A Pieri-type multiplication rule for spherical functions: Bravi–Gandini Conjecture 1.1 for spherical pairs of type A (including F4/Spin(9)) and the Hermitian symmetric cases of types B, C, BC

For an affine spherical homogeneous variety $X=G/K$ (a homogeneous space on which a Borel subgroup of $G$ has a dense orbit), the coordinate ring $\mathbb{C}[X]$ decomposes multiplicity-freely as a direct sum of irreducible $G$-modules $E_X(\lambda)$. Bravi and Gandini (2021; Canad. J. Math. 2023) conjectured a Littlewood–Richardson-type rule determining when $E_X(\nu)$ occurs in the product $E_X(\lambda)\cdot E_X(\mu)$ when the non-degenerate spherical root system of $X$ is of type A: occurrence is equivalent to containment of the corresponding module $V_X(\nu)$ in $V_X(\lambda)\otimes V_X(\mu)$ together with the dominance order condition $\lambda+\mu-\nu\in\mathbb{N}\Delta_X^{\mathrm n}$. They proved the rule whenever the spherical root system decomposes into rank-one subsystems, except the single case $X=\mathrm{F}_4/\mathrm{Spin}(9)$ (supported there only by computational experiments), and reduced the remaining type-A cases to Stanley's 1989 conjecture that the normalized structure constants of Jack symmetric functions are polynomials with non-negative integer coefficients. Stanley's conjecture remains open (only special families and structural properties are known), and no multiplication rule is known for symmetric varieties whose restricted root system is of type C or non-reduced BC (the Hermitian symmetric cases, in which the spherical root system $\Phi_X$ is always of type B), where the conjectural Hermitian criterion (Conjecture 6.2) is unsettled and a related variant criterion is known to fail. The problem is open.

Background

Let $G$ be a complex reductive algebraic group, $K\subset G$ a reductive subgroup, and $X=G/K$ the corresponding homogeneous variety. The pair $(G,K)$ (equivalently $X$) is called spherical if a Borel subgroup $B$ of $G$ (a maximal connected solvable subgroup) has an open dense orbit on $X$; this is equivalent to the coordinate ring $\mathbb{C}[X]$ being multiplicity-free as a $G$-module, that is, $\mathbb{C}[X]=\bigoplus_{\lambda\in\Lambda_X^+}E_X(\lambda)$, where $\Lambda_X^+$ is the weight monoid of $X$ (the semigroup of highest weights of the irreducible $G$-modules occurring in $\mathbb{C}[X]$) and $E_X(\lambda)$ is the irreducible $G$-module of highest weight $\lambda$. Identifying $\mathbb{C}[X]\simeq\mathbb{C}[G]^K$, the elements of $\mathbb{C}[X]$ are the (algebraic) spherical functions on $X$. Since $\mathbb{C}[X]$ is a $G$-algebra, the product of two summands decomposes again into summands, and the basic structural question is: for which $\nu$ does $E_X(\nu)$ occur in the product $E_X(\lambda)\cdot E_X(\mu)$? In the model case $G=H\times H$, $K=\mathrm{diag}(H)$, where $\mathbb{C}[G/K]=\mathbb{C}[H]$ and $E_X(\lambda)\simeq V(\lambda)^*\otimes V(\lambda)$, the answer is the classical Littlewood–Richardson rule (the combinatorial rule describing the decomposition of a product of two Schur functions, the characters of the irreducible representations of the general linear groups, into a sum of Schur functions), so the problem asks for a Littlewood–Richardson-type (Pieri-type) rule for arbitrary spherical pairs.

