2 Notation and preliminaries

If H is a connected algebraic group, we will denote by $\mathcal X(H)$ its character lattice. If H acts on a vector space V, we will denote by $V^H$ the invariant subspace and by $V^{(H)}$ the semi-invariant subset, that is, the union of the semi-invariant subspaces

$$ \begin{align*}V^{(H)}_\chi = \{v \in V \; | \; h.v = \chi(h)v \quad \forall h \in H\} \end{align*} $$

with $\chi \in \mathcal X(H)$ . If $\Xi $ is a lattice, we will denote by $\Xi ^\vee $ its dual lattice, namely $\Xi ^\vee = \operatorname {\mathrm {Hom}}_{\mathbb Z}(\Xi , \mathbb Z)$ .

Let G be a semisimple complex algebraic group. If $K \subset G$ , we will denote by $\mathrm N_G(K)$ the normalizer of K in G, and by $Z_G(K)$ the centralizer of K in G. Fix a maximal torus $T \subset G$ and a Borel subgroup B containing T. We denote by $\Phi $ the root system of G associated to T, and by $\Delta $ the base defined by B and by $\Phi = \Phi ^+ \sqcup \Phi ^-$ the corresponding decomposition into positive and negative roots. When $\Phi $ is an irreducible root system of rank n, we will order the set of simple roots $\Delta = \{\alpha _1, \ldots , \alpha _n\}$ following Bourbaki’s notation.

The monoid of dominant weights of G will be denoted by $\mathcal X(T)^+$ . Given $\lambda \in \mathcal X(T)^+$ , the irreducible representation of G of highest weight $\lambda $ will be denoted by $V_G(\lambda )$ , or simply by $V(\lambda )$ when no ambiguity is possible.

Let $G/K$ be an affine spherical variety. Equivalently, K is a reductive group, and the coordinate ring $\mathbb C[G/K]$ is multiplicity-free as a G-module. By the multiplicity-free property, the description of the G-module structure of $\mathbb C[G/K]$ is equivalent to the description of the monoid of the spherical weights

$$ \begin{align*}\Lambda^+_{G/K} = \{\lambda \in \mathcal X(T)^+ \; | \; V(\lambda)^K \neq 0\}. \end{align*} $$

Because K is reductive, notice that $V(\lambda )$ admits a nonzero K-invariant vector if and only if the same holds for its dual $V(\lambda )^*$ . Therefore, with this notation, we have the decomposition

$$ \begin{align*}\mathbb C[G/K] = \mathbb C[G]^K \simeq \bigoplus_{\lambda \in \mathcal X(T)^+} V(\lambda)^* \otimes V(\lambda)^K \simeq \bigoplus_{\lambda \in \Lambda_{G/K}^+} V(\lambda). \end{align*} $$

For every $\lambda \in \Lambda _{G/K}^+$ , we denote by $E_{G/K}(\lambda )\simeq V(\lambda )$ the corresponding isotypic (irreducible) component in $\mathbb C[G/K]$ .

The previous description can be generalized from the realm of K-invariant functions to that of K-semi-invariant functions, with respect to the right action of G on itself. Indeed, for every character $\chi \in \mathcal X(K)$ , the eigenspace $\mathbb C[G]^{(K)}_\chi $ is also a multiplicity-free G-module. Inside $\mathbb C[G]^{(K)}_\chi $ , every $\lambda \in \Lambda ^+$ with $\big (V(\lambda )^*\big )^{(K)}_\chi \neq 0$ yields an isotypic component $E_{G/K}(\lambda ,\chi ) \simeq V(\lambda )$ . In this setting, the relevant weight monoid is that of the quasi-spherical weights

$$ \begin{align*}\Omega^+_{G/K} = \{\lambda \in \mathcal X(T)^+ \; | \; V(\lambda)^{(K)} \neq 0\}. \end{align*} $$

The algebra structure of the invariant ring $\mathbb C[G]^K$ is encoded by the associated spherical functions. For every $\lambda \in \Lambda ^+_{G/K}$ , fix a nonzero element $f_\lambda \in E_{G/K}(\lambda )^K$ . The K-invariant space in $V(\lambda )$ is one-dimensional; thus, the spherical functions $\{f_\lambda \}_{\lambda \in \Lambda _{G/K}^+}$ form a basis for the space of K-invariant functions

$$ \begin{align*}\mathbb C[G]^{K\times K} = \bigoplus E_{G/K}(\lambda)^K.\end{align*} $$

More explicitly, if $\lambda \in \Lambda _{G/K}^+$ , the spherical function $f_\lambda $ can be constructed (up to a scalar factor) as a matrix coefficient by taking nonzero K-invariants vectors $v_\lambda \in V(\lambda )$ and $\psi _\lambda \in V(\lambda )^*$ and setting

$$ \begin{align*}f_\lambda(gK) = \langle \psi_\lambda,g.v_\lambda \rangle. \end{align*} $$

The decomposition of the product of irreducible G-submodules inside $\mathbb C[G/K]$ is encoded in the structure constants for the multiplication of the corresponding spherical functions (see [Reference Ruitenburg31, Theorem 3.2]).

Proposition 2.1 If $\lambda , \mu , \nu \in \Lambda _{G/K}^+$ and if $f_\lambda f_\mu = \sum a_{\lambda ,\mu }^\nu f_\nu $ , then

$$ \begin{align*}E_{G/K}(\nu) \subset E_{G/K}(\lambda) \cdot E_{G/K}(\mu) \Longleftrightarrow a_{\lambda,\mu}^\nu \neq 0. \end{align*} $$

More generally, a similar characterization can be given in terms of the support of the tensor product of the K-semi-invariant vectors of the quasi-spherical representations (see [Reference Chirivì, Littelmann and Maffei9, Lemma 19], [Reference Bravi, Gandini and Maffei4, Lemma 1.2]).

Proposition 2.2 Let $\lambda , \mu , \nu \in \Omega _{G/K}^+$ , and let $\chi , \chi ' \in \mathcal X(K)$ be such that the semi-invariant subspaces $V(\lambda )^{(K)}_\chi $ and $V(\mu )^{(K)}_{\chi '}$ are nonzero. Let $v_\lambda \in V(\lambda )^{(K)}_\chi $ and $v_\mu \in V(\mu )^{(K)}_{\chi '}$ be nonzero, and let $\pi _\nu : V(\lambda ) \otimes V(\mu ) \rightarrow V[\nu ]$ denote the projection onto the isotypic component $V[\nu ] \subset V(\lambda ) \otimes V(\mu )$ of highest weight $\nu $ . Then,

$$ \begin{align*}E_{G/K}(\nu,\chi+\chi') \subset E_{G/K}(\lambda,\chi) \cdot E_{G/K}(\mu,\chi') \Longleftrightarrow \pi_\nu(v_\lambda \otimes v_\mu)\neq 0. \end{align*} $$

3 Multiplication of spherical functions of symmetric pairs of  type  A

Let G be semisimple and simply connected, and let $\vartheta : G \rightarrow G$ be an algebraic involution. We denote by $K = G^\vartheta $ the set of fixed points. Then, K is a connected reductive subgroup of G, which has finite index in $\mathrm N_G(K)$ .

Let $A \subset G$ be a maximal split torus, that is, $\vartheta (a) = a^{-1}$ , for all $a \in A$ , and A is maximal with this property. Fix a maximal torus T containing A, then $\vartheta (T) = T$ . Let us denote by the same letter $\vartheta $ the induced involution on $\mathfrak t^*$ , then $\vartheta (\Phi ) = \Phi $ and the Killing form is preserved by $\vartheta $ . We also choose the set of positive roots and the Borel subgroup $B \supset T$ in such a way that $\vartheta (\alpha ) \in \Phi ^-$ whenever $\alpha \in \Phi ^+$ and $\alpha \neq \vartheta (\alpha )$ (see [Reference De Concini, Procesi and Gherardelli11, Lemma 1.2]).

Denote $X = G/K$ , and let $A_X \simeq A/A\cap K$ be the image of A in X. Notice that

$$ \begin{align*}A \cap K = \{a \in A \; | \; a^2 = 1\}; \end{align*} $$

thus, $\mathcal X(A_X) = 2\mathcal X(A)$ is identified with the lattice

$$ \begin{align*} \Lambda_X = \{\chi - \vartheta(\chi) \; | \; \chi \in \mathcal X(T)\}. \end{align*} $$

Define

$$ \begin{align*}\widetilde \Phi_X = \{\alpha - \vartheta(\alpha) \; | \; \alpha \in \Phi, \alpha \neq \vartheta(\alpha)\} \end{align*} $$

the set of restricted roots of X. Then, $\widetilde \Phi _X$ is a possibly nonreduced) root system in $\Lambda _X \otimes \mathbb R \simeq \mathcal X(A) \otimes \mathbb R$ , with base

$$ \begin{align*}\widetilde \Delta_X = \{\alpha - \vartheta(\alpha) \; | \; \alpha \in \Delta, \ \vartheta(\alpha) \neq \alpha \} \end{align*} $$

and with Weyl group

$$ \begin{align*}W_X \simeq \mathrm N_G(A)/\mathrm Z_G(A) \simeq \mathrm N_K(A)/\mathrm Z_K(A).\end{align*} $$

The weight lattice and the root lattice of $\widetilde \Phi _X$ are, respectively, identified with $\Lambda _X \simeq \mathcal X(A_X)$ and with $\mathcal X(A/\mathrm N_A(K)) \subset \mathcal X(A_X)$ (see [Reference Vust36, Lemmas 2.3 and 3.1]). Finally, the monoid of the spherical weights $\Lambda _X^+$ is obtained by intersecting the monoid of the dominant weights with the sublattice of $\mathcal X(T)$ defined by $\mathcal X(A_X)$ (see [Reference Vust35, Théorème  3]):

$$ \begin{align*}\Lambda_X^+ = \Lambda_X \cap \mathcal X(T)^+. \end{align*} $$

By a result of Richardson [Reference Richardson30, Corollary 11.5], the restriction of functions yields an algebra isomorphism

$$ \begin{align*}\mathbb C[X]^K \stackrel{\sim}{\longrightarrow} \mathbb C[A_X]^{W_X}. \end{align*} $$

Up to normalization, the restriction of the spherical function $f_\lambda $ to $A_X$ is a specialization of the Jacobi polynomial $P_\lambda ^{(k)}$ associated with the root system $\widetilde \Phi _X$ (see [Reference Heckman, Opdam and Koornwinder17, Section 8.4]). More precisely, let $m = (m_{\widetilde \alpha })$ be the multiplicity function of the restricted roots, that is, for $\widetilde \alpha \in \widetilde \Phi _X$ ,

$$ \begin{align*}m_{\widetilde \alpha} = | \{\beta \in \Phi \; | \; \beta - \vartheta(\beta) = \widetilde \alpha\}|. \end{align*} $$

Then, up to a scalar factor, the restriction ${f_\lambda }_{|A_X}$ coincides with the specialized Jacobi polynomial $P_\lambda ^{(m/2)}$ .

Therefore, the decomposition of the products of the irreducible G-modules inside $\mathbb C[X]$ is encoded in the structure constants for the multiplication of the corresponding specialized Jacobi polynomials:

If $\lambda , \mu , \nu \in \Lambda _X^+$ and if $P_\lambda ^{(m/2)} \, P_\mu ^{(m/2)} = \, \sum a_{\lambda ,\mu }^\nu P_\nu ^{(m/2)}$ , then

$$ \begin{align*}E_X(\nu) \subset E_X(\lambda) \cdot E_X(\mu) \Longleftrightarrow a_{\lambda,\mu}^\nu \neq 0. \end{align*} $$

Let $\alpha \in \Phi $ be such that $\alpha \neq \vartheta (\alpha )$ : then, either $\vartheta (\alpha ) = -\alpha $ , or $\alpha $ and $\vartheta (\alpha )$ are orthogonal, or $\langle \vartheta (\alpha ), \alpha ^\vee \rangle = 1$ (in which case, $\alpha - \vartheta (\alpha ) \in \Phi ^+$ ; see [Reference Vust36, Lemma 2.3]). Thus, if $\sigma \in \widetilde \Delta _X$ , say $\sigma = \alpha - \vartheta (\alpha )$ , we define the coroot $\sigma ^\vee \in \Lambda _X^\vee $ as follows:

1. (i) If $\vartheta (\alpha ) = -\alpha $ , then

   $$ \begin{align*}\sigma^\vee = \tfrac{1}{2}{\alpha^\vee}_{|\Lambda_X}. \end{align*} $$

2. (ii) If $\alpha $ and $\vartheta (\alpha )$ are orthogonal, then

   $$ \begin{align*}\sigma^\vee = {\alpha^\vee}_{|\Lambda_X}. \end{align*} $$

3. (iii) If $\alpha - \vartheta (\alpha ) \in \Phi ^+$ , then

   $$ \begin{align*}\sigma^\vee = {(\alpha- \vartheta(\alpha))^\vee}_{|\Lambda_X}. \end{align*} $$

Notice that the previous definition only depends on $\sigma $ , and not on $\alpha $ . If indeed $\lambda \in \Lambda _X$ , then in all cases, we have

$$ \begin{align*}\langle \lambda, \sigma^\vee\rangle = \frac{2(\lambda, \alpha-\vartheta(\alpha))}{||\alpha-\vartheta(\alpha)||^2}. \end{align*} $$

For later use, we now introduce a slight modification of the restricted root system. Denote

$$ \begin{align*}\Delta_X^\dagger=\{\sigma\in\widetilde \Delta_X \cap 2\Delta \;|\; \langle\sigma,\tau^\vee\rangle\in2\mathbb Z\ \forall \tau\in\widetilde\Delta_X\}, \end{align*} $$

and define

$$ \begin{align*}\Delta_X = (\widetilde \Delta_X \smallsetminus \Delta_X^\dagger) \cup \tfrac{1}{2}\Delta_X^\dagger. \end{align*} $$

Then, $\Delta _X$ is the base of a (reduced) root system $\Phi _X$ (see, e.g., [Reference Knop and Sahi20, Proposition 3.1]): indeed, the scalar product between any two distinct elements in $\Delta _X$ remains nonpositive, and the pairing among roots remains integer. Notice that $\Phi _X$ has also Weyl group $W_X$ . If moreover $\Delta _X$ is irreducible of rank $>1$ and distinct from $\widetilde \Delta _X$ , then $\Delta _X$ and $\widetilde \Delta _X$ have necessarily dual types.

