Simple irreducible graded Lie algebras of finite growth

We classify the simple graded Lie algebras $G=\bigoplus\limits_{i=-\infty}^{+\infty}G_i$, for which the dimension of the space $G_i$ grows as some power of $|i|$, under the additional assumption that the adjoint representation of $G_0$ on $G_{-1}$ is irreducible. From these results we obtain a classification of the primitive infinite-dimensional Cartan pseudogroups of transformations and a classification of symmetric spaces.