The conjectural answer is formulated through invariants of the spherical variety. The spherical roots of $X$, more precisely its n-spherical roots $\Delta_X^{\mathrm n}$, are attached to $X$ by the multiplication itself: the differences $\lambda+\mu-\nu$ for triples with $E_X(\nu)\subset E_X(\lambda)\cdot E_X(\mu)$ generate a submonoid $\mathcal{M}_X^{\mathrm n}$ of the weight lattice, which is free by theorems of Knop and of Avdeev–Cupit-Foutou, and $\Delta_X^{\mathrm n}$ is its unique set of free generators; it is the base of a reduced root system $\Phi_X^{\mathrm n}$, the non-degenerate spherical root system of $X$, whose rank is called the rank of $X$. For a symmetric variety, $\Delta_X^{\mathrm n}$ coincides with the base $\widetilde\Delta_X$ of the restricted root system $\widetilde\Phi_X$ (which may be non-reduced), and $\Phi_X^{\mathrm n}$ is essentially the restricted root system attached to $X$. The base $\Delta_X^{\mathrm n}$ defines a partial order $\nu\leqslant_X\lambda+\mu$ by $\lambda+\mu-\nu\in\mathbb{N}\Delta_X^{\mathrm n}$, and via the based root datum $R_X$ attached to $X$ (the system of characters, cocharacters, roots $\Phi_X$ and coroots of $X$, together with a chosen base, which determines a reductive group up to isomorphism) one gets a complex reductive group $G_X$ whose dominant weights contain $\Lambda_X^+$; here $\Phi_X$ is the spherical root system attached to $X$, the reduced root system obtained from the non-degenerate spherical root system $\Phi_X^{\mathrm n}$ by replacing some of the spherical roots with their half, so that for a symmetric variety of Hermitian type, where the restricted root system $\widetilde\Phi_X$ is of type $\mathrm C$ or non-reduced $\mathrm{BC}$, $\Phi_X$ is always of type $\mathrm B$. Write $V_X(\pi)$ for the irreducible $G_X$-module of highest weight $\pi$. Bravi and Gandini conjectured that for affine $X$ with $\Phi_X^{\mathrm n}$ of type $\mathrm A$, the module $E_X(\nu)$ occurs in $E_X(\lambda)\cdot E_X(\mu)$ if and only if $V_X(\nu)\subset V_X(\lambda)\otimes V_X(\mu)$ and $\nu\leqslant_X\lambda+\mu$.

The evidence, and the known reduction, come from the theory of Jack symmetric functions. The Jack symmetric functions $J^{(k)}_\lambda$ (indexed by partitions $\lambda$ and a parameter $k$) form a basis of the ring of symmetric functions, orthogonal for a scalar product $\langle\cdot,\cdot\rangle_k$; at $k=1$ they specialize to Schur functions $s_\lambda$. Their structure constants $g^\nu_{\lambda\mu}(k)$ are defined by $J^{(k)}_\lambda J^{(k)}_\mu=\sum_\nu g^\nu_{\lambda\mu}(k)\,j_\nu(k)^{-1}J^{(k)}_\nu$ with $j_\nu(k)=\langle J^{(k)}_\nu,J^{(k)}_\nu\rangle_k$, and $g^\nu_{\lambda\mu}(1)$ is (a positive multiple of) the Littlewood–Richardson coefficient $c^\nu_{\lambda\mu}$ (the multiplicity of $s_\nu$ in $s_\lambda s_\mu$). Stanley conjectured in 1989 that each $g^\nu_{\lambda\mu}(k)$ is a polynomial in $k$ with non-negative integer coefficients; polynomiality with integer coefficients is known by work of Knop and Sahi, and non-negativity is known when one of $\lambda,\mu$ is a single row (Stanley's Pieri-type rule; classically, a Pieri rule is an explicit multiplication formula for the product with a single-row or single-column factor), but the conjecture is open in general. The link to spherical varieties is that for a symmetric space $X=G/K$ ($K$ the fixed-point subgroup of an involution of $G$) whose restricted root system (the root system of the pair, computed from a maximal abelian subspace of the complement of the Lie algebra of $K$) is of type $\mathrm A$ and whose restricted roots have multiplicity $m$, the spherical functions on $X$, suitably normalized, are specializations of the $J^{(k)}_\lambda$ at $k=2/m$; consequently $E_X(\nu)\subset E_X(\lambda)\cdot E_X(\mu)\iff g^\nu_{\lambda\mu}(2/m)\neq0$, and Stanley's conjecture would identify the non-zero values with the non-zero Littlewood–Richardson coefficients. Graham and Hunziker had exploited this to prove the analogous rule for Hermitian symmetric spaces (symmetric spaces whose compact dual carries a $G$-invariant complex structure) with restricted root system of type $\mathrm A$. Bravi and Gandini proved their Conjecture 1.1 whenever $\Phi_X^{\mathrm n}$ is a direct sum of rank-one subsystems, via Stanley's Pieri rule, with the single exception of $X=\mathrm F_4/\mathrm{Spin}(9)$ (a symmetric variety with non-reduced restricted root system of type $\mathrm{BC}_1$, whose compact real form is the octonionic projective plane $\mathbb{P}^2(\mathbb{O})$), where the rule is supported only by computational experiments; for the remaining type-A cases they reduced the conjecture to Stanley's conjecture, using factorizations of symmetric overgroups (classified by Onishchik: decompositions of a simple group as an almost direct product of two reductive subgroups) to embed the decomposition problem for $X$ into products of symmetric-variety problems.

