By $q$-multiplicity we mean the generalization of a multiplicity formula for an irreducible representation in a graded space to a generating function for the multiplicity in the graded components. The $q$-multiplicity refines the (non graded) multiplicity formula. The main result of this thesis is a stable range in the space of harmonic polynomials associated to the $(GL(n),O(n))$ case of the Kostant-Rallis theorem. In the stable range the $q$-multiplicity is deduced from certain symmetric function identities and Littlewood's restriction rules. Chapter 3 gives an alternative proof of Littlewood's restriction rules from Howe duality and a classification of unitary highest weight modules due to Enright, Howe and Wallach.

For $n \geq 3$, the $q$-multiplicity for the spherical harmonics is given both for the $(GL(n),O(n))$ and the $(SL(n),SO(n))$ cases. The full $q$ analog of the Kostant-Rallis theorem is described in detail for the symmetric pair $(SL(4),SO(4))$. The significance of this example is that it has implications in the study of entanglement of the mixed 2 qubit states in quantum computation. In chapter 2, a problem from classical invariant theory is addressed. Specifically, the complex orthogonal group acts on the $n \times n$ matrices by restricting the adjoint action of $GL(n,\bbC)$. This gives us an action on the ring of complex valued polynomial functions on the matrices. A combinatorial description of the Hilbert series for the invariant polynomials under this action is given.