degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$  How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.
 A: This was worked out explicitly by A. A. Katanova in her paper Explicit form of certain multivector invariants in Advances in Soviet Mathematics 8 (1992), pp. 87-93.  According to her calculations, the degrees of the generators are 2, 6, 8, 10, 12, 14, and 18, and these freely generate the ring of invariants.  I suggest that you consult this paper for details, which are too long to write out here.
NB:  It is possible that this is also done in the earlier paper by L. V. Antonyan, Classification of $4$-vectors of $8$-dimensional space, Trudy Sem. Vektor. Tenzor. Anal. 20 (1981), pp. 144-161, but I don't have access to this paper and, in any case, I don't read Russian.  It could be that this paper just deals with the normal forms and doesn't discuss the ring of invariants. 
A: You don't really need Vinberg theory to find the degrees of the invariants here. This representation of $SL_8=SL(V)$ can be constructed by considering a maximal rank involution of an algebraic group of type $E_7$, which has fixed points isomorphic to $SL(V)/\{\pm I\}$ and such that the $(-1)$ eigenspace in the Lie algebra is exactly $\Lambda^4(V)$.
Well-known results about representations associated to symmetric spaces, as found for example in Kostant-Rallis' 1971 paper in the American Journal of Mathematics, then tell you that the invariants are given by restricting the invariants for the adjoint representation of the full $E_7$ group, and that these restrictions are non-zero. So the degrees are exactly the degrees of the invariants for the adjoint representation of $E_7$.
If you want to know how to see $SL_8/\{\pm I\}$ inside the group of type $E_7$, just take the subgroup whose Lie algebra is spanned by a Cartan subalgebra and all root spaces which have an even coefficient of $\alpha_2$ (in Bourbaki numbering); the representation $\Lambda^4(V)$ is just the span of all the root-spaces which have an odd coefficient of $\alpha_2$.
