Moment polytopes, semigroup of representations and Kazarnovskii's theorem

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) = ... = f_m(x) = 0, where m=dim(G) and each f_i is a generic function in the space of matrix elements of a representation pi_i of G.