  Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian
$G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector
bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose
equivariant index is a symplectic invariant of $E$. In the present paper, we
study gluing properties of the equivariant index under "symplectic cutting"
operations. Our main application is a proof of the Guillemin-Sternberg
conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a
regular value of $J$, the multiplicity of the trivial representation in the
equivariant index is equal to the Riemann-Roch number of the symplectic
quotient. This generalizes previous results for the case that $G=T$ is abelian.