Accordingly, we extend the weight lattice with the normalized restricted roots, and we define $\Xi _X$ to be the lattice generated by $\Lambda _X$ and $\Delta _X$ (namely by $\Lambda _X$ and $\tfrac 12\Delta _X^\dagger $ ).

For $\sigma \in \Delta _X$ , we define the coroot $\sigma ^\vee \in \Xi _X^\vee $ similarly as we did before: if $\sigma \in \Phi ^+$ , then $\sigma ^\vee \in \Xi _X^\vee $ is the restriction of the associated coroot in $\Lambda ^\vee $ ; if instead $\sigma =2\alpha $ with $\alpha \in \Delta $ , then we define $\sigma ^\vee =\tfrac 12{\alpha ^\vee }_{|\Xi _X}$ ; and finally, if $\sigma = \alpha - \vartheta (\alpha )$ with $\langle \vartheta (\alpha ), \alpha ^\vee \rangle = 0$ , then we define $\sigma ^\vee = {\alpha ^\vee }_{|\Xi _X}$ .

Set

$$ \begin{align*}\Delta_X^\vee=\{\sigma^\vee\ :\ \sigma\in\Delta_X\},\end{align*} $$

regarded as a subset of the dual lattice $\Xi _X^\vee $ . Then, the triple $\mathcal R_X = (\Xi _X, \Delta _X, \Delta _X^\vee )$ is a based root datum, in the following (usual) sense.

A based root datum is a triple $(\Xi ,S,\alpha \mapsto \alpha ^\vee )$ such that:

1. (i) $\Xi $ is a free abelian group of finite rank, S is a finite subset of $\Xi $ , and $\alpha \mapsto \alpha ^\vee $ is a map from S into $\Xi ^\vee $ .

2. (ii) For all $\alpha \in S$ , the pairing $\langle \alpha ,\alpha ^\vee \rangle $ is equal to 2.

3. (iii) The group W of automorphisms of $\Xi $ generated by the reflections

   $$ \begin{align*} s_\alpha\colon \Xi\longrightarrow\Xi, \qquad \qquad \xi \longmapsto \xi-\langle x,\alpha^\vee\rangle\alpha,\end{align*} $$
   with $\alpha \in S$ is finite.

4. (iv) The W-orbit of S is a reduced root system R in the real subspace of $\Xi _{\mathbb R}$ generated by S, and S is a base of R.

Notice that in our case the root datum $\mathcal R_X$ is semisimple, namely the base S spans $\Xi _{\mathbb R}$ as a vector space.

Let

$$ \begin{align*}\Xi_X^+:=\{\lambda\in\Xi_X\ :\ \langle\lambda,\sigma^\vee\rangle\geqslant0\ \forall\ \sigma\in \Delta_X\}\end{align*} $$

be the monoid of the dominant weights of $\mathcal R_X$ . Then, we have a (possibly strict) inclusion $\Lambda _X^+ \subset \Xi _X^+$ , and $\Xi _X^+ = \Xi _X \cap \Lambda ^+$ .

On the weight lattice $\Xi _X$ , we will consider the partial order $\leqslant _X$ defined by $\widetilde \Delta _X$ as follows:

$$ \begin{align*}\pi_1 \leqslant_X \pi_2 \quad\mbox{if}\quad \pi_2 - \pi_1 \in \mathbb N \widetilde \Delta_X. \end{align*} $$

When $\widetilde \Delta _X = \Delta _X$ , this is the usual dominance order on $\Xi _X$ ; otherwise, it is a refinement of the dominance order where the simple roots in $\tfrac 12\Delta _X^\dagger $ are only taken with even coefficients.

3.1 A conjectural rule when $\Phi _X$ is of type $\mathsf A$

When $\widetilde \Phi _X$ (or equivalently $\Phi _X$ ) is of type $\mathsf A$ , then up to a normalization, the Jacobi polynomials $P_\lambda ^{(k)}$ coincide with the Jack symmetric polynomials $J_\lambda ^{(1/k)}$ (see, e.g., [Reference Beerends and Opdam3, Proposition 3.3]). Thus, in this case, the restriction of the spherical function $f_\lambda $ to $A_X$ is a scalar multiple of the specialized Jack polynomial $J_\lambda ^{(2/m)}$ .

As is well known, when dealing with symmetric polynomials, it is often convenient to consider symmetric functions in infinitely many variables. Here, we do the same, and instead of working with the Jack polynomials, we pass to the Jack symmetric functions $J_\lambda ^{(k)}$ , which form a basis, when $\lambda $ vary among all possible partitions, of the ring of symmetric functions in infinitely many variables with coefficients in $\mathbb Q(k)$ . As a general reference on the Jack symmetric functions, see [Reference Stanley33] and [Reference Macdonald28, Section VI.10].

We now recall a conjecture of Stanley concerning the multiplication of the Jack symmetric functions.

Conjecture 3.1 [Reference Stanley33, Conjecture 8.3]

Write $J_\lambda ^{(k)} J_\mu ^{(k)} = \sum _\nu \frac {g_{\lambda ,\mu }^\nu (k)}{j_\nu (k)} J_\nu ^{(k)}$ . Then, the function $g_{\lambda ,\mu }^\nu (k)$ is a polynomial in k, with nonnegative integer coefficients.

To explain the denominator $j_\nu (k)$ , recall that the Jack symmetric functions are pairwise orthogonal with respect to the scalar product $\langle \ ,\ \rangle _k$ defined on the basis of the power-sum symmetric functions $p_\lambda $ ( $\lambda $ being a partition) as follows:

$$ \begin{align*}\langle p_\lambda, p_\mu \rangle_k = \left\{ \begin{array}{cc} z_\lambda \ k^{\ell(\lambda)} & \text{if}\ \lambda =\mu, \\ 0 & \text{if}\ \lambda \neq \mu. \end{array} \right. \end{align*} $$

Here, $\ell (\lambda )$ denotes the length of $\lambda $ , and if $m_i$ is the number of parts in $\lambda $ equal to i, then

$$ \begin{align*}z_\lambda = (1^{m_1} 2^{m_2} \cdots) m_1! \ m_2! \cdots.\end{align*} $$

Therefore, expanding the product of two Jack symmetric functions in terms of Jack symmetric functions, one has

$$ \begin{align*}J_\lambda^{(k)} J_\mu^{(k)} = \sum_\nu \frac{\langle J_\lambda^{(k)} J_\mu^{(k)}, J_\nu^{(k)}\rangle_k}{\langle J_\nu^{(k)},J_\nu^{(k)}\rangle_k}\ J_\nu^{(k)} .\end{align*} $$

Then, $j_\nu (k) = \langle J_\nu ^{(k)},J_\nu ^{(k)}\rangle _k$ , and $g_{\lambda ,\mu }^\nu (k)=\langle J_\lambda ^{(k)} J_\mu ^{(k)}, J_\nu ^{(k)}\rangle _k$ . There is a well-known combinatorial formula which expresses $j_\nu (k)$ as a polynomial in k with nonnegative integer coefficients in terms of hooks of $\nu $ (see [Reference Stanley33, Theorem 5.8]).

It is known that the functions $g_{\lambda ,\mu }^\nu (k)$ are polynomials in k with integer coefficients: this follows from a result of Knop and Sahi [Reference Knop and Sahi20], showing that the coefficients of the expansion of the Jack symmetric functions in terms of the monomial symmetric functions are polynomial in k with nonnegative integer coefficients.

Conjecture 3.1 is known to be true when one of the two partitions $\lambda $ and $\mu $ consists of a single row: this follows from a generalization of the Pieri rule for Jack symmetric functions proved by Stanley himself [Reference Stanley33, Theorem 6.1].

Let us come back to a symmetric space $X=G/K$ with restricted root system of type $\mathsf A$ . If we normalize properly the spherical functions $f_\lambda $ and write $f_\lambda f_\mu = \sum a_{\lambda ,\mu }^\nu \, f_\nu $ , then $a_{\lambda ,\mu }^\nu = g_{\lambda ,\mu }^\nu (2/m)$ . On the other hand, by the very definition of the scalar product $\langle \ ,\ \rangle _k$ , specializing the Jack symmetric function $J_\lambda ^{(k)}$ at $k = 1$ yields a nonzero scalar multiple of the Schur function $s_\lambda $ . Thus, $g_{\lambda ,\mu }^\nu (1)$ is nonzero if and only if the Littlewood–Richardson coefficient $c_{\lambda ,\mu }^\nu $ associated to $\lambda ,\mu ,\nu $ in the restricted root system associated to $G/K$ is nonzero.

For a symmetric space X with restricted root system of type $\mathsf A$ , we get then

(3.1) $$ \begin{align} E_X(\nu) \subset E_X(\lambda) \cdot E_X(\mu) \Longleftrightarrow g_{\lambda, \mu}^\nu(2/m) \neq 0 , \end{align} $$

and Stanley’s conjecture immediately implies

(3.2) $$ \begin{align} E_X(\nu) \subset E_X(\lambda) \cdot E_X(\mu) \Longleftrightarrow c_{\lambda,\mu}^\nu \neq 0. \end{align} $$

When the rank of the restricted root system is 1, only one-row partitions occur as dominant weights; therefore, equation (3.2) holds true thanks to Stanley’s generalization of the Pieri rule mentioned above.

To be more explicit, let $G_X$ be the connected semisimple group defined by the based root datum $\mathcal R_X = (\Xi _X, \Delta _X, \Delta _X^\vee )$ , and correspondingly, let $T_X \subset G_X$ and $B_X \subset G_X$ be a maximal torus and a Borel subgroup containing $T_X$ inducing $\mathcal R_X$ . For $\pi \in \Xi _X^+$ (and in particular for $\pi \in \Lambda _X^+$ ), let $V_X(\pi )$ be the irreducible $G_X$ -module with highest weight $\pi $ . Then, Stanley’s Conjecture 3.1 yields the following (weaker) conjecture.

Conjecture 3.2 Let X be a symmetric variety with restricted root system of type $\mathsf A$ , and let $\lambda , \mu , \nu \in \Lambda _X^+$ . Then,

$$ \begin{align*}E_X(\nu) \subset E_X(\lambda) \cdot E_X(\mu) \Longleftrightarrow \left\{ \begin{array}{c} V_X(\nu) \subset V_X(\lambda) \otimes V_X(\mu), \phantom{\Big|} \\ \nu \leqslant_X \lambda + \mu. \end{array} \right. \end{align*} $$

When the restricted root system of X is of type $\mathsf A_1$ , the previous conjecture holds true by Stanley’s Pieri rule for Jack symmetric functions.

Notice that Conjecture 3.2 is easily shown when $G = H \times H$ and $K = \operatorname {\mathrm {diag}}(H)$ for some reductive algebraic group H (even in types other than $\mathsf A$ ), in which case $X \simeq H$ regarded as $H \times H$ -variety (see, e.g., [Reference De Concini and Popov10, Lemma 3.4]).

Remark 3.3 Notice that in all cases of Table 1 but case Sym.A1 with $n=2$ , it always holds the equality $\Delta _X = \widetilde \Delta _X$ . In particular, in all these cases, Conjecture 3.2 follows immediately from (3.2), and the condition $\nu \leqslant _X \lambda +\mu $ is redundant. On the other hand, to deduce Conjecture 3.2 from (3.2) in the case $X = \operatorname {\mathrm {SL}}(2)/T$ , we need to use the saturation property of the tensor product of $\operatorname {\mathrm {SL}}(2)$ .

Remark 3.4 Rather than $\mathcal R_X$ , in Conjecture 3.2, it would have been more natural to use the based root datum $\widetilde {\mathcal R}_X = (\Lambda _X, \widetilde \Delta _X, \widetilde \Delta _X^\vee )$ , and define the semisimple group $G_X$ accordingly. Indeed, this formulation follows immediately from (3.2), and is also simpler, because the inequality $\nu \leqslant _X \lambda +\mu $ becomes redundant. However, computational experiments suggest that the formulation that we adopted is closer to possible generalizations (see Section 6, where we consider the case of the symmetric subgroups of Hermitian type).

In a slightly different form, Conjecture 3.2 was also considered by Graham and Hunziker [Reference Graham and Hunziker16].

For completeness, we list in Table 1 all the symmetric varieties with a restricted root system of type $\mathsf A$ . In all cases, we also give the simple restricted roots and a minimal set of generators for the weight monoid $\Lambda ^+_X$ .

Table 1: Symmetric pairs with restricted root system of type $\mathsf A_r$

1. (i) $\mathbf {SL(2)/SO(2)}$ : $\Delta _X = \{\alpha _1\}$ , $\Lambda ^+_X$ is generated by $2 \omega _1$ ;

2. (ii) $\mathbf {SL(n)/SO(n)}$ , $n\geqslant 3$ : $\Delta _X = \{2\alpha _i\ : \ 1\leqslant i\leqslant n-1\}$ , $\Lambda ^+_X$ is generated by the weights $2 \omega _i$ with $ 1\leqslant i\leqslant n-1$ ;

3. (iii) $\big (\mathbf {SL(n)} \times \mathbf {SL(n)}\big )/\mathbf {diag(SL(n))}$ , $n \geqslant 2$ : $\Delta _X = \{\alpha _i + \alpha _i'\ :\ 1\leqslant i\leqslant n-1\}$ , $\Lambda ^+_X$ is generated by the weights $\omega _i + \omega _i'$ with $ 1\leqslant i\leqslant n-1$ ;

4. (iv) $\mathbf {SL(2n)/Sp(2n)}$ , $n \geqslant 2$ : $\Delta _X = \{\alpha _{2i-1} + 2\alpha _{2i}+\alpha _{2i+1}\ :\ 1\leqslant i\leqslant n-1\}$ , $\Lambda ^+_X$ is generated by the weights $\omega _{2i}$ with $ 1\leqslant i\leqslant n-1$ ;

5. (v) $\mathbf {Spin(2n+1)/Spin(2n)}$ , $n \geqslant 2$ : $\Delta _X = \{2(\sum _{i=1}^n\alpha _i)\}$ , $\Lambda ^+_X$ is generated by $\omega _1$ ;

6. (vi) $\mathbf {Spin(2n)/Spin(2n-1)}$ , $n \geqslant 3$ : $\Delta _X = \{2(\sum _{i=1}^{n-2}\alpha _i) + \alpha _{n-1} + \alpha _n\}$ , $\Lambda ^+_X$ is generated by  $\omega _1$ ;

7. (vii) $\mathbf {E}_{\mathbf {6}}/\mathbf {F}_{\mathbf {4}}$ : $\Delta _X = \{2\alpha _1 + \alpha _2 + 2\alpha _3 + 2\alpha _4 + \alpha _5, \; \alpha _2 + \alpha _3 + 2\alpha _4 + 2\alpha _5 + 2\alpha _6\}$ , $\Lambda ^+_X$ is generated by $\omega _1$ and $\omega _6$ .

