With the help of operator-theoretic methods, we derive new and powerful criteria for finiteness uniton number a harmonic map from Riemann surface to unitary group $${{\,\mathrm{U}\,}}(n)$$ . These use Grassmannian model where maps are represented by families shift-invariant subspaces $$L^2(S^1,{{\mathbb {C}}}^n)$$ ; give description that model.

For a Veech surface (X, ω), we characterize Aff + (X, ω) invariant subspaces of X n and prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X, ω) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X, ω) prelat...