Let $k$ be an algebraically closed field, of characteristic $0$, $A=\oplus _{n\geq 0} A_n$ a graded $k$-algebra which is finitely generated and a domain, with $A_0=k\cdot 1$ and $A_n\neq 0$ for $n$ big enough. Let $G$ be a reductive group over $k$, $e$ its unit element, $\Lambda $ the set of es of finite dimensional rational simple representations of $G$, 0 the trivial element of dimension $1$ of $\Lambda $. Assume that $G$ operates faithfully and rationally in $A$ by graded automorphisms. For $\lambda \in \Lambda $, let $m_{\lambda ,n}$ be the multiplicity of $\lambda $ in $A_n$. Let $X$ be the affine irreducible variety over $k$ defined by $A$. The group $G$ operates in $X$. To simplify this abstract, assume that $m_{0,n}\neq 0$ for $n$ big enough, that the generic orbit of $G$ in $X$ is closed, and that the stabilizer of a generic point of $X$ is trivial. Then $m_{\lambda ,n}/m_{0,n}\to \dim \lambda $ as $n\rightarrow \infty $. This generalizes a result by R. Howe, which concerns the case where $G$ is finite. Assume that $G$ is the complexification of a connected compact Lie group $K$. Let $\psi _n$ be the character of the representation of $K$ in $A_n$, with a suitable normalization. Then, as $n\rightarrow \infty $, $\psi _n\to \varepsilon _e$ (Dirac mass at $e$) in the space of distributions on $K$.