4 Root systems associated to a spherical homogeneous space

We now recall how to attach a based root system (or more precisely, three based root systems) to a spherical homogeneous space $G/K$ , which for the purposes of the present paper we can assume affine. General references for these constructions are [Reference Brion7, Reference Knop18].

Let $\Lambda _{G/K}$ be the weight lattice of $G/K$ , defined as

$$ \begin{align*}\Lambda_{G/K} = \{\chi \in \mathcal X(T) \; | \; \mathbb C(G/K)^{(B)}_\chi \neq 0\}. \end{align*} $$

Because K is reductive, notice that $\Lambda _{G/K}$ is generated as a lattice by $\Lambda ^+_{G/K}$ . In all the cases that we will consider, we will actually have $\Lambda ^+_{G/K} = \Lambda _{G/K} \cap \Lambda ^+$ .

Looking at the multiplication of spherical modules, we can associate three root systems to $G/K$ . For a fixed $G/K$ , the three root systems will have the same Weyl group $W_{G/K}$ : what changes is only the root normalization.

$\bullet $ The $\mathrm n$ -spherical roots of $G/K$ . The first root system, that we denote by $\Phi ^{\mathrm n}_{G/K}$ , is defined by considering the multiplication of spherical modules inside the invariant space $\mathbb C[G]^K$ . If $\lambda , \mu , \nu \in \Lambda ^+_{G/K}$ is a triple of spherical weights such that $E(\nu ) \subset E(\lambda ) \cdot E(\mu )$ , then $\lambda +\mu -\nu \in \Lambda _{G/K} \cap \mathbb N\Delta $ . Denote by $\mathcal M^{\mathrm n}_{G/K}$ the monoid generated by all possible differences $\lambda +\mu -\nu \in \Lambda _{G/K}$ with $\lambda , \mu , \nu \in \Lambda ^+_{G/K}$ as above. This is a free monoid (see [Reference Knop18, Theorem 1.3], [Reference Avdeev and Cupit-Foutou2, Theorem 4.11]). Its (unique) set of free generators, denoted by $\Delta ^{\mathrm n}_{G/K}$ , is the set of the $\mathrm n$ -spherical roots of $G/K$ . It is the base of a reduced root system, $\Phi ^{\mathrm n}_{G/K}$ (see [Reference Knop18, Theorem 1.3]).

$\bullet $ The $\mathrm {sc}$ -spherical roots of $G/K$ . The second root system, that we denote by $\Phi ^{\mathrm {sc}}_{G/K}$ , is defined by considering the multiplication of quasi-spherical modules inside the semi-invariant space $\mathbb C[G]^{(K)}$ . If $\lambda , \mu , \nu \in \Omega ^+_{G/K}$ is a triple of quasi-spherical weights such that $E_{\chi +\chi '}(\nu ) \subset E_\chi (\lambda ) \cdot E_{\chi '}(\mu )$ , then $\lambda +\mu -\nu \in \Lambda _{G/K} \cap \mathbb N \Delta $ . Let $\mathcal M^{\mathrm {sc}}_{G/K}$ be the monoid generated by all possible differences $\lambda +\mu -\nu $ with $\lambda , \mu , \nu \in \Omega ^+_{G/K}$ as above. Again, this is a free monoid (see [Reference Knop18, Corollary 7.6], [Reference Bravi, Gandini and Maffei4, Proposition 5]). Its (unique) set of free generators, denoted by $\Delta ^{\mathrm {sc}}_{G/K}$ , is the set of the $\mathrm {sc}$ -spherical roots of $G/K$ . It is the base of a reduced root system, $\Phi ^{\mathrm {sc}}_{G/K}$ .

$\bullet $ The minimal spherical roots of $G/K$ . By definition, we have inclusions $\mathcal M^{\mathrm n}_{G/K} \subset \mathcal M^{\mathrm {sc}}_{G/K} \subset \Lambda _{G/K}$ . The two root monoids $\mathcal M^{\mathrm n}_{G/K}$ and $\mathcal M^{\mathrm {sc}}_{G/K}$ actually generate the same cone in $\Lambda _{G/K} \otimes _{\mathbb Z} \mathbb Q$ . However, neither the $\mathrm n$ -spherical roots nor the $\mathrm {sc}$ -spherical roots need to be primitive elements inside $\Lambda _{G/K}$ : the set of the minimal spherical roots of $G/K$ , denoted by $\Delta ^{\min }_{G/K}$ , is the set of the primitive elements in $\Lambda _{G/K}$ associated to the extremal rays of the cone generated by $\mathcal M^{\mathrm n}_{G/K}$ in $\Lambda _{G/K} \otimes _{\mathbb Z} \mathbb Q$ . The minimal spherical roots of $G/K$ form the base of a reduced root system, that we denote by $\Phi ^{\min }_{G/K}$ .

It is clear from their construction that the three root systems $\Phi ^{\min }_{G/K}$ , $\Phi ^{\mathrm {sc}}_{G/K}$ , and $\Phi ^{\mathrm n}_{G/K}$ all have the same Weyl group $W_{G/K}$ .

If $G/K$ is a symmetric variety, we have $\widetilde \Delta _{G/K} = \Delta ^{\mathrm n}_{G/K}$ (see [Reference Knop18, Theorem 6.7]).

Both the $\mathrm n$ -spherical roots and the $\mathrm {sc}$ -spherical roots of $G/K$ arise as minimal spherical roots of a quotient of $G/K$ . Denote indeed by $\bar K$ the spherical closure of K, defined as the kernel of the action of $\mathrm N_G(K)$ on $\mathcal X(K)$ . Then, both $\mathrm N_G(K)$ and $\bar K$ are spherical subgroups of G, which satisfy

$$ \begin{align*}\Delta^{\mathrm n}_{G/K} = \Delta^{\min}_{G/\mathrm N_G(K)} \qquad \qquad \Delta^{\mathrm{sc}}_{G/K} = \Delta^{\min}_{G/\overline K}. \end{align*} $$

This explains our terminology. Moreover, we have equalities

$$ \begin{align*}\Lambda_{G/\mathrm N_G(K)} = \mathbb Z \Delta^{\mathrm n}_{G/K} \qquad \qquad \Lambda_{G/\bar K} = \mathbb Z \Delta^{\mathrm{sc}}_{G/K}. \end{align*} $$

The inclusion $\mathbb Z \Delta ^{\min }_{G/K} \subset \Lambda _{G/K}$ is strict in general: it is an equality precisely when $G/K$ admits a wonderful completion (which always happens for $G/\bar K$ and $G/\mathrm N_G(K)$ ). Notice that, in general, the minimal spherical roots of $G/K$ do not necessarily belong to the root lattice, differently from the $\mathrm n$ -spherical roots and from the $\mathrm {sc}$ -spherical roots which by construction are always in the root lattice of G.

Remark 4.1 The weight lattices of $G/K$ , $G/\bar K$ , and $G/\mathrm N_G(K)$ allow easily to compare K, $\bar K$ , and $\mathrm N_G(K)$ : indeed, the quotient $\mathrm N_G(K)/ K$ is diagonalizable (see [Reference Brion and Pauer8, Corollaire 5.2]), and we have inclusions

$$ \begin{align*}\Lambda_{G/\mathrm N_G(K)} \subset \Lambda_{G/\bar K} \subset \Lambda_{G/K}, \end{align*} $$

inducing isomorphisms (see, e.g., [Reference Gandini12, Lemma 2.4])

$$ \begin{gather*} \mathcal X(\mathrm N_G(K)/K) \simeq \Lambda_{G/K}/\Lambda_{G/\mathrm N_G(K)},\\ \mathcal X\big(\mathrm N_G(K)/\bar K \big) \simeq \Lambda_{G/\bar K}/\Lambda_{G/\mathrm N_G(K)}, \qquad \qquad \mathcal X\big(\bar K/K \big) \simeq \Lambda_{G/K}/\Lambda_{G/\bar K}. \end{gather*} $$

As already anticipated in the case of a symmetric variety, we define a fourth normalization for the spherical roots, setting

$$ \begin{align*}\Delta_{G/K} = (\Delta^{\mathrm n}_{G/K} \smallsetminus \Delta_{G/K}^\dagger) \cup \tfrac{1}{2}\Delta_{G/K}^\dagger, \end{align*} $$

where

$$ \begin{align*}\Delta_{G/K}^\dagger=\{\sigma\in\Delta^{\mathrm n}_{G/K} \cap 2\Delta \;|\; \langle\sigma,\tau^\vee\rangle\in2\mathbb Z\ \; \forall \tau\in\Delta^{\mathrm n}_{G/K}\}. \end{align*} $$

Because $\Delta ^{\mathrm n}_{G/K}$ is obtained from $\Delta ^{\mathrm {sc}}_{G/K}$ by doubling some particular elements in $\Delta ^{\mathrm {sc}}_{G/K} \cap \Delta $ (see [Reference Bravi and Luna5, Section 2.4]), it turns out that we also have

$$ \begin{align*}\Delta_{G/K} = (\Delta^{\mathrm{sc}}_{G/K} \smallsetminus \Delta_{G/K}^\dagger) \cup \tfrac{1}{2}\Delta_{G/K}^\dagger. \end{align*} $$

We denote by $\Phi _{G/K}$ the (reduced) root system generated by $\Delta _{G/K}$ . Notice that $\Phi _{G/K}$ has also Weyl group $W_{G/K}$ . In all cases that we will be interested (see Table 3), we have indeed $\Delta _{G/K} = \Delta ^{\mathrm {sc}}_{G/K}$ : this is always true when K is connected.

If $\sigma \in \Delta ^{\mathrm n}_{G/K}$ , then the following three possibilities occur (see [Reference Brion7, Théorème 2.6]): either $\sigma \in \Phi ^+$ is a positive root, or $\sigma \in 2 \Delta $ is the double of a simple root, or $\sigma = \alpha + \beta $ is the sum of two strongly orthogonal positive roots (that is, neither $\alpha + \beta $ nor $\alpha -\beta $ is in $\Phi $ ). Because $\Delta _{G/K}$ and $\Delta ^{\mathrm {sc}}_{G/K}$ are obtained from $\Delta ^{\mathrm n}_{G/K}$ by replacing some elements in $2\Delta \cap \Delta ^{\mathrm n}_{G/K}$ with the corresponding simple roots, the same description applies to $\Delta ^{\mathrm {sc}}_{G/K}$ and $\Delta _{G/K}$ .

There is actually a finite list of possible cases: the shape of an $\mathrm n$ -spherical root, written as sum of simple roots, is necessarily one of those reported in Table 2 (where we also specify the type of the root subsystem of $\Phi $ generated by the support of the spherical root).

Table 2: $\mathrm {sc}$ -spherical roots

In general, $\Delta _{G/K}$ is not contained in $\Lambda _{G/K}$ . Thus, we extend accordingly the weight lattice, and define $\Xi _{G/K}$ to be the lattice generated by $\Lambda _{G/K}$ together with $\Delta _{G/K}$ , namely by $\Lambda _{G/K}$ and $\tfrac {1}{2}\Delta _{G/K}^\dagger $ .

The following lemma will allow us to define the spherical coroots in $\Xi _{G/K}^\vee $ , in analogy to what we did in the symmetric case.

Proof (i) It is well known that $\langle \lambda , \alpha \rangle \in 2 \mathbb Z$ whenever $2 \alpha \in \Delta ^{\mathrm {sc}}_{G/K}$ (see [Reference Luna26, Section 1.4]). It is indeed being part of Luna’s axioms which classify spherical varieties [Reference Bravi and Luna5, Section 1.2]. Thus, by the definition of $\Delta ^\dagger _{G/K}$ , this property holds for all $\lambda \in \Xi _{G/K}$ .

(ii) We point out that, in [Reference Knop and Schalke21, Lemmas 6.2 and 6.4], a particular decomposition of $\sigma $ as a sum of two strongly orthogonal positive roots is considered, and the statement is proved for that decomposition.

Because $\Lambda ^+_{G/K} = \Lambda ^+_{G/\bar K}$ and $\Delta ^{\mathrm {sc}}_{G/K} = \Delta ^{\mathrm {sc}}_{G/\bar K}$ , we may assume that $K =\bar K$ .

Scrolling through the list of Table 2, we see that the $\mathrm {sc}$ -spherical roots that are neither positive roots nor the double of a simple root are those of the following table (we also report an extra datum, which is defined here below).

The first case, with $\sigma $ of type $\mathsf A_1\times \mathsf A_1$ , is somewhat special, and the statement is well known (see [Reference Luna26, Proposition 3.2]). It is indeed part of Luna’s axioms to classify spherical varieties. Therefore, we will assume that $\operatorname {\mathrm {supp}}(\sigma )$ is connected.

