On the multiplication of spherical functions of reductive spherical pairs of type A

TL;DR

This paper investigates the algebraic structure of spherical functions on certain reductive pairs of type A, proposing a conjectural decomposition rule for their multiplication based on symmetric functions, with partial proofs in specific cases.

Contribution

It introduces a conjectural rule for decomposing products of spherical functions in type A cases, linking to Stanley's conjecture on Jack symmetric functions.

Findings

Proposes a decomposition rule for spherical functions of type A.

Validates the rule in cases where the root system is a direct sum of rank-one subsystems.

Connects the problem to Stanley's conjecture on Jack symmetric functions.

Abstract

Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into irreducible summands of the product of irreducible G-submodules in C[G/K]. When the spherical roots of G/K generate a root system of type A we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of G/K is direct sum of subsystems of rank one.