Invariants of symplectic and orthogonal groups acting on $\text{GL}(n,\mathbb{C})$-modules

Let $\text{GL}(n)=\text{GL}(n,\mathbb{C})$ denote the complex general linear group and let $G\subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$)\mathrm{.}Wetakeapolynomial$$\text{GL}$(n)$-module$W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=\text{SL}(n)$, we develop a method for determining the Hilbert series $H(S(W{)}^G,t)$ of the algebra of invariants $S(W{)}^G$. Then we give explicit examples for computing $H(S(W)^G, t)$. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $\mathrm{\Lambda }(S^{2}V{)}^G$ and $\Lambda(\Lambda^2 V)^G$,where$V = {\mathbb C}^n$denotesthestandard$GL(n)$-module.