Denote

$$ \begin{align*}\Delta_{G/K}^p=\{\alpha\in \Delta\ :\ \langle\lambda,\alpha^\vee \rangle=0\ \forall\ \lambda\in\Omega^+_{G/K}\},\end{align*} $$

and let $\Delta ^{pp}(\sigma ) \subset \operatorname {\mathrm {supp}}(\sigma )$ be the set of simple roots orthogonal to $\sigma $ . Then, we have $\Delta ^{pp}(\sigma )\subset \Delta ^p_{G/K}$ whenever $\sigma \not \in \Phi ^+$ (see [Reference Bravi and Luna5, Section 1.1]).

Notice that in all cases of the previous table with $\operatorname {\mathrm {supp}}(\sigma )$ connected, there exists a unique simple root $\alpha \in \operatorname {\mathrm {supp}}(\sigma )$ not orthogonal to $\sigma $ . Therefore, it is enough to check that, for any spherical root $\sigma \in \Delta ^{\mathrm {sc}}_{G/K} \smallsetminus (\Phi ^+ \cup 2\Delta )$ and for any decomposition $\sigma =\beta +\gamma $ as a sum of two strongly orthogonal positive roots, it holds

$$ \begin{align*}\langle\omega_\alpha,\beta^\vee\rangle=\langle\omega_\alpha,\gamma^\vee\rangle = 1, \end{align*} $$

which can easily be checked case by case.

For $\sigma \in \Delta _{G/K}$ , using Lemma 4.2, we define the coroot $\sigma ^\vee \in \Xi _{G/K}^\vee $ as follows:

1. (i) if $\sigma = 2\alpha $ with $\alpha \in \Delta $ , then

   $$ \begin{align*}\sigma^\vee = \tfrac{1}{2}{\alpha^\vee}_{|\Xi_{G/K}}. \end{align*} $$

2. (ii) if $\sigma =\beta +\gamma $ with $\beta ,\gamma \in \Phi ^+$ strongly orthogonal, then

   $$ \begin{align*}\sigma^\vee = {\beta^\vee}_{|\Xi_{G/K}}. \end{align*} $$

3. (iii) if $\sigma =\beta $ with $\beta \in \Phi ^+$ , then

   $$ \begin{align*}\sigma^\vee = {\beta^\vee}_{|\Xi_{G/K}}. \end{align*} $$

Set $\Delta _{G/K}^\vee = \{\sigma ^\vee \; | \; \sigma \in \Delta _{G/K}\}$ , then by construction we have the following.

Proposition 4.3 The triple $\mathcal R_{G/K} = (\Xi _{G/K}, \Delta _{G/K}, \Delta _{G/K}^\vee )$ is a based root datum. Moreover, the intersection $\Xi _{G/K} \cap \Lambda ^+$ is contained in the monoid of dominant weights of $\mathcal R_{G/K}$

$$ \begin{align*}\Xi_{G/K}^+ :=\{\lambda\in\Xi_{G/K}\ :\ \langle\lambda,\sigma^\vee\rangle\geqslant0\ \forall\ \sigma\in \Delta_{G/K}\}. \end{align*} $$

Differently from the symmetric case, notice that the based root datum $\mathcal R_{G/K}$ is not necessarily semisimple: by Remark 4.1, this happens if and only if K has finite index in its normalizer. When $\mathcal R_{G/K}$ is semisimple, every spherical weight is uniquely determined by its values against the spherical roots; otherwise, the extra information is obtained by restricting to the normalizer.

Proof Denote $K'= \mathrm N_G(K)^\circ $ , then the restriction of characters to $B \cap K'$ induces an exact sequence

$$ \begin{align*}0 \longrightarrow \Lambda_{G/K'} \longrightarrow \Lambda_{G/K} \longrightarrow \mathcal X(B \cap K')^{B \cap K} \longrightarrow 0 \end{align*} $$

(see, e.g., [Reference Gandini12, Lemma 2.4]). On the other hand, $\Lambda _{G/\bar K}\subset \Lambda _{G/K'}$ is a sublattice of finite index, and $\Lambda _{G/\bar K} = \mathbb Z \Delta ^{\mathrm {sc}}_{G/K}$ ; thus, every element in $\Lambda _{G/ K'}$ is uniquely determined by its values against the spherical roots.

5 Affine spherical varieties with root system of type A

We now consider generalizations of Conjecture 3.2 to other affine spherical homogeneous varieties for a simple algebraic group.

Let $(G,K)$ be a reductive spherical pair, and set $X = G/K$ . Let $G_X$ be the connected reductive group defined by the root datum $\mathcal R_X = (\Xi _X, \Delta _X, \Delta _X^\vee )$ , and correspondingly, let $T_X \subset G_X$ and $B_X \subset G_X$ be a maximal torus and a Borel subgroup containing $T_X$ inducing $\mathcal R_X$ . For a spherical weight $\pi \in \Xi _X^+$ (in particular, for $\pi \in \Lambda _X^+$ ), let $V_X(\pi )$ be the irreducible $G_X$ -module with highest weight $\pi $ . In analogy with the symmetric case, we denote by $\leqslant _X$ the partial order on $\Xi _X$ defined by $\Delta ^{\mathrm n}_X$ , namely

$$ \begin{align*}\pi_1 \leqslant_X \pi_2 \stackrel{\mathrm{def}}{\Longleftrightarrow} \pi_2 - \pi_1 \in \mathbb N \Delta^{\mathrm n}_X. \end{align*} $$

Conjecture 5.1 Let $(G,K)$ be a reductive spherical pair with G simple and with $\Phi _{G/K}$ of type $\mathsf A$ , and set $X = G/K$ . Let $\lambda , \mu , \nu \in \Lambda _X^+$ , then

$$ \begin{align*}E_X(\nu) \subset E_X(\lambda) \cdot E_X(\mu) \Longleftrightarrow \left\{ \begin{array}{c} V_X(\nu) \subset V_X(\lambda) \otimes V_X(\mu), \phantom{\Big|} \\ \nu \leqslant_X \lambda + \mu. \end{array} \right. \end{align*} $$

With one exception (case Sph.A15 of Table 3), we will show that the previous conjecture holds true whenever $\Phi _X$ is a direct sum of rank 1 root systems—which happens in most of the cases—as a consequence of Stanley’s Pieri rule for Jack symmetric functions [Reference Stanley33, Theorem 6.1]. More generally, we will show that in all cases but case Sph.A15, the previous conjecture follows from Conjecture 3.2 (and in particular from Stanley’s Conjecture 3.1). Finally, in the remaining case Sph.A15, the conjecture is supported by computational experiments.

For the remainder of the section, G will be a simple and simply connected algebraic group, and $K \subset G$ a connected reductive spherical subgroup such that the spherical root system $\Phi _{G/K}$ is of type $\mathsf A$ . A list of such spherical pairs is given in Table 3 (where we omit the cases of Table 1). For a full list of the reductive spherical pairs $(G,K)$ with G simple and K connected, we refer to [Reference Knop and Röhrle19, Reference Krämer23].

Table 3: Reductive spherical pairs $(G,K)$ with root system of type $\mathsf A$ .

Remark 5.3 Notice that the cases Sph.A7, Sph.A11, and Sph.A15 are actually symmetric, with nonreduced restricted root system of type $\mathsf {BC}_1$ .

Notice also that the unique cases where $\Phi _{G/K}$ is not a direct sum of subsystems of type $\mathsf A_1$ are cases Sph.A8 and Sph.A9. In the first case, $G/K$ is the principal model variety of $\operatorname {\mathrm {SL}}(n)$ : a homogeneous variety $G/H$ is said to be a model variety for G if it is spherical and $\Lambda _{G/K}^+ = \mathcal X(T)^+$ (see [Reference Gelfand and Zelevinsky13, Reference Gelfand and Zelevinsky14]). In the latter case, $G/K$ is the homogeneous space which corresponds to the model variety of $\operatorname {\mathrm {SL}}(n)$ in Luna’s classification [Reference Luna27].

All other cases, where $\Phi _{G/K}$ is a direct sum of subsystems of type $\mathsf A_1$ , are related to some transitive action of a compact Lie group on a projective space, on a sphere, or on a product of spheres. Let indeed $G_{\mathbb R} \subset G$ and $K_{\mathbb R} \subset K$ be the associated compact real forms, then we have the following descriptions for $G_{\mathbb R}/K_{\mathbb R}$ :

1. (i) It is isomorphic to $\mathbb P^n(\mathbb C)$ in case Sph.A7, to $\mathbb P^{2n}(\mathbb C)$ in case Sph.A10, to $\mathbb P^n(\mathbb H)$ in case Sph.A11, and to $\mathbb P^2(\mathbb O)$ in case Sph.A15 (where $\mathbb H$ and $\mathbb O$ denote, respectively, the algebras of the quaternions and of the octonions).

2. (ii) It is isomorphic to $\mathbb S^{2n-1}$ in case Sph.A6, to $\mathbb S^7$ in case Sph.A12, to $\mathbb S^{15}$ in case Sph.A13, to $\mathbb S^6$ in case Sph.A16, and to $\mathbb S^7 \times \mathbb S^7$ in case Sph.A14.

By a theorem of Cartan, if $G_{\mathbb R}/H_{\mathbb R}$ is an irreducible compact Riemannian symmetric space of rank 1, then $H_{\mathbb R}$ acts transitively on the unit sphere in the isotropy representation of $H_{\mathbb R}$ . This gives indeed rise to the transitive actions on spheres which correspond to cases Sph.A6 and Sph.A13 (which, respectively, come from the families of symmetric spaces corresponding to cases Sph.A7 and Sph.A15). More generally, transitive actions of compact Lie groups on spheres were classified by Montgomery–Samelson and Borel: those which are not related to a symmetric space of rank 1 correspond to cases Sph.A12 and Sph.A13 (see, e.g., [Reference Schlichtkrull, Trapa and Vogan32]).

There is indeed another transitive action on a sphere which is associated to a rank 1 symmetric space, which is not included in Table 3: it is the transitive action of $\operatorname {\mathrm {Sp}}(2,\mathbb R) \times \operatorname {\mathrm {Sp}}(2n,\mathbb R)$ on $\mathbb S^{4n-1}$ (which comes from the family of symmetric spaces corresponding to case Sph.A11). This action corresponds to the reductive spherical pair

$$ \begin{align*}(G,K) = \big(\operatorname{\mathrm{Sp}}(2) \times \operatorname{\mathrm{Sp}}(2n), \ \operatorname{\mathrm{diag}}(\operatorname{\mathrm{Sp}}(2)) \times \operatorname{\mathrm{Sp}}(2n-2) \big), \end{align*} $$

in which case $\Phi _{G/K}$ is of type $\mathsf A_1 \times \mathsf A_1$ . Even though G is not simple, this case also fits in the general framework which allows us to deal with the cases of Table 3. We will analyze this case separately in Remark 5.11.

To deal with the cases of Table 3, a first useful remark is that cases Sph.A7, Sph.A9, and Sph.A11, respectively, follow (almost immediately) from the companion cases Sph.A6, Sph.A8, and Sph.A10. This will be explained in Section 5.3.

All the remaining cases of Table 3—with the exception of case Sph.A15—will follow from some cases of Table 1, thanks to a related factorization of a simple algebraic group as a product of two reductive subgroups. Such factorizations have been classified by Onishchik [Reference Onishchik29] in the context of compact Lie groups (see also [Reference Gorbatsevich, Onishchik and Onishchik15, Sections 4.5 and 4.6]) and by Liebeck, Saxl, and Seitz algebraically in arbitrary characteristic [Reference Liebeck, Saxl and Seitz24]. More precisely, in all the considered cases, the multiplication will reduce to that of some symmetric varieties, which correspond to a decomposition of $\Phi _{G/K}$ into two orthogonal components.

In particular, we will find a reductive overgroup $\widehat G \supset G$ and a symmetric subgroup $\widehat K \subset \widehat G$ such that $\widehat G = G \cdot \widehat K$ and $K = G \cap \widehat K$ , yielding a G-equivariant isomorphism

$$ \begin{align*}G/K \simeq \widehat G/\widehat K \end{align*} $$

and inducing a natural projection of weight monoids

$$ \begin{align*}\hat \varphi: \Lambda^+_{G/K} \longrightarrow \Lambda^+_{\widehat G/\widehat K}. \end{align*} $$

As a consequence of a theorem of Luna [Reference Luna25], notice that in the previous setting $G/K$ and $\widehat G/\widehat K$ are isomorphic as G-varieties if and only if $G/K$ embeds G-equivariantly into $\widehat G/\widehat K$ as a dense open subset.

We will also need to bring into the picture a second symmetric pair $(H,K)$ (possibly trivial), where H is a suitable connected reductive subgroup of G containing K and the connected center of $\mathrm N_G(K)$ , yielding an H-equivariant embedding

$$ \begin{align*}H/K \hookrightarrow G/K \end{align*} $$

and inducing a second projection of weight monoids

$$ \begin{align*}\bar \varphi:\Lambda^+_{G/K} \longrightarrow \Lambda^+_{H/K}. \end{align*} $$

The two maps $\hat \varphi $ and $\bar \varphi $ will allow us to identify the spherical functions of $G/K$ , and describe their products, in terms of those of $\widehat G/\widehat K$ and $H/K$ . Before treating the various cases in detail, let us explain here the general argument.

Suppose that we have:

1. (i) a connected reductive overgroup $\widehat G \supset G$ and a spherical subgroup $\widehat K \subset \widehat G$ such that $G/K \simeq \widehat G/\widehat K$ , and

2. (ii) a connected reductive subgroup $H\subset G$ containing K (and the connected center of $\mathrm N_G(K)$ ).

We also assume (it will always be true in our cases) that there is a compatible choice of Borel subgroups, that is, there exists a maximal torus and Borel subgroup $\widehat T \subset \widehat B$ of $\widehat G$ with $\widehat B \widehat K$ open in $\widehat G$ such that:

1. (i) $T = \widehat T \cap G$ is a maximal torus of G, and $B = \widehat B \cap G$ is a Borel subgroup of G with $B K$ is open in G.

2. (ii) $T_H = T \cap H$ is a maximal torus of H, and $B_H = B \cap H$ is a Borel subgroup of H with $B_H K$ is open in H.

