Stable Hilbert series of for classical groups

Given a classical symmetric pair, (G,K), with , we provide descriptions of the Hilbert series of the algebra of K-invariant vectors in the associated graded algebra of 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)→∞.