Delta Operators on Almost Symmetric Functions

Abstract

We construct $\Delta$-operators $F[\Delta ]$ on the space of almost symmetric functions ${\mathcal{P}}_{as}^{+}$. These operators extend the usual $\Delta$-operators on the space of symmetric functions $\Lambda \subset {\mathcal{P}}_{as}^{+}$ 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 ${\tilde{E}}_{(\mu |\lambda )}(x_1,x_2,\dots ;q,t).$ 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 ${\mathcal{P}}_{as}^{+}$ 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 González-Gorsky-Simental.

20 pages, corrected normalization error in section 3.4; this paper answers a conjecture from the author's prior paper arXiv:2307.05864