Delta Operators on Almost Symmetric Functions

We construct Δ-operators F[Δ] on the space of almost symmetric functions Pₐₛ⁺. These operators extend the usual Δ-operators on the space of symmetric functions Λ ⊂ Pₐₛ⁺ central to Macdonald theory. The F[Δ] operators are constructed as certain limits of symmetric functions in the Cherednik operators Yᵢ and act diagonally on the stable-limit non-symmetric Macdonald functions E(μ|λ)(x₁,x₂,…;q,t). Using properties of Ion-Wu limits, we are able to compute commutation relations for the Δ-operators F[Δ] and many of the other operators on Pₐₛ⁺ introduced by Ion-Wu. Using these relations we show that there is an action of Bq,tᵉˣᵗ on almost symmetric functions which we show is isomorphic to the polynomial representation of Bq,tᵉˣᵗ constructed by Gonz\'{a}lez-Gorsky-Simental.