The Stable Limit Daha: The Structure of the Standard Representation

ABSTRACT We prove a number of results about the structure of the standard representation of the stable limit double affine Hecke algebra (DAHA). More precisely, we address the triangularity, spectrum, and eigenfunctions of the limit Cherednik operators, and construct several bases of Poincaré-Birkhoff-Witt (PBW) type for the stable limit DAHA. We establish a remarkable triangularity property concerning the contribution of certain special elements of the PBW basis of a finite rank DAHA of high enough rank to the PBW expansion of a PBW basis element of the stable limit DAHA. The triangularity property implies the faithfulness of the standard representation. This shows that the algebraic structure defined by the limit operators associated to elements of the finite rank DAHAs is precisely the stable limit DAHA.