Two representations of a reductive group G are spectrally equivalent if the
same irreducible representations appear in both of them. The semigroup of
finite dimensional representations of G with tensor product and up to spectral
equivalence is a rather complicated object. We show that the Grothendieck group
of this semigroup is more tractable and give a description of it in terms of
moment polytopes of representations. As a corollary, we give a proof of the
Kazarnovskii theorem on the number of solutions in G of a system f_1(x) = ...

We associate convex bodies to a wide class of graded G-algebras where G is a
connected reductive group. These convex bodies give information about the
Hilbert function as well as the multiplicities of irreducible representations
appearing in the graded algebra. We extend the notion of Duistermaat-Heckman
measure to graded G-algebras and prove a Fujita type approximation theorem as
well as a Brunn-Minkowski inequality for this measure. This in particular
applies to arbitrary G-line bundles giving an equivariant version of the theory
of volumes of line bundles.

Let K(X) be the collection of all non-zero finite dimensional subspaces of
rational functions on an n-dimensional irreducible variety X. For any n-tuple
L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the
number of solutions in X of a system of equations f_1 = ... = f_n = 0 where
each f_i is a generic function from the space L_i. In counting the solutions,
we neglect the solutions x at which all the functions in some space L_i vanish
as well as the solutions at which at least one function from some subspace L_i
has a pole.