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We construct and study an explicit simultaneous $\mathscr{Y}$-eigenbasis ofIon and Wu's standard representation of the $^+$stable-limit double affineHecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basisarises as a generalization of Cherednik's non-symmetric Macdonald polynomialsof type $GL$. We utilize links between $^+$stable-limit double affine Heckealgebra theory of Ion-Wu and the double Dyck path algebra of Carlsson-Mellitthat arose in their proof of the Shuffle Conjecture. As a consequence, thespectral theory for the limit Cherednik operators is understood. The symmetricfunctions comprise the zero weight space. We introduce one extra operator thatcommutes with the $\mathscr{Y}_i$ action and dramatically refines the weightspaces to now be one-dimensional. This operator, up to a change of variables,gives an extension of Haiman's operator $\Delta'$ from $\Lambda$ to$\mathscr{P}_{as}^{+}.$ Additionally, we develop another method to build thisweight basis using limits of trivial idempotents.

Stable-Limit Non-symmetric Macdonald Functions