Beyond type A, Bravi and Gandini formulated Conjecture 6.2 for spherical pairs $(G,K)$ with $K$ a Levi subgroup of $G$ (the centralizer of a torus, i.e. a reductive complement to the unipotent radical of a parabolic subgroup), which reduces to Hermitian symmetric pairs: the same tensor-plus-order criterion, formulated in terms of the $G_X$-modules $V_X(\pi)$ and the order $\leqslant_X$, is conjectured to hold for all Hermitian symmetric pairs except case He.1a, namely $X=\mathrm{SL}(p+q)/\mathrm S(\mathrm{GL}(p)\times\mathrm{GL}(q))$ with $q>p\ge 1$. A related variant question, raised in Section 9 of Bravi–Gandini–Maffei, replaces the tensor condition by $V(\nu)\subset V(\lambda)\otimes V(\mu)$ for the irreducible $G$-modules themselves and the order by the one defined by the sc-spherical roots $\Delta_X^{\mathrm{sc}}$ of $X$; that this variant criterion cannot hold uniformly when the restricted root system is non-reduced is shown by their Example 6.4: on $X=\mathrm{Spin}(10)/\mathrm{GL}(5)$, and more generally on $\mathrm{Spin}(2p)/\mathrm{GL}(p)$ for odd $p$ as well as on $\mathrm E_6/\mathrm D_5\times\mathbb C^*$, for the highest root $\theta$ one has $V(\theta)\subset V(\theta)^{\otimes2}$ and $\theta\leqslant_X2\theta$, yet $E_X(\theta)\not\subset E_X(\theta)^2$, because the unique equivariant projection $V(\theta)\otimes V(\theta)\to V(\theta)$ is the Lie bracket, which annihilates all pure tensors $v\otimes v$; Conjecture 6.2 is a partial attempt to fix that question. Which Hermitian symmetric varieties satisfy the tensor-plus-order criterion of Conjecture 6.2, and what the correct criterion is in the remaining types in general, remains open.

Problem Statement

Settle the conjectural multiplication rule for spherical functions posed by Bravi and Gandini, in the cases left open by their Theorem 1.2 and beyond type A. The two components share one scientific objective: an exact, representation-theoretic and combinatorial characterization of the multiplicative structure of the multiplicity-free $G$-algebra $\mathbb{C}[G/K]$ for reductive spherical pairs, of which component (1) is the type-A core and component (2) is its extension to the Hermitian symmetric cases of the remaining restricted-root types.

(1) Prove or disprove Conjecture 1.1 of Bravi–Gandini: for every reductive spherical pair $(G,K)$ with $G$ simple such that the non-degenerate spherical root system $\Phi_X^{\mathrm n}$ of $X=G/K$ is of type $\mathrm A$, and for all $\lambda,\mu,\nu$ in the weight monoid $\Lambda_X^+$,

$E_X(\nu)\subset E_X(\lambda)\cdot E_X(\mu)\iff\left(V_X(\nu)\subset V_X(\lambda)\otimes V_X(\mu)\ \text{and}\ \lambda+\mu-\nu\in\mathbb{N}\Delta_X^{\mathrm n}\right),$

where $V_X(\pi)$ denotes the irreducible module of highest weight $\pi$ for the reductive group $G_X$ attached to the based root datum of $X$. The open content is precisely: (a) the cases in which $\Phi_X^{\mathrm n}$ is not a direct sum of rank-one subsystems, and (b) the exceptional case $X=\mathrm F_4/\mathrm{Spin}(9)$, so far supported only by computational experiments.

(2) Beyond type A: establish a correct general criterion for $E_X(\nu)\subset E_X(\lambda)\cdot E_X(\mu)$ when $X$ is a Hermitian symmetric variety, whose restricted root system is of type $\mathrm C$ or non-reduced $\mathrm{BC}$ and whose spherical root system $\Phi_X$ is of type $\mathrm B$. The mandatory content is the following three items, each of which is to be settled independently. (i) Prove or disprove Conjecture 6.2 of Bravi–Gandini (the tensor-plus-order criterion in terms of the $G_X$-modules $V_X(\pi)$ and the order $\leqslant_X$) for the Hermitian symmetric pairs other than He.1a ($\mathrm{SL}(p+q)/\mathrm S(\mathrm{GL}(p)\times\mathrm{GL}(q))$, $q>p\ge 1$). (ii) Determine whether the variant criterion formulated in Section 9 of Bravi–Gandini–Maffei holds for case He.1a (there the tensor condition is $V(\nu)\subset V(\lambda)\otimes V(\mu)$ for the irreducible $G$-modules themselves, and the order is defined by the sc-spherical roots $\Delta_X^{\mathrm{sc}}$ of $X$, a canonical base of a reduced root system attached to $X$). (iii) If Conjecture 6.2 is refuted in (i), identify the Hermitian symmetric pairs other than He.1a on which its tensor-plus-order criterion fails, and provide the criterion demanded by this component for those pairs. An affirmative answer to (i) settles (iii); a refutation of (i) is discharged by exhibiting the pairs on which the criterion fails.

References

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