Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials

Abstract: We study the $q$-analogue of the Haldane-Shastry model, a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of Uglov, via 'freezing' from the affine Hecke algebra. We obtain explicit expressions for the spin-analogue of Macdonald operators. Through freezing these yield the higher Hamiltonians of the spin chain, including a Hamiltonian of the opposite chirality. The sum of the two chiral Hamiltonians has a real spectrum also for $q$ a root of unity. We clarify the relation between patterns labelling the eigenspaces, known as 'motifs', and the corresponding degeneracies in the crystal limit $q\to\infty$. For each motif we obtain an explicit expression for the exact eigenvector, valid for generic $q$, that has ('pseudo' or 'l-') highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of $A$ and $D$ and annihilated by $C$. It has a simple component featuring the 'symmetric square' of the $q$-Vandermonde times a Macdonald polynomial - or more precisely its quantum spherical zonal special case. Its other components are obtained through the action of the Hecke algebra, followed by 'evaluation' of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the spectrum is complete. The model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the $Y$-operators of the affine Hecke algebra. The key step in our diagonalisation is that on a subspace of suitable polynomials the first $M$ 'classical' (i.e. no difference part) $Y$-operators in $N$ variables reduce, upon evaluation as above, to $Y$-operators in $M$ variables with parameters at the quantum zonal spherical point.