Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians

In this paper, we prove a combinatorial rule describing the restriction of any irreducible representation of U(n+m) to the subgroup U(n)×U(m). We also derive similar rules for the reductions from SU(n+m) to S(U(n)×U(m)), and from SU(n+m) to SU(n)×SU(m). As applications of these representation-theoretic results, we compute the spectra of the Bochner–Laplacian on powers of the determinant bundle over the complex Grassmannian Grn(Cn+m). The spectrum of the Dirac operator acting on the spin Grassmannian Grn(Cn+m) is also partially computed. A further application is given by the determination of the spectrum of the Hodge–Laplacian acting on the space of smooth functions on the unit determinant bundle over Grn(Cn+m).