Moment Maps and Equivariant Szegö Kernels

Let M be a connected n-dimensional complex projective manifold and consider an Hermitian ample holomorphic line bundle (L, hL) on M . Suppose that the unique compatible covariant derivative ∇L on L has curvature −2πiΩ, where Ω is a Kähler form. Let G be a compact connected semisimple Lie group and μ : G×M → M a holomorphic Hamiltonian action on (M,Ω). Let g be the Lie algebra of G, and denote by Φ : M → g the moment map. Let us also assume that the action of G on M linearizes to a holomorphic action on L; given that the action is Hamiltonian, the obstruction for this is of topological nature [GS1]. We may then also assume that the Hermitian structure hL of L, and consequently the connection as well, are G-invariant. Therefore for every k ∈ N there is an induced linear representation ofG on the space H(M,L) of global holomorphic sections of L. This representation is unitary with respect to the natural Hermitian structure of H(M,L) (associated to Ω and hL in the standard manner). We may thus decompose H(M,L) equivariantly according to the irreducible representations of G. The subject of this paper is the local and global asymptotic behaviour of certain linear series defined in terms this decomposition. Namely, we shall first consider the asymptotic behaviour as k → +∞ of the linear subseries of H(M,L) associated to a single irreducible representation, and then of the linear subseries associated to a whole ladder of irreducible representations. To this end, we shall estimate the asymptoptic growth, in an appropriate local sense, of these linear series on some loci in M defined in terms of the moment map Φ. To express the problem in point more precisely, fix a Cartan subalgebra h ⊂ g, and let H ⊂ G be the corresponding Cartan subgroup. Let R ⊂ h be the root system for (g, h), and fix a basis S = {αi} ⊂ R for R. Let X(H) be the group of characters of H , identified with a subgroup of h (a character