Construction of Commuting Difference Operators for Multiplicity Free Spaces

The analysis of invariant differential operators on certain multiplicity free spaces led recently to the introduction of a family of symmetric polynomials that is more general than Jack polynomials (see [KS], but also [OO1], [OO2]). They are called interpolation Jack polynomials, shifted Jack polynomials, or Capelli polynomials. Apart from being inhomogeneous, they are distinguished from classical Jack polynomials by their very simple definition in terms of vanishing conditions. One of the most important and non-obvious properties of Capelli polynomials is that they are eigenfunctions of certain explicitly given difference (as opposed to differential) operators (see [KS]). This readily implies that their top homogeneous term is in fact a (classical) Jack polynomial. Other consequences include a binomial theorem, a Pieri formula, and much more. It is well-known that Jack polynomials are tied to root systems of type A and that they have natural analogues for other root systems (see e.g., [He]). Therefore, it is a natural problem as to whether this holds for the Capelli polynomials as well. Okounkov [Ok] proposed such an analogue for root systems of type BC and proved that these share some of the nice properties of Capelli polynomials. But, unfortunately, Okounkov’s polynomials do not satisfy difference equations. Also their representation-theoretic significance is not clear. In this paper we go back to the origin and let ourselves be guided by the theory of multiplicity free actions. It is known (see section 7 for details) that these actions give rise to combinatorial structures consisting of four data (Γ,Σ,W, `). Here Γ is a lattice, Σ ⊂ Γ a basis, W ⊆ Aut Γ a finite reflection group, and ` ∈ Γ some element. These data alone suffice to formulate the definition of a (generalized) Capelli polynomial but, in that generality, neither existence nor uniqueness will hold, let alone any other good property.