On the one hand, the restriction of the $\widehat G$ -action to G induces an isomorphism of G-modules

$$ \begin{align*}\bigoplus_{\mu \in \Lambda^+_{\widehat G/\widehat K}} E_{\widehat G/\widehat K}(\mu) \; = \; \mathbb C[\widehat G/\widehat K] \stackrel{\sim}{\longrightarrow} \mathbb C[G/K] \quad = \bigoplus_{\lambda \in \Lambda^+_{G/K}} E_{G/K}(\lambda). \end{align*} $$

Thus, for all $\lambda \in \Lambda ^+_{G/K}$ , there exists a unique $\hat \lambda \in \Lambda ^+_{\widehat G/\widehat K}$ such that

$$ \begin{align*}\operatorname{\mathrm{Hom}}_G\big(V_{\widehat G}\big(\hat \lambda\big),V_G(\lambda)\big) \neq 0, \end{align*} $$

and we define $\hat \varphi (\lambda ) = \hat \lambda $ .

On the other hand, by definition, for all $\lambda \in \Lambda _{G/K}^+$ , the invariant space $V_G(\lambda )^K$ is one-dimensional. Looking at the decomposition of $V_G(\lambda )$ into simple H-modules, it follows that there exists a unique $\bar \lambda \in \Lambda _{H/K}^+$ such that

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{H}\big(V_G(\lambda), V_{H}\big(\bar \lambda\big)\big) \neq 0 \end{align*} $$

(and the dimension of the latter is necessarily 1). Thus, we define $\bar \varphi (\lambda ) = \bar \lambda $ .

Consider now the restriction of functions from $G/K$ to $H/K$

$$ \begin{align*}\bigoplus_{\lambda \in \Lambda^+_{G/K}} E_{G/K}(\lambda) \; = \; \mathbb C[G/K] \stackrel{\rho}{\longrightarrow} \mathbb C[H/K] \quad = \bigoplus_{\mu \in \Lambda^+_{H/K}} E_{H/K}(\mu). \end{align*} $$

In the following lemma, we show that $\rho (E_{G/K}(\lambda ))$ is nonzero; hence, it coincides with $E_{H/K}(\bar \lambda )$ .

Proof By the discussion above, we only need to show that $\rho (V) \neq 0$ .

Suppose that $V \simeq V_G(\lambda )$ . Fix K-invariant vectors $v_\lambda \in V_G(\lambda )$ and $\psi _\lambda \in V_G(\lambda )^*$ , and define

$$ \begin{align*}f_{G,\lambda}(gK) = \langle \psi_\lambda, g.v_\lambda \rangle. \end{align*} $$

Then, $f_{G,\lambda } \in V^K$ is a nonzero K-invariant function.

Similarly, if $v_{\bar \lambda } \in V_{H}(\bar \lambda )$ and $\psi _{\bar \lambda } \in V_{H}(\bar \lambda )^*$ are K-invariant vectors, setting

$$ \begin{align*}f_{H,\bar \lambda}(hK) = \langle \psi_{\bar \lambda},h.v_{\bar \lambda} \rangle, \end{align*} $$

we get a nonzero K-invariant function on $H/K$ .

On the other hand, there is a unique H-equivariant embedding of $V_{H}(\bar \lambda )$ inside $V_G(\lambda )$ , and a unique H-equivariant embedding of $V_{H}(\bar \lambda )^*$ inside $V_G(\lambda )^*$ . With respect to such embeddings, we have equalities

$$ \begin{align*}V_G(\lambda)^K = V_{H}(\bar \lambda)^K \qquad \qquad (V_G(\lambda)^*)^K = (V_{H}(\bar \lambda)^*)^K. \end{align*} $$

Because these invariant subspaces are one-dimensional, it follows that the K-invariant vectors $v_\lambda $ and $v_{\bar \lambda }$ are proportional, and similarly for the K-invariant linear functions $\psi _\lambda $ and $\psi _{\bar \lambda }$ . Thus, the restriction $\rho (f_{G,\lambda })$ is a nonzero scalar multiple of $f_{H,\bar \lambda }$ .

We now show that the two maps $\hat \varphi $ and $\bar \varphi $ , defined above, are actually morphisms of monoids.

Proposition 5.5 With notations and hypotheses as above, both the maps

$$ \begin{align*}\hat \varphi : \Lambda^+_{G/K} \longrightarrow \Lambda^+_{\widehat G/\widehat K} \quad\mbox{and}\quad \bar \varphi : \Lambda^+_{G/K} \longrightarrow \Lambda^+_{H/K}\end{align*} $$

are additive maps.

Proof (i) Let $\Gamma (\widehat G,G)$ be the branching monoid associated to the pair $(\widehat G,G)$ , namely

$$ \begin{align*}\Gamma(\widehat G,G) = \{(\pi,\lambda) \in \mathcal X(\widehat T)^+ \times \mathcal X(T)^+ \; | \; \operatorname{\mathrm{Hom}}_{G}(V_G(\lambda), V_{\widehat G}(\pi)) \neq 0 \}. \end{align*} $$

This is indeed a finitely generated monoid (see [Reference Akhiezer and Panyushev1]).

Consider the projections

$$ \begin{align*}p : \Gamma(\widehat G,G) \rightarrow \mathcal X(\widehat T)^+, \qquad \qquad q : \Gamma(\widehat G,G) \rightarrow \mathcal X(T)^+. \end{align*} $$

Because K is spherical in G, notice that $p^{-1}(\Lambda _{\widehat G/\widehat K}^+)$ is a submonoid of $\Gamma (\widehat G,G)$ , and that q induces an isomorphism $p^{-1}(\Lambda _{\widehat G/\widehat K}^+) \rightarrow \Lambda _{G/K}^+$ .

If indeed $(\pi ,\lambda ) \in \Gamma (\widehat G,G)$ and $\pi \in \Lambda _{\widehat G/\widehat K}^+$ , then by the isomorphism $\mathbb C[G/K] \simeq \mathbb C[\widehat G/\widehat K]$ , it follows $\lambda \in \Lambda _{G/K}^+$ . Conversely, if $\lambda \in \Lambda _{G/K}^+$ , because $\mathbb C[G/K] \simeq \mathbb C[\widehat G/\widehat K]$ , the multiplicity-free property implies that there is a unique $\pi \in \Lambda _{\widehat G/\widehat K}^+$ with $(\pi ,\lambda ) \in \Gamma (\widehat G,G)$ , and by definition $\pi = \hat \varphi (\lambda )$ .

It is now clear that, under the isomorphism $\Lambda _{G/K}^+ \simeq p^{-1}(\Lambda _{\widehat G/\widehat K}^+)$ , the map $\hat \varphi $ coincides with the restriction of the projection p. In particular, we see that $\hat \varphi $ is an additive map.

(ii) The proof that $\bar \varphi $ is additive is similar to the previous one, by considering the branching monoid associated to the pair $(G,H)$

$$ \begin{align*}\Gamma(G,H) = \{(\lambda, \pi) \in \mathcal X(T)^+ \times \mathcal X(T_H)^+ \; | \; \operatorname{\mathrm{Hom}}_{H}(V_H(\pi), V_{G}(\lambda)) \neq 0 \} \end{align*} $$

together with the projections

$$ \begin{align*}p : \Gamma(G,H) \rightarrow \mathcal X(T)^+, \qquad \qquad q : \Gamma(G,H) \rightarrow \mathcal X(T_H)^+. \end{align*} $$

Then, p induces an isomorphism of monoids $q^{-1}(\Lambda _{H/K}^+) \rightarrow \Lambda _{G/K}^+$ , and $\bar \varphi $ corresponds to the restriction of q.

Therefore, we can extend $\hat \varphi $ and $\bar \varphi $ to morphisms of the associated weight lattices. When the lattice $\Xi _{G/K}$ is bigger than $\Lambda _{G/K}$ , we will see in the cases under consideration that the extensions of $\hat \varphi $ and $\bar \varphi $ to $\Xi _{G/K}$ , respectively, take values in $\Xi _{\widehat G/\widehat K}$ and $\Xi _{H/K}$ . The description of the maps $\hat \varphi $ and $\bar \varphi $ will immediately imply the following.

Proposition 5.6 Let $(G,K)$ be a reductive spherical pair with $\Phi _{G/K}$ of type $\mathsf A$ , and suppose that we are in one of the following cases of Table 3:

$$ \begin{align*}\mathrm{Sph.A6, \; Sph.A8, \; Sph.A10, \; Sph.A12, \; Sph.A13, \; Sph.A14, \; and \;Sph.A16}.\end{align*} $$

Then, there exist a connected reductive subgroup $H \subset G$ containing K (possibly equal to K), a connected reductive overgroup $\widehat G \supset G$ , and a symmetric subgroup $\widehat K \subset \widehat G$ with $\widehat K \cap G = K$ such that the map

$$ \begin{align*}(\hat \varphi,\bar \varphi) \; \colon \; \Xi_{G/K} \longrightarrow \Xi_{\widehat G/\widehat K} \oplus \Xi_{H/K}\end{align*} $$

is an isogeny of based root data

$$ \begin{align*} \mathcal R_{G/K} \longrightarrow \mathcal R_{\widehat G/\widehat K} \oplus \mathcal R_{H/K}. \end{align*} $$

To be clear about notation and terminology, $\mathcal R_{\widehat G/\widehat K} \oplus \mathcal R_{H/K}$ is the based root datum defined by

$$ \begin{align*}\big(\Xi_{\widehat G/\widehat K} \oplus \Xi_{H/K},\ \Delta_{\widehat G/\widehat T}\sqcup \Delta_{H/K},\ \Delta_{\widehat G/\widehat T}^\vee\sqcup \Delta_{H/K}^\vee\big),\end{align*} $$

and $\varphi =(\hat \varphi ,\bar \varphi )$ is an isogeny of based root data in the following (usual) sense:

1. (i) $\varphi : \Xi _{G/K} \rightarrow \Xi _{\widehat G/\widehat K} \oplus \Xi _{H/K}$ is injective with finite cokernel.

2. (ii) $\varphi $ restricts to a bijection between $\Delta _{G/K}$ and $\Delta _{\widehat G/\widehat T}\sqcup \Delta _{H/K}$ .

3. (iii) The dual map satisfies $\varphi ^\vee (\varphi (\sigma )^\vee )=\sigma ^\vee $ , for all $\sigma \in \Delta _{G/K}$ .

In particular, we stress the following property.

In the next two subsections, we will deal with the various cases of Proposition 5.6.

For every spherical pair $(G,K)$ , we provide the groups $\widehat G$ , $\widehat K$ , and H with the required properties. The inclusions among the given groups are well known, and all the mentioned properties can be read off from the classification of the factorizations of simple algebraic groups [Reference Liebeck, Saxl and Seitz24, Reference Onishchik29] (see also [Reference Gorbatsevich, Onishchik and Onishchik15, Sections 4.5 and 4.6]).

For each affine spherical variety $G/K$ , $\widehat G/\widehat K$ , and $H/K$ , we describe the sets of spherical roots $\Delta _{G/K}$ , $\Delta _{\widehat G/\widehat K}$ , and $\Delta _{H/K}$ (which we call normalized spherical roots), and minimal set of generators of the monoids $\Lambda ^+_{G/K}$ , $\Lambda ^+_{\widehat G/\widehat K}$ , and $\Lambda ^+_{H/K}$ (which we call fundamental spherical weights). These data are also well known, and can be read off, for example, from [Reference Bravi and Pezzini6].

Finally, in each case, we describe the maps $\hat \varphi $ and $\bar \varphi $ . Being additive, it is enough to describe the images of the fundamental spherical weights of $G/K$ .

5.1 Indecomposable cases

We consider here the cases Sph.A12 and Sph.A16 where we take the subgroup $H \subset G$ as K itself (thus, $\Xi _{H/K} = 0$ and $\Delta _{H/K} = \varnothing $ ). The overgroup $\widehat G \supset G$ and the symmetric subgroup $\widehat K \subset \widehat G$ with the prescribed properties are given below.

From its explicit description, it will be clear that the map

$$ \begin{align*}\hat \varphi : \Lambda_{G/K} \longrightarrow \Lambda_{\widehat G/\widehat K} \end{align*} $$

induces an isomorphism of root data

$$ \begin{align*} \mathcal R_{G/K} \stackrel{\sim}{\longrightarrow} \mathcal R_{\widehat G/\widehat K}. \end{align*} $$

1. (Sph.A12) $G/K = \operatorname {\mathrm {Spin}}(7) / \mathsf G_2$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SO}}(8) / \operatorname{\mathrm{SO}}(7).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \mbox{Norm. spherical roots} & \mbox{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K\phantom{\Big|} & \alpha_1 + 2 \alpha_2 + 3 \alpha_3 & \omega_3 \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & 2(\alpha'_1 + \alpha'_2) + \alpha'_3 + \alpha'_4 & \omega'_1 \\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccc|} \hline \phantom{\Big|} \hat \varphi & : & a\omega_3 & \longmapsto & a\omega_1' \\ \hline \end{array}\end{align*} $$

2. (Sph.A16) $G / K = \mathsf G_2 / \operatorname {\mathrm {SL}}(3)$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SO}}(7) / \operatorname{\mathrm{SO}}(6).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \mbox{Norm. spherical roots} & \mbox{sph. weights}\\ \hline \hline \phantom{\Big|} G/K \phantom{\Big|} & 4\alpha_1 + 2 \alpha_2 & \omega_1 \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & 2(\alpha'_1 + \alpha'_2 + \alpha'_3) & \omega'_1 \\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccc|} \hline \phantom{\Big|} \hat \varphi & : & a\omega_1 & \longmapsto & a\omega_1' \\ \hline \end{array} \end{align*} $$

Proposition 5.8 Conjecture 5.1 holds true if $(G,K)$ is one of the spherical pairs Sph.A12 and Sph.A16 of Table 3.

Proof By restricting the action from $\widehat G$ to G, every spherical module for the pair $(\widehat G, \widehat K)$ remains irreducible and yields a spherical module for the pair $(G, K)$ . This gives a canonical bijection between the two: the restriction of every irreducible $\widehat G$ -module containing a nontrivial $\widehat K$ -invariant subspace is irreducible as a G-module as well, and it is obviously K-spherical.

