Delta Operators on Almost Symmetric Functions

Abstract

We construct $\Delta$-operators $F[\Delta]$ on the space of almost symmetric functions $\mathscr{P}_{as}^{+}$. These operators extend the usual $\Delta$-operators on the space of symmetric functions $\Lambda \subset \mathscr{P}_{as}^{+}$ central to Macdonald theory. The $F[\Delta]$ operators are constructed as certain limits of symmetric functions in the Cherednik operators $Y_i$ and act diagonally on the stable-limit non-symmetric Macdonald functions $\widetilde{E}_{(\mu|\lambda)}(x_1,x_2,\ldots;q,t).$ Using properties of Ion-Wu limits, we are able to compute commutation relations for the $\Delta$-operators $F[\Delta]$ and many of the other operators on $\mathscr{P}_{as}^{+}$ introduced by Ion-Wu. Using these relations we show that there is an action of $\mathbb{B}_{q,t}^{\text{ext}}$ on almost symmetric functions which we show is isomorphic to the polynomial representation of $\mathbb{B}_{q,t}^{\text{ext}}$ constructed by Gonz\'{a}lez-Gorsky-Simental.