Baker-Akhiezer function for the deformed root system $BC(l,1)$ and bispectrality

Abstract

We show that a Sergeev-Veselov difference operator of rational Macdonald-Ruijsenaars (MR) type for the deformed root system $BC(l,1)$ preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker-Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero-Moser-Sutherland (CMS) Hamiltonian for $BC(l,1)$. By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type $BC(l,1)$.