A restriction theorem and the Poincaré series for \(U\)-invariants (Q1891218)

Let \(X\) be a normal irreducible affine variety acted on by a connected semisimple group \(G\). We study regular functions on \(X\), which are invariant relative to a maximal unipotent subgroup \(U\) of \(G\). We prove a theorem that allows us to reduce the finding of the algebra of \(U\)- invariants to an action of a smaller group on a subvariety of \(X\). The algebra of \(U\)-invariants on \(X\) always has a natural grading and we give several inequalities for the degree of the related Poincaré series. Applications of these results to \(U\)-invariants of affine double cones and to decompositions of tensor products of irreducible representations of \(G\) are given.