Harmonically lifting linear series from curves to K3 surfaces

We investigate the question of which Brill--Noether special linear series on a curve on a K3 surface lift to the surface. Work of Reid, Donagi--Morrison, Green--Lazarsfeld, Lelli-Chiesa, and others shows that Brill--Noether special linear series admits lifts in many cases, in particular when the linear series computes the Clifford index of the curve. Using the recently defined notion of harmonic filtrations of Lazarsfeld--Mukai bundles, we show the same is true for a linear series computing the Brill--Noether special Clifford index, an analogue of the classical Clifford index considering only Brill--Noether special linear series, and show that the Brill--Noether special Clifford index is constant in linear series on K3 surfaces.