Szegö kernel of a symplectic quotient

The object of this paper is the relation between the Szegö kernel of an ample line bundle on a complex projective manifold, M , and the Szegö kernel of the induced polarization on the quotient of M by the holomorphic action of a compact Lie group, G. Let M be an n-dimensional complex projective manifold and L an ample line bundle on it. Suppose a connected compact Lie group G acts on M as a group of holomorphic automorphisms, and that the action linearizes to L. Without loss of generality, we may then choose a G-invariant Kähler form Ω onM representing c1(L). We may also assume given a G-invariant Hermitian metric h on L such that the unique Hermitian connection on L compatible with the holomorphic structure has normalized curvature −2πiΩ. Let Φ : M → g, where g is the Lie algebra of G, be the moment map for the action (this is essentially equivalent to the assignment of a linearization). Suppose that 0 ∈ g lies in the image of Φ, that it is a regular value of Φ, and furthermore that G acts freely on Φ(0). In this situation, the Hilbert-Mumford quotient for the action is nonsingular and may be naturally identified with the symplectic reduction M0 = Φ (0)/G. The latter, furthermore, inherits a naturally induced polarization L0, with an Hermitian metric h0 and a Kähler form Ω0. The quotient structures L0, h0,Ω0 are induced simply by descending the restrictions of L, h,Ω to Φ(0) down to M0 [GS2]. Let H(M,L) and H(M0, L ⊗k 0 ) denote the spaces of holomorphic sections of powers of L on M and of powers of L0 on M0, respectively. Then G acts linearly on H(M,L), and in the given hypothesis by the theory of [GS2] there is for every integer k ≥ 0 a natural isomorphism H(M,L) ∼=