  This paper applies the decomposition theorem in intersection cohomology to
geometric invariant theory quotients, relating the intersection cohomology of
the quotient to that of the semistable points for the action.
  Suppose a connected reductive complex algebraic group $G$ acts linearly on a
complex projective variety $X$. We prove that if $1 \to N \to G \to H \to 1$ is
a short exact sequence of connected reductive groups, and $X^{ss}$ the set of
semistable points for the action of $N$ on $X$, then the $H$-equivariant
intersection cohomology of the geometric invariant theory quotient $X^{ss}//N$
is a direct summand of the $G$-equivariant intersection cohomology of $X^{ss}$.
