Quantization Formula for Symplectic Manifolds with Boundary

We extend our earlier work in TiZ1], where an analytic approach to the Guillemin-Sternberg geometric quantization conjecture GuSt] was developed, to the case of manifolds with boundary. We also give a general quantization formula that works for both regular and singular reductions. As simple applications, we prove an analytic analogue of the relative residue formula of Guillemin-Kalkman GuK] and Martin M], as well as a Guillemin-Sternberg type formula for singular reductions under circle actions. Submitted: February 1997, revised: January 1998 and July 1998, final version: March 1999.