Method of computations of Hilbert series

I will report on a joint work with V. Baldoni-Silva.
Let $G$ be a semi-simple group acting linearly on a vector space $V$. It is important to understand the graded ring of invariants polynomials under the action of $G$, a subject intensively studied by V. Popov; I will present a method to compute the corresponding Hilbert series based on a residue calculus in several variables.
I will give some examples related to $N$ qubits.