KP Trigonometric Solitons and an Adelic Flag Manifold

We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1–41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero–Moser system and the rational Ruijsenaars–Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127–165].