In particular, the decomposition of the product of two spherical functions of $G/K$ immediately reduces to the decomposition of the product of the corresponding spherical functions of $\widehat G/\widehat K$ .

By Stanley’s Pieri rule for Jack symmetric functions, we know that Conjecture 3.2 holds true for the symmetric pair $(\widehat G,\widehat K)$ . On the other hand, $\hat \varphi $ induces an isomorphism of root data $\mathcal R_{G/K} \simeq \mathcal R_{\widehat G/\widehat K}$ . Thus, Conjecture 5.1 holds true for $(G,K)$ as well.

5.2 Decomposable cases

We now consider the cases Sph.A6, Sph.A8, Sph.A10, Sph.A13, and Sph.A14 of Table  3: we define the subgroup $H \subset G$ , the overgroup $\widehat G \supset G$ , and the symmetric subgroup $\widehat K \subset \widehat G$ with the required properties. Notice that H will always contain the connected center of $\mathrm N_G(K)$ . As already mentioned, for the given inclusions among groups and their related properties, one can see [Reference Liebeck, Saxl and Seitz24, Reference Onishchik29] and also [Reference Gorbatsevich, Onishchik and Onishchik15, Sections 4.5 and 4.6].

We also provide an explicit description of the maps

$$ \begin{align*}\hat \varphi: \Lambda_{G/K} \longrightarrow \Lambda_{\widehat G/\widehat K} \quad\mbox{and}\quad \bar \varphi: \Lambda_{G/K} \longrightarrow \Lambda_{H/K}. \end{align*} $$

Proposition 5.6 will be an immediate consequence of these descriptions.

1. (Sph.A6) $G / K = \operatorname {\mathrm {SL}}(n) / \operatorname {\mathrm {SL}}(n-1)$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SO}}(2n) / \operatorname{\mathrm{SO}}(2n-1),\end{align*} $$
   $$ \begin{align*}H / K = \operatorname{\mathrm{GL}}(n-1) / \operatorname{\mathrm{SL}}(n-1).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|} G/K \phantom{\Big|} & \alpha_1 + \cdots + \alpha_{n-1} & \omega_1, \ \omega_{n-1} \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & 2(\alpha'_1 + \cdots + \alpha'_{n-2}) + \alpha'_{n-1} + \alpha'_n & \omega'_1 \\ \hline \phantom{\Big|}H/K \phantom{\Big|} & \text{ none } & \pm\omega''_{n-1}\\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|} \hat \varphi & : & a \omega_1 + b\omega_{n-1} & \longmapsto & (a+b)\omega'_1 \\ \hline \phantom{\Big|} \bar \varphi & : & a \omega_1 + b\omega_{n-1} & \longmapsto & (b-a) \omega''_{n-1} \\ \hline \end{array} \end{align*} $$

2. (Sph.A8) $G / K = \operatorname {\mathrm {SL}}(2n+1) / \operatorname {\mathrm {Sp}}(2n)$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SL}}(2n+2) / \operatorname{\mathrm{Sp}}(2n+2),\end{align*} $$
   $$ \begin{align*}H / K = \operatorname{\mathrm{GL}}(2n) / \operatorname{\mathrm{Sp}}(2n).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. spherical weights}\\ \hline \hline \phantom{\Big|} G/K \phantom{\Big|} & \alpha_i + \alpha_{i+1}\ (i=1,\ldots,2n-1) & \omega_i\ (i=1,\ldots,2n) \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & \alpha'_{2i-1} + 2\alpha'_{2i} + \alpha'_{2i+1}\ (i=1,\ldots,n) & \omega'_{2i}\ (i=1,\ldots, n) \\ \hline \phantom{\Big|}H/K \phantom{\Big|} & \alpha''_{2i-1} + 2\alpha''_{2i} + \alpha''_{2i+1}\ (i=1,\ldots, n-1) & \omega''_{2i}\ (i=1,\ldots,n-1),\ \pm\omega''_{2n} \\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|}\hat \varphi & : & \sum_{i=1}^{2n} a_i \omega_i & \longmapsto & \sum_{i=1}^n (a_{2i-1} + a_{2i}) \omega'_{2i} \\ \hline \phantom{\Big|}\bar \varphi & : & \sum_{i=1}^{2n} a_i \omega_i & \longmapsto & \sum_{i=1}^{n-1} (a_{2i}+a_{2i+1}) \omega''_{2i} + (a_{2n} - \sum_{i=1}^n a_{2i-1}) \omega''_{2n} \\ \hline \end{array} \end{align*} $$

3. (Sph.A10) $G / K = \operatorname {\mathrm {Sp}}(2n) / [\operatorname {\mathrm {GL}}(1) \times \operatorname {\mathrm {Sp}}(2n-2)]$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SL}}(2n) / \operatorname{\mathrm{GL}}(2n-1),\end{align*} $$
   $$ \begin{align*}H / K = \operatorname{\mathrm{Sp}}(2) / \operatorname{\mathrm{GL}}(1).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K\phantom{\Big|} & \alpha_1, \ \alpha_1 + 2(\alpha_2 + \cdots + \alpha_{n-1}) + \alpha_n & 2\omega_1, \ \omega_2 \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & \alpha'_1 + \cdots + \alpha'_{2n-1} & \omega'_1 + \omega'_{2n-1} \\ \hline \phantom{\Big|}H/K \phantom{\Big|} & \alpha''_1 & 2\omega''_1\\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|} \hat \varphi & : & 2a \omega_1+ b \omega_2 & \longmapsto & (a+b) (\omega'_1 + \omega'_{2n-1}) \\ \hline \phantom{\Big|} \bar \varphi & : & 2a \omega_1 + b \omega_2 &\longmapsto & 2a \omega''_1 \\ \hline \end{array} \end{align*} $$

4. (Sph.A13) $G / K = \operatorname {\mathrm {Spin}}(9) / \operatorname {\mathrm {Spin}}(7)$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{SO}}(16) / \operatorname{\mathrm{SO}}(15),\end{align*} $$
   $$ \begin{align*}H/K = \operatorname{\mathrm{Spin}}(8) / \operatorname{\mathrm{Spin}}(7).\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K \phantom{\Big|} & \alpha_1+\alpha_2+\alpha_3+\alpha_4, \ \alpha_2 + 2\alpha_3 + 3\alpha_4 & \omega_1, \ \omega_4 \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & 2(\alpha'_1 + \cdots + \alpha'_6) + \alpha'_7+\alpha'_8 & \omega'_1 \\ \hline \phantom{\Big|} H/K \phantom{\Big|} & \alpha''_1+2\alpha''_2+\alpha''_3+2\alpha''_4 & \omega''_4\\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|} \hat \varphi & : & a \omega_1 + b \omega_4 & \longrightarrow & (2a+b) \omega'_1 \\ \hline \phantom{\Big|} \bar \varphi & : & a \omega_1 + b \omega_4 & \longmapsto & b \omega''_4 \\ \hline \end{array} \end{align*} $$

5. (Sph.A14) $G / K = \operatorname {\mathrm {Spin}}(8) / \mathsf G_2$

   $$ \begin{align*}\widehat G / \widehat K = \operatorname{\mathrm{Spin}}(8)/\operatorname{\mathrm{Spin}}(7) \times \operatorname{\mathrm{Spin}}(8)/\operatorname{\mathrm{Spin}}(7)\end{align*} $$
   (where the two copies of $\operatorname {\mathrm {Spin}}(7)$ are not conjugated in $\operatorname {\mathrm {Spin}}(8)$ ),
   $$ \begin{align*}H/K = \operatorname{\mathrm{Spin}}(7)/ \mathsf G_2.\end{align*} $$
   $$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K \phantom{\Big|} & \alpha_1+\alpha_2+\alpha_3, \ \alpha_1+\alpha_2+\alpha_4, \ \alpha_2 + \alpha_3 + \alpha_4 & \omega_1, \ \omega_3, \ \omega_4 \\ \hline \phantom{\bigg|}\widehat G / \widehat K\phantom{\bigg|} & 2\alpha'_1 + 2\alpha'_2 + \alpha'_3 + \alpha'_4, \ \alpha''_1 + 2\alpha''_2+ 2\alpha''_3 + \alpha''_4 & \omega'_1, \ \omega''_3 \\ \hline \phantom{\Big|} H/K \phantom{\Big|} & \alpha'''_1+2\alpha'''_2+3\alpha'''_3 & \omega'''_3\\ \hline \end{array} \end{align*} $$
   $$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|} \hat \varphi & : & a \omega_1 + b \omega_3 + c \omega_4 & \longmapsto & (a+c)\omega'_1 + (b+c)\omega''_3 \\ \hline \phantom{\Big|} \bar \varphi & : & a \omega_1 + b \omega_3 + c \omega_4 & \longmapsto & (a+b)\omega'''_3 \\ \hline \end{array} \end{align*} $$

Given $\lambda \in \Lambda _{G/K}^+$ , recall our notations $\hat \lambda =\hat \varphi (\lambda )$ and $\bar \lambda =\bar \varphi (\lambda )$ . If $\mu \in \Lambda ^+_{G/K}$ , we write $\mu \sim \lambda $ if $\hat \mu =\hat \lambda $ . Identifying $\mathbb C[\widehat G/ \widehat K] \simeq \mathbb C[G/K]$ , by definition, we have then

$$ \begin{align*}E_{\widehat G/\widehat K}(\hat \lambda) = \bigoplus_{\lambda' \sim \lambda} E_{G/K}(\lambda'). \end{align*} $$

Theorem 5.9 Let $(G,K)$ be one of the following spherical pairs of Table 3:

$$ \begin{align*}\mathrm{Sph.A6, \ Sph.A8, \ Sph.A10, \ Sph.A13, \ and \; Sph.A14. }\end{align*} $$

Then, for all $\lambda ,\mu ,\nu \in \Lambda ^+_{G/K}$ , we have

$$ \begin{align*}\begin{array}{ccc} E_{G/K}(\nu) \subset E_{G/K}(\lambda) \cdot E_{G/K}(\mu) & \Longleftrightarrow & \left\{ \begin{array}{c} E_{\widehat G/\widehat K}\big(\hat\nu\big) \subset E_{\widehat G/\widehat K}\big(\hat\lambda\big) \cdot E_{\widehat G/\widehat K}\big(\hat\mu\big), \phantom{\Big|} \\ E_{H/K}(\bar \nu) \subset E_{H/K}(\bar \lambda) \cdot E_{H/K}(\bar\mu). \end{array}\right. \end{array} \end{align*} $$

Proof Suppose that $E_{G/K}(\nu ) \subset E_{G/K}(\lambda ) \cdot E_{G/K}(\mu )$ . Looking at the associated $\widehat G$ -modules generated in the coordinate ring $\mathbb C[\widehat G/\widehat K]\ \simeq \mathbb C[G/K]$ , it follows that $E_{\widehat G/\widehat K}\big (\hat \nu \big ) \subset E_{\widehat G/\widehat K}\big (\hat \lambda \big ) \cdot E_{\widehat G/\widehat K}\big (\hat \mu \big )$ as well. Similarly, looking at the images in $\mathbb C[H/K]$ , we see that $E_{H/K}(\bar \nu ) \subset E_{H/K}(\bar \lambda ) \cdot E_{H/K}(\bar \mu )$ .

Suppose conversely that both the inclusions $E_{\widehat G/\widehat K}(\hat \nu ) \subset E_{\widehat G/\widehat K}(\hat \lambda ) \cdot E_{\widehat G/\widehat K}(\hat \mu )$ and $E_{H/K}(\bar \nu ) \subset E_{H/K}(\bar \lambda ) \cdot E_{H/K}(\bar \mu )$ hold. By the first inclusion, the restriction of functions from $G/K$ to $H/K$ induces a commutative diagram as follows:

(3.2)

By the second inclusion, there are $\lambda ' \sim \lambda $ , $\mu '\sim \mu $ , and $\nu ' \sim \nu $ such that $\bar \lambda ' = \bar \lambda $ , $\bar \mu ' = \bar \mu $ , $\bar \nu ' = \bar \nu $ , and $E_{G/K}(\nu ') \subset E_{G/K}(\lambda ') \cdot E_{G/K}(\mu ')$ . On the other hand, by Corollary 5.7, the restriction of $\bar \varphi $ to the fibers of $\hat \varphi $ is injective. Thus, $\lambda ' = \lambda $ , $\mu ' = \mu $ , and $\nu ' = \nu $ , and we get $E_{G/K}(\nu ) \subset E_{G/K}(\lambda ) \cdot E_{G/K}(\mu )$ .

Corollary 5.10 Let $(G,K)$ be one of the following spherical pairs of  Table 3:

$$ \begin{align*}\mathrm{Sph.A6, \ Sph.A8, \ Sph.A10, \ Sph.A13, \ and \; Sph.A14. }\end{align*} $$

If Conjecture 3.2 holds true, then Conjecture 5.1 holds true for $(G,K)$ . In particular, Conjecture 5.1 holds true in the cases

$$ \begin{align*}\mathrm{Sph.A6, \ Sph.A10, \ Sph.A13, \; and \; Sph.A14. }\end{align*} $$

Proof Set $X = G/K$ , $Y = \widehat G/\widehat K$ , and $Z = H/K$ . Then, by Proposition 5.6, we have an isogeny of reductive groups

$$ \begin{align*}G_Y \times G_Z \rightarrow G_X. \end{align*} $$

Notice that a $G_X$ -module is irreducible if and only if it is irreducible as a $(G_Y \times G_Z)$ -module under the previous map.

