spectra of certain Gelfand pairs associated with Heisenberg groups. Ann. Mat. Pura Appl. (4) 192(4), 719–740 (with Erratum on pp. 741–743) (2013)
Benson, C., Ratcliff, G.: Maple 15 worksheet for the action $Sp(4)\times Sp(4)\times U(2)\times {\mathbb{T}} :M_{8,2}({\mathbb{C}})$ (2013). http://core.ecu.edu/math/bensonf/research/research.html
Carcano, G.: A commutativity condition for algebras of invariant functions. Boll. Un. Mat. Ital. B (7) 1(4), 1091–1105 (1987)
MATH MathSciNet Google Scholar
Daszkiewicz, A., Przebinda, T.: On the moment map of a multiplicity free action. Colloq. Math. 71(1), 107–110 (1996)
MATH MathSciNet Google Scholar
Fulton, W., Harris, J.: Representation Theory, Volume 129 of Graduate Texts in Mathematics. A First Course, Readings in Mathematics. Springer, New York (1991)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, Volume 26 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1991)
Book Google Scholar
Howe, R., Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions. Math. Ann. 290(3), 565–619 (1991)
Article MATH MathSciNet Google Scholar
Kac, V.G.: Some remarks on nilpotent orbits. J. Algebra 64(1), 190–213 (1980)
Article MATH MathSciNet Google Scholar
Knop, F.: Some remarks on multiplicity free spaces. In: Broer, A., Sabidussi, G. (eds.) Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), Volume 514 of NATO Advanced Science Institutes Series C. Mathematical and Physical Sciencs, pp. 301–317. Kluwer, Dordrecht (1998)
Leahy, A.S.: A classification of multiplicity free representations. J. Lie Theory 8(2), 367–391 (1998)
MATH MathSciNet Google Scholar
Stanley, R.P.: Enumerative Combinatorics. Vol. 2, Volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (with a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) (1999)

Download references

Author information

Authors and Affiliations

Department of Mathematics, East Carolina University, Greenville, NC, 27858, USA
Chal Benson & Gail Ratcliff

Authors

Chal Benson
View author publications
You can also search for this author in PubMed Google Scholar
Gail Ratcliff
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chal Benson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benson, C., Ratcliff, G. Spaces of bounded spherical functions on Heisenberg groups: part II. Annali di Matematica 194, 533–561 (2015). https://doi.org/10.1007/s10231-013-0387-x

Download citation

Received: 21 August 2013
Accepted: 28 October 2013
Published: 12 November 2013
Issue Date: April 2015
DOI: https://doi.org/10.1007/s10231-013-0387-x

Spaces of bounded spherical functions on Heisenberg groups: part II

Abstract

Similar content being viewed by others

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

AN ORBIT MODEL FOR THE SPECTRA OF NILPOTENT GELFAND PAIRS

On the Korányi spherical maximal function on Heisenberg groups

1 Introduction

Theorem 1.1

Theorem 1.2

2 Preliminaries and background results

2.1 Spherical points and well-behaved multiplicity free actions

Definition 2.1

Lemma 2.2

Definition 2.3

Lemma 2.4

2.2 A limiting procedure

Lemma 2.5

Definition 2.6

3 Product actions

Lemma 3.1

Proof

4 Reducible but indecomposable multiplicity free actions

5 Case-by-case analysis: fixed rank examples

5.1 Notational conventions

5.2 \( \mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\mathbf{SU(n)}\ (n\ge 2)\)

5.3 \(\mathbf{SU(n)}\oplus _\mathbf{SU(n)}\mathbf{SU(n)}^*\ ({n}\ge 3)\)

5.4 \((\mathbf{SU(n)}\otimes \mathbf{SU}(2))\oplus _\mathbf{SU}(2)(\mathbf{SU}(2)\otimes \mathbf{SU(m)}) \ (n,m\ge 2)\)

5.5 \((\mathbf{Sp(2n)}\otimes \mathbf{{SU}(2)})\oplus _\mathbf{SU(2)}(\mathbf{{SU}(2)}\otimes \mathbf{SU(m)}) \ (n,m\ge 2)\)

5.6 \((\mathbf{Sp(2n)}\otimes \mathbf{SU(2)})\oplus _{\mathbf{SU(2)}}(\mathbf{SU(2)}\otimes \mathbf{Sp(2m)}) \ (n,m\ge 2)\)

5.7 \(\mathbf{SU(2)}\oplus _{\mathbf{SU(2)}}(\mathbf{SU(2)}\otimes \mathbf{Sp(2n)}) \ (n\ge 2)\)

5.8 \(\mathbf{Sp(2n)}\oplus _{\mathbf{Sp(2n)}}\mathbf{Sp(2n)} \ (n\ge 2)\)

5.9 \(\mathbf{Spin(8)}\oplus _{\mathbf{Spin(8)}}\mathbf{SO(8)}\)

6 Case-by-case analysis: variable rank examples

6.1 \(\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})\) (Table 2)

6.2 \(\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})\) (Table 3)

6.3 \(\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})\) (Table 4)

Remark 6.1

6.4 \(\mathbf{SU(n)}^{*}\oplus _{\mathbf{SU(n)}}\Lambda ^2\mathbf{(SU(n))}\) (Table 5)

6.5 Spherical points

6.5.1 \({\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})}\) (Table 2)

6.5.2 \({\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}(\mathbf{SU(n)}\otimes \mathbf{SU(m)})}\) (Table 3)

6.5.3 \({\mathbf{SU(n)}\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})}\) (Table 4)

6.5.4 \({\mathbf{SU(n)}^*\oplus _{\mathbf{SU(n)}}\Lambda ^2(\mathbf{SU(n)})}\) (Table 5)

6.6 A combinatorial lemma

Lemma 6.2

Proof

Remark 6.3

6.7 Justification of the spherical point formulas

6.7.1 Row norms

6.7.2 Column norms

6.7.3 Row and column inner products

6.8 Evaluation of the highest weight vectors

6.8.1 Calculation of \({\det }_r(z')\) for \(z=v(\varvec{\lambda },\varvec{\mu })\) in case \(m=n\)

Remark 6.4

6.8.2 Calculation of \({\det }_r(z)\) for \(z=v(\varvec{\lambda },\varvec{\mu })\) in case \(m=n\)

6.8.3 Calculation of \({\det }_r(z)\) and \({\det }_r(z')\) for \(z=v(\varvec{\lambda },\varvec{\mu })\) in case \(m=n-1\)

6.9 Limit conditions

Notes

References

Author information

Authors and Affiliations

Corresponding author

Rights and permissions

About this article

Cite this article

Share this article

Keywords

Mathematics Subject Classification (1991)

Search

Navigation