Stable Hilbert series of $\mathcal{S}(\mathfrak{g}{)}^K$ for classical groups

Given a classical symmetric pair, $(G,K)$, with $\mathfrak{g}=Lie(G)$, we provide descriptions of the Hilbert series of the algebra of $K$-invariant vectors in the associated graded algebra of $\mathcal{U}(\mathfrak{g})$ viewed as a $K$-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note that the stable Hilbert series of one symmetric pair often coincides with others. Also, for the case of the real form $U(p,q)$ we derive a closed expression for the Hilbert series when $\min (p,q)\to \mathrm{\infty }$.