If $\lambda \in \Lambda _X^+$ , regarding $V_X(\lambda )$ as a $(G_Y \times G_Z)$ -module, by the definition of $\hat \lambda $ and $\bar \lambda $ , we see that

$$ \begin{align*}V_X(\lambda) \simeq V_Y \big(\hat \lambda\big) \boxtimes V_Z\big(\bar \lambda\big). \end{align*} $$

In particular, given $\lambda , \mu , \nu \in \Lambda ^+_X$ , we have

$$ \begin{align*}\begin{array}{ccc} V_X(\nu) \subset V_X(\lambda) \otimes V_X(\mu) \Longleftrightarrow \left\{\!\! \begin{array}{c} V_Y\big(\hat\nu\big) \subset V_Y \big(\hat\lambda\big) \otimes V_Y\big(\hat\mu\big), \phantom{\Big|} \\ V_Z(\bar \nu) \subset V_Z(\bar \lambda) \otimes V_Z(\bar\mu). \end{array}\right. \end{array} \end{align*} $$

Thus, by Theorem 5.9, we see that Conjecture 5.1 holds for X if and only if it holds both for Y and Z.

If both Y and Z are direct products of symmetric varieties with restricted root system of type $\mathsf A_1$ , then by Stanley’s Pieri rule for Jack symmetric functions, we see that Conjecture 3.2 holds true both for Y and Z. Thus, in this case, the claim follows for X as well.

Remark 5.11 There is another case which can be put in the framework outlined at the beginning of this subsection, even though G is not simple. In this case,

$$ \begin{align*}G/K = [\operatorname{\mathrm{Sp}}(2) \times \operatorname{\mathrm{Sp}}(2n)] / [\operatorname{\mathrm{diag}}(\operatorname{\mathrm{Sp}}(2)) \times \operatorname{\mathrm{Sp}}(2n-2)],\end{align*} $$

and $\Phi _{G/K}$ is of type $\mathsf A_1 \times \sf A_1$ , and Conjecture 5.1 holds true thanks to the same arguments of Theorem 5.9 and Corollary 5.10. The corresponding symmetric varieties are

$$ \begin{align*}\widehat G/\widehat K = \operatorname{\mathrm{SO}}(4n)/\operatorname{\mathrm{SO}}(4n-1), \end{align*} $$

$$ \begin{align*} H/K = [\operatorname{\mathrm{Sp}}(2) \times \operatorname{\mathrm{Sp}}(2) \times \operatorname{\mathrm{Sp}}(2n-2)]/[\operatorname{\mathrm{diag}}(\operatorname{\mathrm{Sp}}(2)) \times \operatorname{\mathrm{Sp}}(2n-2)] & \\ \simeq [\operatorname{\mathrm{Sp}}(2) \times \operatorname{\mathrm{Sp}}(2) ]/\operatorname{\mathrm{diag}}(\operatorname{\mathrm{Sp}}(2))&, \end{align*} $$

which have both restricted root system of type $\mathsf A_1$ . The associated spherical roots and spherical weights are described in the tables here below, together with the maps $\hat \varphi $ and  $\bar \varphi $ .

$$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. spherical weights}\\ \hline \hline \phantom{\Big|} G/K & \alpha_0 + \alpha_1, \, \alpha_1 + 2(\alpha_2 + \cdots + \alpha_{n-1}) + \alpha_n & \omega_0+\omega_1, \ \omega_2 \\ \hline \phantom{\bigg|} \widehat G/\widehat K \phantom{\bigg|} & 2(\alpha'_1 + \cdots + \alpha'_{2n-2}) + \alpha'_{2n-1} + \alpha'_{2n} & \omega_1' \\ \hline \phantom{\Big|} H/K & \alpha_0'' + \alpha_1'' & \omega_0'' + \omega_1'' \\ \hline \end{array} \end{align*} $$

$$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|} \hat \varphi & : & a (\omega_0+\omega_1) + b\omega_2 & \longmapsto & (a+2b)\omega'_1 \\ \hline \phantom{\Big|} \bar \varphi & : & a (\omega_0+\omega_1) + b\omega_2 & \longmapsto & a (\omega_0''+\omega_1'') \\ \hline \end{array} \end{align*} $$

5.3 Other cases with restricted root system of type $\mathsf A$

We now explain how the cases Sph.A7, Sph.A9, and Sph.A11 of Table 3 follow (almost immediately) from their companion cases Sph.A6, Sph.A8, and Sph.A10.

1. (Sph.A7) $G = \operatorname {\mathrm {SL}}(n), \ K = \operatorname {\mathrm {GL}}(n-1)$ . This case is immediately reduced to Sph.A6. Denote indeed $K' = \operatorname {\mathrm {SL}}(n-1)$ : then $K = \mathrm N_G(K')$ , and the spherical roots of $G/K$ are the same as those of $G/K'$ . On the other hand, $\Omega _{G/K} = \Lambda _{G/K'}$ , and $V(\lambda )^{(K)} = V(\lambda )^{K'}$ for all $\lambda \in \Omega _{G/K}^+$ ; therefore, $\mathbb C[G]^{(K)} = \mathbb C[G]^{K'}$ .

2. (Sph.A9) $G = \operatorname {\mathrm {SL}}(2n+1), \ K = \operatorname {\mathrm {GL}}(1) \times \operatorname {\mathrm {Sp}}(2n)$ . This case is immediately reduced to Sph.A8. Consider indeed $K' = \operatorname {\mathrm {Sp}}(2n)$ : then $K = \mathrm N_G(K')$ , and the spherical roots of $G/K$ are the same as those of $G/K'$ . Moreover, as in the previous case, we have $\mathbb C[G]^{(K)} = \mathbb C[G]^{K'}$ .

3. (Sph.A11) $G = \operatorname {\mathrm {Sp}}(2n), \ K = \operatorname {\mathrm {Sp}}(2) \times \operatorname {\mathrm {Sp}}(2n-2)$ . This case is immediately reduced to Sph.A10. Set indeed $K' = \operatorname {\mathrm {GL}}(1) \times \operatorname {\mathrm {Sp}}(2n-2)$ : then $K' \subset K$ ; thus, we have an inclusion $\mathbb C[G]^K \subset \mathbb C[G]^{K'}$ . On the other hand, the unique spherical root of $G/K$ is a spherical root of $G/K'$ as well, and if $\lambda , \mu \in \Lambda _{G/K}$ , then $\mu \leqslant _{G/K} \lambda $ if and only if $\mu \leqslant _{G/K'} \lambda $ .

As a consequence of the previous analysis, we get the following.

Proposition 5.12 Let $(G,K)$ be one of the spherical pairs Sph.A7, Sph.A9, and Sph.A11 of Table 3. If Conjecture 3.2 holds true, then Conjecture 5.1 holds true for $(G,K)$ . In particular, Conjecture 5.1 holds true for Sph.A7 and Sph.A11.

6 A conjectural rule in the case of spherical Levi subgroups

There is another case where we can apply the techniques of Section 5, and find two morphisms which decompose the root system of an affine spherical variety into those of two symmetric varieties. However, in this case, the root system generated by the spherical roots is of type $\mathsf B$ , and as far as we know, there is no rule (not even conjectural) to multiply spherical functions on a symmetric variety in this case.

6.1 The spherical pair $(\operatorname {\mathrm {SO}}(2p+1),\operatorname {\mathrm {GL}}(p))$

Set $G = \operatorname {\mathrm {SO}}(2p+1)$ and $K = \operatorname {\mathrm {GL}}(p)$ with $p\geqslant 3$ . Then, the homogeneous space $G/K$ is spherical, and it is a model space for G (see [Reference Gelfand and Zelevinsky13, Reference Gelfand and Zelevinsky14, Reference Luna27]).

The spherical root system $\Phi _{G/K}$ is of type $\mathsf B_{\lfloor p/2\rfloor } \times \mathsf B_{\lceil p/2\rceil }$ if $p\geqslant 4$ , and of type $\mathsf A_1\times \mathsf B_2$ if $p=3$ . In this case as well, it is possible to define reductive groups $H \subset G \subset \widehat G$ and two symmetric varieties $H/K$ and $\widehat G/\widehat K$ endowed with equivariant morphisms

$$ \begin{align*}H/K \hookrightarrow G/K \stackrel{\sim}{\longrightarrow} \widehat G/\widehat K, \end{align*} $$

which produce an isogeny of root data such as in Proposition 5.6. This will allow to extend Theorem 5.9 to this case as well, and to reduce the multiplication of the spherical functions on $G/K$ to the case of two symmetric varieties.

More precisely, define

$$ \begin{align*}\widehat G = \operatorname{\mathrm{SO}}(2p+2), \quad \widehat K = \operatorname{\mathrm{GL}}(p+1), \quad H = \operatorname{\mathrm{SO}}(2p).\end{align*} $$

The root systems $\Phi _{\widehat G/\widehat K}$ and $\Phi _{H/K}$ are of type $\mathsf B$ , whereas the restricted root systems $\widetilde \Phi _{\widehat G/\widehat K}$ and $\widetilde \Phi _{H/K}$ are either of type $\mathsf B\mathsf C$ or of type $\mathsf C$ , depending on the parity of p.

As in Section 5, the equivariant morphisms $H/K \hookrightarrow G/K \stackrel {\sim }{\to } \widehat G/\widehat K$ induce linear maps of the respective weight monoids

$$ \begin{align*}\hat \varphi : \Lambda^+_{G/K} \longrightarrow \Lambda^+_{\widehat G/\widehat K} , \qquad \qquad \bar \varphi : \Lambda^+_{G/K} \longrightarrow \Lambda^+_{H/K}. \end{align*} $$

For $\lambda \in \Lambda _{G/K}^+$ , we set $\hat \lambda = \hat \varphi (\lambda )$ and $\bar \lambda = \bar \varphi (\lambda )$ .

The spherical roots and the weight lattices, as well as the maps $\hat \varphi $ and $\bar \varphi $ , are described in the tables below. We separate the two cases p even and p odd.

( $p=2n$ even)

$$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K \phantom{\Big|} & \begin{array}{l} \alpha_i+\alpha_{i+1} \ (i=1, \ldots, 2n-1),\ \alpha_{2n} \end{array} & \begin{array}{l}\omega_i\ (i=1, \ldots ,2n-1),\\ 2\omega_{2n} \end{array}\\ \hline \phantom{\Big|}\widehat G / \widehat K\phantom{\Big|} & \begin{array}{l} \alpha'_{2i-1} + 2\alpha'_{2i}+\alpha'_{2i+1}\ (i=1, \ldots, n-1) , \\ \alpha'_{2n-1} + \alpha'_{2n} + \alpha'_{2n+1} \end{array} & \begin{array}{l}\omega'_{2i}\ (i=1, \ldots ,n-1), \\ \omega'_{2n} + \omega'_{2n+1} \end{array}\\ \hline \phantom{\Big|}H/K\phantom{\Big|} & \begin{array}{l} \alpha''_{2i-1} + 2\alpha''_{2i}+\alpha''_{2i+1} \ (i=1, \ldots, n-1), \\ \alpha''_{2n} \end{array} & \begin{array}{l}\omega''_{2i}\ (i=1, \ldots ,n-1), \\ 2\omega''_{2n} \end{array}\\ \hline \end{array} \end{align*} $$

$$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|}\hat \varphi & : & \sum_{i=1}^{2n} a_i \omega_i & \longmapsto & \big(\sum_{i=1}^{n-1} (a_{2i-1} + a_{2i}) \omega'_{2i}\big) + (a_{2n-1}+\frac{a_{2n}}{2}) (\omega'_{2n} + \omega'_{2n+1})\\ \hline \phantom{\Big|}\bar \varphi & : & \sum_{i=1}^{2n} a_i \omega_i & \longmapsto & \big(\sum_{i=1}^{n-1} (a_{2i}+a_{2i+1}) \omega''_{2i}\big) + a_{2n} \omega''_{2n}\\ \hline \end{array} \end{align*} $$

( $p=2n+1$ odd)

$$ \begin{align*}\begin{array}{|c|c|c|} \hline \phantom{\Big|} & \text{Normalized spherical roots} & \text{Fund. sph. weights}\\ \hline \hline \phantom{\Big|}G/K \phantom{\Big|} & \alpha_i+\alpha_{i+1} \ (i=1, \ldots, 2n),\ \alpha_{2n+1} & \begin{array}{l}\omega_i \ (i=1,\ldots ,2n), \\ 2\omega_{2n+1}\end{array} \\ \hline \phantom{\Big|}\widehat G / \widehat K\phantom{\Big|} & \begin{array}{l} \alpha'_{2i-1} + 2\alpha'_{2i}+\alpha'_{2i+1} \ (i=1, \ldots, n), \\ \alpha'_{2n+2} \end{array} & \begin{array}{l}\omega'_{2i} (i=1, \ldots, n), \\ 2\omega'_{2n+2}\end{array} \\ \hline \phantom{\Big|}H/K\phantom{\Big|} & \begin{array}{l} \alpha''_{2i-1} + 2\alpha''_{2i}+\alpha''_{2i+1} \ (i=1, \ldots, n-1), \\ \alpha''_{2n-1} + \alpha''_{2n} + \alpha''_{2n+1} \end{array} & \begin{array}{l}\omega''_{2i} (i=1, \ldots , n-1), \\ \omega''_{2n} + \omega''_{2n+1}\end{array} \\ \hline \end{array} \end{align*} $$

$$ \begin{align*}\begin{array}{|ccccl|} \hline \phantom{\Big|}\hat \varphi & : & \sum_{i=1}^{2n+1} a_i \omega_i & \longmapsto & \big(\sum_{i=1}^n (a_{2i-1} + a_{2i}) \omega'_{2i}\big) + a_{2n+1} \omega'_{2n+2}\\ \hline \phantom{\Big|}\bar \varphi & : & \sum_{i=1}^{2n+1} a_i \omega_i & \longmapsto & \big(\sum_{i=1}^{n-1} (a_{2i}+a_{2i+1}) \omega''_{2i}\big) + (a_{2n}+\frac{a_{2n+1}}{2}) (\omega''_{2n} + \omega''_{2n+1})\\ \hline \end{array} \end{align*} $$

Therefore, in this case as well, we obtain the similar results as those in Proposition 5.6 and Theorem 5.9.

Theorem 6.1 Let $(G,K)$ be the spherical pair $(\operatorname {\mathrm {SO}}(2p+1),\operatorname {\mathrm {GL}}(p))$ , then with the above definitions:

1. (i) The map

   $$ \begin{align*}(\hat \varphi, \bar \varphi) : \Xi_{G/K} \longrightarrow \Xi_{\widehat G/\widehat K} \oplus \Xi_{H/K} \end{align*} $$
   is an isogeny of based root data
   $$ \begin{align*} \mathcal R_{G/K} \longrightarrow \mathcal R_{\widehat G/\widehat K} \oplus \mathcal R_{H/K}. \end{align*} $$

2. (ii) For all $\lambda ,\mu ,\nu \in \Lambda ^+_{G/K}$ , we have

   $$ \begin{align*}\begin{array}{ccc} E_{G/K}(\nu) \subset E_{G/K}(\lambda) \cdot E_{G/K}(\mu) & \Longleftrightarrow & \left\{ \begin{array}{c} E_{\widehat G/\widehat K}(\hat \nu) \subset E_{\widehat G/\widehat K}(\hat \lambda) \cdot E_{\widehat G/\widehat K}(\hat \mu), \phantom{\Big|} \\ E_{H/K}(\bar \nu) \subset E_{H/K}(\bar \lambda) \cdot E_{H/K}(\bar \mu). \end{array}\right. \end{array} \end{align*} $$

The symmetric variety $\operatorname {\mathrm {SO}}(2n)/\operatorname {\mathrm {GL}}(n)$ , however, is not covered by Conjecture 3.2. This prompts us to wonder how the conjecture should be rephrased in this case, and more generally in the case of the symmetric varieties of Hermitian type (that is, the symmetric varieties $G/K$ with K a Levi subgroup).

Notice that if $(G,K)$ is a spherical pair with K a Levi subgroup of G, then either $G/K$ is a symmetric space of Hermitian type, or it is the pair $(\operatorname {\mathrm {SO}}(2p+1),\operatorname {\mathrm {GL}}(p))$ considered above, or it is the pair $(\operatorname {\mathrm {Sp}}(2p),\operatorname {\mathrm {GL}}(1) \times \operatorname {\mathrm {Sp}}(2p-2))$ which appeared as case Sph.A10 in Section 5.2. In particular, the analysis in this section reduces the case of the spherical Levi subgroups to the case of the Hermitian symmetric subgroups.

6.2 The Hermitian symmetric case

Let $X = G/K$ be a symmetric variety of Hermitian type, with G simple and simply connected. This means that the center of K has positive dimension, or equivalently that K is a Levi subgroup of G. In this case, the restricted root system $\widetilde \Phi _X$ is either of type $\mathsf C$ or $\mathsf {BC}$ , whereas the root system $\Phi _X$ is always of type $\mathsf B$ .

For completeness, we list in Table 4 all the symmetric varieties of Hermitian type. In all cases, we also give the simple restricted roots and a minimal set of generators for the weight monoid $\Lambda ^+_X$ .

1. (i) $\mathbf {SL(p+q)/S(GL(p)}L \times \mathbf {GL(q))}$ , $q> p \geqslant 1$ : $\Delta _X = \{\alpha _i + \alpha _{p+q-i}\ :\ 1\leqslant i < p \}\cup \{\alpha _p + \cdots + \alpha _q \}$ , $\Lambda ^+_X$ is generated by the weights $\omega _i + \omega _{p+q-i}$ with $1\leqslant i\leqslant p$ .

2. (ii) $\mathbf {SL(2p)/S(GL(p)} \times \mathbf {GL(p))}$ , $p \geqslant 1$ : $\Delta _X = \{\alpha _i + \alpha _{2p-i}\ :\ 1\leqslant i < p\}\cup \{\alpha _p\}$ , $\Lambda ^+_X$ is generated by the weights $\omega _i + \omega _{2p-i}$ with $1\leqslant i < p$ , together with $2 \omega _p$ .

3. (iii) $\mathbf {Spin(2p+1)/Spin(2)} \times \mathbf {Spin(2p-1)}$ , $p \geqslant 2$ : $\Delta _X = \{\alpha _1, \; 2 (\alpha _2 + \cdots + \alpha _p)\}$ , $\Lambda ^+_X$ is generated by $2\omega _1$ and $\omega _2$ .

4. (iv) $\mathbf {Spin(2p)/Spin(2)} \times \mathbf {Spin(2p-2)}$ , $p \geqslant 3$ : $\Delta _X = \{\alpha _1, \; 2 (\alpha _2 + \cdots + \alpha _{p-2}) + \alpha _{p-1}+\alpha _p \}$ , $\Lambda ^+_X$ is generated by $2\omega _1$ and $\omega _2$ .

5. (v) $\mathbf {Sp(2p)/GL(p)}$ , $p \geqslant 2$ : $\Delta _X =\{2\alpha _i\ :\ 1\leqslant i\leqslant p-1\}\cup \{\alpha _p\}$ , $\Lambda ^+_X$ is generated by the weights $2\omega _i$ with $1 \leqslant i \leqslant p$ .

6. (vi) $\mathbf {Spin(4p+2)/GL(2p+1)}$ , $p \geqslant 1$ : $\Delta _X =\{\alpha _{2i-1}+2\alpha _{2i}+\alpha _{2i+1}\, :\, 1\leqslant i < p\}\cup \{\alpha _{2p-1} + \alpha _{2p} + \alpha _{2p+1} \}$ , $\Lambda ^+_X$ is generated by the weights $\omega _{2i}$ with $1 \leqslant i < p$ together with $\omega _{2p} + \omega _{2p+1}$ .

7. (vii) $\mathbf {Spin(4p)/GL(2p)}$ , $p \geqslant 2$ : $\Delta _X =\{\alpha _{2i-1}+2\alpha _{2i}+\alpha _{2i+1}\ :\ 1\leqslant i\leqslant p-1\}\cup \{ \alpha _{2p} \}$ , $\Lambda ^+_X$ is generated by the weights $\omega _{2i}$ with $1 \leqslant i < p$ , together with $2 \omega _{2p}$ .

8. (viii) $\mathbf {E}_{\mathbf {6}}/{\mathbf {D}}_{\mathbf {5}} \times \mathbb C^*$ : $\Delta _X = \{\alpha _1 + \alpha _3 +\alpha _4+\alpha _5 + \alpha _6, \; 2\alpha _2 + \alpha _3 + 2\alpha _4 + \alpha _5 \}$ , $\Lambda ^+_X$ is generated by $\omega _1+\omega _6$ and $\omega _2$ .

9. (ix) $\mathbf {E}_{\mathbf {7}}/\mathbf {E}_{\mathbf {6}}\times \mathbb C^*$ : $\Delta _X = \{2\alpha _1 + \alpha _2 + 2\alpha _3 +2\alpha _4 + \alpha _5, \; \alpha _2 + \alpha _3 + 2(\alpha _4 + \alpha _5 + \alpha _6), \; \alpha _7\}$ , $\Lambda ^+_X$ is generated by $\omega _1$ , $\omega _6$ , and $2\omega _7$ .

Table 4: Symmetric varieties of Hermitian type.

If $\widetilde \Phi _X$ is nonreduced, then K is self-normalizing, and the derived subgroup $K'$ is also spherical in G. If instead $\widetilde \Phi _X$ is reduced, then K is not self-normalizing, and its derived subgroup is not spherical in G. Therefore, there are some Hermitian symmetric subgroups which have a nonsymmetric semisimple spherical counterpart, similarly to the cases Sph.A7 and Sph.A9 of Table 3.

We list such spherical pairs in Table 5. In all cases, we also give the simple restricted roots and a minimal set of generators for the weight monoid $\Lambda ^+_X$ .

1. (i’) $\mathbf {SL(p+q)/SL(p)} \times \mathbf {SL(q))}$ , $q> p \geqslant 1$ : $\Delta _X = \{\alpha _i + \alpha _{p+q-i}\ :\ 1\leqslant i < p \}\cup \{\alpha _p + \cdots + \alpha _q \}$ , $\Lambda ^+_X$ is generated by the weights $\omega _i + \omega _{p+q-i}$ with $1\leqslant i < p$ together with $\omega _p$ and $\omega _q$ .

2. (vi’) $\mathbf {Spin(4p+2)/SL(2p+1)}$ , $p \geqslant 1$ : $\Delta _X =\{\alpha _{2i-1}+2\alpha _{2i}+\alpha _{2i+1}\ :\ 1\leqslant i < p\}\cup \{\alpha _{2p-1} + \alpha _{2p} + \alpha _{2p+1} \}$ , $\Lambda ^+_X$ is generated by the weights $\omega _{2i}$ with $1 \leqslant i < p$ together with $\omega _{2p}$ and $\omega _{2p+1}$ .

3. (viii’) $\mathbf {E}_{\mathbf {6}}/\mathbf {D}_{\mathbf {5}}$ : $\Delta _X = \{\alpha _1 + \alpha _3 +\alpha _4+\alpha _5 + \alpha _6, \; 2\alpha _2 + \alpha _3 + 2\alpha _4 + \alpha _5 \}$ , $\Lambda ^+_X$ is generated by $\omega _1$ , $\omega _2$ , and $\omega _6$ .

Table 5: Nonsymmetric reductive spherical pairs $(G,K)$ with $G/\mathrm N_G(K)$ symmetric of Hermitian type.

If $(G,K')$ is a spherical pair appearing in Table 5 and $K = \mathrm N_G(K')$ is the corresponding symmetric subgroup of Hermitian type, then we have $\Omega _{G/K}^+ =\Lambda _{G/K'}^+$ and $\mathbb C[G]^{(K)} = \mathbb C[G]^{K'}$ , as in the cases Sph.A7 and Sph.A9. Thus, the problem of decomposing the product of two spherical functions on $G/K$ reduces to the analogous problem on $G/K'$ .

In the same spirit of Conjectures 3.2 and 5.1, we formulate the following for the multiplication of spherical functions in terms of the tensor product of the restricted root system, which by computational experiments seems to hold for all the Hermitian symmetric pairs but in case He.1a.

Conjecture 6.2 Let $(G,K)$ be a reductive spherical pair with G simple and K a Levi subgroup of G different from case He.1a. For all $\lambda , \mu , \nu \in \Lambda _X^+$ , it holds

$$ \begin{align*}E_{G/K}(\nu) \subset E_{G/K}(\lambda) \cdot E_{G/K}(\mu) \Longleftrightarrow \left\{ \begin{array}{c} V_X(\nu) \subset V_X(\lambda) \otimes V_X(\mu), \phantom{\Big|} \\ \nu \leqslant_X \lambda + \mu. \end{array} \right. \end{align*} $$

By Theorem 6.1 and Corollary 5.10, the previous conjecture immediately reduces to the case where $G/K$ is a symmetric variety of Hermitian type.

Remark 6.3 In [Reference Bravi, Gandini and Maffei4, Section 9], the following question was considered. Suppose that G is simply laced, and let $X = G/K$ be a spherical variety with $\Delta ^{\mathrm {sc}}_X \cap \Delta = \varnothing $ . Let $\lambda , \mu , \nu \in \Lambda _X^+$ , is it true that $E_X(\nu ) \subset E_X(\lambda ) \cdot E_X(\mu )$ if and only if $V(\nu ) \subset V(\lambda ) \otimes V(\mu )$ and $\nu \leqslant _{\Delta ^{\mathrm {sc}}_X} \lambda + \mu $ ?

This indeed seems to be true in several cases, and in particular in case He.1a. Assuming the validity of Conjecture 3.2, it is easy to see that when X is symmetric and both $\Phi $ and $\Phi _X$ are of type $\mathsf A$ , a positive answer to the previous question follows from the saturation theorem [Reference Knutson and Tao22]. See also [Reference Graham and Hunziker16], where a similar question is considered for the symmetric varieties with restricted root system of type $\mathsf A$ . However, the following example shows that the previous question does not always have a positive answer, at least when $\widetilde \Phi _X$ is nonreduced. Conjecture 6.2 is a partial attempt to fix the question.

Example 6.4 Consider the symmetric variety $X = \operatorname {\mathrm {Spin}}(10)/\operatorname {\mathrm {GL}}(5)$ . The restricted root system $\widetilde \Phi _X$ is of type $\mathsf {BC}_2$ , with base

$$ \begin{align*}\widetilde \Delta_X = \{\alpha_1 + 2\alpha_2 + \alpha_3, \; \alpha_3 + \alpha_4 + \alpha_5 \}. \end{align*} $$

Let us consider the highest root $\theta = \alpha _1 + 2\alpha _2 + 2\alpha _3 + \alpha _4 + \alpha _5$ : then $\theta \in \Lambda _X^+$ and $\theta \leqslant _X 2 \theta $ . Moreover, $V(\theta ) \subset V(\theta )^{\otimes 2}$ ; however, taking the product inside $\mathbb C[X]$ , we have $E_X(\theta ) \not \subset E_X(\theta )^2$ : indeed, the unique nonzero equivariant projection

$$ \begin{align*}\pi : V(\theta) \otimes V(\theta) \rightarrow V(\theta) \end{align*} $$

is given by the bracket on the Lie algebra $\mathfrak g \simeq V(\theta )$ . It follows that $\pi (v \otimes v) = 0$ , for all $v \in V(\theta )$ , and we conclude by Proposition 2.2.

The same happens more generally for $\operatorname {\mathrm {Spin}}(2p)/\operatorname {\mathrm {GL}}(p)$ whenever p is odd, and for $\mathsf E_6/\mathsf D_5 \times \mathbb C^*$ : indeed, in all these cases, the highest root is a spherical weight, and $V(\theta )$ occurs in $ V(\theta )^{\otimes 2}$ with multiplicity 1.

Both these examples are instances of exceptional symmetric varieties of Hermitian type, that is, symmetric varieties of Hermitian type with nonreduced restricted root system. There is another instance of such a symmetric variety, namely $X = \operatorname {\mathrm {SL}}(p+q)/\mathrm S(\operatorname {\mathrm {GL}}(p) \times \operatorname {\mathrm {GL}}(q))$ when $p \neq q$ : however, in this case, $V(\theta )$ occurs in $ V(\theta )^{\otimes 2}$ with multiplicity 2, and the inclusion $E_X(\theta ) \subset E_X(\theta )^2$ holds true inside $\mathbb C[X]$ .